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5900 | https://theradavist.com/iso-astm-bicycle-testing | The Radavist
Radar
The Dust-Up: Should Brands Clearly Classify Bicycles With ISO or ASTM Testing?
John Watson
May 22, 2025
Do you ever jump a road bike off a curb? Or perhaps you’ve taken a gravel bike onto singletrack for some fun. Have you ever thought about whether that bike was designed for “underbiking?” Or do you even care? Our latest installment of The Dust-Up examines ISO and ASTM bicycle testing and its implications for consumer and brand transparency…
The Monopole is very clear about weight limits for its No1 cargo bike and ISO testing…
ASTM Testing
ASTM, the organization responsible for impact testing of consumer goods, released a press release in 2009 that classified the conditions within which common bicycles were operated. The organization did this to protect the consumer and create a framework in which companies can categorize the sort of riding their bicycles have been engineered for.
Close your eyes and imagine the number of bicycles being ridden right at this moment across the globe. That number has to be in the millions, right? Now think about the conditions these bicycles are being ridden in, from a toddler learning how to operate a balance bike, to a road group ride, perhaps a gravel race, or a dirt jump competition. Now think about each of the bicycle frames being ridden. What denotes their riding style? Is it the tire clearance? Or suspension? Frame material or weight?
Subcommittee F08.10on Bicycles, part of ASTM International Committee F08 on Sports Equipment and Facilities, is responsible for developing and keeping current standards of bicycle use up to date.
And it’s been a while since those standards have been updated.
In 2009, this subcommittee F08.10 published an important revision to one of its key standards. This revision was originally intended as a document to guide developers of other bicycle standards like ASTM F2043 – aka Classification for Bicycle Usage – and has now been revised for use as an educational tool for both those in the bicycle industry and for consumers purchasing bikes.
David Mitchell, an engineering consultant from MET Ltd., a UK-based materials testing and engineering lab, and an F08.10 subcommittee member, states the revision of the ASTM F2043 classification was prompted by the U.S. Consumer Product Safety Commission who expressed an interest in the standard as a means to guide bicycle purchasers in selecting a proper bike for its intended use.
The byproduct of these subcommittee revisions is a five-part Condition Classification expressing intended bicycle uses and incorporating a standard naming scheme that would apply to all manufacturers.
Wait a second, did we just write “standard” and “all manufacturers” in the same sentence? In the bike industry? Pffffffffffft.
ASTM Conditions
The five uses are:
Condition 0 — Adult supervision required, no traffic;
Condition 1 — Suitable for road riding (only);
Condition 2 — For off-road riding and jumps less than 12 in. (30 cm.);
Condition 3 — For rough off-road riding and jumps less than 24 in. (61 cm.); and
Condition 4 — For extreme off-road riding.
The problem is that there is no transparency on the consumer side of these products. Hell, even on the media side of this topic, we have to ask brands directly if it’s okay to showcase a bike being ridden a certain way.
What is interesting about these Conditions is it goes from a 24″ jump to “extreme off-road riding” – which we have learned is akin to RedBull Rampage-level riding. Yet, the sorts of rigors BMX frames face are unclassified.
“There is not currently a designation for BMX-style riding,” says Mitchell. “As bicycles evolve, the subcommittee intends to add additional designations for BMX use and other areas.” Mitchell notes that engineering personnel who can define new levels of bicycle use are welcome to join in the subcommittee’s work on future revisions to ASTM F2043.
This was a press release from 2009, and there haven’t been any substantial updates to ASTM testing since, only for recommended BMX helmet use.
This begs the question: when gravel bikes are ridden on mountain bike singletrack and people review those bikes as being capable machines for such terrain, would it be beneficial for the brands to be transparent about ASTM testing? We’re less concerned about mountain bikes being used as mountain bikes, but as “road,” “all-road,” “gravel,” and “drop bar mountain bikes” or “adventure bikes” are offered to consumers with very little delineation of use case, we have to wonder if these bikes would benefit from appropriate categorizations.
ISO Testing: Engineering Specificity
We reached out to a number of brands, all in the “off-road, drop-bar” sector, which we primarily review here at The Radavist, asking them which Conditions they rate their bikes for, of which all deflected the question with a simple statement:
“We adhere to our manufacturer’s ISO testing standards.”
What they are referring to ISO standards, which offer more exact testing requirements for frames and components to pass/fail. ISO 4210-2:2023 includes the requirements for city/trekking, young adult, mountain, and racing bicycles.
ISO 4210-2:2023 gets into the weeds and specifies the exact safety and performance requirements for the design, assembly, and testing of bike frames and complete bicycles. It lays down guidelines for manufacturer’s instructions on the use and care of such bicycles. As such, ISO is deemed as more of an “internal” language for the testing of bicycles and less of a “public facing” testing standardization.
If you’re buying a bike and it says it adheres to ISO 4210, do you even know what that means?
These specs are written in engineering talk from a product, research, and development aspect, and less of a marketing use-case scenario as with the case of ASTM standards. As such, they are incredibly nuanced and complex, making it hard for consumers to understand exactly how these products are to be used.
Even ISO standards do not apply to “stunt bicycles,” as the opening article of ISO 4210-2:2023 states:
“This document does not apply to specialized types of bicycle, such as delivery bicycles, recumbent bicycles, tandems, BMX bicycles, and bicycles designed and equipped for use in severe applications such as sanctioned competition events, stunting, or aerobatic maneuvers.”
So what gives? If brands aren’t using ASTM to classify their products and ISO standards don’t encapsulate “stunting,” where does that leave us? To fill in the gaps here, Taiwanese manufacturers adhere to the Taiwan Bicycle Industry Standard, which was launched in 2015.
One brand that also wished to remain anonymous mentioned, “Testing is difficult to discuss, which is perhaps why CPSC standards exist.”
CPSC is Vague, Too
More broadly, CPSC standards help dictate a certain degree of requirements but lack any real meat worthy of classification for use cases:
1: Adults of normal intelligence (lol) and ability (lol) must be able to assemble a bicycle that requires assembly.
2: A bicycle may not have unfinished sheared metal edges or other sharp parts that may cut a rider’s hands or legs. Sheared metal edges must be rolled or finished to remove burrs or feathering.
3: When the bicycle is tested for braking (§1512.18(d) and/or (e)) or road performance (§1512.18(p) or (q)), neither the frame, nor any steering part, wheel, pedal, crank, or braking system part may show a visible break.
4: Screws, bolts, and nuts used to fasten parts may not loosen, break, or fail during testing.
5: Control cables must be routed so that they do not fray from contact with fixed parts of a bicycle or with the ends of the cable sheaths. The ends of control cables must be capped or treated so that they do not unravel.
6: A bicycle may not have any protrusions within the shaded area of Diagram 1. However, control cables up to ¼ inch thick and cable clamps made of material not thicker than 3/16 inch may be attached to the top tube.
Whomp, whomp.
Unfortunately, it comes down to the fact that we live in a litigious society, and brands/manufacturers are afraid of liability lawsuits. One company that, surprise surprise, wishes to remain anonymous told us that if “we tell people our forks are rated to 24″ or Condition 03 riding, then they are in fact engineered to 40″ drops just to cover our butt.”Another company stated: “So officially unofficial off the record statement is we don’t use ASTM because ISO exceeds ASTM rating and we, like all reputable brands, produce components that exceed ISO.”
Media’s Responsibility and ISO ASTM Testing
Editorially, over here at The Radavist, this puts us in a unique position to begin prying for ISO or ASTM standards testing Conditions – assuming the brands classify their products within the ISO or ASTM framework – in our product reviews. Perhaps through more transparency in product reviews, these discussions can help steer brands (and media) to uphold the realities of bicycle standards testing on the ground… and maybe keep our staff from jumping bikes that don’t meet Condition 02 or 03.
This creates a conundrum, as we’ve found; what if brands aren’t willing to discuss this openly?
I once shot a photo of a rider launching a big wall ride, which resulted in the company that made a component on the bike saying that component wasn’t rated to that sort of riding, which involved a 48″ drop. They said the component was “over-engineered” to resist this sort of riding, but the photo’s circulation on a website made them squeamish.
“Stunting” on drop bar bikes isn’t the only place we should be mindful of. There are a few use cases in particular that we’ve been wary of:
Nitto front racks installed on touring bikes or commuters without the “security strap,” which leads to failure. A simple toe strap around the handlebar and rack tongue loop can save you if the fork crown hardware snaps.
“Bikepacking”-style minimalist racks are being installed (improperly) on bikes without rack mounts, sometimes leading to frame failure as they are overloaded. This particularly becomes problematic with the ideology that “any bike is a bikepacking bike.” We’ve never done this editorially, but we saw it happen online during last year’s Tour Divide Race with a few misused racks.
Overloading of bikes with rack mounts and then riding out of their designed terrain. I.e., a “gravel road” bike being overloaded and ridden down rocky doubletrack and jeep roads.
Road stems and handlebars on drop bar mountain bikes or adventure bikes, ridden in mountain bike terrain, when they have not been tested to the rigors of MTB riding.
Many, if not most, framebuilder bikes are not ISO tested. However, framebuilders work one-on-one with their customers to ensure the rider’s safety by engineering the appropriate tubing specifications. Many aren’t building a bike for the masses, only for that individual customer.
Even though factories in Asia adhere to ISO, many overseas ISO testing facilities are also owned by the very factories that make the frames.
What do you think? As a consumer, do these ASTM and ISO standards mean anything to you? Should bikes have a clear rating for their intended use? Would you like to see more transparency from brands on the degree of testing their bikes have received?
Or perhaps the point of this entire article is whether there should be more transparent communication between a brand and its customer base about how its bikes ought to be ridden.
Or… hear me out. Just run what you brung, follow installation instructions as specified by the company… and bikes sometimes break.
Let us know in the comments.
If you’re new to this series, welcome to The Dust-Up. This will be a semi-regular platform for Radavist editors and contributors to make bold, sometimes controversial claims about cycling. A way to challenge long-held assumptions that deserve a second look. Sometimes, they will be global issues with important, far-reaching consequences. Other times, they will shed light on little nerdy corners of our world that don’t get enough attention.
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5901 | https://byjus.com/maths/factors-of-59/ | The factors of 59 are the numbers that divide 59 exactly without leaving any remainder. In other words, the numbers that are multiplied together resulting in the number 59 are the factors of 59. The factors of 59 and its pair factors can be expressed in positive and negative form, but they cannot be represented in the fraction or decimal form. In this article, we will learn the factors of 59, positive and negative pair factors of 59, how to find the prime factors of 59 using the prime factorization method, and many solved examples.
Table of Contents:
What are the Factors of 59?
Pair Factors of 59
Factors of 59 by Division Method
Prime Factorization of 59
Solved Examples
FAQs
What are the Factors of 59?
In Mathematics, the factors of a number are the numbers that divide the given number evenly without leaving a remainder value. Thus, the factors of 59 are the numbers that are multiplied in pairs resulting in the original number 59. As 59 is a prime number, it has only two factors, such as one and the number itself. Hence, the factors of 59 are 1 and 59. Similarly, the negative factors of 59 are -1 and -59.
| |
| Factors of 59: 1 and 59 Prime Factorization of 59: 1 × 59 or 59. |
Pair Factors of 59
As discussed above, the pair factor of 59 is expressed in both positive and negative form. A pair of numbers that are multiplied together, and it results in 59 are the pair factors of 59. Since the number 59 is a prime number, it has only one positive pair factor and one negative pair factor. The following are the positive and negative pair factors of 59.
Positive Pair Factor of 59:
| | |
--- |
| Positive Factors of 59 | Positive Pair Factors of 59 |
| 1 × 59 | (1, 59) |
Negative Pair Factor of 59:
| | |
--- |
| Negative Factors of 59 | Negative Pair Factors of 59 |
| -1 × -59 | (-1, -59) |
Therefore, the positive and negative pair factors of 59 are (1, 59) and (-1, -59), respectively.
Factors of 59 by Division Method
The factors of 59 using the division method are found by dividing the number 59 by different integers. If the integers exactly divide the number 59, then those integers are the factors of 59. The factors of 59 using the division method are found as follows:
59/1 = 59 (Factor is 1 and Remainder is 0)
59/59 =1 (Factor is 59 and Remainder is 0)
If any numbers other than 1 and 59 divides 59, it leaves a remainder value. Hence, the factors of 59 are 1 and 59.
Prime Factorization of 59
In the prime factorization of 59, the number 59 is written in the form of the product of its prime factors. Thus, the process of finding prime factors of 59 using the prime factorization method is given as follows:
Take a pair factor of 59, say (1, 59)
Now, check the factors of 59, whether it is prime or composite. If a number is a prime number, leave the number as it is. If a number is a composite number, split the number into its prime factors.
Here, the factors of 59, such as 1 and 59, are prime numbers.
Thus, 59 is written as the product of 1 and 59.
Therefore, the prime factorization of 59 is 1 × 59 or 59.
Video Lesson on Prime Factors
38,903
| | |
--- |
| Learn More on Factors of a Number: | |
| Factors of 49 | Factors of 50 |
| Factors of 54 | Factors of 57 |
Solved Examples
Example 1:
What are the common factors of 59 and 58?
Solution:
The factors of 59 are 1 and 59.
The factors of 58 are 1, 2, 29 and 58.
Thus, the common factor of 59 and 58 is 1.
Example 2:
Find the common factors of 59 and 60.
Solution:
Factors of 59 = 1 and 59.
Factors of 60 = 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60.
Hence, the common factor of 59 and 60 is 1.
Example 3:
Find the common factors of 59 and 53.
Solution:
The factors of 59 are 1 and 59.
The factors of 53 are 1 and 53.
As both the numbers are prime numbers, the common factor of 59 and 53 is 1.
Stay tuned with BYJU’S – The Learning App and download the app to learn all the important Maths concepts.
Frequently Asked Questions on Factors of 59
Q1
What are the factors of 59?
As the number 59 is a prime number, the factors of 59 are 1 and 59.
Q2
What is the prime factorization of 59?
The prime factorization of 59 is 1 × 59 or 59.
Q3
What are the positive and negative pair factors of 59?
The positive and negative pair factors of 59 are (1, 59) and (-1, -59), respectively.
Q4
Is 28 a factor of 59?
No, 28 is not a factor of 59. As the number 59 is a prime number, it has only two factors: one and the number itself.
Q5
What is the sum of factors of 59?
The sum of factors of 59 is 60. We know that the factors of 59 are 1 and 59.
Hence, the sum of factors of 59 = 1+59 = 60.
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5902 | https://pubmed.ncbi.nlm.nih.gov/38006314/ | Role of the Degree of Vascular Invasion in Predicting Prognosis of Follicular Thyroid Carcinoma - PubMed
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. 2024 Apr 19;109(5):1291-1300.
doi: 10.1210/clinem/dgad689.
Role of the Degree of Vascular Invasion in Predicting Prognosis of Follicular Thyroid Carcinoma
Haruhiko Yamazaki12,Kiminori Sugino2,Ryohei Katoh3,Kenichi Matsuzu2,Wataru Kitagawa2,Mitsuji Nagahama2,Yasushi Rino4,Aya Saito4,Koichi Ito2
Affiliations Expand
Affiliations
1 Department of Breast and Thyroid Surgery, Yokohama City University Medical Center, 4-57 Urafunecho, Minami-ku, Yokohama City, Kanagawa, 232-0024, Japan.
2 Department of Surgery, Ito Hospital, 4-3-6, Jingumae, Shibuya-ku, Tokyo, 150-8308, Japan.
3 Department of Pathology, Ito Hospital, 4-3-6, Jingumae, Shibuya-ku, Tokyo, 150-8308, Japan.
4 Department of Surgery, Yokohama City University School of Medicine, 3-9 Fukuura, Kanazawa-ku, Yokohama City, Kanagawa, 236-0004, Japan.
PMID: 38006314
DOI: 10.1210/clinem/dgad689
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Role of the Degree of Vascular Invasion in Predicting Prognosis of Follicular Thyroid Carcinoma
Haruhiko Yamazaki et al. J Clin Endocrinol Metab.2024.
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. 2024 Apr 19;109(5):1291-1300.
doi: 10.1210/clinem/dgad689.
Authors
Haruhiko Yamazaki12,Kiminori Sugino2,Ryohei Katoh3,Kenichi Matsuzu2,Wataru Kitagawa2,Mitsuji Nagahama2,Yasushi Rino4,Aya Saito4,Koichi Ito2
Affiliations
1 Department of Breast and Thyroid Surgery, Yokohama City University Medical Center, 4-57 Urafunecho, Minami-ku, Yokohama City, Kanagawa, 232-0024, Japan.
2 Department of Surgery, Ito Hospital, 4-3-6, Jingumae, Shibuya-ku, Tokyo, 150-8308, Japan.
3 Department of Pathology, Ito Hospital, 4-3-6, Jingumae, Shibuya-ku, Tokyo, 150-8308, Japan.
4 Department of Surgery, Yokohama City University School of Medicine, 3-9 Fukuura, Kanazawa-ku, Yokohama City, Kanagawa, 236-0004, Japan.
PMID: 38006314
DOI: 10.1210/clinem/dgad689
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Abstract
Objective: The present study investigated the prognostic factors for follicular thyroid carcinoma (FTC) with the incorporation of the histologic subtype and degree of vascular invasion (VI).
Patients: The records of 474 patients with FTC confirmed by surgical specimens at Ito Hospital from January 2005 to December 2014 were reviewed in this retrospective cohort study. The Cox proportional hazard model was used to determine factors associated with disease-free survival (DFS) and distant metastasis-free survival.
Results: Of the 474 patients, 140 (30%) had minimally invasive FTC, 260 (55%) had encapsulated angio-invasive FTC, and 74 (16%) had widely invasive FTC. Among the 428 patients with M0 FTC, the 10-year DFS rates of patients with minimally invasive FTC (n = 133), encapsulated angio-invasive FTC (n = 247), and widely invasive FTC (n = 48) were 97.3%, 84.2%, and 69.9% (P < .001), respectively. A multivariate analysis identified aged ≥55 years (hazard ratio [HR], 2.204; 95% CI, 1.223-3.969; P = .009), histologic subtype (HR, 2.068; 95% CI, 1.064-4.021; P = .032), VI of ≥2 (HR, 6.814; 95% CI, 3.157-14.710; P < .001), and tumor size >40 mm (HR, 2.014; 95% CI, 1.089-3.727; P = .026) as independent negative prognostic factors for DFS.
Conclusion: Our study results may enable us to stratify the prognosis of FTC more accurately by combining the histologic subtype with the degree of VI ≥2, aged ≥55 years, and tumor size >40 mm.
Keywords: follicular thyroid carcinoma; prognosis; vascular invasion.
© The Author(s) 2023. Published by Oxford University Press on behalf of the Endocrine Society. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com.
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Adenocarcinoma, Follicular / diagnosis
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Let a,b,n a,b,n be positive integers in which a,b≤n a,b≤n. You are locked in a room, with n distinguishable keys and n distinguishable locks in it. You know that each lock can be unlocked by a unique key, but you don't know which key unlocks which lock. You also know each key initially has a durablility of a and each lock initially has a durability of b. The only way out of the room is unlock at least one of the locks, but each time you try to unlock a lock using a key, the durablility of both that lock and key will decrease by 1. A key or lock cannot be used for any further unlocking attempts if its durability is 0. For each n, what pairs of a,b guarentees a strategy which you can certainly leave the room?
I haven't posted in a while, let's see how this goes. This problem is made by me a few years ago.
Nice that someone has the correct condition and proving that it is sufficient. To prove that it is necessary, you can
Notice that you can fix what lock-key pairs that you will try before the start (Why?) and then you should prove for all valid lists of lock-key pairs, there exists a lock-key pairing such that none of the attempts are correct.
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edited May 27, 2024 at 0:00
Culver KwanCulver Kwan
asked May 26, 2024 at 9:41
Culver KwanCulver Kwan
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The exact condition is
a+b>n.
Actually, something slightly stronger is true: A strategy is winning if and only if
there is some collection of k keys and ℓ locks, satisfying k+ℓ>n, that are tested in all combinations. (As then k≥b and ℓ≥a, it is clear that such a collection can exist if and only if a+b>n.) In particular, this will show that the strategy suggested by Florian F (with k=b, ℓ=a) is not only optimal but actually a necessary part of any winning strategy!
We can prove this equivalence by an application of
Hall's Marriage Theorem.
Suppose we are handed any strategy.
As there is no nontrivial information gain throughout the game, the strategy can reasonably be considered as a list S of pairs of keys and locks to be tested that is fixed from the outset. To this strategy, we associate the graph G S that has all keys and locks as its 2 n vertices and contains an edge between a key and a lock if these are not tested by the strategy.
Now, observe that
a perfect matching in G S is exactly the same thing as an association of keys to locks that evades the strategy S. Hence, the strategy S is winning if and only if G S does not have a perfect matching. By HMT, this is the case if and only if there is some subset A of k keys (wlog) that has altogether fewer than k neighbouring locks. Now, a lock is neighbouring A if and only if it is not tested against every key in A. Hence, the complement of the neighbourhood in the locks is exactly the set of locks that are tested by all the keys in A. But this now is exactly our condition above − since the neighbourhood has size less than k, the complement will have size ℓ greater than n−k. Thus, a violating set for HMT is exactly the same thing as such a collection, which shows the claim.
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edited May 27, 2024 at 7:41
answered May 27, 2024 at 7:31
Tim SeifertTim Seifert
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1 Nice work applying HMT! Necessity is the hardest part of this puzzle, and I like your proof of necessity better, so I will give the checkmark to you.Culver Kwan –Culver Kwan 2024-05-27 12:11:36 +00:00 Commented May 27, 2024 at 12:11
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I believe the condition is:
a+b>n
This condition is sufficient.
Choose a locks and b keys and try all combinations. If no lock opens, that means the b keys belong to the n−a remaining locks. But that is impossible because b>n−a.
Is this condition necessary?
I think it is also necessary, i.e. if a+b≤n then there is always a possibility of "using up" all locks and keys without finding a match. But I have trouble finding a convincing argument. The only justification is that the strategy above (choose a key and b locks) seems to be the best but has the possibility to fail when a+b=n or less.
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edited May 26, 2024 at 19:19
answered May 26, 2024 at 14:31
Florian FFlorian F
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Oops... You are obviously right. Thanks for the correction. I fixed the answer.Florian F –Florian F 2024-05-26 19:18:41 +00:00 Commented May 26, 2024 at 19:18
Hi you may want to look at the hint I posted Culver Kwan –Culver Kwan 2024-05-27 00:01:06 +00:00 Commented May 27, 2024 at 0:01
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I thought of it in terms of a
directed graph. Suppose the locks and keys are both numbered 1,…,n, but that key 3 doesn't necessarily open lock 3. Every time you try a key in a lock and it doesn't work, you can draw a red directed edge (or arrow) on the graph. For example, if you try key 3 in lock 6, you draw a directed edge from node 3 to node 6. Because of the durability requirement, we know the out-degree (number of edges coming out of a node) of the red edges is at most a, and the in-degree is at most b.
Suppose
a+b≤n. We'd like to show that, no matter what keys and locks we've tried so far (red edges), that there is some assignment of keys to locks (green edges) that misses all the red edges. The green edges form a permutation, and thus we can think of the green edges a forming a bunch of directed cycles that cover all the nodes in the graph.
Can this always be done?
We will build up the cycles one node at a time. Consider a node v. If x→y→z→x is an already established green cycle, then we could potentially add v anywhere in that cycle. For example, if y→v→z are two edges that are not red, then we could create the cycle x→y→v→z→x, and we would have incorporated v into the green cycles. However, if we can't add v here, that means either y→v or v→z is red. If we can't add v to this cycle at all, we see (x→v or v→y) and (y→v or v→z) and (z→v or v→x) are red. These are six seperate edges, so that means at least three edges incident to v are red. This is the same for every cycle in our list so far. If w is just another node not in a cycle, then we can see that either (v→w or w→v) is red, so we get another red edge from w. We also must have v→v is red, and this actually counts twice. So that means v must be involved in n+1 red edges, but this is impossible if a+b≤n. Therefore, there is a way to incorporate v into the decomposition of cycles, and thus one by one we can incorporate all the nodes. That means there is some way the locks and keys could be paired up so that they avoid everything that we guessed so far, proving that Florian F's condition is necessary.
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edited May 27, 2024 at 1:26
answered May 27, 2024 at 1:21
Tyler SeacrestTyler Seacrest
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Your solution kinda seems correct but I am not sure (lazy to verify details), and I like the other proof of necessity better (it is cleaner, and is my solution also).Culver Kwan –Culver Kwan 2024-05-27 12:10:24 +00:00 Commented May 27, 2024 at 12:10
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5904 | https://math.stackexchange.com/questions/2941118/express-fx-tanx-frac-pi4-as-a-sum-of-an-even-function-and-an-odd-f | analysis - Express $f(x)= \tan(x+ \frac{\pi}{4})$ as a sum of an even function and an odd function - Mathematics Stack Exchange
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Express f(x)=tan(x+π 4)f(x)=tan(x+π 4) as a sum of an even function and an odd function [closed]
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Express f(x)=tan(x+π 4)f(x)=tan(x+π 4) as sum of two functions and one of them has to be even and the other odd like f(x)=e+o f(x)=e+o, so that e e is an even function and o o is an odd function.
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edited Oct 3, 2018 at 19:11
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Altaïr Ibn-La'AhadAltaïr Ibn-La'Ahad
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5 f(x)=f(x)+f(−x)2+f(x)−f(−x)2 f(x)=f(x)+f(−x)2+f(x)−f(−x)2.metamorphy –metamorphy 2018-10-03 18:45:01 +00:00 Commented Oct 3, 2018 at 18:45
I don't understand Altaïr Ibn-La'Ahad –Altaïr Ibn-La'Ahad 2018-10-03 18:45:34 +00:00 Commented Oct 3, 2018 at 18:45
3 The first term is an even function, the second is an odd one.metamorphy –metamorphy 2018-10-03 18:46:10 +00:00 Commented Oct 3, 2018 at 18:46
I get it now and thank you. Also can you prove that the first term is even and second term is odd?Altaïr Ibn-La'Ahad –Altaïr Ibn-La'Ahad 2018-10-03 19:02:53 +00:00 Commented Oct 3, 2018 at 19:02
2 Substitute x←−x x←−x. Moreover, this representation is unique (if f(x)=e(x)+o(x)f(x)=e(x)+o(x), then f(−x)=e(x)−o(x)f(−x)=e(x)−o(x) - now add and subtract...).metamorphy –metamorphy 2018-10-03 19:04:51 +00:00 Commented Oct 3, 2018 at 19:04
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Let's go to this step by step.
First, express the tangent function in terms of the sine function and cosine function:
f(x)=sin(x+π 4)cos(x+π 4)f(x)=sin(x+π 4)cos(x+π 4)
Make use of angle-sum identities for the sine function and cosine function:
f(x)=sin(x)⋅cos(π 4)+cos(x)⋅sin(π 4)cos(x)⋅cos(π 4)−sin(x)⋅sin(π 4)f(x)=sin(x)⋅cos(π 4)+cos(x)⋅sin(π 4)cos(x)⋅cos(π 4)−sin(x)⋅sin(π 4)
Recognize that sin(π 4)=cos(π 4)=1 2√sin(π 4)=cos(π 4)=1 2:
f(x)=sin(x)+cos(x)cos(x)−sin(x)f(x)=sin(x)+cos(x)cos(x)−sin(x)
Multiply both the denominator and numerator by cos(x)+sin(x)cos(x)+sin(x):
f(x)=sin(x)+cos(x)cos(x)−sin(x)⋅cos(x)+sin(x)cos(x)+sin(x)f(x)=sin(x)+cos(x)cos(x)−sin(x)⋅cos(x)+sin(x)cos(x)+sin(x)
It follows that
f(x)=sin 2(x)+cos 2(x)+2⋅sin(x)⋅cos(x)cos 2(x)−sin 2(x)f(x)=sin 2(x)+cos 2(x)+2⋅sin(x)⋅cos(x)cos 2(x)−sin 2(x)
Regarding the numerator, make use of a Pythagorean identity and a double-angle identity for the sine function. Regarding the denominator, make use of a double-angle identity for the cosine function:
f(x)=1+sin(2 x)cos(2 x)f(x)=1+sin(2 x)cos(2 x)
Split the fraction:
f(x)=1 cos(2 x)+sin(2 x)cos(2 x)f(x)=1 cos(2 x)+sin(2 x)cos(2 x)
Express the function in terms of the secant function and tangent function:
f(x)=sec(2 x)+tan(2 x)f(x)=sec(2 x)+tan(2 x)
It can be derived that f(x)=e(x)+o(x)f(x)=e(x)+o(x) holds since sec(−z)=s e c(z)sec(−z)=s e c(z) and tan(−z)=−tan(z)tan(−z)=−tan(z).
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answered Oct 3, 2018 at 19:35
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Here is another way to do it:
Use the formula for the tangent of a sum to render
tan(x+π/4)=1+tan x 1−tan x tan(x+π/4)=1+tanx 1−tanx
Multiply the numerator and denominator by 1+tan x 1+tanx forcing the denominator to an even function:
tan(x+π/4)=(1+tan x)2 1−tan 2 x tan(x+π/4)=(1+tanx)2 1−tan 2x
Now just expand the binomial square in the numerator and identify the even and odd parts. You can even simplify the odd portion using the double angle formula for tangent, just for kicks.
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edited Oct 4, 2018 at 15:35
amWhy
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Thanks for catching that @amWhy. The "boring" part was a play on the comments, but the joke is on me because somehow that comment got edited. :-S Oscar Lanzi –Oscar Lanzi 2018-10-04 16:20:34 +00:00 Commented Oct 4, 2018 at 16:20
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5905 | https://www.youtube.com/watch?v=x2M2xGbL9DY | Proof of "limit (x^n - a^n)/(x - a) = na^(n - 1)"using algebraic division | ZJ learning | Limits#6
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In our previous video we proved this theorem
using the geometric series. Today we will be proving it using algebraic division. If you remember the remainder theorem that
we learnt before, it says that the remainder of a function f(x) when divided by (x - a)
is f(a). So here f(x) is our dividend that is f(x) = x to the power n - a to the power
n. So here f(a) = a to the power n - a to the
power n. So this gives us 0. So this means that there is no remainder when x to the power
n - a to the power n is divided by (x - a). Now lets get into our division. So here we
have x to the power n - a to the power n divided by (x - a). So when we divided many students
think that a to the power 'n' is the 2nd term here. In fact a to the power 'n' here is our
last term, that means it is our nth term. So we write it at the end, meaning the rest
in between are zero. So our 1st step will be to remove this x to
the power n. So to remove this x to the power n we need to subtract it by x to the power
'n'. So to get this x to the power n what will our quotient be? For this we divide x
to the power n by this x. So x to the power n divided by x will give us x to the power
(n - 1). So that will be our quotient value; x to the power (n - 1). So now x to the power
n - x to the power n will give us zero. So next we multiply this (-a) into x to the
power (n - 1). So this will gives us -ax to the power (n - 1). Now we subtract 0 by this
values. So 0 - (-ax to the power (n - 1)) will give us positive ax to the power (n - 1).
So to check what we have done correct here, we again multiply and see. So x into x to
the power (n - 1) will give us x to the power (n - 1 + 1). So that is x to the power n.
So here we have got our 'x' to the power 'n'. Similarly this is also correct. Now we can
go to our 2nd step. So now our 2nd step will be to remove this
ax to the power (n - 1). So for that we need to subtract it by ax to the power (n - 1).
So what should the quotient value be here to get this value. So for that we take this
ax to the power (n - 1) and divide it by this x value. So that will be equal to ax to the
power (n - 2). So this is our quotient value. Now that we got this, then we multiply this
(-a) with ax to the power (n - 2). So that will be - a squaredx to the power (n - 2).
So here we will have zero. 0 - (-a squaredx to the power (n - 2) will be plus a squaredx
to the power (n - 2). So here this and this will cancel off and only this a squared into
x to the power (n - 2) will remain. So our next step will be to cancel off this
a squaredx to the power (n - 2). So for this we divide a squared x to the power (n - 2)
by x giving a squared into x to the power (n - 3). So basically the same process will
go for 'n' number of terms. Now lets analyze this quotient here. So here
we can see that the power of 'x' reduces 1 by 1 in the consecutive term. So here the
1st term we have x to the power (n - 1) so the next one we have x to the power (n - 2),
the 3rd term we have x to the power (n - 3). So the nth term will have (n - n) that means
x to the power zero which will be 1. Similarly when we analyze the power of a we
can see that the power of a increases by 1 in the next term. So here we have a to the
power zero that is 1. So here we have a to the power 1, the 2nd term. Then the 3rd term
will be a to the power 2. So the nth term will be a to the power (n - 1). So now when
we do this long division, our nth term will have the final remainder as zero. How do we
know that this is zero, we proved it in the beginning of our video here, that when we
divide x to the power n - a to the power n by (x - a) we get the remainder as 0 So now lets look into our theorem. So here
we have limit of (x to the power n - a to the power n) divided by (x - a) as x tends
to a. So instead of this function, we write our quotient here. So that would be limit
of (x to the power (n - 1) + ax to the power (n - 2) similarly until we go for 'n' number
of terms as x tends to a. Now we use our direct substitution here. So
when we use our direct substitution we get 'a' to the power (n - 1) + aa to the power
(n - 2) + a squareda to the power (n - 3) until we go for 'n' number of terms. So finally
we get 'a' to the power (n - 1) + 'a' to the power (n - 1) so all the terms will have 'a'
to the power (n - 1). So this will continue for 'n' number of terms. That means we have
na to the power (n - 1). So finally we get the answer that we want. So this proves that
the limit of x to the power n - a to the power n divided by (x - a) as x tends to a = na
to the power (n - 1). So that is the proof of this theorem using
algebraic division. In our next video we will show that this theorem is true for even for
negative values of 'n' Comment down below if you have any questions. Until we meet again,
take care. |
5906 | https://study.com/skill/learn/how-to-find-horizontal-and-vertical-asymptotes-of-a-rational-function-with-a-quadratic-numerator-or-denominator-explanation.html | How to Find Horizontal and Vertical Asymptotes of a Rational Function with a Quadratic Numerator or Denominator | Precalculus | Study.com
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How to Find Horizontal and Vertical Asymptotes of a Rational Function with a Quadratic Numerator or Denominator
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00:05 How to find horizontal…
01:56 How to find horizontal…
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Kathryn Boddie, Amy McKenney
Instructors Kathryn Boddie
Kathryn has taught high school or university mathematics for over 10 years. She has a Ph.D. in Applied Mathematics from the University of Wisconsin-Milwaukee, an M.S. in Mathematics from Florida State University, and a B.S. in Mathematics from the University of Wisconsin-Madison.
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Amy McKenney
Amy has taught high school mathematics for over 14 years. She has a master's degree in education from Plymouth State University and her undergraduate degree in mathematics. She is certified to teach grades 7-12 mathematics.
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Example SolutionsPractice Questions
Steps for Finding Horizontal and Vertical Asymptotes of a Rational Function with a Quadratic Numerator or Denominator.
Step 1: Find the horizontal asymptote by comparing the degrees of the numerator and denominator.
If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.
If the degree of the denominator is greater than the degree of the numerator, the horizontal asymptote is y=0.
If the degrees are equal, then the horizontal asymptote is y=a b, where a is the leading coefficient of the numerator and b is the leading coefficient of the denominator.
Step 2: Fully factor the numerator and denominator. Cancel any common factors. Any factors that cancel completely from the denominator correspond to holes in the graph rather than vertical asymptotes.
Step 3: Find any vertical asymptotes of the rational function by setting each remaining factor of the denominator equal to zero. The solutions to the resulting equations are the vertical asymptotes of the function.
Vocabulary and Equations for Finding Horizontal and Vertical Asymptotes of a Rational Function
Rational Function: A rational function is a function that is made of a ratio of polynomials. Polynomial functions are functions of the form p(x)=a n x n+a n−1 x n−1+…+a 2 x 2+a 1 x+a 0 where each a i is a real number, a n≠0 is called the leading coefficient, and n≥0 is called the degree.
Vertical Asymptote: A vertical asymptote is a vertical line x=c that the graph of a function cannot touch. The function is undefined at x=c and the graph either goes up forever or down forever as it approaches the vertical asymptote.
Horizontal Asymptote: A horizontal asymptote is a horizontal line y=d that the graph of a function approaches as x gets larger in the positive or negative direction.
We will use these steps, definitions, and equations to find horizontal and vertical asymptotes of a rational function with a quadratic numerator or denominator in the following two examples.
Example Problem 1 - Horizontal and Vertical Asymptotes - Quadratic Numerator
Find the horizontal and vertical asymptotes of f(x)=3 x 2+6 x x−1.
Step 1: Find the horizontal asymptote by comparing the degrees of the numerator and denominator.
If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.
If the degree of the denominator is greater than the degree of the numerator, the horizontal asymptote is y=0.
If the degrees are equal, then the horizontal asymptote is y=a b, where a is the leading coefficient of the numerator and b is the leading coefficient of the denominator.
The degree of the numerator is 2 since the highest exponent in the numerator is 2. The degree of the denominator is 1 since the highest exponent in the denominator is 1.
Since the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.
Step 2: Fully factor the numerator and denominator. Cancel any common factors. Any factors that cancel completely from the denominator correspond to holes in the graph rather than vertical asymptotes.
We can factor the numerator by factoring out the greatest common factor 3 x.
f(x)=3 x 2+6 x x−1 f(x)=3 x(x+2)x−1
There are no common factors.
Step 3: Find any vertical asymptotes of the rational function by setting each remaining factor of the denominator equal to zero. The solutions to the resulting equations are the vertical asymptotes of the function.
To find any vertical asymptotes, we need to set any factor remaining in the denominator equal to zero. We only have one factor, x−1 in the denominator.
x−1=0 x=1
The vertical asymptote is x=1.
Example Problem 2 - Horizontal and Vertical Asymptotes - Quadratic Denominator
Find the horizontal and vertical asymptotes of f(x)=4 x−8 3 x 2−5 x−2.
Step 1: Find the horizontal asymptote by comparing the degrees of the numerator and denominator.
If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.
If the degree of the denominator is greater than the degree of the numerator, the horizontal asymptote is y=0.
If the degrees are equal, then the horizontal asymptote is y=a b, where a is the leading coefficient of the numerator and b is the leading coefficient of the denominator.
The degree of the denominator (2) is greater than the degree of the numerator (1), and so the horizontal asymptote is:
y=0
Step 2: Fully factor the numerator and denominator. Cancel any common factors. Any factors that cancel completely from the denominator correspond to holes in the graph rather than vertical asymptotes.
Factoring the numerator and denominator:
f(x)=4 x−8 3 x 2−5 x−2 f(x)=4(x−2)(3 x+1)(x−2)
Simplifying by canceling common factors,
f(x)=4 3 x+1
We have a hole when x−2=0, so at x=2, but this is not a vertical asymptote.
Step 3: Find any vertical asymptotes of the rational function by setting each remaining factor of the denominator equal to zero. The solutions to the resulting equations are the vertical asymptotes of the function.
3 x+1=0 x=−1 3
The vertical asymptote is x=−1 3.
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Table of Contents
Steps for Finding Horizontal and Vertical Asymptotes of a Rational Function with a Quadratic Numerator or Denominator.
Vocabulary and Equations for Finding Horizontal and Vertical Asymptotes of a Rational Function
Example Problem 1 - Horizontal and Vertical Asymptotes - Quadratic Numerator
Example Problem 2 - Horizontal and Vertical Asymptotes - Quadratic Denominator
Test your current knowledge
Practice
Finding Horizontal and Vertical Asymptotes of a Rational Function with Quadratic Numerator or Denominator
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Word Problems
24 Ratio Word Problems for Grades 6-7 With Tips On Supporting Students’ Progress
Emma Johnson
Ratio word problems are introduced for the first time in 6th grade. The earliest mention of ‘Ratio’ is in the 6th grade Ratio and Proportional Relationships strand of the Common Core State Standards for math.
At this early stage, it is essential to concentrate on the language and vocabulary of ratio relationships. Children need to be clear on the meaning of the ratio symbol right from the start of the topic. Word problems really help children understand this concept since they make it much more relevant and meaningful than a ratio question with no context.
Concrete resources and visual representations are key to the success of children’s early understanding of ratio. These resources are often used in 2nd grade word problems, 3rd grade word problems and 4th grade word problems. There is often a misconception amongst upper elementary teachers and students, that mathematical manipulatives are only for children who struggle in math. However, all students should be introduced to this new concept through resources, such as two-sided counters and visual representations, such as bar models as this can help with understanding basic mathematical concepts such as addition and subtraction word problems.
As students progress across grades, they continue to build on their knowledge and understanding of ratio. This means that students gradually move away from the practical and visual resources, while word problems continue to be a key element to any lessons involving ratios. This foundational ratio knowledge is used as a base for proportional relationships and then linear relationships in later grades.
Ratio word problems are also an essential component of any lessons on ratio, to help children understand how ratios are used in real life.
Ratio Check for Understanding Quiz
10 questions with answers covering ratio to test your 6th and 7th grade student understanding of ratios.
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Ratio word problems
Schools following CCSS
Ratios in 6th grade
Children are first introduced to ratio and ratio problems in 6th grade:
Recognize and write ratio relationships using conventional math notation
Recognize and write unit rate relationships using conventional math notation
Create tables of equivalent ratios and use the tables to solve problems, including measurement problems
Calculate unit rates and use unit rates to solve problems
Recognize and write percents (a special type of ratio)
Calculate a whole given a part or a percent
Note that all original ratio relationships in 6th grade involve only whole numbers – however students are expected to create equivalent ratios that involve fractions or decimals (particularly for unit rates)
Ratios in 7th grade
Students in 7th grade continue to build on their knowledge of ratio from 6th grade.
Calculate unit rates and solve problems for ratios comparing two fractions
Solve problems involving proportional relationships
Solve problems involving percentage change, including: percentage increase, decrease and original value problems and simple interest in financial mathematics.
Use scale factors to draw smaller or larger geometric figures
Calculate the scale factor between two similar geometric figures
Schools not following CCSS
Schools that do not follow the Common Core will most likely cover the topics above but may do so in a different order or a different grade levels. Consult with your school’s specific curriculum map for clarification.
Why are word problems important for children’s understanding of ratio?
Solving word problems is important for helping children develop their understanding of ratio and the different ways ratio is used in everyday life. Without this context, ratio can be quite an abstract concept, which children find difficult to understand. Word problems bring ratios to life and enable students to see how they will make use of this skill outside the classroom.
Third Space Learning’s online one-on-one tutoring programs relate math concepts to real-world situations to deepen conceptual understanding. The online lessons are personalized to fill the gaps in each individual student’s math knowledge, helping them to build skills and confidence.
How to teach ratio word problem-solving in middle school
It is important that children learn the skills needed to solve ratio word problems. As with any math problem, children need to make sure they have read the questions carefully and thought about exactly what is being asked and whether they have fully understood this. The next step is to identify what they will need to do to solve the problem and whether there are any concrete resources or visual representations that will help them. Even older students can benefit from drawing a quick sketch to understand what a problem is asking.
Here is an example:
Jamie has a bag of red and yellow candies.
For every red candy, there are 2 yellow candies.
If the bag has 6 red candies, how many candies are in the whole bag?
How to solve:
What do you already know?
We know that for every red candy, there are 2 yellow candies.
If there are 6 red candies, we need to find how many yellow candies there must be.
If there are 2 yellow candies for each 1 red candy, then we can multiply 2 by 6, to calculate how many yellow candies there are with 6 red candies.
Once we have calculated the total number of yellow candies (12), we then need to add this to the 6 red candies, to work out how many candies are in the bag altogether.
If there are 6 red candies and 12 yellow candies, there must be 18 candies in the bag altogether.
How can this be represented visually?
We can use the two-sided counters to represent the red and yellow candies.
We put down 1 red counter and 2 yellow counters.
We then need to repeat this 6 times, until there are 6 red counters and 12 yellow counters.
We can now visually see the answer to the word problem and that there are now a total of 18 counters (18 candies in the bag).
Meet Skye, the voice-based AI tutor making math success possible for every student.
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Ratio word problems for 6th grade
Word problems for 6th grade often incorporate multiple skills: a ratio word problem may also include elements from multiplication word problems, division word problems, percentage word problems and fraction word problems.
Question 1
Sophie was trying to calculate the number of students in her school.
She found the ratio of boys to girls across the school was 3:2.
If there were 120 boys in the school…
How many girls were there?
How many students were there altogether?
Answer: a) 80 b) 200
a. 80 girls
This can be shown as a bar model.
120 ÷ 3 = 40
40 x 2 = 80
b. 200 students altogether
120 boys + 80 girls = 200
Question 2
Students in the Eco Club in 6th Grade wanted to investigate how many worksheets were being printed each week in Math and English.
They found that there were 160 Math worksheets and 80 English worksheets being printed each week.
What is the ratio of Math to English worksheets? Write the ratio in simplest form.
Answer: 2:1
The ratio 160:80 can be simplified by dividing both sides by 80.
Question 3
The 6th-grade rugby club has 30 members. The ratio of boys to girls is 4:1. How many boys and girls are in the club?
Answer: 24 boys and 6 girls
The ratio of 4:1 has 5 parts:
30 ÷ 5 = 6
Boys: 4 x 6 = 24
Girls: 1 x 6 = 6
Question 4
Yasmine has a necklace with purple and blue beads.
The ratio of purple:blue beads = 1:3
There are 24 beads on the necklace. How many purple and blue beads are there?
Answer: 6 purple beads and 18 blue beads
The ratio of 1:3 has 4 parts and 24 ÷ 4 = 6 beads per part.
Purple: 1 x 6 = 6
Blue: 3 x 6 = 18
Question 5
Maisie drives past a field of sheep and cows.
She figures out that the ratio of sheep to cows is 3:1.
If there are 5 cows in the field, how many sheep are there?
Answer: 15 sheep
If there are 5 cows in the field, the 1 has been multiplied by 5.
We need to also multiply the 3 by 5, which is 15.
Question 6
At a party, there is a choice of 2 flavors of jelly beans – orange and lemon.
The ratio of the jelly beans is 3:1 (orange: lemon).
What percent of the jelly beans are orange?
Answer: 75%
To find the percent, we need to write the ratio of orange jelly beans to total jelly beans. Since for every 3 orange jelly beans, there are 4 jelly beans in total (3 orange + 1 lemon = 4 total), the ratio is 3 to 4 or ¾.
To convert the ratio to a percent, the denominator needs to be 100.
Multiply both parts of the fraction by 25.
75/100 is equal to 75%.
Question 7
The school photocopier prints out 150 sheets in 3 minutes.
How many sheets can it print out in 15 minutes?
Answer: 750 sheets in 15 minutes
We need to multiply 3 by 5 to get 15 minutes. This means we also need to multiply 150 by 5 = 750.
Question 8
Rowen wants to buy $80 worth of books. He will have to pay a 5% tax. How much will Rowen pay in total for the books?
Answer: $84
Since $8 is 10% of $8, then half will be 5%, which is $4.
$80 + $4 = $84
Ratio word problems for 6th grade
Question 1
David has 2 grandchildren: Olivia (age 6) and Mia(age 3)
He decides to share $60 between the 2 children in a ratio of their ages.
How much does each child get?
Answer: Olivia gets $40, Mia gets $20
Ratio of 2:1 = 3 part and 60 ÷ 3 = $20 per part.
Olivia: 2 x 20 = $40
Mia: 1 x 20 = $20
Question 2
A rectangle has the ratio of width to length 2:3. If the perimeter of the rectangle is 50cm, what’s the area?
Answer: Area: 150cm2
The ratio has 5 parts:
Divide 50 by 5 to work out 1 part = 10
The 2 widths must be 2 x 10 = 20
The 2 lengths must be 3 x 10 = 30
To work out the width of 1 side, divide the 20 by 2 = 10
To work out the length of 1 side, divide the 30 by 2 = 15
Width: 10cm and Length: 15cm
Area: 10 x 15 = 150cm2
Question 3
For a class field trip, 45% of students in Kinley’s class want to go to the zoo. The other students want to go to see a play. If 11 students want to go see a play, how many total students are in Kinley’s class?
Answer: 20 students
Since 5% is 1 student, then 45% is 9 students.
Adding the students that want to go to the zoo and the students that want to see a play, is the whole class, or 100% of the students.
11 + 9 = 20 students
Question 4
A piece of ribbon is 45cm long.
It has been cut into 3 smaller pieces in a ratio of 4:5.
How long is each piece?
Answer: 20cm, 25cm
4:3:2 =9 parts: 45 ÷ 9 = 5cm per part.
4 x 5 = 20cm
3 x 5 = 25cm
Question 5
Chloe is making a smoothie for her and her 3 friends.
She has the recipe for making a smoothie for 4 people: 240ml yogurt, 120 ml milk, 2 bananas, 180g strawberries and 1 tablespoon of sugar.
How much yogurt would be needed to make a smoothie for 8 people.
How many g of strawberries are needed to make the smoothie for 2 people?
Answer:
480 ml yogurt
240ml x 2 = 480
90g strawberries
180 ÷ 2 = 90
Question 6
The ratio of cups of flour: cups of water in the recipe for making the dough for a pizza base is 7:4.
The pizza restaurant needs to make a large number of pizzas and is using 42 cups of flour. How much water will be needed?
Answer: 24 cups of water
Multiply 7 by 6 to get 42 cups of flour.
We, therefore, need to also multiply the 4 by 6 to calculate how many cups of water are needed.
4 x 6 = 24
Question 7
Muhammad shared $56 between him and his brother Hamza in a ratio of 3:5 (3 for Hamza and 5 for him).
How much did each get?
Answer: Muhammad got $35, his brother got $21
Ratio of 3:5 = 8 parts
56 ÷ 8 = $7 per part
3 x 7 =$ 21
5 x 7 = $35
Question 8
Scott has read 40% of his book. If he has 168 pages left, how many total pages does the book have?
Answer: 280 pages
100% – 40% = 60%
The 168 pages represent the 60% that Scott has not read.
Since 60% is 168, dividing both by 6 shows that 10% is 28. And multiplying by 10 shows how many pages are in 100% of the book.
Ratio word problems for 7th grade
Question 1
For every 250 tickets sold, the theater receives $687.50.
How much money did the theater charge for 1 ticket?
Answer: $2.75
The original ratio of tickets sold to dollars the theater made is 250 : 687.50.
To get liters to 1, divide both sides by 250.
Question 2
A faucet drips ⅓ of a liter of water in ½ of an hour. How long does it take the faucet to drip 1 liter?
Answer: 1 and ½ hours
The original ratio of liters to hours ⅓ : ½.
To get liters to 1, multiply by sides by 3.
When liters is 1, hours is equal to 3/2 or 1 and ½.
Question 3
The angles in a triangle are in the ratio of 3:4:5 for angles A, B and C.
Calculate the size of each angle.
Answer:
Angle A: 45°
Angle B: 60°
Angle C: 75°
3:4:5 = 12 parts. 180÷ 12 = 15 (each part is worth 15°)
3 x 15 = 45
4 x 15 = 60
5 x 15 = 75
Question 4
A drink is made by mixing pineapple and lemonade in the ratio of ⅔ of a cup to ⅘ of a cup. If Jayla has 1 cup of lemonade, how much pineapple should she add to keep the same drink ratio?
Answer: ⅚ of a cup
The original ratio of cups pineapple to lemonade is ⅔ :⅘.
To get cups of lemonade to 1, multiply by sides by 5/4.
When lemonade is 1 cup, pineapple is 10/12 or 5/6.
Question 5
A $74 pair of shoes is on sale for 60% off. If Kalani buys two pairs of the shoes and pays the 7% tax on the total cost, how much money did Kalani spend in all?
Answer: $63.34
100% – 60% = 40%, so Kalani will pay 40% of the original price.
$74 0.4 = $29.60
The two pairs, before tax, cost: $29.60 + $29.60 = $59.20
Since 7% = 0.07, multiply $59.20 times 0.07 to calculate the tax.
$59.20 0.07 = $4.144 (round down to $4.14).
$59.20 + $4.14 = $63.34
Note: You can also solve by multiplying $59.20 1.07 = $63.34
Question 6
Jaxton bought a video game console for $650 two years ago. He just sold it for $475. What is the percent decrease, to the nearest percent, in the price of the game console from when Jaxton bought it to when he sold it?
Answer: 27%
$650 – $475 = $175
Divide the difference by the original price.
$175 $650 = 0.269…
To convert the decimal to a percent, multiply by 100.
0.269 100 = 26.9%, which rounds up to 27%.
Question 7
Two companies are making an orange-colored paint.
Company A makes the orange paint by mixing red and yellow paint in a ratio of 5:7.
Company B makes the orange paint by mixing red and yellow paint in a ratio of 3:4.
Which company uses a higher proportion of red paint to make the orange?
Answer: Company B uses more red paint.
Company A: 5:7 = 5/12 is red
Company B: 3:4 = 3/7 is red
We can compare the fractions by giving them the same denominator, to find the equivalent fractions.
5/12 = 35/84
3/7 = 36/84
Question 8
$8,142 is invested in a savings account with a 0.2% simple interest rate per month. What is the total interest earned after 4 years?
Answer: $781.63
Use the equation I=Prt:
P = $8,142
r = 0.002
t = 4 years x 12 = 48 months
$8,142 (0.002 x 48) = $781.632
$781.632 rounds to $781.63
More word problems
Looking for word problems on more topics? Take a look at our practice problems for Years 3-6 including money word problems, time word problems, addition word problems and subtraction word problems.
Do you have students who need extra support in math?
Skye—our AI math tutor built by experienced teachers—provides students with personalized one-on-one, spoken instruction that helps them master concepts, close skill gaps, and gain confidence.
Since 2013, we’ve delivered over 2 million hours of math lessons to more than 170,000 students, guiding them toward higher math achievement.
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The content in this article was originally written by former UK Deputy Headteacher and has since been revised and adapted for US schools by math curriculum specialist and former elementary math teacher Katie Keeton.
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12: Acids and Bases
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12.5: pH and Kw
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12.4: The Strengths of Acids and Bases
12.6: Buffers
Page ID 17495
( \newcommand{\kernel}{\mathrm{null}\,})
Table of contents
1. pH
2. The Effective Range of the pH Scale
3. Living Systems
4. Contributors
Skills to Develop
To define the pH scale as a measure of acidity of a solution
Because of its amphoteric nature (i.e., acts as both an acid or a base), water does not always remain as H 2O molecules. In fact, two water molecules react to form hydronium and hydroxide ions:
(12.5.1)2H 2O(l)⇌H 3O+(aq)+OH−(aq)
This is also called the self-ionization of water. The concentration of H 3O+ and OH− are equal in pure water because of the 1:1 stoichiometric ratio of Equation 12.5.1. The molarity of H 3 O+ and OH- in water are also both 1.0×10−7 M at 25° C. Therefore, a constant of water (K w) is created to show the equilibrium condition for the self-ionization of water. The product of the molarity of hydronium and hydroxide ion is always 1.0×10−14.
(12.5.2)K w=[H 3O+][OH−]=1.0×10−14
This equations also applies to all aqueous solutions. However, K w does change at different temperatures, which affects the pH range discussed below.
Note
H+ and H 3O+ is often used interchangeably to represent the hydrated proton, commonly call the hydronium ion.
pH
Because K w is constant (1.0×10−14 at 25 °C, the pK w is 14, the constant of water determines the range of the pH scale. To understand what the pK w is, it is important to understand first what the "p" means in pOH, and pH. The Danish biochemist Søren Sørenson proposed the term pH to refer to the "potential of hydrogen ion." He defined the "p" as the negative of the logarithm, -log, of [H+]. Therefore the pH is the negative logarithm of the molarity of H. The pOH is the negative logarithm of the molarity of OH- and the pK w is the negative logarithm of the constant of water. These definitions give the following equations:
(12.5.3)pH=−log[H+]
(12.5.4)pOH=−log[OH−]
(12.5.5)pK w=−log[K w]
A logarithm, used in the above equations, of a number is how much a power is raised to a particular base in order to produce that number. To simplify this, look at the equation: log b a=x. This correlates to b x=a. A simple example of this would be log 10100=2, or 10 2=100. It is assumed that the base of Logarithms is ten if it is not stated. So for the sake of pH and pOH problems it will always be ten. When x is a negative number that means you are dividing it by the power. So, if log 100.01=−2 which can be written 10−2=0.01, then 10−2 also means 1/10 2. The log function can be found on your scientific calculator. Now if we apply this to pH and pOH we can better understand how we calculate the values.
At room temperature,
(12.5.6)K w=1.0×10−14
So
(12.5.7)pK w=−log[1.0×10−14]
Using the properties of logarithms, Equation 12.5.7 can be rewritten as
(12.5.8)10−pK w=10−14.
By substituting, we see th at pK w is 14. The equation also shows that each increasing unit on the scale decreases by the factor of ten on the concentration of H+. For example, a pH of 1 has a molarity ten times more concentrated than a solution of pH 2.
Since
(12.5.9)pK w=14
(12.5.10)pK w=pH+pOH=14
Note
The pH scale is logarithmic, meaning that an increase or decrease of an integer value changes the concentration by a tenfold. For example, a pH of 3 is ten times more acidic than a pH of 4. Likewise, a pH of 3 is one hundred times more acidic than a pH of 5. Similarly a pH of 11 is ten times more basic than a pH of 10.
The Effective Range of the pH Scale
It is common that the pH scale is argued to range from 0-14 or perhaps 1-14 but neither is correct. The pH range does not have an upper nor lower bound, since as defined above, the pH is an indication of concentration of H+. For example, at a pH of zero the hydronium ion concentration is one molar, while at pH 14 the hydroxide ion concentration is one molar. Typically the concentrations of H+ in water in most solutions fall between a range of 1 M (pH=0) and 10-14 M (pH=14). Hence a range of 0 to 14 provides sensible (but not absolute) "bookends" for the scale (Figure 12.5.1). One can go somewhat below zero and somewhat above 14 in water, because the concentrations of hydronium ions or hydroxide ions can exceed one molar. Figure 1 depicts the pH scale with common solutions and where they are on the scale.
Figure 12.5.1: Solutions and the placement of them on pH scale
From the range 7-14, a solution is basic. The pOH should be looked in the perspective of OH- instead lf H+. Whenever the value of pOH is greater than 7, then it is considered basic. And therefore there are more OH- than H+ in the solution
At pH 7, the substance or solution is at neutral and means that the concentration of H+ and OH- ion is the same.
From the range 1-7, a solution is acidic. So, whenever the value of a pH is less than 7, it is considered acidic. There are more H+ than OH- in an acidic solution.
Note
The pH scale does not have an upper nor lower bound. Negative pH values are possible
The following PhET simulation helps to build an intuition of the pH scale.
Example 12.5.1
If the concentration of NaOH in a solution is 2.5×10−4 M, what is the concentration of H 3O+?
SOLUTION
Because
1.0×10−14=[H 3O+][OH−]
to find the concentration of H 3 O+, solve for the [H 3 O+].
1.0×10−14[OH−]=[H 3O+]
1.0×10−14 2.5×10−4=[H 3O+]=4.0×10−11 M
Example 12.5.2
Find the pH of a solution of 0.002 M of HCl.
Find the pH of a solution of 0.00005 M NaOH.
SOLUTION
The equation for pH is -log [H+]
[H+]=2.0×10−3 M
pH=−log[2.0×10−3]=2.70
The equation for pOH is -log [OH-]
[OH−]=5.0×10−5 M
pOH=−log[5.0×10−5]=4.30
pK w=pH+pOH
and
pH=pK w−pOH
then
pH=14−4.30=9.70
Living Systems
Molecules that make up or are produced by living organisms usually function within a narrow pH range (near neutral) and a narrow temperature range (body temperature). Many biological solutions, such as blood, have a pH near neutral. pH influences the structure and the function of many enzymes (protein catalysts) in living systems. Many of these enzymes have narrow ranges of pH activity (Table 12.5.1).
Table 12.5.1: pH in Living SystemsCompartmentpH
Gastric Acid 1
Lysosomes 4.5
Granules of Chromaffin Cells 5.5
Human Skin 5.5
Urine 6
Neutral H 2 O at 37 °C 6.81
Cytosol 7.2
Cerebrospinal Fluid 7.3
Blood 7.43-7.45
Mitochondrial Matrix 7.5
Pancreas Secretions 8.1
Cellular pH is so important that death may occur within hours if a person becomes acidotic (having increased acidity in the blood). As one can see pH is critical to life, biochemistry, and important chemical reactions. Common examples of how pH plays a very important role in our daily lives are given below:
Water in swimming pool is maintained by checking its pH. Acidic or basic chemicals can be added if the water becomes too acidic or too basic.
Whenever we get a heartburn, more acid build up in the stomach and causes pain. We needs to take antacid tablets (a base) to neutralize excess acid in the stomach.
The pH of blood is slightly basic. A fluctuation in the pH of the blood can cause in serious harm to vital organs in the body.
Certain diseases are diagnosed only by checking the pH of blood and urine.
Certain crops thrive better at certain pH range.
Enzymes activate at a certain pH in our body.
Contributors
Emmellin Tung (UCD), Sharon Tsao (UCD), Divya Singh (UCD), Patrick Gormley (Lapeer Community School District)
12.5: pH and Kw is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by LibreTexts.
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12.6: Buffers
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12.1: Arrhenius Definition of Acids and Bases
12.2: Brønsted-Lowry Definition of Acids and Bases
12.3: Water is both an Acid and a Base
12.4: The Strengths of Acids and Bases
12.6: BuffersA buffer is a solution that resists sudden changes in pH.
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What is gerrymandering in U.S. politics?
Gerrymandering is the practice of drawing electoral district boundaries to give one political party an advantage (political gerrymandering) or to dilute the voting power of racial or ethnic minority groups (racial gerrymandering).
Is gerrymandering legal?
Political gerrymandering is legal in states that permit their legislatures to redraw congressional districts to increase the number of congressional representatives of the legislative majority or to limit the number of congressional representatives of the legislative minority. Political gerrymandering cannot be challenged in federal courts, but racial gerrymandering can.
Where does the term “gerrymandering” originate from?
The term originates from the name of Gov. Elbridge Gerry of Massachusetts, whose 1812 law created districts benefiting his party. The shape of one district resembled a salamander, inspiring the term.
What Supreme Court case established the “one person, one vote” principle?
Gray v. Sanders (1963) established the “one person, one vote” principle by striking down Georgia’s county-based voting system.
What did the Supreme Court rule in Thornburg v. Gingles (1986)?
The Supreme Court ruled that racial gerrymandering violates Section 2 of the Voting Rights Act, which prohibits practices that effectively reduce the voting power of racial minority groups.
What did the Supreme Court rule in Rucho v. Common Cause (2019)?
The Supreme Court ruled that “partisan gerrymandering claims present political questions beyond the reach of the federal courts.”
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gerrymandering, in U.S. politics, the practice of drawing the boundaries of electoral districts in a way that gives one political party an advantage over its rivals (political or partisan gerrymandering) or that dilutes the voting power of members of racial or ethnic minority groups (racial gerrymandering). The term is derived from the name of Gov. Elbridge Gerry of Massachusetts, whose administration enacted a law in 1812 defining new state senatorial districts. The law consolidated the Federalist Party vote in a few districts and thus gave disproportionate representation to Democratic-Republicans. The outline of one of these districts was thought to resemble a salamander. A satirical cartoon by Elkanah Tisdale that appeared in the Boston Gazette graphically transformed the districts into a fabulous animal, “The Gerry-mander,” fixing the term in the popular imagination.
See also What is gerrymandering?
A basic objection to gerrymandering of any kind is that it tends to violate two tenets of electoral apportionment—compactness and equality of size of constituencies. The constitutional significance of the latter principle was set forth in a U.S. Supreme Court ruling issued in 1962, Baker v. Carr, in which the Court held that the failure of the legislature of Tennessee to reapportion state legislative districts to take into account significant changes in district populations had effectively reduced the weight of votes cast in more populous districts, amounting to a violation of the equal protection clause of the Fourteenth Amendment. In 1963, in Gray v. Sanders, the Court first articulated the principle of “one person, one vote” in striking down Georgia’s county-based system for counting votes in Democratic primary elections for the office of U.S. senator. One year later, in Wesberry v. Sanders, the Court declared that congressional electoral districts must be drawn in such a way that, “as nearly as is practicable, one man’s vote in a congressional election is to be worth as much as another’s.” And in the same year, the Court affirmed, in Reynolds v. Sims, that “the Equal Protection Clause requires that the seats in both houses of a bicameral state legislature must be apportioned on a population basis.”
Regarding cases of gerrymandering based on race, the Supreme Court has held (in Thornburg v. Gingles, 1986) that such practices are incompatible with Section 2 of the 1965 Voting Rights Act (as amended in 1982), which generally prohibits voting standards or practices whose practical effect is that members of racial minority groups “have less opportunity than other members of the electorate to…elect representatives of their choice.” In Shaw v. Reno (1993), the Court ruled that electoral districts whose boundaries cannot be explained except on the basis of race can be challenged as potential violations of the equal protection clause, and in Miller v. Johnson (1995) it held that the equal protection clause also prohibits the use of race as the “predominant factor” in drawing electoral-district boundaries.
Until the 1980s, disputes regarding political gerrymandering were generally considered nonjusticiable (not decidable by federal courts) on the presumption that they presented “political questions” that are properly decided by the legislative or the executive branch. In Davis v. Bandemer (1986), however, a plurality of the Supreme Court held that political gerrymanders could be found unconstitutional (under the equal protection clause) if the resulting electoral system “is arranged in a manner that will consistently degrade a voter’s or a group of voters’ influence in the political process as a whole.” A majority of the Court also agreed that the instance of gerrymandering before it did not display any of the “identifying characteristics of a nonjusticiable political question” that had been laid out in Baker v. Carr, including, as the Baker Court had put it, “a lack of judicially discoverable and manageable standards for resolving it.” Although the majority in Bandemer could not agree on what standards should be used to adjudicate challenges to political gerrymanders, it refused to accept that none existed, declaring on that basis that “we decline to hold that such claims are never justiciable.”
In 2004, in Vieth v. Jubelirer, a plurality of the Court pointedly embraced what the Bandemer Court had declined to hold, on the grounds that “no judicially discernible and manageable standards for adjudicating political gerrymandering claims have emerged” since the Bandemer decision. Although siding with the plurality in rejecting the challenge to the political gerrymander in question, Justice Anthony Kennedy asserted that it had not been long enough since the Bandemer decision to conclude that no suitable standards could ever emerge (“by the timeline of the law 18 years is rather a short period”). Pointing to the rapid development and routine use of computer-assisted districting, he argued that such technologies “may produce new methods of analysis that…would facilitate court efforts to identify and remedy the burdens” imposed by political gerrymanders, “with judicial intervention limited by the derived standards.”
Just such a standard was proposed in Gill v. Whitford (2018), a challenge to a Wisconsin redistricting law enacted by the Republican-controlled state legislature following the 2010 decennial census. In that case, the plaintiffs argued that the discriminatory effects of the redistricting plan could be measured objectively by comparing the “efficiency” of votes cast for Republican or Democratic candidates in state legislative elections since 2012. Political gerrymandering characteristically results in a greater number of “wasted” votes for the disfavoured party (i.e., votes for a losing candidate or votes for a winning candidate in excess of the number needed to win), a discrepancy that can be represented as an “efficiency gap” between the parties when the difference between wasted votes is divided by the total number of votes cast. The plaintiffs argued that efficiency gaps of 7 percent or greater were legally significant because they were more likely than smaller gaps to persist through the 10-year life of a redistricting plan. The Court’s ruling, however, did not consider whether the efficiency gap amounted to the “judicially discernible and manageable” standard it had been waiting for. Instead, the justices held unanimously (9–0) that the plaintiffs lacked standing to sue, and the case was remanded (7–2) to the district court for further argument.
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Following Kennedy’s retirement in 2018, the Supreme Court once again took up the issue of the justiciability of political gerrymandering claims in Rucho v. Common Cause (2019). There the Court’s conservative majority, over the bitter objections of its more liberal members, declared (5–4) that “partisan gerrymandering claims present political questions beyond the reach of the federal courts.”
Following the decennial census of 2020, the dominant parties in many states took advantage of the Rucho ruling to openly gerrymander congressional districts to their own advantage. In 2025, at the request of Republican President Donald Trump, the state of Texas departed from convention by holding a special redistricting session well before the next decennial census. The acknowledged purpose of the political gerrymandering was to ensure the addition of five more Republican congressional representatives in the midterm elections of 2026, thus preventing Democrats from overcoming the Republicans’ narrow majority in the House of Representatives. Trump also urged other Republican-controlled states to conduct similar gerrymanders. In response, California’s Democratic governor Gavin Newsom announced that his state would conduct its own redistricting to prevent Republicans from unjustly retaining their House majority. A few other Democratic governors, accusing Republicans of undermining American democracy, also indicated their willingness to facilitate political gerrymandering in their states.
Brian Duignan The Editors of Encyclopaedia Britannica |
5910 | http://localwww.math.unipd.it/~silvio/papers/ModalLogic/DModalLogic.pdf | 11/5/01 - 1 -The sequent calculus for the modal logic D by Silvio Valentini Dep. of Matematica Pura ed Applicata via Belzoni 7 - 35131 Padova (Italy) Summary We present a sequent calculus for the deontic logic D and prove its main syntactic and semantic properties, i.e. cut-elimination, interpolation, completeness with respect to serial frames, finite model property and decidability.
1. Introduction The modal logic D (for deontic) is usually presented as the extension of the minimal normal modal logic K by the axiom schema KA→¬K¬A [Seg], i.e. D is the minimal modal logic obtained by adding to the classical propositional calculus the axioms K-Ax: K(A→B)→(KA→KB) D-Ax: KA→¬K¬A and closing under MP: A A→B B and Nec: A KA It is easy to see that D can equivalently be obtained by adding to a standard sequent calculus for the classical propositional logic, for instance LK in [Tak], the modal rules: KR: X|—A KX|—KA and DR’: X|—A KX|—¬K¬A where KX stands for the set of formulas {KB:B∈X} if X is a set of formulas.
In fact the sequent A1,…,An| —B1,…,Bm is provable in this sequent calculus if and only if the formula A1∧…∧An→(B1∨…∨Bm) is a theorem of D and in particular we have | —B if and only if B is a theorem of D. Let us here show only the modal steps of the obvious proof by induction on the depth of the considered derivation since the non-modal ones are completely standard1 .
On one hand we immediately have 1A more detailed proof of this theorem is shown in [Val] for the case of the modal logic K and the sequent calculus obtained by adding to LK only the rule KR.
11/5/01 - 2 -K-Ax: A→B,A|—B K(A→B),KA|—KB K(A→B)|—KA→KB |—K(A→B)→(KA→KB) D-Ax: A|—A KA|—¬K¬A |—KA→¬K¬A Nec: |—A |—KA and, on the other one, D is closed under KR and DR’ since from C→A using K-Ax and MP we obtain KC→KA and hence using D-Ax and MP we have KC→¬K¬A and it is well known that, in K, KC1∧…∧KCn→K(C1∧…∧Cn).
Even if these rules are very natural they are not the simplest ones since the conclusion of DR’ can have more than one derivation, for instance by a ¬-introduction rule. This fact suggests the new rule DR: X|— KX|— where, according with the intended meaning of a sequent, the empty set on the right hand side both in the premise and in the conclusion stands for falsum, i.e. the empty disjunction. In a language which contains also the symbol ⊥ (to be interpreted in falsum) DR becomes X|—⊥ KX|—⊥ . It is easy to see that DR’ and DR are equivalent over a calculus for K, i.e. which contains KR [Val]. In fact on one side we have X|—A X,¬A|— KX,K¬A|— KX|—¬K¬A and on the other X|— X|—⊥ KX|—¬K¬⊥ ⊥|— |—¬⊥ |—K¬⊥ ¬K¬⊥|— KX|— .
In this way we have also proved that D is the modal logic obtained from K by adding only the axiom ¬K⊥, i.e. a particular instance of the characteristic axiom of D for A≡ ⊥ because in K ¬K¬⊥ is logically equivalent to ⊥, since in this case DR is a consequence of KR and an occurrence of the cut-rule.
In the following we will refer to the modal system defined by KR and DR by DS.
2. Cut-elimination for DS The theorem of cut-elimination can be easily proved for the sequent calculus DS by a standard double induction on the degree (principal induction) and the length of the thread (secondary induction) of the cut-formula. The steps to lower the thread and the non modal reductions are completely standard [Tak], while the modal reductions, characteristic of DS, are X|—A KX|—KA A,Y|—B KA,KY|—KB KX,KY|—KB ⇒ X|—A A,Y|—B X,Y|—B KX,KY|—KB and X|—A KX|—KA A,Y|— KA,KY|— KX,KY|— ⇒ X|—A A,Y|— X,Y|— KX,KY|— 11/5/01 - 3 -A standard consequence of cut-elimination is the interpolation theorem, which can be proved for DS by the well-known technique of Maehara-Takeuti [Tak]; i.e. we prove that if the sequent X| —Y is derivable in DS then, for any partition X1, X2 of X and any partition Y1, Y2 of Y there is a formula C, the interpolant, which contains only the propositional variables common both to the formulas in X1∪Y 1 and X2∪Y 2 such that the sequents X1| —Y 1,C and C,X2| —Y 1 are provable. Here we show only the modal steps of the usual proof by induction on the depth of a cut-free derivation of X| —Y in DS: (KR-1) Let us suppose that the sequents X1| —C and C,X2| —A are provable. Then obviously also the sequents KX1| —KC and KC,KX2| —KA are provable.
(KR-2) Let us suppose that the sequents X1| —A,C and C,X2| — are provable. Then it is not difficult to see that the sequents KX1| —KA,¬K¬C and ¬K¬C,KX2| — are also provable.
(DR) let us suppose that the sequents X1| —C and C,X2| — are provable then the sequents KX1| —KC and KC,KX2| — are provable.
3. Semantics Let us call serial [Seg] a Kripke frame such that for any x∈F there is a y∈F such that xRy, i.e. a frame such that “there is always a future”; then any theorem of D is true in any serial frame. In fact, since any frame verifies K-Ax and is closed under MP and Nec [Seg], we must only show that KA→¬K¬A holds in any serial frame; this is obvious since, for any point w of a Kripke frame, || —wKA,K¬A if and only if w has no successor. Since KR is valid in any Kripke frame [Val], we can equivalently show that DR is valid in any serial frame; in fact if || —wKX and wRy then || —yX, i.e. a D-countermodel for the sequent KX| — is also a countermodel for the sequent X| —. We have hence shown the validity of DS with respect to the class of the serial frames. We give now a proof of the completeness theorem which shows at the same time also cut redundancy, decidability and the finite model property for D.
We can set up a proof procedure for D which looks for the provability of a sequent X| —Y as follows. Let us write DRR: O X|—C1 … X|—Cn X|— P,KX|—KC1,…,KCn,Q O where P and Q are sets of propositional variables and X is a set of formulas, to mean that the conclusion is derivable if at least one of the premises is derivable (DRR stands for D Ramification Rule).
D is obviously closed under DRR (use weakening) and hence it is a valid rule, but much more interesting is that DRR is sufficient to derive any theorem of D. In fact, consider a sequent calculus whose rules are the standard propositional rules and DRR and whose axioms are the sequents X| —Y such that X∩Y≠∅, then the procedure we look for is simply “apply any applicable rule (cut excluded!!) and stop on the axioms”. First note that this procedure stops on any sequent since the premises of every rule contain only proper subformulas of the formulas present in the conclusion. Hence, provided the procedure is correct, we have proved that the logic 11/5/01 - 4 -D is decidable. Moreover since DRR is a valid rule any sequent the procedure declares to be provable is really provable without using any cut. On the other hand if the procedure states a sequent not to be provable then we can inductively construct a D-countermodel for that sequent following the “proof-tentative” produced by the procedure starting from the leafs and going toward the root.
In fact any leaf in the proof-tentative of a non provable sequent is obviously a sequent P| —Q, where P and Q are sets of propositional variables such that no pi∈P is a qj∈Q; hence it can be falsified at a reflexive point with a valuation which forces any pi∈P and no qj∈Q.
If any premise of a propositional rule is falsified by one point then also the conclusion of that rule is falsified by the same point.
Finally if we have an occurrence of DRR then, by inductive hypothesis, we already have constructed n+1 D-countermodels for the n+1 sequents X| —C1 … X| —Cn, X| —, and hence we can obtain a D-countermodel for P,KX| —KC1,…,KCn,Q simply by adding a new irreflexive point such that it forces any pi∈P and no qj∈Q and is linked by an intransitive relation to all the n+1 countermodels.
Hence we have proved that a sequent that the procedure states not to be provable can really be falsified in a D-countermodel, which is obviously finite since the procedure always stops in a finite number of steps, constructed using only irreflexive points except for the top-most ones.
Finally we observe that the cut-rule is redundant since the proof of a provable sequent that can be easily extracted by a successful proof tentative produced by the procedure uses no cut.
References [Seg] K. Segerberg, An essay in classical modal logic, Ph.D. dissertation, Uppsala 1971 [Tak] G. Takeuti, Proof Theory, North Holland 1975 [Val] S. Valentini, Cut-elimination in a modal sequent calculus for K, Boll. Un. Mat. It., (6), 1-B, (1982), pp. 119-130 |
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Understanding cube roots is essential for anyone dealing with mathematics, whether in academia, professional fields, or everyday life. In this comprehensive guide, we will explore the concept of cube roots, delve into the mathematics behind them, solve them, and introduce an invaluable tool to solve them: the cube root calculator. This tool simplifies the process of finding cube roots, making it accessible to students, professionals, and hobbyists alike.
What is a Cube Root?
In mathematics, the cube root of a number is defined as the fraction of the number that, when cubed (multiplied by itself three times), yields the original number. For example, the cube root of 27 is 3 because 3×3×3=27. This operation is the inverse of cubing a number.
Symbol and Notation
The exact cube root part of a number 𝑥 x is denoted as ³√𝑥. This notation is used in mathematical expressions and can also be input into our cube root calculator for quick calculations.
How Does the Cube Root Calculator Work?
Step-by-Step Usage
Input the Number: Enter the number for which you want to find the cube root into the calculator’s input field.
Calculate: Click the “Calculate” button to process the input.
View Results: The calculator displays the cube root of the given number in the output field. If the number is a perfect cube, it will display an integer; otherwise, it will show a decimal approximation.
Detailed Examples of Cube Roots
Example 1: Cube Root of a Perfect Cube
Number: 64
Cube Root: ³√64=4
Calculation: Since 4×4×4=64, the cube root of 64 is 4.
Example 2: Cube Root of a Non-Perfect Cube
Number: 10
Cube Root: ³√≈2.154
Calculation note: 10 is not a perfect cube, and its cube root does not result in an integer. The calculator approximates it to three decimal places.
Understanding Cube Root Calculations
Perfect Cubes
A perfect cube is a number that can be expressed as the cube of an integer. Examples include:
8=2³
27=3³
125=5³
Perfect cubes are important because their cube roots are integers, which are easier to understand, write and visualize.
Cube Roots of Negative Numbers
The cube root of a negative number is the negative of the cube root of the corresponding positive number. For example:
³√-27=−3 because (−3)×(−3)×(−3)=−27
Practical Applications of Cube Roots
Cube roots are not just academic; they have practical applications in various fields:
Geometry and Architecture: Calculating the dimensions of cubes and cuboid structures.
Physics and Engineering: Determining the volume and density of materials.
Everyday Problems: Figuring out storage requirements or container sizes.
Real-Life Example of Using Cube Roots
Example: Estimating Container Sizes
If a company needs to store 512 cubic feet of goods and wants to use a cubical storage container, the side length of the cubed root container can be calculated using the cube root:
³√512=8 feet per side
Conclusion
The cube root calculator is free online tool, a powerful tool that simplifies the process of finding cube roots. Whether you are a student, a professional, or just curious about mathematics, understanding how to calculate cube roots and their applications can greatly enhance your numerical literacy. This guide provides you with the knowledge, method, and tools to tackle any challenge involving cube roots, making complex calculations straightforward and accessible.
Frequently Asked Questions
What is a cube root?
A cube root of a number is defined as a value that, when cubed (multiplied by itself three times), returns the original number. For instance, the cube root of given number 27 is 3 because 3×3×3=27. Cube roots can apply to both positive and negative numbers, or digits, making them versatile in various mathematical calculations.
How can I calculate the cube root of a number using your cube root calculator?
To find the cube root using our root calculator, simply enter the given number into the input field and press the "Calculate" button. The calculator will quickly display the formula for the cube root, providing both exact solutions and approximate values depending on whether the number is a perfect cube.
What is the difference between cube roots and square roots?
Cube roots and the square root of roots are similar in that they both involve finding a number that produces a specific result when raised to a power (cubed or squared, respectively). However, while the square root of roots are concerned with finding a number that produces the original number when multiplied by itself once, cube roots involve multiplication of the number three times. Our tool focuses on cube roots, but understanding both concepts is beneficial for comprehensive mathematical knowledge.
Can the cube root calculator find cube roots of negative numbers?
Yes, the cube root calculator can determine the cube root of negative numbers. For example, the cube root of -27 is the exact cube root of -3, as (−3)×(−3)×(−3)=−27. This feature is particularly useful because it respects the mathematical property that cube roots of real numbers can indeed be negative.
How does the calculator handle non-perfect cubes?
When you enter a non-perfect cube into the form of the cube root calculator, it computes an answer using an approximate decimal value of the cube root. Non-perfect cubes do not have integer cube roots, so the calculator provides a precise decimal answer to help users understand the approximate value of the cube root.
What is meant by the "nth root," and does your calculator support nth root calculations?
The nth root of a number refers to a value that, when raised to the power of n, results in the original number. While our focus here is on cube roots (which are a special case of nth roots where n=3), understanding general nth roots can be beneficial for broader mathematical contexts. Our specific nth root using tool is designed for cube roots, but an nth root calculator might be useful for more general needs.
How accurate is the cube root calculator?
Our cube root calculator is designed to provide highly accurate results for both perfect and non-perfect cubes. For perfect cubes, it will return the exact integer cube root. For non-perfect cubes, it calculates a precise decimal approximation, ensuring users have reliable answer information for their mathematical queries.
Alize Mendez
June 25, 2024
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When are the sum, difference, product, and quotient of two monomials also a monomial?
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Sum/Difference of 2 monomials is (in simplest terms) a monomial when the 2 monomials are 'like terms
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The sum and difference of two monomials are monomials only when the terms are like terms. The product and quotient of two monomials will always yield a monomial, provided the denominator is not zero. Thus, understanding the nature of the terms involved is key to determining the outcome.
Explanation
In algebra, a monomial is a mathematical expression consisting of only one term. The terms are made up of numbers and variables combined using multiplication. To determine when the sum, difference, product, and quotient of two monomials is also a monomial, let’s look at each case individually:
Sum of Two Monomials: The sum of two monomials is a monomial only when the two monomials are like terms. Like terms have the same variable factors raised to the same powers. For example, 3 x 2+5 x 2=8 x 2 is a monomial, but 3 x 2+4 x=7 x 2+4 x is not a monomial.
Difference of Two Monomials: Similar to the sum, the difference of two monomials is a monomial when the terms are like terms. An example would be 6 y−2 y=4 y which is a monomial, whereas 6 y−3 y 2=3 y−3 y 2 is not a monomial.
Product of Two Monomials: The product of two monomials is always a monomial. This is because multiplying two expressions combines their coefficients and variables according to the rules of exponents. For example, 2 x⋅3 x 2=6 x 3 is a monomial.
Quotient of Two Monomials: The quotient of two monomials is also a monomial, provided that the denominator is not zero. For example, 2 x 8 x 3=4 x 2 is a monomial. If the quotient results in variables being canceled out, as in x 5 x 2=5 x it remains a monomial as well.
In summary, for sums and differences, the monomials must be like terms; for products and quotients, they will always yield a monomial as long as the denominator is not zero.
Examples & Evidence
Examples include 3 x 2+5 x 2=8 x 2 (sum) which is a monomial, and 2 x⋅3 x 2=6 x 3 (product) is also a monomial.
Algebraic definitions state that a monomial consists of a single term, and the operations on monomials have established rules regarding like terms and products/quotients.
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Factorise fully: a x−a y−c x+cy 7 x−7 y−k x+k y ak+a t−5 k−5 t pq+p s−9 q−9 s
m−9=3 2 m−12
Divide. 5×1 0−1 2×1 0 1
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Determinants of Matrices and Their Properties
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I tried gaussian elimination and ended up with:
⎡⎣⎢⎢⎢⎢v 1 v 2 v 3 5 2 v 1+v 4⎤⎦⎥⎥⎥⎥[v 1 v 2 v 3 5 2 v 1+v 4]
Then I used the rule that says If B is obtained from A by adding a multiple of a row of A to another row, then det(B)=det(A) but it's saying that the answer detB=detA=8 is incorrect.
Where did I go wrong?
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asked Apr 17, 2012 at 6:12
Kyle V.Kyle V.
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You can get a subscript using an underscore, e.g. v_1 produces v 1 v 1.joriki –joriki 2012-04-17 06:42:10 +00:00 Commented Apr 17, 2012 at 6:42
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Use the fact that the determinant is multilinear.
det⎛⎝⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜r 1⋮r i−1 r i+α s i r i+1⋮r n⎞⎠⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟=det⎛⎝⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜r 1⋮r i−1 r i r i+1⋮r n⎞⎠⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟+α det⎛⎝⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜r 1⋮r i−1 s i r i+1⋮r n⎞⎠⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟det(r 1⋮r i−1 r i+α s i r i+1⋮r n)=det(r 1⋮r i−1 r i r i+1⋮r n)+α det(r 1⋮r i−1 s i r i+1⋮r n)
where r 1,…,r n r 1,…,r n are rows, s i s i is a row, α α is a scalar, and i i is arbitrary.
Also use the fact that a determinant of a matrix with two identical rows is equal to 0 0.
(Since you don't show your Gaussian elimination, I can't tell whether you made a mistake or performed an operation that would change the value of the determinant.)
Here's my computation of this, using multilinearity; when we exchange two rows, it multiplies the determinant by −1−1:
det⎛⎝⎜⎜⎜4 v 1+2 v 4 v 2 v 3 5 v 1+2 v 4⎞⎠⎟⎟⎟=det⎛⎝⎜⎜⎜4 v 1 v 2 v 3 5 v 1+2 v 4⎞⎠⎟⎟⎟+det⎛⎝⎜⎜⎜2 v 4 v 2 v 3 5 v 1+2 v 4⎞⎠⎟⎟⎟=det⎛⎝⎜⎜⎜4 v 1 v 2 v 3 5 v 1⎞⎠⎟⎟⎟+det⎛⎝⎜⎜⎜4 v 1 v 2 v 3 2 v 4⎞⎠⎟⎟⎟+det⎛⎝⎜⎜⎜2 v 4 v 2 v 3 5 v 1⎞⎠⎟⎟⎟+det⎛⎝⎜⎜⎜2 v 4 v 2 v 3 2 v 4⎞⎠⎟⎟⎟=4⋅5⋅det⎛⎝⎜⎜⎜v 1 v 2 v 3 v 1⎞⎠⎟⎟⎟+4⋅2⋅det⎛⎝⎜⎜⎜v 1 v 2 v 3 v 4⎞⎠⎟⎟⎟+2⋅5⋅det⎛⎝⎜⎜⎜v 4 v 2 v 3 v 1⎞⎠⎟⎟⎟+2⋅2⋅det⎛⎝⎜⎜⎜v 4 v 2 v 3 v 4⎞⎠⎟⎟⎟=20(0)+8 det(A)+10(−1)det⎛⎝⎜⎜⎜v 1 v 2 v 3 v 4⎞⎠⎟⎟⎟+4(0)=8 det(A)−10 det(A)=−2 det(A)=−2(8)=−16.det(4 v 1+2 v 4 v 2 v 3 5 v 1+2 v 4)=det(4 v 1 v 2 v 3 5 v 1+2 v 4)+det(2 v 4 v 2 v 3 5 v 1+2 v 4)=det(4 v 1 v 2 v 3 5 v 1)+det(4 v 1 v 2 v 3 2 v 4)+det(2 v 4 v 2 v 3 5 v 1)+det(2 v 4 v 2 v 3 2 v 4)=4⋅5⋅det(v 1 v 2 v 3 v 1)+4⋅2⋅det(v 1 v 2 v 3 v 4)+2⋅5⋅det(v 4 v 2 v 3 v 1)+2⋅2⋅det(v 4 v 2 v 3 v 4)=20(0)+8 det(A)+10(−1)det(v 1 v 2 v 3 v 4)+4(0)=8 det(A)−10 det(A)=−2 det(A)=−2(8)=−16.
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edited Apr 17, 2012 at 16:50
answered Apr 17, 2012 at 6:21
Arturo MagidinArturo Magidin
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Ok I tried the gaussian elimination again like so: add -1row1 to row4, add -4row4 to row1, divide row1 by 2, swapped row4 and row1. So I ended up with [v1, v2, v3, v4] and I figured the operations I made would result in the determinant being -1 2 d e t(A)=−4 1 2 d e t(A)=−4 but it says that answer is incorrect.Kyle V. –Kyle V. 2012-04-17 16:35:50 +00:00 Commented Apr 17, 2012 at 16:35
1 @StickFigs: You should have put that information on the main question, not buried here in a comment. I think you have the constant wrong. Note that det⎛⎝⎜⎜⎜2 v 4 v 2 v 3 v 1⎞⎠⎟⎟⎟=2 det⎛⎝⎜⎜⎜v 4 v 2 v 3 v 1⎞⎠⎟⎟⎟=−2 det⎛⎝⎜⎜⎜v 1 v 2 v 3 v 4⎞⎠⎟⎟⎟=−2 det(A).det(2 v 4 v 2 v 3 v 1)=2 det(v 4 v 2 v 3 v 1)=−2 det(v 1 v 2 v 3 v 4)=−2 det(A). That is, when you multiply the first row by 1 2 1 2, you don't get a 1 2 1 2 "outside", you get a 2 2 (because it must cancel the factor of 1 2 1 2 you had).Arturo Magidin –Arturo Magidin 2012-04-17 16:42:52 +00:00 Commented Apr 17, 2012 at 16:42
Ah, I see what I did wrong now. I confused B and A and thought B was obtained from A by multiplying a row by 1/2 but B is actually obtained from A by multiplying a row by 2.Kyle V. –Kyle V. 2012-04-17 16:50:39 +00:00 Commented Apr 17, 2012 at 16:50
@StickFigs: Yes; in my experience, this is the common "pitfall" in computing determinants like this. It's best to keep track of the operations as you do them (as I did in the comment above), rather than try to figure out the appropriate constant to multiply by after the fact by going over the list of operations done.Arturo Magidin –Arturo Magidin 2012-04-17 16:58:27 +00:00 Commented Apr 17, 2012 at 16:58
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Starting from A A, to get to your target, you might first multiply the first row by 4 4, then add 2 2 times the fourth row to the first row. I'll leave it to you to decide how to get from there to the target. Adding 2 2 times the fourth row to the first row doesn't change the determinant, but multiplying the first row by 4 4 multiplies the determinant by 4 4.
Another way to do it is to note that your target matrix is B A B A where
B=⎛⎝⎜⎜⎜4 0 0 5 0 1 0 0 0 0 1 0 2 0 0 2⎞⎠⎟⎟⎟B=(4 0 0 2 0 1 0 0 0 0 1 0 5 0 0 2)
(do you see why?), and that det(B A)=det(B)det(A)det(B A)=det(B)det(A).
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answered Apr 17, 2012 at 6:28
Robert IsraelRobert Israel
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Gaussian elimination may involve swapping rows, and swapping rows changes the sign of the determinant; thus Gaussian elimination doesn't leave the determinant unchanged. However, it seems you also performed other operations beyond the ones minimally required for Gaussian elimination.
The most direct way to arrive at your result would be to subtract the fourth row from the first, then multiply the first row by −1−1 and the last row by 1/2 1/2. Since the determinant is multilinear, multiplying a row by a factor multiplies the determinant by that factor. Thus the determinant of your result is −1/2−1/2 times the determinant of the matrix you started from.
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answered Apr 17, 2012 at 6:39
jorikijoriki
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5914 | https://www.math.cmu.edu/~mlavrov/arml/16-17/geometry-11-06-16.pdf | Similar Triangles Western PA ARML 2016-2017 Page 1 Cyclic Quadrilaterals David Altizio, Andrew Kwon 1 Lecture • A quadrilateral is said to be cyclic if it can be inscribed inside a circle.
• Let ABCD be a cyclic quadrilateral. Then we have the following properties: – ∠ABC + ∠ADC = ∠BCD + ∠BAD = 180◦ – ∠ABD = ∠ACD, etc.
– (Ptolemy) AB · CD + AD · BC = AC · BD.
– (Braghmaputa) Suppose a, b, c, and d are the side lengths of a cyclic quadrilateral K, and set s = a+b+c+d 2 . Then [K] = p (s −a)(s −b)(s −c)(s −d).
• The catch here is that these rules all go the other way around as well! So for example, if you can prove that ∠ABD = ∠ACD, then you know ABCD is cyclic! This is very helpful, since it allows for transferring between different types of angle equalities.
• You can also use Power of a Point to determine whether four points lie on the same circle as well.
2 Problems 1. Suppose ABC is a right triangle with a right angle at B. Point D lies on side AB, and E is the foot of the perpendicular from D to AC. If ∠BAC = 17◦and ∠ABE = 23◦, compute ∠DCB.
2. [AMC 10B 2011] In the given circle, the diameter EB is parallel to DC, and AB is parallel to ED. The angles AEB and ABE are in the ratio 4 : 5. What is the degree measure of angle BCD?
3. Suppose that P is a point on minor arc d BC of the circumcircle of equilateral triangle ABC. If PB = 3 and PC = 7, compute PA.
4. Let ABC be a right triangle with ∠B = 90◦. Points D and E are placed such that ACDE is a square. No part of the interior of the square lies inside △ABC. Let O be the center of this square.
Find ∠OBC.
A B C D E 5. Two related problems about angle bisectors.
(a) [AIME 2016] In △ABC let I be the center of the inscribed circle, and let the bisector of ∠ACB intersect AB at L. The line through C and L intersects the circumscribed circle of △ABC at the two points C and D. If LI = 2 and LD = 3, then IC = p q, where p and q are relatively prime positive integers. Find p + q.
Similar Triangles Western PA ARML 2016-2017 Page 2 (b) [Ray Li] In triangle ABC, AB = 36, BC = 40, CA = 44. The bisector of angle A meets BC at D and the circumcircle at E different from A. Calculate the value of DE2.
6. [Bulgaria 1993] A parallelogram ABCD with an acute angle BAD is given. The bisector of ∠BAD intersects CD at point L, and the line BC at point K. Prove that the circumcenter of △LCK lies on the circumcircle of △BCD.
7. [AMC 10B 2013] In triangle ABC, AB = 13, BC = 14, and CA = 15. Distinct points D, E, and F lie on segments BC, CA, and DE, respectively, such that AD ⊥BC, DE ⊥AC, and AF ⊥BF. The length of segment DF can be written as m n , where m and n are relatively prime positive integers. What is m + n?
8. [USAMO 1992] Let ABCD be a convex quadrilateral such that the diagonals AC and BD are perpendicular, and let P be their intersection. Prove that the reflections of P with respect to AB, BC, CD, DA lie on a circle.
9. Two more related problems.
(a) [AIME 1991] A hexagon is inscribed in a circle. Five of the sides have length 81 and the sixth, denoted by AB, has length 31. Find the sum of the lengths of the three diagonals that can be drawn from A.
(b) [AMC 12B 2014] Let ABCDE be a pentagon inscribed in a circle such that AB = CD = 3, BC = DE = 10, and AE = 14. The sum of the lengths of all diagonals of ABCDE is equal to m n , where m and n are relatively prime positive integers. What is m + n?
10. [APMO 2007] Let ABC be an acute angled triangle with ∠BAC = 60◦and AB > AC. Let I be the incenter and H the orthocenter of the triangle ABC. Prove that 2∠AHI = 3∠ABC.
11. [David Altizio] Let A1A2A3A4A5A6 be a hexagon inscribed inside a circle of radius r. Fur-thermore, for each positive integer 1 ≤i ≤6 let Mi be the midpoint of the segment AiAi+1, where A7 ≡A1. Suppose that hexagon M1M2M3M4M5M6 can also be inscribed inside a circle. If A1A2 = A3A4 = 5 and A5A6 = 23, then r2 can be written in the form m n where m and n are positive relatively prime integers. Find m + n.
12. [AIME 2016] Circles ω1 and ω2 intersect at points X and Y . Line ℓis tangent to ω1 and ω2 at A and B, respectively, with line AB closer to point X than to Y . Circle ω passes through A and B intersecting ω1 again at D ̸= A and intersecting ω2 again at C ̸= B. The three points C, Y , D are collinear, XC = 67, XY = 47, and XD = 37. Find AB2.
13. [Balkan MO 1992] Let D, E, F be points on the sides BC, CA, AB respectively of a triangle ABC (distinct from the vertices). If the quadrilateral AFDE is cyclic, prove that 4A[DEF] A[ABC] ≤ EF AD 2 .
14. [USAMO 2008] Let ABC be an acute, scalene triangle, and let M, N, and P be the midpoints of BC, CA, and AB, respectively. Let the perpendicular bisectors of AB and AC intersect ray AM in points D and E respectively, and let lines BD and CE intersect in point F, inside of triangle ABC. Prove that points A, N, F, and P all lie on one circle. |
5915 | https://math.stackexchange.com/questions/4557757/find-angle-x-in-the-given-composite-figure-of-triangle-bac-and-triangle-b | geometry - Find angle $x$ in the given composite figure of $\triangle BAC$ and $\triangle BDC$. - Mathematics Stack Exchange
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Find angle x x in the given composite figure of △B A C△B A C and △B D C△B D C.
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A very unique question featuring a composite diagram of two triangles, with a missing angle and two equal sides. I am posting this here to see what kind of different approaches there could be to solve it. Please feel free to leave your own answers!
(I have posted my own approach as an answer below)
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edited Oct 20, 2022 at 22:02
冥王 Hades冥王 Hades
asked Oct 20, 2022 at 22:00
冥王 Hades冥王 Hades
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Yet another geometrical alternative. Since ∠B A C+∠B D C=180∘∠B A C+∠B D C=180∘, we rotate and translate △B C D△B C D into △F B A△F B A as shown in the figure below:
(Note that A A lies on C F C F. However, we cannot not assume a priori that D D lies on B F B F, so we cannot simply observe that △B C F△B C F is equilateral.)
Now
∠F=∠C B D=180∘−∠B C D−∠B D C=75∘−x.∠F=∠C B D=180∘−∠B C D−∠B D C=75∘−x.
But B C=B F B C=B F, so
∠F=∠B C F=∠B C D+∠A C D=45∘+x,∠F=∠B C F=∠B C D+∠A C D=45∘+x,
so
75∘−x=45∘+x.75∘−x=45∘+x.
Therefore, x=15∘x=15∘.
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answered Oct 20, 2022 at 22:48
L. F.L. F.
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Amazing approach, in fact I did also consider this approach as well, however since I couldn't fit it on my answer, I decided not to include it.冥王 Hades –冥王 Hades 2022-10-20 22:50:32 +00:00 Commented Oct 20, 2022 at 22:50
1 @Goku Your approach is interesting too. Rotating is simpler in this case since conveniently A B=C D A B=C D, but I'm sure constructing cyclic quadrilaterals is a fruitful approach in many similar situations.L. F. –L. F. 2022-10-20 22:54:55 +00:00 Commented Oct 20, 2022 at 22:54
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This is my approach to this. I'll add an explanation as well.
Here's how I go about it:
1.) Mark all the appropriate points and lines in the figure. Notice that ∠B A C∠B A C and ∠B D C∠B D C are supplementary. This gives us some motivation to construct a cyclic quadrilateral. By mapping point B B onto point E E and rotating △B D C△B D C about the line segment B C B C gives us △B E C△B E C which is congruent to △B D C△B D C where ∠B E C=105∠B E C=105.
2.) Notice that this not only gives us a quadrilateral A B E C A B E C, but in fact we get a cyclic quadrilateral as ∠B E C+∠B A C=180∠B E C+∠B A C=180. Also note that segment A B=C D=C E A B=C D=C E. As well as B D=B E B D=B E. Join point A A and E E via segment A E A E. Using the properties of cyclic quadrilaterals, we can see that ∠B A E=∠B C E=x∠B A E=∠B C E=x.
3.) Notice that quadrilateral A B E C A B E C is a cyclic quadrilateral with equal non-parallel sides, this implies that A B E C A B E C is in fact an isosceles trapezoid (this can also very easily be proven via some basic angle chasing and properties of congruence). This means that the diagonals of this quadrilateral are also equal, therefore segment A E=B C A E=B C. Notice that proves that △B A E△B A E is congruent to △B C E△B C E via the SSS property. Therefore ∠A B E=∠B E C=105∠A B E=∠B E C=105. It follows that ∠A C E=75∠A C E=75 as it is supplementary to ∠A B E∠A B E. Therefore we can conclude that 2 x+45=75 2 x+45=75, therefore x=15 x=15.
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answered Oct 20, 2022 at 22:01
冥王 Hades冥王 Hades
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Simple trigonometric solution. Let a=|A B|=|C D|a=|A B|=|C D|. By the law of sines,
|B C|=sin 75∘sin(x+45∘)⋅a|B C|=sin75∘sin(x+45∘)⋅a
in △A B C△A B C and
|B C|=sin 105∘sin(75∘−x)⋅a|B C|=sin105∘sin(75∘−x)⋅a
in △B C D△B C D, so
sin 75∘sin(75∘−x)=sin 105∘sin(x+45∘).sin75∘sin(75∘−x)=sin105∘sin(x+45∘).
But sin 75∘=sin 105∘sin75∘=sin105∘, so
sin(75∘−x)=sin(x+45∘).sin(75∘−x)=sin(x+45∘).
Therefore, x=15∘x=15∘.
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answered Oct 20, 2022 at 22:34
L. F.L. F.
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Excellent trigonometric approach, nice and simple. I did try a trigonometric approach too, but mine was a lot longer and required more computation.冥王 Hades –冥王 Hades 2022-10-20 22:36:10 +00:00 Commented Oct 20, 2022 at 22:36
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5916 | https://pmc.ncbi.nlm.nih.gov/articles/PMC11866701/ | Late-onset Sheehan’s syndrome: a major diagnostic challenge—a case report - PMC
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. 2025 Feb 26;19:75. doi: 10.1186/s13256-024-05000-5
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Late-onset Sheehan’s syndrome: a major diagnostic challenge—a case report
Luis Miguel Osorio-Toro
Luis Miguel Osorio-Toro
1 Specialization in Internal Medicine, Department of Health, Universidad Santiago de Cali, Calle 5 # 62-00, Santiago de Cali, Colombia
2 Department of Research and Education, Clínica de Occidente S.A., Santiago de Cali, Colombia
3 Genetics, Physiology, and Metabolism Research Group (GEFIME), Universidad Santiago de Cali, Santiago de Cali, Colombia
Find articles by Luis Miguel Osorio-Toro
1,2,3, Yessica Alejandra Ordoñez-Guzman
Yessica Alejandra Ordoñez-Guzman
2 Department of Research and Education, Clínica de Occidente S.A., Santiago de Cali, Colombia
Find articles by Yessica Alejandra Ordoñez-Guzman
2, Jhon Fernando Montenegro-Palacios
Jhon Fernando Montenegro-Palacios
1 Specialization in Internal Medicine, Department of Health, Universidad Santiago de Cali, Calle 5 # 62-00, Santiago de Cali, Colombia
2 Department of Research and Education, Clínica de Occidente S.A., Santiago de Cali, Colombia
3 Genetics, Physiology, and Metabolism Research Group (GEFIME), Universidad Santiago de Cali, Santiago de Cali, Colombia
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1,2,3, Jhon Herney Quintana-Ospina
Jhon Herney Quintana-Ospina
1 Specialization in Internal Medicine, Department of Health, Universidad Santiago de Cali, Calle 5 # 62-00, Santiago de Cali, Colombia
2 Department of Research and Education, Clínica de Occidente S.A., Santiago de Cali, Colombia
3 Genetics, Physiology, and Metabolism Research Group (GEFIME), Universidad Santiago de Cali, Santiago de Cali, Colombia
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1,2,3, Julian Andres Pacichana-Abadia
Julian Andres Pacichana-Abadia
1 Specialization in Internal Medicine, Department of Health, Universidad Santiago de Cali, Calle 5 # 62-00, Santiago de Cali, Colombia
2 Department of Research and Education, Clínica de Occidente S.A., Santiago de Cali, Colombia
3 Genetics, Physiology, and Metabolism Research Group (GEFIME), Universidad Santiago de Cali, Santiago de Cali, Colombia
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1,2,3, Jorge Enrique Daza-Arana
Jorge Enrique Daza-Arana
1 Specialization in Internal Medicine, Department of Health, Universidad Santiago de Cali, Calle 5 # 62-00, Santiago de Cali, Colombia
4 Health and Movement Research Group, Universidad Santiago de Cali, Santiago de Cali, Colombia
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1,4,✉, Hector Fabio Escobar-Vargas
Hector Fabio Escobar-Vargas
1 Specialization in Internal Medicine, Department of Health, Universidad Santiago de Cali, Calle 5 # 62-00, Santiago de Cali, Colombia
2 Department of Research and Education, Clínica de Occidente S.A., Santiago de Cali, Colombia
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1,2, Katherine Restrepo-Erazo
Katherine Restrepo-Erazo
1 Specialization in Internal Medicine, Department of Health, Universidad Santiago de Cali, Calle 5 # 62-00, Santiago de Cali, Colombia
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1, Andrés Felipe García-Ramos
Andrés Felipe García-Ramos
1 Specialization in Internal Medicine, Department of Health, Universidad Santiago de Cali, Calle 5 # 62-00, Santiago de Cali, Colombia
2 Department of Research and Education, Clínica de Occidente S.A., Santiago de Cali, Colombia
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1,2
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1 Specialization in Internal Medicine, Department of Health, Universidad Santiago de Cali, Calle 5 # 62-00, Santiago de Cali, Colombia
2 Department of Research and Education, Clínica de Occidente S.A., Santiago de Cali, Colombia
3 Genetics, Physiology, and Metabolism Research Group (GEFIME), Universidad Santiago de Cali, Santiago de Cali, Colombia
4 Health and Movement Research Group, Universidad Santiago de Cali, Santiago de Cali, Colombia
✉
Corresponding author.
Received 2024 Sep 23; Accepted 2024 Nov 19; Collection date 2025.
© The Author(s) 2025
Open Access This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material. You do not have permission under this licence to share adapted material derived from this article or parts of it. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit
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PMCID: PMC11866701 PMID: 40011977
Abstract
Background
Sheehan’s syndrome is a form of maternal hypopituitarism resulting from excessive blood loss during or after childbirth. This extensive bleeding may reduce blood flow to the pituitary gland, causing pituitary cell damage and death (necrosis). The incidence of Sheehan’s syndrome has decreased in developed countries, whereas in developing countries, it remains a substantial cause of morbidity and mortality among at-risk populations.
Case presentation
We describe the case of a 59-year-old patient of mestizo ethnicity, with an unusual presentation of Sheehan’s syndrome 38 years after postpartum hemorrhage that affected hormone secretion at the adenohypophysis. During hospitalization, central adrenal insufficiency, low free thyroxine levels, decreased pituitary gland size, hypogonadotropic hypogonadism, and growth hormone deficiency were noted. The patient was treated with hydrocortisone and levothyroxine, with satisfactory clinical progress and improvement in her quality of life.
Conclusion
Late-onset Sheehan’s syndrome is a progressive disease, with nonspecific symptoms, which leads to delayed diagnosis and, if not treated in time, may have fatal consequences.
Keywords: Sheehan’s syndrome, Hormones, Postpartum hemorrhage, Adrenal insufficiency, Central hypothyroidism
Background
Sheehan’s syndrome (SS) is a postpartum hypopituitarism resulting from pituitary gland necrosis due to postpartum hemorrhage . First described by the British pathologist Sheehan in 1937 , SS is a rare cause of hypopituitarism in women, accounting for 0.5% of all cases . The incidence of this condition is decreasing in developed countries; however, in less developed countries, it is a significant cause of morbidity and mortality . SS may occur during the postpartum period or even several months or years after delivery; some studies have reported up to a 20.37 ± 8.34 year delay in diagnosis , which is attributable to incomplete pituitary damage with slow progression . SS is characterized by varying degrees of anterior and sometimes posterior pituitary gland disorders. Hormone deficiencies originating from the anterior pituitary gland may lead to adrenal insufficiency, central hypothyroidism, hypogonadism, growth hormone deficiency, and lactation failure. In addition, the posterior pituitary gland may rarely be affected, and when it does, it may cause central diabetes insipidus .
In this report, we present the case of a woman with SS of atypical presentation, which occurred 38 years after having experienced postpartum hemorrhage, with involvement of the hormones produced by the adenohypophysis. Late-onset presentation is rarely described in literature.
Case presentation
The patient is a 59-year-old woman of mestizo ethnicity, with no history of disease, a history of postpartum hemorrhage at 21 years of age, and no clinical repercussions after the event, such as agalactia or amenorrhea. She presented to the emergency department on 15 June 2023, with clinical symptoms for 8 days consisting of asthenia, adynamia, vomiting, and disorientation. Upon admission, hypotension, hyponatremia, and hypoglycemia were documented, and management was initially started with hypertonic saline and dextrose. The internal medicine unit described the clinical symptoms as hypo-osmolar euvolemic hyponatremia, with laboratory tests showing the presence of central adrenal insufficiency and thyrotropin levels within the normal range at the time. The patient was started on intravenous bolus hydrocortisone (100 mg), followed by 50 mg every 6 hours. Because no significant clinical improvement was observed, the patient was assessed by endocrinologists, who performed additional laboratory tests (Table1). The patient was diagnosed with central hypothyroidism; thus, treatment was started with levothyroxine at a loading dose of 300 µg, followed by a maintenance dose of 100 µg per day, in addition to the previously prescribed hydrocortisone. A contrast-enhanced magnetic resonance imaging of the sella turcica showed decreased pituitary gland size and a prominent suprasellar cistern relative to the sella turcica, showing an aspect of a partially empty sella (Fig.1). The remaining pituitary axes revealed hypogonadotropic hypogonadism and growth hormone deficiency. On the basis of hypopituitarism and decreased pituitary gland size, late-onset SS was diagnosed. Clinical and paraclinical improvements were evident 4 days after starting treatment, and the patient was discharged with hydrocortisone, 10 mg in the morning and 5 mg in the afternoon, and levothyroxine (100 µg) every day. The patient has shown satisfactory progress, with no new episodes of decompensation at 3 months after hospital discharge, and has a good quality of life.
Table 1.
Laboratory tests
| Laboratory tests | Admission | Day 4 after treatment | Reference value |
:--- :--- |
| Potassium | 4.3 | 4.4 | 3.5–5.1 mEq/L |
| Sodium | 116 | 137 | 132–146 mEq/L |
| Ureic nitrogen | 10 | 12 | 9–23 mg/dl |
| Creatinine | 0.7 | 0.8 | 0.5–1.1 mg/dl |
| Glucose | 54 | 95 | 70–100 mg/dl |
| Leukocytes | 3.3 | 5.9 | 4.0–11.8 × 10 9/l |
| Neutrophils | 1.7 | 3.3 | 2.0–7.7 × 10 9/l |
| Hemoglobin | 11.2 | 12.8 | 12.3–15.3 g/dL |
| Platelets | 128.000 | 227.000 | 203.000–445.000 cells/mm 3 |
| Thyrotropin | 0.95 | 0.97 | 0.35–5.50 µUI/ml |
| Free thyroxine | 0.26 | 0.94 | 0.89–1.48 ng/dl |
| Estradiol | < 10.0 | — | 28 pg/ml |
| Luteinizing hormone | 2.47 | — | 15.9–54.0 mUI/ml |
| Follicle-stimulating hormone | 8.1 | — | 23–116.3 mUI/ml |
| Somatomedin C | 4.06 | — | 44–240 ng/ml |
| Prolactin | 0.51 | — | 1.8–20.3 ng/ml |
| Serum cortisol at 8 a.m | 2.78 | — | 5.27–22.45 µg/dl |
| Adrenocorticotropic hormone | 3.2 | — | 4.7–48.8 pg/ml |
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Fig.1.
Open in a new tab
Decreased pituitary gland showing an aspect of a “partially empty” sella. A prominent suprasellar cistern relative to the sella turcica is visible (red arrows). A T2-weighted sagittal cut image. B T2-weighted coronal cut image
Discussion
Diagnosing SS requires high clinical suspicion because, in most cases, the initial symptoms are mild and often insignificant. Consequently, diagnosis is likely to be delayed, thereby hindering identification and placing the patient’s life at risk. The onset of symptoms can be acute in the postpartum period or up to several years after delivery [1, 7]. Moreover, symptoms of hormonal deficiency may appear 3–32 years after delivery and up to 38 years after delivery, as in our case. This case reflects the substantial challenge associated with SS diagnosis and highlights the importance of asking patients about their gestational history, such as postpartum hemorrhage, which in most cases requires blood products.
According to the anatomical location of the secretory cells with respect to the vasculature, prolactin and growth hormone secretion is most commonly affected, followed by luteinizing hormone and follicle-stimulating hormone secretion. When pituitary gland necrosis is severe, adrenocorticotropic hormone and thyroid-stimulating hormone secretion may also be affected . In our case, there was evidence of involvement of the five hormonal lines (Table1). Although agalactia has been documented in up to 100% of reported cases , it was not observed in our case, which is unusual and makes diagnosis more challenging.
In some previously reported cases (Table2), the clinical presentation of SS was as rapid as in the immediate postpartum period or took up to five decades postpartum . The manifestations of SS may vary, which hinders diagnosis. In addition, its characteristics are based on the time elapsed between delivery and SS onset. In the short term (< 5 years), agalactia, amenorrhea, and asthenia are the most frequent; in the intermediate term (6–15 years), syncope, pericardial effusion, and abdominal pain are common; finally, in the long term (> 15 years), hypoglycemia, hyponatremia, dilated cardiomyopathy, hypotension, and confusion are reported.
Table 2.
Previously reported cases
| Presentation Sheehan’s syndrome | Initial clinical manifestations | Hormones affected | References |
:--- :--- |
| Postpartum up to < 5 years | 1. Agalactia 2. Amenorrhea 3. Asthenia 4. Diabetes insipidus 5. Psychosis 6. Pancytopenia | 1. Prolactin 2. Gonadotropins 3. Corticotropin 4. Thyrotropin 5. Somatotropin 6. Antidiuretic | Olmes et al. Genetu et al. Sethuram et al. De silva et al. Rabee et al. |
| 6–15 years | 1. Syncope 2. Asthenia 3. Pericardial effusion 4. Abdominal pain | 1. Thyrotropin 2. Corticotropin 3. Gonadotropin 4. Somatotropin | Mishra et al. Sadiq et al. Powers et al. |
| > 15 years | 1. Hypoglycemia 2. Hyponatremia 3. Dilated cardiomyopathy 4. Hypoglycemia 5. Confusion 6. Asthenia 7. Amenorrhea 8. Agalactia | 1. Corticotropin 2. Thyrotropin 3. Gonadotropin 4. Prolactin 5. Somatotropin | Dourado et al. Makharia et al. Romero et al. Rabee et al. |
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When initially approaching the disease, thyrotropin and free thyroxine can define the diagnosis of hypothyroidism and help locate the alteration, indicating whether it is primary or central hypothyroidism . These tests should be performed together to prevent incorrect interpretations when the thyrotropin level is within the normal range. Furthermore, pancytopenia is uncommon in patients with SS, but full recovery is achieved with thyroid hormone and glucocorticoid treatment after reaching a euthyroid and eucortisolemic state . Hyponatremia is the most common electrolyte alteration in patients with SS, with reported rates ranging from 21% to 59% [9, 13], being a differential diagnosis in patients admitted to the emergency unit with hyponatremia .
Late-onset SS is a progressive disease with nonspecific symptoms that can delay diagnosis. It can have fatal consequences if not treated early. However, excellent responses can be obtained with appropriate treatment, thereby benefiting the patient’s quality of life.
Conclusion
Late-onset SS is a progressive disease with nonspecific symptoms whose diagnosis is usually delayed. If not treated early, the disease can have fatal consequences. Therefore, patients’ obstetric clinical history is essential to identify a history of postpartum hemorrhage and the need for blood products. Treatment consists of hormone replacement with the aim of reducing morbidity and mortality and improving quality of life.
Acknowledgements
Acknowledgements to Universidad Santiago de Cali and Clínica de Occidente S.A. for allowing the development of this research.
Author contributions
Conception: LMOT, YAOG, KRE, and AFGR. Design of the work: LMOT, JEDA, HFEV, and KRE. Analysis or interpretation of data: JFMO, JHQO, JAPA, and AFGR. All authors read and approved the final manuscript.
Funding
This research has been funded by Dirección General de Investigaciones of Universidad Santiago de Cali under call No. 01-2025. The research team's activities were sponsored by the clinical institution and the Universidad Santiago de Cali. However, the authors declare their full autonomy during all phases of the study.
Availability of data and materials
Not applicable.
Declarations
Ethics approval and consent to participate
The institution does not require ethical approval for the publication as a single case report. In this case report, all procedures were performed in accordance with the ethical and bioethical standards of the Scientific Committee of the medical institution, and the 1964 Declaration of Helsinki and its subsequent versions.
Consent for publication
Written informed consent was obtained from the patient for publication of this case report and any accompanying images. A copy of the written consent is available for review by the Editor-in-Chief of this journal.
Competing interests
The authors declare that they have no competing interests.
Footnotes
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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5917 | https://www.onemathematicalcat.org/algebra_book/online_problems/Pythagorean_Theorem.htm | The Pythagorean Theorem
(Click for cat book)
A $\,90^\circ$ angle is called a right angle. A right triangle is a triangle with a $\,90^\circ\,$ angle.
In a right triangle, the side opposite the $\,90^\circ$ angle is called the hypotenuse and the remaining two sides are called the legs.
The angles in any triangle add up to $\,180^\circ\,.$
In any triangle, the longest side is opposite the largest angle, and the shortest side is opposite the smallest angle. Thus, in a right triangle, the hypotenuse is always the longest side.
The Pythagorean Theorem gives a beautiful relationship between the lengths of the sides in a right triangle: the sum of the squares of the shorter sides is equal to the square of the hypotenuse. Furthermore, if a triangle has this kind of relationship between the lengths of its sides, then it must be a right triangle!
THEOREM The Pythagorean Theorem
Let $\,\,T\,\,$ be a triangle with sides of lengths $\,a\,,$ $\,b\,,$ and $\,c\,,$ where $\,c\,$ is the longest side (if there is a longest side). Then: $$ \begin{gather} \cssId{s15}{\text{$T\,$ is a right triangle}}\cr\cr \cssId{s16}{\text{if and only if}}\cr\cr \cssId{s17}{a^2 + b^2 = c^2} \end{gather} $$
Have fun with many proofs of the Pythagorean Theorem! This applet (you'll need Java) is one of my favorites. Let it load, then keep pressing ‘Next’.
Examples
Question: Suppose that two angles in a triangle are $\,60^\circ$ and $\,30^\circ.$ Is it a right triangle? Answer YES, NO, or MAYBE.
Solution: Yes. The third angle must be $\,180^\circ - 60^\circ - 30^\circ = 90^\circ\,.$
Question: Suppose that a triangle has a $\,100^\circ$ angle. Is it a right triangle? Answer YES, NO, or MAYBE.
Solution: No. The remaining two angles must sum to $\,80^\circ,$ so neither remaining angle is a $\,90^\circ$ angle.
Question: Suppose that a triangle has a $\,70^\circ$ angle. Is it a right triangle? Answer YES, NO, or MAYBE.
Solution: Maybe. The remaining two angles must sum to $\,110^\circ,$ so one of the remaining angles could be a $\,90^\circ$ angle.
Question: Suppose the legs of a right triangle have lengths $\,3\,$ and $\,x\,,$ and the hypotenuse has length $\,5\,.$ Find $\,x\,.$
Solution: $$ \begin{gather} \cssId{s48}{3^2 + x^2 = 5^2}\cr \cssId{s49}{9 + x^2 = 25}\cr \cssId{s50}{x^2 = 16}\cr \cssId{s51}{x = 4} \end{gather} $$
Note: $x\,$ cannot equal $\,-4\,,$ because lengths are always positive.
The $\,3{-}4{-}5\,$ triangle is a well-known right triangle. Multiplying all the sides of a triangle by the same positive number does not change the angles. Thus, if you multiply the sides of a $\,3{-}4{-}5\,$ triangle by any positive real number $\,k\,,$ then you will still have a right triangle. For example, these are all right triangles:
| | |
--- |
| $6{-}8{-}10$ | ($\,k = 2\,$) |
| $9{-}12{-}15$ | ($\,k = 3\,$) |
| $1.5{-}2{-}2.5$ | ( $\,k = 0.5\,$ ) |
| $3\pi{-}4\pi{-}5\pi$ | ($\,k = \pi\,$ ) |
| and so on! |
Question: Suppose a triangle has sides of lengths $\,1\,,$ $\,\sqrt{3}\,,$ and $\,2\,.$ Is it a right triangle?
Solution: Yes. Since $\,2 \gt \sqrt{3}\,,$ the longest side has length $\,2\,.$ And: $$ \cssId{s76}{1^2 + {(\sqrt{3})}^2} \cssId{s77}{= 1 + 3} \cssId{s78}{= 4} \cssId{s79}{= 2^2} $$
Concept Practice |
5918 | https://www.svsu.edu/media/mathampphysicsresourcecenter/docs/mathphysicsresourcecenterpdf/Refresher%20for%20Elementary%20Algebra%20Practice.pdf | Math Placement Test Practice: Elementary Algebra Practice Quick Refresher for Selected Topics Purpose: This refresher is not intended to teach new concepts. Its purpose is rather to remind students of concepts once learned in a previous college or high school course. Operations with Integers • Absolute value is the distance from the origin. So, the absolute value is a non-negative number. • The order of mathematical operations is not left to right, but instead it follows a hierarchy described by the acronym PEMDAS. The order of operations is… o “P” represents operations within parentheses or grouping symbols o “E” represents exponentiation operations o “MD” represents the operations of multiplication and division. Both multiplication and division are operations at same level and are therefore done left-to-right in the order that they appear o “AS” represents operations of addition and subtraction. Addition and subtraction are operations at same level and are therefore done left-to-right in the order that they appear. • An exponent tells how many times the base is multiplied by itself. For example, 𝟐𝟐𝟑𝟑= 𝟐𝟐∙𝟐𝟐∙𝟐𝟐. • Remember that a number directly preceding an open parenthesis means the quantity prior is multiplied by the quantity within, for example 𝟓𝟓(𝟕𝟕) = 𝟓𝟓∙𝟕𝟕 • Subtraction is just a special case of addition, just adding a negative, for example 𝟑𝟑−𝟐𝟐= 𝟑𝟑+ (−𝟐𝟐) • Division is just a special case of multiplication, just multiplying by the inverse of the divisor, for example 3 ÷ 2 = 3 ∙ 1 2 Operations with Fractions • The denominator of a fraction is the denomination of a whole (how many equal parts are in a whole.) The numerator is the number of those parts in the fraction. So 5 6 is five slices of a pie that has been cut into six equal-sized pieces. • An integer has a denominator of 1, because for an integer each whole has 1 part. • Simplest form means the numerator and denominator of a fraction have no common factors • The process of multiplying fractions involves 1) canceling any factors common to the numerators and denominators, 2) multiplying the numerators, 3) multiplying the denominators, and 4) reducing the answer to its simplest form • The process of dividing fractions involves 1) inverting the fraction after the division symbol and then 2) multiplying the fraction before the division symbol by the inverted fraction afterwards • To be added or subtracted, fractions must have identical denominators. Common denominators can be created by multiplying each fraction by the appropriate form of 1 in disguise. For example: 7 18 − 3 15 = 7 18 ∙ 5 5 − 3 15 ∙ 6 6 = 35 90 − 18 90 = 17 90. Percentages, Decimals and Fractions • Any fraction with a numerator that is more than one-half of its denominator is greater than one-half. Likewise, any fraction with a numerator that is less than one-half of its denominator is less than one-half. • Each piece of a pie cut into more equal sized pieces is smaller. So, fractions with identical numerators become smaller as their denominators become larger. Since the numerator represents the number of pieces of a pie, fractions with equal denominators are larger, if their numerators are larger. • Since the fraction bar is equivalent to a division symbol, a fraction can be converted to a decimal by dividing the numerator by its denominator • A decimal can be converted to a fraction by 1) putting the decimal over 1, 2) multiplying the numerator and denominator by a multiple of a 10 that eliminates the decimal, and 3) simplifying the resulting fraction • A percent is defined as a ratio out of 100, % 100 = 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑤𝑤ℎ𝑜𝑜𝑜𝑜𝑜𝑜 • Since a decimal and its percent are multiples of 100 of each other, the percent can be found by multiplying the decimal by 100. The decimal can be found by diving its percent by 100. The percent is numerically larger than its decimal form. • When writing algebraic equations, the word “is” means equals. The word “of” often signifies multiplication. Geometric Calculations • Perimeter is distance along the outer edges of a figure. Circumference is the distance around a circle. Perimeter and circumference can be thought of as the length of a fence around the figure. Since they are lengths, perimeter and circumference have dimensions of length. • Area is the amount of surface within the two-dimensional figure. It has dimensions of length squared • Since a triangle is equivalent to a parallelogram divided by two, its area is one-half of that of the parallelogram, so for a triangle 𝐴𝐴= 1 2 𝑏𝑏ℎ. • Often students confuse the equations for area and circumference of a circle. One way of preventing this confusion is thinking about the dimensions of area and circumference. Since area has units of length squared, the radius must be squared, 𝐴𝐴= 𝜋𝜋𝑟𝑟2. Since circumference has units of length, the radius must be to the first power, 𝐶𝐶= 2𝜋𝜋𝜋𝜋. Operations with Algebraic Expressions • Do not confuse addition or subtraction, such as (3𝑥𝑥2 −5𝑥𝑥+ 7) −(4𝑥𝑥2 + 𝑥𝑥+ 1), with multiplication, such as (3𝑥𝑥2 −5𝑥𝑥+ 7)(4𝑥𝑥2 + 𝑥𝑥+ 1) • Squaring a polynomial creates additional middle terms, such as (2𝑥𝑥+ 5)2 = (2𝑥𝑥+ 5)(2𝑥𝑥+ 5) = (4𝑥𝑥2 + 20𝑥𝑥+ 25). An exponent cannot not be simply distributed through the parentheses when there is addition or subtraction within the parentheses • When adding or subtracting like terms, the exponents of the variables do not change • When multiplying terms, the exponents for variables may change • Follow order of operations when performing arithmetic with polynomials Operations with Exponents and Roots • Only like roots can be combined by addition or multiplication, such as √7 + 3√7 = 4√7 • Scientific notation has the form of 𝑏𝑏 𝑥𝑥 10𝑛𝑛, where 1 ≤𝑏𝑏< 10 and 𝑛𝑛 is an integer • The exponents rules are… o 𝑥𝑥0 = 1 o 𝑥𝑥−𝑛𝑛= 1 𝑥𝑥𝑛𝑛 o 𝑥𝑥𝑎𝑎𝑥𝑥𝑏𝑏= 𝑥𝑥𝑎𝑎+𝑏𝑏 o 𝑥𝑥𝑎𝑎 𝑥𝑥𝑏𝑏= 𝑥𝑥𝑎𝑎−𝑏𝑏 o (𝑘𝑘𝑘𝑘)𝑎𝑎= 𝑘𝑘𝑎𝑎𝑥𝑥𝑎𝑎 o ቀ𝑥𝑥𝑎𝑎 𝑦𝑦𝑏𝑏ቁ 𝑐𝑐 = 𝑥𝑥𝑎𝑎𝑎𝑎 𝑦𝑦𝑏𝑏𝑏𝑏 Factoring • Always try to remove a greatest common factor first • If the highest order term is negative, remove a factor of −𝟏𝟏 • If four terms, try factoring by grouping • If two terms, try factoring by difference of squares • If three term and the leading coefficient is 1, find the factors of the constant whose sum equals the coefficient of the middle term • If three terms and leading term and constant are both perfect squares, first try using their square roots. • Remember all factoring can be checked by multiplying the factors Operations with Rational Expressions • Leave the final answer in factored form • When multiplying, factor the numerators and denominators, cancel, and then multiply • When dividing, invert the fraction after the division symbol, replace the division symbol with multiplication, and follow the steps for multiplication • When simplifying complex fractions, flip the bottom fraction and multiply (This assumes the original denominator of complex fraction is a single fraction) • When adding or subtracting, factor the denominators, find the common denominator, make all denominators common by multiplying by one in disguise, and combine any like terms in the numerators Solving Equations and Inequalities • Answers can be checked by substituting the result into each side of the equation or inequality and determining if the equality or inequality is true. • When solving problems involving fractions, make all non-fractions into fractions by placing them over 1. Then eliminate the denominators by either making them all common or multiplying each term by the common denominator. • If an equation contains polynomials in the denominators, factor the polynomials to find the common denominator. Solving Systems of Linear Equations • Each equation in the system represents a line. The two lines can intersect, be parallel, or be the same line. The solution set is where the lines coincide. Intersecting lines have one solution at the coordinates of intersection. Parallel lines have no solution, since they never cross. Identical lines have infinite solutions since the lines coincide at every point on the line. • The substitution method is easier to use if in one equation a variable has a coefficient of 1 or -1, because no fractions are then created when solving for that variable in that equation • The elimination method is the second method Linear Functions and Their Graphs • Key equations: o Slope: 𝑚𝑚= 𝑦𝑦2−𝑦𝑦1 𝑥𝑥2−𝑥𝑥1, for points (𝑥𝑥1, 𝑦𝑦1) and (𝑥𝑥2, 𝑦𝑦2) o Slope-intercept: 𝑦𝑦= 𝑚𝑚𝑚𝑚+ 𝑏𝑏, where m is the slope and (0, 𝑏𝑏) is the y-intercept o Point-slope: 𝑦𝑦−𝑦𝑦1 = 𝑚𝑚(𝑥𝑥−𝑥𝑥1), where m is the slope and (𝑥𝑥1, 𝑦𝑦1) is a point on the line • Intercepts occur where the line crosses the axis, so the opposite coordinate of an intercept is zero. This means that the y-coordinate of the x-intercept is zero and the x-coordinate of the y-intercept is zero. • To put an equation of a line in slope-intercept form, solve for y • To find the equation of a line, find the slope either from two points or from a parallel or perpendicular line, substitute the slope and a point into the point-slope form, and solve for y • Parallel lines have the same slope. Perpendicular lines have slopes that are the negative reciprocal of each other. Applications • Key words for operations in algebraic expressions and equations are… o Addition: sum, more than, added to, … o Subtraction: difference, less than, subtracted from, … o Multiplication: of, times, product, … o Division: quotient, ratio, … o Equal sign: is, equals, amounts to, … • Be aware that order matters in subtraction and division. The order is sometimes in the mathematical phrase in the opposite order of the written phrase. For instance, a number less than 3 is written algebraically as 3 −𝑥𝑥. • When solving application problems, define the meaning of your variables in writing • Most often the quantity you know nothing about should be assigned a variable. For example, for the phrase “the length is five less than twice the width,” nothing is known about the width and the length is defined in terms of the width. So, the width should be assigned variable, say 𝑤𝑤. Then the length would equal the expression 2𝑤𝑤−5. • Often it is helpful to organize the information from the problem in a diagram or table. Also, list the knowns and unknowns. • Pythagorean’s Theorem, 𝑎𝑎2 + 𝑏𝑏2 = 𝑐𝑐2, applies to a right triangle where a and b are the lengths of the legs and c is the length of the hypotenuse, which is across from the right angle • Problems involving travel often make use of 𝒅𝒅= 𝒓𝒓𝒓𝒓, distance = (speed)(time) |
5919 | https://www.quora.com/How-do-we-compare-numbers-with-big-exponents-and-different-bases-for-example-how-do-we-compare-3-210-and-17-140 | Something went wrong. Wait a moment and try again.
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How do we compare numbers with big exponents and different bases (for example, how do we compare 3^210 and 17^140)?
Rupesh Ashok Kumar
Studied at Kendriya Vidyalaya HVF, Avadi, Chennai
·
7y
Taking logarithm on both sides
210 log(3) = Approximately 100 digits
140 log(17) = Approximately 172 digits
So obviously from this 17^140 is larger
Jit Mitra
Chemical Engineer with interest in Mathematics
·
7y
3^210 = (3^3)^70 = 27^70
17^140 = (17^2)^70 = 289^70
In this, it is clear that 17^140 is greater.
Logarithm would be a more general and sure way of doing this. But I believe you intend to do this without using a calculator.
Generally you will have to find a way to equate either the base, or the powers of the two numbers using some trivial results (which might be completely arbitrary).
Like this one (I am using the fact that 9<17) :
17^140 > 9^140 = 3^280 which is greater than 3^210.
Deb P. Choudhury
Former Professor at University of Allahabad
·
Author has 10K answers and 8M answer views
·
7y
It may be difficult in general, but in this case the numbers are 3^(3×70)= (3^3)^70 = 27^70 and 17^(2×70) = (17^2)^70 = 289^70, so clearly second one is much larger.
Related questions
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Can you give some examples of exponents with different bases?
What's the math behind merging different exponent and base?
Nandeesh H N (ನಂದೀಶ್ ಎಚ್ ಎನ್)
An engineer who likes Maths, English, Kannada, linguistics
·
Author has 2.3K answers and 3.4M answer views
·
7y
Convert both the bases to 10.
3^210=10^(210log 3)
17^140=10^(140log 17)
Compare the exponents.
210 log 3= 100.195
140 log 17 = 170.26
The second number is obviously bigger.
Assistant
Bot
·
1y
To compare numbers with big exponents and different bases, like 3210 and 17140, we can use logarithms. This method allows us to transform the comparison of large numbers into a more manageable form.
Steps to Compare 3210 and 17140:
Take the logarithm of both sides: We can use natural logarithms (or any logarithm base), but natural logarithms are commonly used. We will compare:
ln(3210)andln(17140)
Apply the power rule of logarithms: This states that ln(ab)=b⋅ln(a). Therefore:
ln(3210)=210⋅ln(3)
ln(17140)=140⋅ln
To compare numbers with big exponents and different bases, like 3210 and 17140, we can use logarithms. This method allows us to transform the comparison of large numbers into a more manageable form.
Steps to Compare 3210 and 17140:
Take the logarithm of both sides: We can use natural logarithms (or any logarithm base), but natural logarithms are commonly used. We will compare:
ln(3210)andln(17140)
Apply the power rule of logarithms: This states that ln(ab)=b⋅ln(a). Therefore:
ln(3210)=210⋅ln(3)
ln(17140)=140⋅ln(17)
Now we need to compare 210⋅ln(3) and 140⋅ln(17).
Calculate the logarithms:
Using approximate values:
ln(3)≈1.0986
ln(17)≈2.8332
Substitute the values:
210⋅ln(3)≈210⋅1.0986≈230.826
140⋅ln(17)≈140⋅2.8332≈396.648
Comparison:
230.826<396.648
Since 210⋅ln(3)<140⋅ln(17), it follows that:
3210<17140
Conclusion:
Thus, 3210 is less than 17140. This method of using logarithms is very effective for comparing numbers with large exponents and different bases.
Related questions
How do you add numbers that have the same base but have different exponents?
Can a number to fractional exponent be expressed as the product of two identical numbers?
How can I add exponents with different bases?
What are exponents?
Which is the largest, 3 power 210, 7 power 140, 17 power 105, or 34 power 84?
Kurt Mager
Enjoys solving math problems
·
Author has 17.6K answers and 7.3M answer views
·
2y
3210=(33)70=2770–––––
17140=(172)70=28970––––––
289>27
So 17140>3210
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Swati Rawat
Bachelor of Science in Statistics (academic discipline) & Mathematics, Ccsu Meerut (Graduated 2018)
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7y
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What is the difference between an exponent & a power of a number?
It is a good question. Actually there is a difference between the two.
In mathematics, exponent implies the small number, positioned at the up-right of the base number. Actually, it represents, the number of times the base number is used as a factor, i.e. multiplied by itself. It can be numbers, constants and even variables. Whenever exponents are used to expressing a large number, the process is termed as raising to power.
For example: Let see for 2⁵
In this 5 is exponent and 2 is base.
Whereas, on the other hand,
The term ‘power’, is used to mean, the number arrived at, by raising a base number t
It is a good question. Actually there is a difference between the two.
In mathematics, exponent implies the small number, positioned at the up-right of the base number. Actually, it represents, the number of times the base number is used as a factor, i.e. multiplied by itself. It can be numbers, constants and even variables. Whenever exponents are used to expressing a large number, the process is termed as raising to power.
For example: Let see for 2⁵
In this 5 is exponent and 2 is base.
Whereas, on the other hand,
The term ‘power’, is used to mean, the number arrived at, by raising a base number to the exponent. It consist of two elements i.e. base and exponent, wherein the base number is the number that is multiplied by itself and exponent is the number of times base number is multiplied. Power is nothing but a number expressed with the help of exponent. It is the product of repeated multiplication, of the same factor.
For example: Let's take 2⁵,
In this whole quantity 2⁵ is power.
Hope this helps.
Adithya Lanka
Worked at Bloomberg (company)
·
Author has 92 answers and 468K answer views
·
9y
Related
How can I easily tell which is greater, 3^210 or 7^140?
3^3=27; 7^2=49
3^3 <7^2 Raise both the sides to the power 70 you get
3^210<7^140
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Carlos Eŭ Th
Triple IMO bronze medalist
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Author has 5.8K answers and 4.4M answer views
·
6y
Related
How do we define exponents that have an irrational base and power?
I like to make a distinction between powers and exponents. They happen to be related but one is a discrete approach while the other works better on continuous analysis.
Powers. Given a ring ⟨A,⋅⟩, let's define a naturalpower of a∈A as:
a1=aan+1=a⋅an
Examples of rings are the natural numbers and their product, the integers as their product, the rationals and their product, the reals and their product, etc. When we call multiplication the ring operation, power can be seen as repeated multiplication.
From this definition we can draw
I like to make a distinction between powers and exponents. They happen to be related but one is a discrete approach while the other works better on continuous analysis.
Powers. Given a ring ⟨A,⋅⟩, let's define a naturalpower of a∈A as:
a1=aan+1=a⋅an
Examples of rings are the natural numbers and their product, the integers as their product, the rationals and their product, the reals and their product, etc. When we call multiplication the ring operation, power can be seen as repeated multiplication.
From this definition we can draw some properties such as an+m=an⋅am, and an×m=(am)n.
We can generalize. If the ring A has an identity 1∈A, then we can start with a0=1 rather than a1=a. If ring A is inversable, we can define a−n as the inverse of an, and particularly a−1 as the inverse of a. Actually the most common notation of inverse is by using −1 as power. The properties regarding addition and multiplication of powers are preserved.
An example of an inversible ring is the strictly positive rational numbers and their product. Or the rational numbers sans zero with their product.
Now. We have defined natural powers, and generalized for integer powers. We could further ask if we can keep generalizing powers.
I can define that apq=b if and only if ap=bq. That preserves the identity ap=aq×pq=(apq)q=bq.
However in the ring of positive rational numbers, apq is not always defined. That's one of the reasons real numbers were defined.
Real numbers are not just the completion of rationals so that apq could exist. Real numbers are the completion that allows Cauchy sequence rational numbers to be defined.
If we take the sequence {1+1n}n∈N we can see that the sequence concentrates around 1. We will say that the sequence converges in 1. But let's take another sequence:
p0=1,q0=1pn=n⋅pn−1+1,qn=n⋅qn1an=pnqn
That sequence accumulate at some point. Differences between consecutive numbers of series is smaller and smaller and each difference is significantly smaller than the previous difference. It does accumulate but it does not accumulate at any rational number.
So the real numbers can be defined as the space where all rational sequences that accumulate (Cauchy sequences) accumulate into. And it is also the space where all real series that accumulate accumulate into. This is known as the continuous completion of the rational numbers.
And now, given any rational power of a positive rational (or real) number: apq I can find a rational Cauchy sequence {bn}n such as bqn gets closer and closer to ap.
This way I can define apq for any positive real number a.
But: how about ax for x an irrational real number. If x is real there exists a rational sequence {xn}n such as xn converges to x. So I can define bn=axn. Given that xn is rational, then we have already defined axn for any positive real a, so bn is well defined on the reals. Now, if {bn}n converges, then ax is defined.
So we have generalized power from natural powers, to integer powers, to rational powers, and, if we can warrantee that certain sequence is a Cauchy sequence, then we have generalized power ax for x real, for the ring of positive real numbers.
Exponentiation. The other problem is exponentiation.
It is related powers. For example if we take a loan of S money, with composite interest of z per period T, and I will pay in nT. I will have to pay back S(1+z)n. We have this power function as an answer.
Let's take the same nominal interest z per T, but I will promise to pay back in half a period. So the interest is z/2 per T/2. But if I would pay in the original period, I would have to pay back S(1+z/2)2.
If instead of half the period I had chosen a third of the period for the same nominal interest of z per T the effective interest (plus capital) would be S(1+z/3)3.
I wonder what is the limit for the same nominal interest if I take an infinitesimal fraction of the period. The total effective payback per period for the n-th fraction is S(1+z/n)n
And the limit is
So we might define a function , and our answer is .
The function does exist and is called exponentiation, and it is usually noted .
Now, or can be proven that and from that identity we can prove that and that for any rational sequence that converges to , then the sequence does converge, and converges to , so we can freely write that .
So, if , then we can say that exists and .
Now. For we get the uninteresting result that . But an interesting case is . Then we can say that , so if we give a name to , then we can use that name. Let's define , and .
And now we define where means the inverse function of .
Why we call the inverse function as an power of ? Because functions with composition are a ring, so we can use the power notation to mean composition. And the inverse of a function composed with the function is the identity.
The concept of exponentiation is a continuous concept. It exists in any space where the definition
makes sense. Particularly this definition can extend very well from real numbers to complex numbers. The definition does not work, however, for rationals or integers: isn't an integer when is integer (except zero). Integers are a discrete space. is well defined for the integers as a power (as longer as belongs to an inversible ring), but is not defined for integers as an exponentiation: we need to expand the dominion to as the integer subset of real numbers so I can define it as exponentiation.
But then, can be defined for real exponent/power and positive real base as either using the real exponentiation function (for ), or as a the convergency of the sequence where is a rational sequence that converges to .
Notice that I insist on the base to be part of the ring of positive reals and the real product.
With the exponentiation approach, if is real, then is positive, so there is no real solution for if is negative
With the generalizing power approach, and power irredictuble denominator even, there would be two real solutions for if is positive, and no solution if is negative. But in the space of strictly positive reals, there is always a unique solution.
Charlotte
Author of "Something Blue by Charlotte Harris"
·
1y
Related
How do you add and subtract exponents with different bases?
It is possible to add and subtract exponents with different bases and you can also add and subtract Exponents with different powers. These are the simplified equations.
It is possible to add and subtract exponents with different bases and you can also add and subtract Exponents with different powers. These are the simplified equations.
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David Smith
BSc (Hons) in Mathematics & Computer Science, University of Bristol (Graduated 1986)
·
Author has 3.6K answers and 4M answer views
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3y
Related
How do I evaluate complex numbers with complex exponents, for example, (3+5i) ^2+i?
Complex exponentiation is, on the face of it, simple. If we have then we calculate:
We calculate by expressing in polar form:
(1)
then we can calculate its logarithm easily:
(2)
this gives us in terms of real and imaginary components.
Now we can calculate with normal complex multiplication to give us the real and imaginary components:
then the exponent becomes
Complex exponentiation is, on the face of it, simple. If we have then we calculate:
We calculate by expressing in polar form:
(1)
then we can calculate its logarithm easily:
(2)
this gives us in terms of real and imaginary components.
Now we can calculate with normal complex multiplication to give us the real and imaginary components:
then the exponent becomes
BEWARE !
That’s the very simple explanation. The problem is that the complex logarithm is multi-valued. There is more than one value for whose exponent gives back the original number, . This is rather like there being two different choices for a square root.
Look at equation (1).
We can add to the argument without changing the value (because ). So we have
(1A)
(2A)
When we calculate , those extra contributions in the imaginary component of change both the real and imaginary components of the result and thus the final value we calculate for .
If happens to be an integer then those contributions make no difference to the final result (which is what we would hope for).
If w is not an integer (e.g. or is imaginary or complex) then we have multiple solutions for .
Catherine Celice
Former Former Developmental Math and Statistics Lecturer at Wayne State University (1997–2008)
·
Author has 2K answers and 1.4M answer views
·
1y
Related
Why do we use exponents to represent repeated addition? What would happen if we didn't use exponents? And how did this come to be used in mathematics?
Answering: “Why do we use exponents to represent repeated addition? What would happen if we didn't use exponents? And how did this come to be used in mathematics?”
We do not use exponents to represent repeated addition. We use exponents to represent repeated multiplication.
We use multiplication to represent repeated addition.
One reason for using alternate notation is to make things easier to work with in certain situations. In the case of addition, if we had to indicate that we were adding 100 of the number 5 together, we would send too long writi
Answering: “Why do we use exponents to represent repeated addition? What would happen if we didn't use exponents? And how did this come to be used in mathematics?”
We do not use exponents to represent repeated addition. We use exponents to represent repeated multiplication.
We use multiplication to represent repeated addition.
One reason for using alternate notation is to make things easier to work with in certain situations. In the case of addition, if we had to indicate that we were adding 100 of the number 5 together, we would send too long writing that out and it would be difficult to read. Also, it would result in more mistakes both in writing it and in reading it.
Similarly, exponents can save time writing out repeated addition and make it easier to both read and write and result in fewer mistakes doing both. Multiplying 100 of the number 5 together would be hard to write out using multiplication notation.
versus
By the way, when one wants to write out a very large number such as 1 trillion or 1 trillion times 1 trillion, exponents make this easier. They are especially handy for writing out powers of 10 we don’t even have names for.
1 trillion = 1,000,000,000,000 =
1 trillion times 1 trillion = 1,000,000,000,000,000,000,000,000 =
What is ? It would be a mess writing those numbers out in normal notation then performing the multiplication, but since we used exponent notation this multiplication is much easier to do and to write: .
Charlotte Harris
More than 5 years of research on the topic.
·
1y
Related
How do I subtract exponents with different powers but with the same base?
How? - Use the Exponents equation for subtracting Exponents with different powers but the same base. The equation below with example gives an answer as a number with a fraction applied to a base raised to an Exponent.
How? - Use the Exponents equation for subtracting Exponents with different powers but the same base. The equation below with example gives an answer as a number with a fraction applied to a base raised to an Exponent.
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6. 2.5 Charged Pendulum 1 point A ball of mass my and charge q is suspended from a light string in the presence of a horizontal electric field, Ē, near the surface of the earth. At equilibrium it makes an angle of 30° with the vertical. When it is replaced by a mass of m2 and charge q, it makes and angle of 60° with the vertical in equilibrium as shown
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Question: 6. 2.5 Charged Pendulum 1 point A ball of mass my and charge q is suspended from a light string in the presence of a horizontal electric field, Ē, near the surface of the earth. At equilibrium it makes an angle of 30° with the vertical. When it is replaced by a mass of m2 and charge q, it makes and angle of 60° with the vertical in equilibrium as shown
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2.5 Charged Pendulum 1 point A ball of mass my and charge q is suspended from a light string in the presence of a horizontal electric field, Ē, near the surface of the earth. At equilibrium it makes an angle of 30° with the vertical. When it is replaced by a mass of m2 and charge q, it makes and angle of 60° with the vertical in equilibrium as shown below. E What is the ratio m/m2 of the masses? Enter answer here 7. 2.6 Charged Rods 1 point Two rods that are each 1 m in length are arranged on an axis so that their ends are 1 m apart as shown. The left rod has a charge density = +3C/m and the right rod has a charge density = -4°C/m. What is the magnitude of the electric field at the point shown, which is 0.7 m from the end of the left rod? Answer in N/C. Enter answer here
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Tools & Reference>Rheumatology
Hypertrophic Osteoarthropathy
Updated: Feb 22, 2024
Author: Vishnuteja Devalla, MD; Chief Editor: Herbert S Diamond, MD more...;)
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Hypertrophic Osteoarthropathy
Sections Hypertrophic Osteoarthropathy
Overview
Practice Essentials
Background
Pathophysiology
Etiology
Epidemiology
Prognosis
Patient Education
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Presentation
History
Physical Examination
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DDx
Workup
Laboratory Studies
Imaging Studies
Histologic Findings
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Treatment
Medical Care
Surgical Care
Long-Term Monitoring
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Medication
Medication Summary
Nonsteroidal Anti-inflammatory Drugs (NSAIDs)
Adrenergic beta-blockers
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Media Gallery;)
References;)
Overview
Practice Essentials
Hypertrophic osteoarthropathy (HOA) is a syndrome characterized by clubbing of the digits, periostitis of the long (tubular) bones, and arthritis. It is also known as pachydermoperiostosis (PDP), in which thickening of the skin occurs.
HOA can be primary (hereditary or idiopathic) or secondary. Secondary HOA, which accounts for about 80% of HOA cases, is associated with an underlying pulmonary, cardiac, hepatic, or intestinal disease and often has a more rapid course. As a paraneoplastic syndrome, it most commonly occurs with pulmonary or pleural tumors; however, other tumors (eg, nasopharyngeal carcinoma and esophageal cancer) may also be involved.
An evaluation for the primary condition is warranted in patients with possible secondary HOA; for example, a search for an intrathoracic malignancy or chronic infection. See Workup.
No direct therapy for primary HOA is available. In secondary HOA, treatment involves addressing the underlying cause. For both primary and secondary HOA, symptomatic treatment is indicated.
Examples of treatment of the underlying cause include the following:
Oncologic therapy for cancer (eg, surgical resection of tumor, chemotherapy, radiotherapy)
Surgical correction of cardiac anomalies
Antibiotics for infection
Symptomatic treatments include the following:
NSAIDs
Bisphosphonates
Octreotide
Vagotomy
See Treatment.
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Background
The clinical triad of digital clubbing, arthralgias, and ossifying periostitis that characterizes hypertrophic osteoarthropathy (HOA) has been recognized since the late 1800s and was previously known as hypertrophic pulmonary osteoarthropathy (HPOA). Hippocrates first described digital clubbing 2500 years ago, hence the use of the term Hippocratic fingers.
Observations made in modern times by Eugen von Bamberger (1889) and Pierre Marie (1890) led to the term MarieBamberger disease. Work by other investigators led to identification of various causes of this digital anomaly, which can be the first manifestation of a severe organic disorder such as chronic pulmonary and cardiac diseases,
Primary hypertrophic osteoarthropathy (PHO; also termed primary pachydermoperiostosis or Touraine-Solente-Gole syndrome) was initially described by Friedreich in 1868 and then by Touraine et al in 1935, who recognized its familiar features and proposed the following classification [9, 10] :
Complete - Pachydermia, digital clubbing, and periostosis
Incomplete - No pachydermia
Forme fruste - Prominent pachydermia with few skeletal manifestations
In some cases, the diagnosis of HOA as primary can be challenged with the development of a disease that is known to be associated with secondary HOA. This may occur as late as 6-20 years after the appearance of HOA. [11, 12]
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Pathophysiology
The development of hypertrophic osteoarthropathy (HOA) has been linked to several mechanisms, including excessive collagen deposition, endothelial hyperplasia, edema, and new bone formation. It has been hypothesized that these mechanisims are driven by paraneoplastic growth factors, such as prostaglandin E and other cytokines; and neurologic, hormonal, and immune mechanisms. [16, 17] All, or at least many, likely contribute to its development in different clinical situations. A popular theory involves the interaction between activated platelets and the endothelium, which is discussed further below. [14, 16, 18] Primary and secondary HOA have distinct pathophysiologies despite similar clinical presentations.
Primary HOA involves mutations in the HPGD or SLCO2A1 gene (see Etiology), which code for enzymes involved in prostaglandin metabolism. The mutations result in elevated levels of prostaglandin E2 (PGE2), with decreases in the level of its metabolites. Under normal conditions, PGE2 is degraded into unstable 13, 14-dihydro-15-keto PGE2 and then into stable 13, 14-dihydro-15-keto PGA2, with 15-hydroxyprostaglandin dehydrogenase (15-PGDH) as a key enzyme in the catabolic pathway. Increased PGE2 levels have been shown to stimulate activity of both osteoclasts and osteoblasts, which may contribute to the skeletal manifestations of primary HOA, including periostosis and acro-osteolysis. [20, 21] The exact effects of PGE2 on the skin are not completely understood.
Kozak et al tested the hypothesis that elevated systemic levels of PGE2 was 2.3-fold higher in patients with clubbing than in patients without clubbing. Several other pathophysiologic mechanisms have also been observed in HOA. The most important of these mechanisms involve circulating signaling molecules and growth factors that are normally cleared from the blood by the pulmonary endothelium.
Secondary HOA is most often associated with an underlying pulmonary disease, mainly bronchogenic carcinoma; hence the older term hypertrophic pulmonary osteoarthropathy. HOA has been observed in up to 17% of bronchogenic carcinoma patients. Secondary HOA can also be associated with non-pulmonary conditions, including cardiovascular, gastrointestinal, hepatobilliary, and endocrine diseases.
Normally, megakaryoctes released from bone marrow into the general circulation travel to the pulmonary microvasculature, where they fragment into platelets. If that fails to occur, the platelet precursors can become trapped in the peripheral vasculature, where they release platelet-derived growth factor (PDGF) and vascular endothelial growth factor (VEGF), which promote vascularity. This has been demonstrated in patients with cyanotic heart diseases, in which large circulating platelets with abnormal, and at times bizarre, morphology have been found. Those macrothrombocytes are responsible for the aberrant platelet volume distribution curves. [27, 16]
Several physiologic and anatomic processes have been defined in which these large particles reach the fingertip capillaries and impact release of growth factors. Megakaryocytes or megakaryocyte fragments have been observed bypassing the lung capillary network (eg, in patients with right-to-left intracardiac shunts, carcinoma of the bronchus, anatomic malformation of the vasculature, patent ductus arteriosus complicated by pulmonary hypertension and a right-to-left shunt) and forming large platelet clumps on the left side of the heart or in large arteries (eg, subacute bacterial endocarditis, subclavian aneurysm), or chronic platelet excess (eg, in chronic inflammatory bowel disease).
For the reasons above, cyanotic heart diseases are an excellent model for studying HOA pathogenesis because more than one third of patients with lifelong clubbing secondary to cyanotic heart disease eventually display the full HOA syndrome. HOA caused by intrapulmonary shunting of blood becomes evident only in the limbs that receive unsaturated blood, for example, in patients with patent ductus arteriosus complicated by pulmonary hypertension and a right-to-left shunt.
Megakaryocytes or megakaryocyte fragments impacted at distal sites release growth factors that include bradykinin, slow-reacting substance of anaphylaxis, transforming growth factorβ1 (TGF-β1), VEGF, and PDGF stored in the platelet alpha granules. Those are all angiogenic, with trophic effects on capillary beds. In addition, they all enhance the activity of osteoblasts and fibroblasts. This initiates finger clubbing by inducing connective-tissue matrix synthesis periostosis. [16, 18]
Increased circulating growth factor levels thus would explain all of the features of HOA. PDGF and VEGF are thought to contribute significantly to the development of HOA. VEGF is a platelet-derived factor whose action is induced by hypoxia. It is a potent angiogenic and permeability-enhancing factor, as well as a bone-forming agent. VEGF receptors are expressed in subperiosteal bone-forming cells. Both PDGF and VEGF induce vascular hyperplasia, new bone formation, and edema.
In keeping with this hypothesis, Matucci-Cerinic et al have shown elevated levels of von Willebrand factor antigen (vWF:Ag) in persons with primary HOA and those with HOA secondary to cyanotic heart disease. vWF:Ag is a surrogate marker of endothelial activation and damage, as shown by the fact that high plasma levels of vWF:Ag are also found in the vasculitides, myocardial infarction, diabetic microangiopathy, and scleroderma. Other substances that are found at increased levels in the plasma of patients with HOA and could have a role in disease progression and periosteal proliferation include endothelin-1 and β-thromboglobulin.
Thus, a common pathogenetic pathway for HOA possibly involves localized activation of endothelial cells by an abnormal platelet population. Macrothrombocyte and endothelial cell activation can also be present in cases of HOA associated with other disease entities such as liver cirrhosis, in which a prominent intrapulmonary shunting of blood occurs.
Stimulation of fibroblasts by PDGF, epidermal growth factor (EGF) and TGF-β along with overexpression of VEGF have also been linked to the extensive myelofibrosis seen in a few cases of pachydermoperiostosis.
A second proposed mechanism for the development of HOA is a vagally-mediated alteration in limb perfusion. Interestingly, the anatomic distribution of vagal nerve fibers correlates to the area of clubbing. Vagotomy and sympatholytic drugs have been reported to reverse or to improve HOA, suggesting a role for reflex vagal stimulation. Bazaar et al proposed that sympathetic override of the normal protective function of vagal innervation is the basis of HOA. Sympathetic activity has been noted to induce cytokine changes consistent with inflammation.
Among these, epinephrine has been shown to induce production of interleukin (IL)-11 in human osteoblasts. Recombinant IL-11 has been shown to cause reversible symmetric periostitis in the extremities. In diseased states, autonomic stimulation may occur as a result of chemoreceptor activation in response to acidosis, hypoxia, or hypercapnia. Examples include sleep apnea, congestive heart failure, kidney failure, and tumor-induced hypoxia. Reversal of those conditions with removal of the associated lung neoplasm or correction of a cyanotic heart malformation suggests that alteration of lung function plays an important role.
A third mechanism is the possibility of ectopic production of hormonelike substances (such as VEGF) by tumor or inflammatory tissue, resulting in excessive circulating levels of angiogenic substances that would cause capillary bed hypertrophy and periosteal reaction, as noted earlier. Elevated circulating concentrations of VEGF and evidence of tumor production of VEGF have been found in lung cancer. Following tumor resection, the concentrations of VEGF markedly decline, which also correlates with clinical improvement. Increased levels of VEGF and IL-6 caused by the genetic mutation of K-ras might play a role in the pathogenesis of HOA with lung cancer.
Diverse types of cancers produce VEGF as a mechanism of tumor dissemination. Abnormal expression of VEGF is also known to occur in non-neoplastic diseases associated with HOA, such as Graves disease and inflammatory bowel disease. These diseases are characterized by prominent endothelial cell involvement, leading to overproduction of VEGF and thus acropachy. In HOA related to vascular prosthesis infection, Alonso-Bartolome et al suggested involvement of the humoral pathway giving rise to graft infectionassociated HOA syndrome by endotoxin or vasoactive compound activated or released by bacteria adherent to the graft.
Chronic activation of macrophages secondary to pulmonary pathologies may lead to digital clubbing by continual production of profibrotic tissue repair factors (eg, growth factors, fibrogenic cytokines, angiogenic factors, remodelling collagenases). These factors act systemically, but their effect is greatest at those parts of the vasculature which are most sensitive to these actions, such as the nail beds. Hypoxia also triggers the activation of macrophages.
The role of different cytokines and cell receptors, including IL-6 and the osteoprotegerin or RANKL (receptor activator of nuclear factor kappa-Î ligand) system have been described on the development of the disease. Higher serum levels of IL-6 and RANKL are associated with increased values in markers of bone resorption (degradation products of C-terminal telopeptides of type-I collagen and urinary hydroxyproline/creatinine ratio) and reduced serum levels of bone alkaline phosphatase, a marker of bone formation, suggesting that HOA is characterized by increased bone resorption, probably mediated by IL-6 and RANKL.
The pathogenesis underlying the higher risk of HOA in males, as proposed by Bianchi et al, relates to the high levels of nuclear steroid receptors, increased cytosolic estrogen receptors, and absence of detectable progesterone and androgen cytosolic receptors in HOA. Those suggest increased tissue sensitivity to different circulating sex steroids, which could enhance tissue epidermal growth factor or transforming growth factor alpha production and use.
HOA can be associated with pregnancy and aging secondary to platelet abnormalities, hormonal disturbances, and cytokine dysfunction.
Enhanced Wnt genetic signaling contributes to the development of pachydermia skin changes in primary HOA by enhancing dermal fibroblast functions. The Wnt signaling consists of canonical and noncanonical pathways. These signaling pathways are mediated by Wnt protein, which binds to a frizzled Wnt receptor. Wnt signaling is modulated by several different families of secreted down-regulators. Among them, Dickkopf (DKK) is a family of cysteine-rich proteins comprising at least four different forms (DKK1, DKK2, DKK3, and DKK4), which are coordinately expressed in mesodermal lineages.
The best studied of these is DKK1, which blocks the canonical Wnt signaling by inducing endocytosis of lipoprotein receptorrelated protein 5/6 (LRP5/6) complex 12 without affecting the frizzled Wnt receptor. High mRNA levels of DKK1 in human dermal fibroblasts of the palms and soles inhibit the function and proliferation of melanocytes via the suppression of catenin and microphthalmia-associated transcription factor. These findings suggest that DKK1 is deeply involved in the formation and differentiation of the skin. Decreased expression of DKK1 in fibroblasts and enhanced expression of catenin in the skin of patients with PDP, suggest that Wnt signaling is enhanced in PDP. These results suggest that enhanced Wnt signaling contributes to the development of pachydermia.
Various rare associations have been described, including hypertrophic gastropathy, peptic ulcers, gynecomastia, acro-osteolysis of fingers and toes, Crohn disease, atherothrombotic brain infarction, renal amyloid A (AA) amyloidosis, and bone marrow failure due to myelofibrosis. Only six cases of myelofibrosis in primary HOA have been described to date. The development of myelofibrosis makes primary HOA a disease with unfavorable outcome. Several factors including increased collagen fibers, infiltration and overgrowth of fibroblasts in bone marrow, and overactivity of platelet-derived growth factor may play a role in this complication.
Digital clubbing and hypertrophic osteoarthropathy are linked, and many authors postulate a single pathological entity, regardless of the etiology, which evolves in a centripetal fashion, with finger or toe clubbing appearing first and thickening of the tubular bones of the extremities occurring at later stages of the process. Hypertrophic osteoarthropathy without clubbed nails appears to be rare and few cases have been reported.
Clubbed digits
Clubbing is characterized by elevation of the nail and widening of the distal phalanx caused by swelling of the subungual capillary bed resulting from increased collagen deposition, interstitial inflammation with edema, and proliferation of the capillaries themselves. Perivascular infiltrates of lymphocytes and vascular hyperplasia are responsible for thickening of the vessel walls. Electron microscopy reveals Weibel-Palade bodies and prominent Golgi complexes, confirming structural vessel wall damage. Vast numbers of arteriovenous anastomoses may also be seen in the nail bed.
Two types of bone changes can be found in the distal phalanges, hypertrophic and osteolytic. Hypertrophy or bony overgrowth predominates in patients with HOA secondary to lung cancer, whereas acro-osteolysis predominates in patients with HOA secondary to cyanotic congenital heart disease. The type of bone remodeling process depends on the age when clubbing develops. If clubbing appears in childhood, osteolysis is more prominent; however, if it develops after puberty, hypertrophic changes take place. Pineda et al hypothesize that a putative circulating growth factor destroys immature bone.
Periosteum
Periosteal new bone formation is a hallmark of hypertrophic osteoarthropathy. It mostly affects the appendicular skeleton, usually bilaterally and symmetrically along the metadiaphyseal regions of the bones.
Neoangiogenesis, edema, and osteoblast proliferation in distal tubular bones lead to subperiosteal new bone formation in HOA. Subperiosteal new bone formation occurs along the distal diaphysis of tubular bones, progressing proximally over time. The irregular periosteal proliferation affects predominantly the distal ends of long bones, including the epiphysis in 80-97% of patients. Usually the metacarpals, metatarsals, tibia, fibula, radius, ulna, femur, humerus, and clavicle are involved. The tibia is almost invariably involved. [17, 50] Involvement of the epiphysis distinguishes it from the secondary form, which typically spares the epiphysis.
Initially, excessive connective tissue and subperiosteal edema elevate the periosteum; then, new osteoid matrix is deposited beneath the periosteum. As this mineralizes, a new layer of bone is formed, and, eventually, the distal long bones may become sheathed with a cuff of new bone.
Synovium
Synovial involvement may occur with subperiosteal changes. Thickening of the subsynovial blood vessels and mild lining-layer hyperplasia may occur. [23, 17] The edematous synovium becomes mildly infiltrated with lymphocytes, plasma cells, and occasional polymorphonuclear leukocytes, but the results of immunohistologic studies are negative. Electron-dense subendothelial deposits are present in vessel walls. [52, 53, 54] In a study of a patient with primary HOA and chronic arthritis, Lauter et al found multilayered basement laminae around small subsynovial blood vessels consistent with the late stages of vascular injury. Synovial fluid is usually noninflammatory with low leukocyte counts and few neutrophils. [52, 54]
Skin
Skin changes are more evident in primary HOA and are caused by dysregulation of mesenchymal cells. Characteristic cutaneous manifestations include pachydermia (thickening of the skin) of the face and the scalp, cutis verticis gyrata, and bilateral ptosis over the eyes resulting in blepharoptosis. These changes yield a characteristic leonine or bulldog appearance.
Other dermatologic manifestations are acne, eczema, seborrhea, and palmoplantar hyperhidrosis. The skin of the hands and feet are also thickened, but usually not folded.
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Etiology
Hypertrophic osteoarthropathy (HOA) may be either primary (hereditary or idiopathic) or secondary to a variety of malignant and nonmalignant conditions. Primary HOA comprises about 3-5% of all cases of hypertrophic osteoarthropathy.
Primary hypertrophic osteoarthropathy
Primary HOA is also known as pachydermoperiostosis (PDP). Primary HOA has been linked to mutations in two genes: 15-hydroxyprostaglandin dehydrogenase (HPGD) and solute carrier organic anion transporter family, member 2A1 (SLCO2A1). Both autosomal dominant inheritance with incomplete penetrance and recessive inheritance have been reported. Hereditary primary HOA consists of three subtypes based on inheritance pattern and genetic mutation :
Primary hypertrophic osteoarthropathy, autosomal recessive 1 (PHOAR1) caused by HPGD mutation
Primary hypertrophic osteoarthropathy, autosomal recessive 2 (PHOAR2) caused by SLCO2A1 mutation
Primary hypertrophic osteoarthropathy, autosomal dominant (PHOAD) caused by SLCO2A1mutation
A family history of the disease can be traced in only about 25-38% of primary HOA cases. Familial recurrence of PDP has been reported in 33-100% of pedigrees.
PHOAR1 involves homozygous and compound heterozygous germline mutations in HPGD, which encodes 15-hydroxyprostaglandin dehydrogenase, an NAD+ dependent enzyme that catalyzes prostaglandins. Homozygous HPGD mutations have so far been reported in 10 families; all but one displayed parental consanguinity. Only two of those families were of European origin. The c.175_176delCT frameshift mutation appears to be recurrent and to be the most common HPGD mutation in White families.
So far, seven HPGD alterations are known. The allelic spectrum of the HPGD gene includes a novel c.217+1G>A mutation. Seven coding HPGD exons encode the 266 amino acid 15-hydroxyprostaglandin dehydrogenase, which is ubiquitously expressed. All HPGD mutations constitute loss-of-function alleles due to protein truncation or missense changes that affect hydrogen bonds lining the 15-PGDH enzyme reaction cavity. Individuals with homozygous mutations have chronically elevated prostaglandin E2 (PGE2) levels.
Secondary hypertrophic osteoarthropathy
Secondary HOA is also called Pierre Marie-Bamberger syndrome. In adulthood, 90% of generalized hypertrophic osteoarthropathy is associated with an intrathoracic infectious or neoplastic condition. The association with malignancy is relatively common in adults. [62, 63] HOA may precede the diagnosis of the underlying disease.
Conditions underlying secondary hypertrophic osteoarthropathy can be easily separated into malignant and nonmalignant diseases. Paraneoplastic HOA is more common in subjects aged 5070 years. Among malignancy-related hypertrophic osteoarthropathy, pulmonary malignancies compose 80% of reported cases, most of which are nonsmall cell lung cancer such as squamous cell or adenocarcinoma.
As many as 5% of adults with lung cancer demonstrate signs of HOA. However, in a Japanese study of 1226 patients with lung cancer patients, 54.5% demonstrated abnormally high uptake on bone scintigraphy, suggesting possible HPO, but only 0.8% had clubbed fingers and joint pain and were eventually confirmed as having pulmonary HOA. Of the patients with confirmed lung cancer and HOA, most were males and heavy smokers and had advanced disease.
Lung cancer accounts for almost 20% of isolated digital clubbing and over 60% of HOA in adults. These data suggest that the development of HOA or simple digital clubbing in adults should prompt lung cancer screening, even in the absence of detectable respiratory symptoms. This would allow earlier diagnosis and so permit early treatment, leading to better outcome.
Other malignancies reported in the literature to be associated with HOA include the following :
Nasopharyngeal cancer
Mesothelioma
Renal cell carcinoma
Esophageal cancer
Gastric tumor
Pancreatic cancer
Breast phyllodes tumor
Melanoma
Thymic cancer
Hodgkin lymphoma
Nonmalignant causes of hypertrophic osteoarthropathy include a number of GI and other diseases, including neoplastic, pulmonary, cardiac, infectious, endocrine, psychiatric, and multisystem diseases.
Chronic respiratory diseases include cystic fibrosis, pulmonary fibrosis, sarcoidosis, chronic obstructive pulmonary disease, pulmonary tuberculosis, pulmonary primary intestinal lymphangiectasia (Waldmann disease), pulmonary epithelioid hemangioendothelioma, bronchiectasis, diffuse inflammatory lung disease, pulmonary arteriovenous malformations, and chronic hypoxemia. HOA has been reported in association with chronic rejection of a lung transplant. A case report describes HOA in a patient who had undergone bilateral lung transplantation because of severe pulmonary sarcoidosis.
Inflammatory bowel disease (Crohn disease and ulcerative colitis), celiac sprue, gastric hypertrophy, laxative abuse, polyposis, intestinal acute cellular rejection, primary intestinal lymphoma, juvenile polyps of the stomach, and gastric adenocarcinoma are associated with HOA. Chronic enteropathy has been reported in patients with primary HOA due to SLCO2A1 mutation.
Liver disease and cirrhosis resulting from cholestasis, chronic active hepatitis, biliary atresia, primary sclerosing cholangitis, Wilson disease, primary biliary cirrhosis, and alcoholic cirrhosis are also causes. These also include hepatocellular carcinoma and primary liver rhabdomyosarcoma.To our knowledge, hypertrophic osteoarthropathy has not been reported to occur with liver steatosis in the English literature. Hypertrophic osteoarthropathy has been associated with organ transplant in one isolated liver transplant recipient with chronic liver rejection. No association with transplant medications has been noted.
Congenital cyanotic congenital heart diseases, rheumatic diseases, and left ventricular tumors have been implicated as well.
Neurologic causes include primitive neuroectodermal tumors (PNETs).
Other causes include chronic infections associated with cystic fibrosis, HIV, tuberculosis, aspergillus, infective endocarditis, subacute bacterial endocarditis, vascular prosthesis infections, syphilis, and immune deficiency syndrome and amyloidosis.
Mediastinal causes include esophageal carcinoma, thymoma, and achalasia.
Miscellaneous causes include the following:
Graves disease
Thalassemia
Diverse malignancies
Polyneuropathy, organomegaly, endocrinopathy, M protein, and skin changes (POEMS) syndrome
Metastatic phyllodes tumor of breast
Epithelioid hemangioendothelioma
Nasopharyngeal carcinoma with lung metastasis
Thymic carcinoma
Renal cell carcinoma with lung metastasis
Osler-Weber-Rendu syndrome
Deep infections such as vascular graft infection and perianal abscess
Primary HOA and POEMS syndrome overlap; both conditions are associated with digital clubbing, pachyderma, hyperhidrosis, gynecomastia, and bone proliferation.
The causes of localized hypertrophic osteoarthropathy include hemiplegia, patent ductus arteriosus with pulmonary hypertension, infected arterial grafts, endothelial infections, and extensive endothelial injury of a limb. In patients with patent ductus arteriosus, pulmonary hypertension causes right-to-left shunting of blood, which may cause HOA of the toes and fingers on the left side. [73, 18] Development of hypertrophic osteoarthropathy localized to areas distal to a vascular prosthesis may allow early diagnosis of graft infection. Cases of bilateral or monomelic hypertrophic osteoarthropathy of the lower limbs (or isolated clubbing of the toes) revealing an aortic prosthesis infection have been reported in the last 40 years. Therefore, unilateral clubbing always suggests a condition affecting the vessels or nerves of the arm, leg, or thoracic outlet.
Thomas first described thyroid acropachy in 1933. It is a rare condition associated with prior or active Graves disease. Thyroid acropachy is characterized by the triad of (1) clubbing; (2) noninflammatory swelling of the soft tissues of the hands and feet; and (3) asymptomatic, asymmetrical, exuberant, periosteal proliferation preferentially affecting the diaphysis of the metacarpal and metatarsal bones. It usually coexists with exophthalmos and pretibial myxoedema, and patients can be hypothyroid, euthyroid, or hyperthyroid.
Medication
Various medications, including prostaglandin, vitamin A, and fluoride, can produce periostitis and bony changes resembling hypertrophic osteoarthropathy. An association between senna misuse and finger clubbing (reversible with cessation of senna) has been reported.
Voriconazole has been reported to probably induce periostitis, but no apparent inflammatory arthritis was noted in the case series report. The presentation more closely resembles nodular periostitis or periostitis deformans than hypertrophic osteoarthropathy.
Unlike patients with hypertrophic osteoarthropathy, patients with voriconazole-associated periostitis lack the cardinal features of digital clubbing and noninflammatory joint effusions. The periosteal reaction was dense and irregular, as opposed to the smooth and single layer periostitis described in lung-cancerassociated hypertrophic osteoarthropathy. In addition to the involvement of tubular bones characteristic of classic hypertrophic osteoarthropathy, the patients also had variable involvement of the clavicles, ribs, scapulae, and pelvis. Chen and Mulligan suggested that fluoride toxicity may be the cause of voriconazole-associated periostitis.
Hypertrophic osteoarthropathy was also noted in one case after a long-term use of bevacizumab for metastatic colorectal cancer.
Pediatric cases
Hypertrophic osteoarthropathy is an uncommon disease in the pediatric age group. It is characterized by noninflammatory joint effusions, terminal digit clubbing, and radiographic evidence of periosteal new bone formation affecting the hands, feet, and distal limbs.
In children, most cases of generalized hypertrophic osteoarthropathy are due to non-neoplastic causes such as pulmonary infections, cystic fibrosis, and congenital cyanotic heart disease. Cyanotic heart disease is the prototype of hypertrophic osteoarthropathy because almost all patients have clubbing and more than a third of patients have the full-blown syndrome.
Case reports have described an association with biliary atresia, including an adolescent patient with a history of liver transplantation at 4 months for biliary atresia who was initially diagnosed with juvenile rheumatoid arthritis. This patient was also found to have hepatopulmonary syndrome.
Malignancy-associated hypertrophic osteoarthropathy in children and young adults is not well documented but numerous case reports describe the association with carcinoma of the nasopharynx, osteosarcoma with lung metastasis, rhabdomyosarcoma, Hodgkin lymphoma, thymic carcinoma, and pleural mesothelioma. A case report has described hypertrophic osteoarthropathy presenting as the first symptom of recurrent infantile fibrosarcoma.
The authors did not identify any reported cases of hypertrophic osteoarthropathy associated with lung carcinoma in children or young adults in the literature. Intrathoracic disease should be considered when hypertrophic osteoarthropathy is detected in a child with a known or suspected malignant disease, and the occurrence of hypertrophic osteoarthropathy during follow-up should alert the physicians for possible recurrence of the neoplastic disease or intrathoracic involvement. To the authors' knowledge, to date only 34 cases of hypertrophic osteoarthropathy have been reported in pediatric patients with neoplastic diseases. Of those, 12 had carcinoma of the nasopharynx, 8 had osteosarcoma, 8 had Hodgkin lymphoma, 3 had thymic carcinoma, 1 had periosteal sarcoma, 1 had pleural mesothelioma, and 1 had recurrent infantile fibrosarcoma. [83, 82]
An atypical form of hypertrophic osteoarthropathy has presentation limited to lower extremities.
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Epidemiology
Primary hypertrophic osteoarthropathy (HOA) is a rare condition. The precise incidence of this syndrome is unknown. According to one study, it has an estimated prevalence of 0.16%.
No systematic prevalence studies have been performed for secondary HOA, but cases are associated with many illnesses. According to Rassam et al, HOA occurred in about 3% (9 of 280) of consecutive lung cancer cases seen between 1970-1975. Other literature has described higher rates in primary lung cancer, ranging from about 4% to 32%.
In a study of consecutive patients with congenital cardiac disease, MartÃnez-LavÃn et al identified HOA in 10 of 32 patients (31%). HOA associated with respiratory failure is reported to occur in 27% of patients.
HOA affects persons of all races but is more common in African Americans. Primary HOA due to HPGD mutation has no sexual predominance. Primary HOA due toSLCO2A1 mutation has a marked predominance in males, with a male-to-female ratio of 9:1, and males usually show a more severe phenotype. Secondary osteoarthropathy has the same sex ratio as the associated illnesses.
Onset of primary HOA due to HPGD mutation is more common in children. Onset of cases due to SLCO2A1 mutation is more likely during puberty. [60, 87] Secondary HOA is rarely encountered in children and adolescents; it most commonly affects individuals aged 55-75 years.
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Prognosis
Primary HOA has a self-limiting course, with progression stopping at the end of adolescence. There is no cure for the skeletal abnormalities. The mortality and morbidity of secondary HOA vary with the associated illness. Secondary osteoarthritis may complicate long-standing HOA.
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Patient Education
Patients first diagnosed with hypertrophic osteoarthropathy should be reassured regarding its good prognosis as a musculoskeletal condition. That being established, they should be informed of its significance and the need for further investigation to rule out any treatable associated disease. These investigations are guided by results from thorough clinical evaluations, including questions specifically targeting intrathoracic diseases.
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Clinical Presentation
References
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Media Gallery
Clubbing associated with hypertrophic osteoarthropathy can be classified into 3 topographical groups (ie, symmetrical, unilateral, unidigital). This is symmetrical clubbing; it involves all the fingers.
Joint symptoms of hypertrophic osteoarthropathy range from mild to severe arthralgias that involve the metacarpal joints, wrists, elbows, knees, and ankles. The range of motion of affected joints may be slightly decreased. When effusions are present, they usually involve the large joints (eg, knees, ankles, wrists).
For hypertrophic osteoarthropathy diagnosis, radionuclide bone scan using technetium Tc 99m polyphosphate shows increased uptake of the tracer in the periosteum, often appearing pericortical and linear in nature. These findings can be present even when findings from plain radiographs are doubtful. The clubbed digits may also show increased uptake in early passage flow studies.
In adulthood, 90% of generalized hypertropic osteoarthropathy cases are associated with an intrathoracic infectious or neoplastic condition.
Radiograph of both hands in a 42-year-old man with a family history of primary hypertrophic osteoarthropathy who had coarsened facial features and thickness of the scalp. Note the soft-tissue clubbing and acro-osteolysis of the terminal phalanges.
Macroradiograph of the left hand in a patient known to have long-standing bronchiectasis shows extensive lamellar periosteal new bone formation around the shafts of the distal radius, ulna, metacarpals, and proximal phalanges (also see the next image).
Radiograph in a patient with long-standing bronchiectasis shows extensive lamellar periosteal reaction around the lower parts of the femora.
A 53-year-old male smoker presented with lower-limb pain around the hips, knees, and ankles. A chest radiograph was obtained as a part of the workup and demonstrated an opacity in the left apical region (arrow) suggestive of a bronchial neoplasm. Results of percutaneous needle biopsy confirmed a squamous carcinoma (see also the next image).
Radiograph in a 53-year-old male smoker with lower-limb pain around the hips, knees, and ankles shows a subtle periosteal reaction around the upper parts of the femora on the medial aspects (see also the next image).
Anteroposterior radiograph of the right ankle in a 53-year-old male smoker with lower-limb pain around the hips, knees, and ankles shows lamellar periosteal new bone formation around the lower shafts of the tibia and fibula.
Radiograph in a 32-year-old woman treated for Graves disease (thyrotoxicosis) who presented with a vague discomfort in the hands. Radiograph shows a mixture of hair-on-end and lamellar periosteal reaction around the distal shafts of the second metacarpal bones caused by thyroid acropachy.
Radiograph of the lower legs in a patient presenting with infected ulceration of the right lower leg caused by venous insufficiency. Note the extensive lamellar periosteal new bone around the shafts of the tibia and fibula.
Lateral radiograph of the tibia and fibula in a patient with chronic venous insufficiency shows periosteal new bone formation around the tibia and fibula. Note the arterial and venous calcifications.
Anteroposterior radiograph of the femur in an athlete with a previous history of trauma to the thigh shows a traumatic periostitis of the mid femur. Note the calcific myositis.
Radiograph of the arm in a 3-month-old male infant presenting with fever and irritability shows massive periosteal new bone formation around the humerus, radius, and ulna associated with infantile cortical hyperostosis (Caffey disease). Note the sparing of the proximal phalanges.
Radionuclide scans show the typical appearance of secondary hypertrophic osteoarthropathy caused by a bronchogenic carcinoma.
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Contributor Information and Disclosures
Vishnuteja Devalla, MD Fellow, Department of Rheumatology, Dartmouth-Hitchcock Medical CenterVishnuteja Devalla, MD is a member of the following medical societies: American College of Physicians, American College of RheumatologyDisclosure: Nothing to disclose.
Stephanie Danielle Mathew, DO Fellowship Program Director, Dartmouth-Hitchcock Medical CenterStephanie Danielle Mathew, DO is a member of the following medical societies: American College of Physicians, American College of Rheumatology, American Osteopathic AssociationDisclosure: Nothing to disclose.
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5922 | https://www.quora.com/What-word-is-rightly-pronounced-wrong-but-wrongly-pronounced-right | Something went wrong. Wait a moment and try again.
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What word is rightly pronounced wrong but wrongly pronounced right?
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The word you are looking for is "wrong." It's a play on words: it is "rightly pronounced" as "wrong," but if you pronounce it "right," you are "wrong." This riddle plays with the meanings and the sounds of the words.
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This is called a “trick question.” This happens when a question is filled with so many oxymorons that it can not be answered legitimately. Monsieur Out.
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Thanks for this great question, which I think illustrates and revolves around characteristic wordplay in English.
The answer is “wrong,” as:
it is rightly (formal Eng.) pronounced as the word “wrong.”
it can be wrongly pronounced (a playful way of saying “pronounced just as or similarly to the way in which the word ‘wrong’ is pronounced) in the right (formal Eng.) way as the word “wrong”
You know what I mean!!! Happy lexicon-ing, everyone.
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Some foreign words are like that. For instance
crepe
. In English we pronounce it “crape”, but in French it is pronounced “crep”. But if you pronounce it correctly, then people won’t know what you are talking about. That is, it will be wrongly pronounced right, as you put it. If you pronounce it incorrectly, then people will understand you. It is rightly pronounced wrong.
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5923 | https://math.stackexchange.com/questions/2441524/solving-for-t-in-an-exponential-equation | algebra precalculus - Solving for t in an exponential equation - Mathematics Stack Exchange
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Solving for t in an exponential equation
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Solve for t t in the equation:
(1+t)e−t=0.05(1+t)e−t=0.05
This is where I have reached.
(1+t)e−t=0.05(1+t)e−t=0.05
1+t=0.05 e t 1+t=0.05 e t
ln(1+t)=ln 0.05 e t ln(1+t)=ln0.05 e t
ln 1+ln t=ln 0.05+ln e t ln1+lnt=ln0.05+lne t
ln t=ln 0.05+t lnt=ln0.05+t
I am stucked
algebra-precalculus
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edited Sep 23, 2017 at 11:25
Parcly Taxel
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Ashalley SamuelAshalley Samuel
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1 Please use MathJax.José Carlos Santos –José Carlos Santos 2017-09-23 10:37:54 +00:00 Commented Sep 23, 2017 at 10:37
is this e−T(1+T)=0.05 e−T(1+T)=0.05 ?Dr. Sonnhard Graubner –Dr. Sonnhard Graubner 2017-09-23 10:59:46 +00:00 Commented Sep 23, 2017 at 10:59
No. I have made the necessary corrections. It's e^ (-T)Ashalley Samuel –Ashalley Samuel 2017-09-23 11:09:45 +00:00 Commented Sep 23, 2017 at 11:09
1 the logarithm of the sum is not the sum of the logarithms M. Van –M. Van 2017-09-23 11:27:54 +00:00 Commented Sep 23, 2017 at 11:27
1 there is no closed solution for this, use numerical methods Vasili –Vasili 2017-09-23 11:28:14 +00:00 Commented Sep 23, 2017 at 11:28
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Solution in terms of the Lambert W function.
(1+t)e−t=1 20(−1−t)e−t=−1 20(−1−t)e−1−t=−1 20 e−1−t=W(−1 20 e)1+t=−W(−1 20 e)t=−W(−1 20 e)−1(1+t)e−t=1 20(−1−t)e−t=−1 20(−1−t)e−1−t=−1 20 e−1−t=W(−1 20 e)1+t=−W(−1 20 e)t=−W(−1 20 e)−1
All complex solutions are obtained by taking all branches of W. The real solutions are
−W 0(−1 20 e)−1≈−.981258037995−W−1(−1 20 e)−1≈4.74386451839−W 0(−1 20 e)−1≈−.981258037995−W−1(−1 20 e)−1≈4.74386451839
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edited Sep 23, 2017 at 11:51
answered Sep 23, 2017 at 11:45
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(+1) nice use of Lambert function, its kinda obvious, but i think not to the OP.user411780 –user411780 2017-09-23 11:52:41 +00:00 Commented Sep 23, 2017 at 11:52
So how do you get the figures in using Lambert W function. I understand only to where the figures were obtained Ashalley Samuel –Ashalley Samuel 2017-09-29 13:51:50 +00:00 Commented Sep 29, 2017 at 13:51
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If you do not use the beautiful Lambert function, just consider the most general problem where you look for the zero(s) of function
f(t)=(1+t)e−t−a f(t)=(1+t)e−t−a
f′(t)=−t e−t f′(t)=−t e−t
f′′(t)=(t−1)e−t f″(t)=(t−1)e−t
So, the function increases when t<0 t<0, goes through a maximum when t=0 t=0 (by the second derivative test), decreases when t>0 t>0.
At the maximum, we have f(0)=1−a f(0)=1−a; so, if a<1 a<1, there will be two real roots (one negative and one positive). If a=1 a=1, a double root t=0 t=0 and if a>1 a>1, no root at all.
If you graph the function for a=0.05 a=0.05, you will see that there is one root close to t=−1 t=−1 and another one close to t=5 t=5. You then have all the elements to start Newton method which will give the following iterates
⎛⎝⎜⎜⎜⎜⎜⎜⎜⎜n 0 1 2 3 4 t n−1.00000000000000000000−0.98160602794142788392−0.98125816020529694931−0.98125803799504305042−0.98125803799502797244⎞⎠⎟⎟⎟⎟⎟⎟⎟⎟(n t n 0−1.00000000000000000000 1−0.98160602794142788392 2−0.98125816020529694931 3−0.98125803799504305042 4−0.98125803799502797244)
⎛⎝⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜n 0 1 2 3 4 5 t n 5.0000000000000000000 4.7158684089742339658 4.7435578246859564746 4.7438644812774523864 4.7438645183905778323 4.7438645183905783759⎞⎠⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟(n t n 0 5.0000000000000000000 1 4.7158684089742339658 2 4.7435578246859564746 3 4.7438644812774523864 4 4.7438645183905778323 5 4.7438645183905783759)
These are the solutions for twenty significants figures.
Concerning the estimates of the solutions (for a<1 a<1), we can gnerate estimates approximating the function f(x)f(x) using, say, a [2,3][2,3] Padé approximant. Solving for the roots of numerator, we should get
t 1,2=2(42−24 a−18 a 2±−131 a 4−3386 a 3−4356 a 2+3194 a+4679−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−√)13 a 2+114 a+53 t 1,2=2(42−24 a−18 a 2±−131 a 4−3386 a 3−4356 a 2+3194 a+4679)13 a 2+114 a+53
For a=0.05 a=0.05, this will give as estimates t 1=−0.978 t 1=−0.978 and t 2=3.754 t 2=3.754.
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edited Sep 30, 2017 at 3:13
answered Sep 23, 2017 at 14:27
Claude LeiboviciClaude Leibovici
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Am trying to use the numerical methods approach. But how do i get the initial value for "t"?Ashalley Samuel –Ashalley Samuel 2017-09-29 15:15:07 +00:00 Commented Sep 29, 2017 at 15:15
@AshalleySamuel. See my edit.Claude Leibovici –Claude Leibovici 2017-09-30 03:13:29 +00:00 Commented Sep 30, 2017 at 3:13
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Question about notation on the NIST DLMF
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Many relations on the NIST DLMF have certain restriction on parameters that must be satisfied in order for the relation to hold. Take for example equation 15.8.5 15.8.5 which lists multiple constraints on the variable z z, namely, |p h z|<π,|p h(1−z)|<π|p h z|<π,|p h(1−z)|<π.
My question is if the constraints should be read as an "and" or "or" statement. Going back to the example, should I interpret the constraints as
|p h z|<π and|p h(1−z)|<π|p h z|<π and|p h(1−z)|<π
or
|p h z|<π or|p h(1−z)|<π?|p h z|<π or|p h(1−z)|<π?
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asked Aug 27, 2019 at 16:14
Aaron HendricksonAaron Hendrickson
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1 This is just a guess, but I think it means "or". If you look at the formulas in 15.6, they use a semicolon in contexts where "and" is intended. (Or at least I think so.)saulspatz –saulspatz 2019-08-27 16:29:39 +00:00 Commented Aug 27, 2019 at 16:29
@saulspatz I am starting to think is does mean "and" see dlmf.nist.gov/5.9.E3. It would seem the integral requires both criteria to be satisfied to converge. That said, I am still not fully convinced yet.Aaron Hendrickson –Aaron Hendrickson 2019-08-27 17:24:42 +00:00 Commented Aug 27, 2019 at 17:24
It looks like they weren't fully consistent then. What a shame.saulspatz –saulspatz 2019-08-27 22:24:18 +00:00 Commented Aug 27, 2019 at 22:24
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The condition |p h z|<π|p h z|<π means that there is a branch cut for reals less than 0 0. The condition |p h(1−z)|<π|p h(1−z)|<π means that there is a branch cut for reals greater than 1 1. The context is that both of these branch cuts are to be excluded and thus the "," is to be interpreted as "and".
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answered Aug 27, 2019 at 18:12
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5925 | https://math.libretexts.org/Courses/Truckee_Meadows_Community_College/TMCC%3A_Precalculus_I_and_II/Under_Construction_test2_03%3A_Polynomial_and_Rational_Functions/Under_Construction_test2_03%3A_Polynomial_and_Rational_Functions_3.8%3A_Inverses_and_Radical_Functions | Skip to main content
3.8: Inverses and Radical Functions
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3.7: Rational Functions
3.9: Modeling Using Variation
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( \newcommand{\kernel}{\mathrm{null}\,})
Learning Objectives
In this section, you will:
Find the inverse of an invertible polynomial function.
Restrict the domain to find the inverse of a polynomial function.
A mound of gravel is in the shape of a cone with the height equal to twice the radius.
The volume is found using a formula from elementary geometry.
We have written the volume in terms of the radius . However, in some cases, we may start out with the volume and want to find the radius. For example: A customer purchases 100 cubic feet of gravel to construct a cone shape mound with a height twice the radius. What are the radius and height of the new cone? To answer this question, we use the formula
This function is the inverse of the formula for in terms of .
In this section, we will explore the inverses of polynomial and rational functions and in particular the radical functions we encounter in the process.
Finding the Inverse of a Polynomial Function
Two functions and are inverse functions if for every coordinate pair in , , there exists a corresponding coordinate pair in the inverse function, , . In other words, the coordinate pairs of the inverse functions have the input and output interchanged. Only one-to-one functions have inverses. Recall that a one-to-one function has a unique output value for each input value and passes the horizontal line test.
For example, suppose a water runoff collector is built in the shape of a parabolic trough as shown in Figure . We can use the information in the figure to find the surface area of the water in the trough as a function of the depth of the water.
Because it will be helpful to have an equation for the parabolic cross-sectional shape, we will impose a coordinate system at the cross section, with measured horizontally and measured vertically, with the origin at the vertex of the parabola (Figure ).
From this we find an equation for the parabolic shape. We placed the origin at the vertex of the parabola, so we know the equation will have form . Our equation will need to pass through the point , from which we can solve for the stretch factor .
Our parabolic cross section has the equation
We are interested in the surface area of the water, so we must determine the width at the top of the water as a function of the water depth. For any depth , the width will be given by , so we need to solve the equation above for and find the inverse function. However, notice that the original function is not one-to-one, and indeed, given any output there are two inputs that produce the same output, one positive and one negative.
To find an inverse, we can restrict our original function to a limited domain on which it is one-to-one. In this case, it makes sense to restrict ourselves to positive values. On this domain, we can find an inverse by solving for the input variable:
This is not a function as written. We are limiting ourselves to positive values, so we eliminate the negative solution, giving us the inverse function we’re looking for.
,
Because is the distance from the center of the parabola to either side, the entire width of the water at the top will be . The trough is 3 feet (36 inches) long, so the surface area will then be:
This example illustrates two important points:
When finding the inverse of a quadratic, we have to limit ourselves to a domain on which the function is one-to-one.
The inverse of a quadratic function is a square root function. Both are toolkit functions and different types of power functions.
Functions involving roots are often called radical functions. While it is not possible to find an inverse of most polynomial functions, some basic polynomials do have inverses. Such functions are called invertible functions, and we use the notation .
Warning: is not the same as the reciprocal of the function . This use of “–1” is reserved to denote inverse functions. To denote the reciprocal of a function , we would need to write:
An important relationship between inverse functions is that they “undo” each other. If is the inverse of a function , then is the inverse of the function . In other words, whatever the function does to , undoes it—and vice-versa.
, for all in the domain of
and
, for all in the domain of
Note that the inverse switches the domain and range of the original function.
VERIFYING TWO FUNCTIONS ARE INVERSES OF ONE ANOTHER
Two functions, and , are inverses of one another if for all in the domain of and ,
Howto: Given a polynomial function, find the inverse of the function by restricting the domain in such a way that the new function is one-to-one
Replace with .
Interchange and .
Solve for , and rename the function .
Example : Verifying Inverse Functions
Show that and are inverses, for .
Solution
We must show that and .
and
Therefore, and are inverses.
Exercise
Show that and are inverses.
Answer a
Answer b
Example : Finding the Inverse of a Cubic Function
Find the inverse of the function .
Solution
This is a transformation of the basic cubic toolkit function, and based on our knowledge of that function, we know it is one-to-one. Solving for the inverse by solving for .
Analysis
Look at the graph of and . Notice that one graph is the reflection of the other about the line . This is always the case when graphing a function and its inverse function.
Also, since the method involved interchanging and , notice corresponding points. If is on the graph of ,then is on the graph of . Since is on the graph of , then is on the graph of . Similarly, since is on the graph of ,then is on the graph of (Figure ).
Exercise
Find the inverse function of .
Answer
Restricting the Domain to Find the Inverse of a Polynomial Function
So far, we have been able to find the inverse functions of cubic functions without having to restrict their domains. However, as we know, not all cubic polynomials are one-to-one. Some functions that are not one-to-one may have their domain restricted so that they are one-to-one, but only over that domain. The function over the restricted domain would then have an inverse function. Since quadratic functions are not one-to-one, we must restrict their domain in order to find their inverses.
RESTRICTING THE DOMAIN
If a function is not one-to-one, it cannot have an inverse. If we restrict the domain of the function so that it becomes one-to-one, thus creating a new function, this new function will have an inverse.
How to: Given a polynomial function, restrict the domain of a function that is not one-to-one and then find the inverse
Restrict the domain by determining a domain on which the original function is one-to-one.
Replace with .
Interchange and .
Solve for , and rename the function or pair of function .
Revise the formula for by ensuring that the outputs of the inverse function correspond to the restricted domain of the original function.
Example : Restricting the Domain to Find the Inverse of a Polynomial Function
Find the inverse function of :
,
,
Solution
The original function is not one-to-one, but the function is restricted to a domain of or on which it is one-to-one (Figure ).
To find the inverse, start by replacing with the simple variable .
Interchange and .
Take the square root.
Add to both sides.
This is not a function as written. We need to examine the restrictions on the domain of the original function to determine the inverse. Since we reversed the roles of and for the original , we looked at the domain: the values could assume. When we reversed the roles of and , this gave us the values could assume. For this function, , so for the inverse, we should have , which is what our inverse function gives.
The domain of the original function was restricted to , so the outputs of the inverse need to be the same, , and we must use the + case:
The domain of the original function was restricted to , so the outputs of the inverse need to be the same, , and we must use the – case:
Analysis
On the graphs in Figure , we see the original function graphed on the same set of axes as its inverse function. Notice that together the graphs show symmetry about the line . The coordinate pair is on the graph off f and the coordinate pair is on the graph of . For any coordinate pair, if is on the graph of , then is on the graph of . Finally, observe that the graph of intersects the graph of on the line . Points of intersection for the graphs of and will always lie on the line .
Example : Finding the Inverse of a Quadratic Function When the Restriction Is Not Specified
Restrict the domain and then find the inverse of
.
Solution
We can see this is a parabola with vertex at that opens upward. Because the graph will be decreasing on one side of the vertex and increasing on the other side, we can restrict this function to a domain on which it will be one-to-one by limiting the domain to .
To find the inverse, we will use the vertex form of the quadratic. We start by replacing with a simple variable, , then solve for .
Interchange and .
Add 3 to both sides.
Take the square root.
Add 2 to both sides.
Rename the function.
Now we need to determine which case to use. Because we restricted our original function to a domain of , the outputs of the inverse should be the same, telling us to utilize the + case
If the quadratic had not been given in vertex form, rewriting it into vertex form would be the first step. This way we may easily observe the coordinates of the vertex to help us restrict the domain.
Analysis
Notice that we arbitrarily decided to restrict the domain on . We could just as easily have opted to restrict the domain to , in which case . Observe the original function graphed on the same set of axes as its inverse function in Figure . Notice that both graphs show symmetry about the line . The coordinate pair is on the graph of and the coordinate pair is on the graph of . Observe from the graph of both functions on the same set of axes that
domain of range of
and
domain of range of .
Finally, observe that the graph of intersects the graph of along the line .
Exercise
Find the inverse of the function , on the domain .
Answer
Finding Inverses
Notice that the functions from previous examples were all polynomials, and their inverses were radical functions. If we want to find the inverse of a radical function, we will need to restrict the domain of the answer because the range of the original function is limited.
How to: Given a radical function, find the inverse
Determine the range of the original function.
Replace with , then solve for .
If necessary, restrict the domain of the inverse function to the range of the original function.
Example : Finding the Inverse of a Radical Function
Restrict the domain of the function and then find the inverse.
Solution
Note that the original function has range . Replace with , then solve for .
Replace with .
Interchange and .
Square each side.
Add 4.
Rename the function .
Recall that the domain of this function must be limited to the range of the original function.
,
Analysis
Notice in Figure that the inverse is a reflection of the original function over the line . Because the original function has only positive outputs, the inverse function has only nonnegative inputs.
Exercise
Restrict the domain and then find the inverse of the function .
Answer
: ,
Solving Applications of Radical Functions
Radical functions are common in physical models, as we saw in the section opener. We now have enough tools to be able to solve the problem posed at the start of the section.
Example : Solving an Application with a Cubic Function
A mound of gravel is in the shape of a cone with the height equal to twice the radius. The volume of the cone in terms of the radius is given by
Find the inverse of the function that determines the volume of a cone and is a function of the radius . Then use the inverse function to calculate the radius of such a mound of gravel measuring 100 cubic feet. Use .
Solution
Start with the given function for . Notice that the meaningful domain for the function is since negative radii would not make sense in this context nor would a radius of . Also note the range of the function (hence, the domain of the inverse function) is . Solve for in terms of , using the method outlined previously. Note that in real-world applications, we do not swap the variables when finding inverses. Instead, we change which variable is considered to be the independent variable.
Solve for .
Solve for .
This is the result stated in the section opener. Now evaluate this for and .
Therefore, the radius is about 3.63 ft.
Determining the Domain of a Radical Function Composed with Other Functions
When radical functions are composed with other functions, determining domain can become more complicated.
Example : Finding the Domain of a Radical Function Composed with a Rational Function
Find the domain of the function:
Solution
Because a square root is only defined when the quantity under the radical is non-negative, we need to determine where
The output of a rational function can change signs (change from positive to negative or vice versa) at x-intercepts and at vertical asymptotes. For this equation, the graph could change signs at , , and .
To determine the intervals on which the rational expression is positive, we could test some values in the expression or sketch a graph. While both approaches work equally well, for this example we will use a graph as shown in Figure .
This function has two x-intercepts, both of which exhibit linear behavior near the x-intercepts. There is one vertical asymptote, corresponding to a linear factor; this behavior is similar to the basic reciprocal toolkit function, and there is no horizontal asymptote because the degree of the numerator is larger than the degree of the denominator. There is a y-intercept at .
From the y-intercept and x-intercept at , we can sketch the left side of the graph. From the behavior at the asymptote, we can sketch the right side of the graph.
From the graph, we can now tell on which intervals the outputs will be non-negative, so that we can be sure that the original function will be defined. has domain or , or in interval notation, .
Finding Inverses of Rational Functions
As with finding inverses of quadratic functions, it is sometimes desirable to find the inverse of a rational function, particularly of rational functions that are the ratio of linear functions, such as in concentration applications.
Example : Finding the Inverse of a Rational Function
The function
represents the concentration of an acid solution after mL of 40% solution has been added to 100 mL of a 20% solution. First, find the inverse of the function; that is, find an expression for in terms of . Then use your result to determine how much of the 40% solution should be added so that the final mixture is a 35% solution.
Solution
We first want the inverse of the function in order to determine how many mL we need for a given concentration. We will solve for in terms of .
Now evaluate this function at 35%, which is .
We can conclude that 300 mL of the 40% solution should be added.
Exercise
Find the inverse of the function .
Answer
Access these online resources for additional instruction and practice with inverses and radical functions.
Graphing the Basic Square Root Function
Find the Inverse of a Square Root Function
Find the Inverse of a Rational Function
Find the Inverse of a Rational Function and an Inverse Function Value
Inverse Functions
Key Concepts
The inverse of a quadratic function is a square root function.
If is the inverse of a function , then is the inverse of the function . See Example .
While it is not possible to find an inverse of most polynomial functions, some basic polynomials are invertible. See Example .
To find the inverse of certain functions, we must restrict the function to a domain on which it will be one-to-one. See Examples and
When finding the inverse of a radical function, we need a restriction on the domain of the answer. See Example and .
Inverse and radical and functions can be used to solve application problems. See Examples and .
3.7: Rational Functions
3.9: Modeling Using Variation |
5926 | https://www.savemyexams.com/a-level/physics/aqa/17/revision-notes/4-mechanics-and-materials/4-6-work-energy-and-power/4-6-2-area-under-a-force-displacement-graph/ | A LevelPhysicsAQARevision NotesMechanics & MaterialsWork, Energy & PowerArea Under a Force-Displacement Graph
Area Under a Force-Displacement Graph (AQA A Level Physics): Revision Note
Exam code: 7408
Author
Ashika
Last updated
Area Under a Force-Displacement Graph
The work done by a force acting over a distance can also be found from a force-displacement graph
If the force is not constant and is plotted against the displacement of the object:
The work done is equal to the area under the force-displacement graph
This is because:
Work done = Force × Displacement
The work done is therefore equivalent whether there is:
A small force over a long displacement
A large force over a small displacement
The graph may need to be split up into sections. The total area is the sum of the areas of each section
The area underneath the force-displacement graph is the work done
Worked Example
The graph shows how a force varies over a displacement of 80 m.
Calculate the work done.
Answer:
Step 1: Split the graph into sections
The work done is the area under the graph
The total area can be found by splitting the graph into sections A and B
Step 2: Calculate the area of section A
Section A is a right-angled triangle where the area is 0.5 × base × height
0.5 × 80 × (250 – 100) = 6000 J
Step 3: Calculate the area of section B
Section B is a rectangle where the area is base × height
80 × 100 = 8000 J
Step 4: Calculate the total work done
The total work done is the sum of both areas
Work done = 6000 + 8000 = 14 000 J
Examiner Tips and Tricks
Always check the units on the axes when calculating values from a graph. Sometimes the force will be given in kN or the displacement in km. These must be converted into SI units to calculate the work done in J.
Variable Forces
The force on an object may not always be constant, this is known as a variable force
This is more representative of a force in real life
If a force is constant, then the following equations can be used:
W = Fs
P = Fv
If a force is varying, the above equations cannot be used, instead, work done must be found from the area under the force-displacement graph
If a varying force increases, then an object’s acceleration increases and vice versa
Worked Example
A person is pulling a suitcase through an airport with a rough surface. They apply a force of 150 N over a distance of 12 m. Afterwards, the person gets progressively tired and the applied force is linearly reduced to 60 N. The total distance through which the suitcase has moved is 25 m.
Calculate the work done by the force applied by the person over 25 m.
Answer:
Step 1: Sketch a force-displacement graph and split it into sections
Step 2: Split the graph into sections
The work done is the area under the graph
The total area can be found by splitting the graph into sections A and B
Step 3: Calculate the area of section A
Section A is a rectangle where the area is base × height
AA = 12 × 150 = 1800 J
Step 4: Calculate the area of section B
Section B is a trapezium, which can be split into a right-angled triangle and a rectangle
Step 5: Calculate the total work done
The total work done is the sum of both areas
Work done = AA + AB = 1800 + 1365 = 3165 J
Examiner Tips and Tricks
When sketching graphs, they don’t have to be to scale. However, it is important to label the key points on the x and y-axis from the question to calculate the areas underneath the graph.
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5927 | https://www.ae.msstate.edu/tupas/SA2/def.proportional_limit.html | P ROPORTIONAL L IMIT
Proportional limit is the point on the stress-strain diagram where the curve becomes nonlinear. The proportional limit stress is the value of stress corresponding to the elastic limit of the material. For strain levels below the elastic limit strain, Hooke's law may be used to relate stress to strain. The proportional limit is commonly assumed to coincide with the yield point unless otherwise stated in the problem statement.
This is a typical shear stress-strain diagram.
This is an ideal stress-strain diagram where the proportional limit and the yield point coincide and stress remains constant beyond the elastic limit. This type of material is known as elasto-plastic. |
5928 | https://howellkb.uah.edu/DE2/Lecture%20Notes/DE1Review.pdf | 1 A Brief Review of Elementary Ordinary Differential Equations At various points in the material we will be covering, we will need to recall and use material normally covered in an elementary course on ordinary differential equations. In these notes, we will very briefly review the main topics that will be needed later. For more complete discussions of these topics, see your current text on differential equations (by Boyce and DiPrima), or your old introductory text on ordinary differential equations, or see your instructor’s treatment of these topics.1 1.1 Basic Terminology Recall: A differential equation (often called a “de”) is an equation involving derivatives of an unknown function. If the unknown can be assumed to be a function of only one variable (so the derivatives are the “ordinary” derivatives from Calc. I), then we say the differential equation is an ordinary differential equation (ode). Otherwise, the equation is a partial differential equation (pde). Our interest will just be in odes. In these notes, the variable will usually be denoted by x and the unknown function by y or y(x) .
Recall, also, for any given ordinary differential equation: 1.
The order is the order of the highest order derivative of the unknown function explicitly appearing in the equation.
2.
A solution is any function (or formula for a function) that satisfies the equation.
3.
A general solution is a formula that describes all solutions to the equation. Typically, the general solution to a kth order ode contains k arbitrary/undetermined constants.
4.
Typically, a “differential equation problem” consists of a differential equation along with some auxiliary conditions the solution must also satisfy (e.g., “initial values” for the solution). In practice you usually find the general solution first, and then choose values for the “undetermined constants” so that the auxiliary conditions are satisfied.
1 available online at www.math.uah.edu/howell/DEtext.
1/3/2013 1 Review of Elementary ODEs 2 1.2 Some “Analytic” Methods for Solving First-Order ODEs (Warning: Here, the word “analytic” just means that the method leads to exact formulas for solutions, as opposed to, say, a numerical algorithm that gives good approximations to particular solutions at fixed points. Later in this course, the word “analytic” will mean something else.) Separable Equations∗ A first-order ode is separable if it can be written as dy dx = g(x)h(y) .
Such a de can be solved by the following procedure: 1.
Get it into the above form (i.e., the derivative equaling the product of a function of x (the g(x) above), with a function of y (the above h(y) ).
2.
Divide through by h(y) (but also consider the possibility that h(y) = 0 ).
3.
Integrate both sides with respect to x (don’t forget an arbitrary constant).
4.
Solve the last equation for y(x) .
!◮Example 1.1: Consider finding the general solution to dy dx = 2x y2 + 1 .
Going through the above steps: 1 y2 + 1 dy dx = 2x H ⇒ Z 1 y2 + 1 dy dx dx = Z 2x dx H ⇒ arctan(y) = x2 + c H ⇒ y = tan x2 + c .
Linear Equations† A first-order ode is said to be linear if it can be written in the form dy dx + p(x)y = q(x) where p(x) and q(x) are known functions of x . Such a differential equation can be solved by the following procedure: ∗see, also, chapter 4 of the online text † see, also, chapter 5 of the online text version: 1/3/2013 Review of Elementary ODEs 3 1.
Get it into the above form.
2.
Compute the integrating factor µ(x) = e R p(x) dx (don’t worry about arbitrary constants here).
3.
(a) Multiply the equation from the first step by the integrating factor.
(b) Observe that, by the product rule, the left side of the resulting equation can be rewritten as d dx [µy] , thus giving you the equation d dx [µ(x) y(x)] = µ(x) q(x) .
4.
Integrate both sides of your last equation with respect to x , and solve for y(x) . Don’t forget the arbitrary constant.
!◮Example 1.2: Consider finding the general solution to x dy dx + 4y = 21x3 .
Dividing through by x gives dy dx + 4 x y = 21x2 .
So the integrating factor is µ(x) = e R p(x) dx = e R 4/x dx = e4 ln x = x4 .
Multiplying the last differential equation above by this integrating factor and then continuing as described in the procedure: x4 dy dx + 4 x y = x4 21x2 H ⇒ x4 dy dx + 4x3y = 21x6 .
But, by the product rule, d dx x4y(x) = x4 dy dx + 4x3y , and so we can rewrite our last differential equation as d dx x4y(x) = 21x6 .
This can be easily integrated and solved: Z d dx x4y(x) dx = Z 21x6 dx H ⇒ x4y(x) = 3x7 + c H ⇒ y(x) = 3x7 + c x4 H ⇒ y(x) = 3x3 + cx−4 .
version: 1/3/2013 Review of Elementary ODEs 4 Two notes on this method: 1.
The formula for the integrating factor µ(x) is actually derived from the requirement that d dx [µ(x) y(x)] = µdy dx + dµ dx y = µdy dx + µpy , which is the “observation” made in step 3b of the procedure. This means that µ must satisfy the simple differential equation dµ dx = µp .
2.
Many texts state a formula for y(x) in terms of p(x) and q(x) . The better texts also state that memorizing and using this formula is stupid.
Other Methods Other methods for solving first-order ordinary differential equations include the integration of exact equations, and the use of either clever substitutions or more general integrating factors to reduce “difficult” equations to either separable, linear or exact equations. See a good de text if you are interested.
1.3 Higher-Order Linear Differential Equations Basics‡ An N th order differential equation is said to be linear if it can be written in the form a0y(N) + a1y(N−1) + · · · + aN−2y′′ + aN−1y′ + aN y = f where f and the ak’s are known functions of x (with a0(x) not being the zero function). The equation is said to be homogeneous if and only if f is the zero function (i.e., is always 0 ).
Recall that, if the equation is homogeneous, then we have “linearity” , that is, whenever y1 and y2 are two solutions to a homogeneous linear differential equation, and a and b are any two constants, then y = ay1 + by2 is another solution to the differential equation. In other words, the set of solutions to a homogeneous linear differential equations is a vector space of functions.
(Isn’t it nice to see vector spaces again?) Recall further, that 1.
The general solution to an N th order linear homogeneous ordinary differential equation is given by y(x) = c1y1(x) + c2y2(x) + · · · + cN yN(x) where the ck’s are arbitrary constants and {y1, y2, . . . , yN} ‡ see, also, chapter 12, sections 1 – 3, and chapter 14 of the online text version: 1/3/2013 Review of Elementary ODEs 5 is a linearly independent set of solutions to the homogeneous de. (i.e., {y1, y2, . . . , yN} is a basis for the N-dimensional space of solutions to the homogeneous differential equation.) 2.
A general solution to an N th order linear nonhomogeneous ordinary differential equation is given by y(x) = yp(x) + yh(x) where yp is any particular solution to the nonhomogeneous ordinary differential equation and yh is a general solution to the corresponding homogeneous ode.
In “real” applications, N is usually 1 or 2 . On rare occasions, it may be 4 , and, even more rarely, it is 3 . Higher order differential equations can arise, but usually only in courses on differential equations. Do note that if N = 1 , then the differential equation can be solved using the method describe for first order linear equations (see page 2).
Notes About Linear Independence Recall that a set of functions {y1(x), y2(x), . . . , yN(x)} is linearly independent if and only if none of the yk’s can be written as a linear combination of the other yk’s . There are several ways to test for linear independence. The one usually discussed in de texts involves the corresponding Wronskian W(x) , given by W = y1 y2 y3 · · · yn y1′ y2′ y3′ · · · yn′ y1′′ y2′′ y3′′ · · · yn′′ .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
y1(n−1) y2(n−1) y3(n−1) · · · yn(n−1) .
The test is that the set of N solutions {y1(x), y2(x), . . . , yN(x)} to some given N th-order homogeneous linear differential equation is linearly independent if and only if W(x0) ̸= 0 for any point in the interval over which these yk’s are solutions.
This is a highly recommended test when N > 2 , but, frankly, it is silly to use it when N = 2 . Then, we just have a pair of solutions {y1(x), y2(x)} and any such pair is linearly independent if and only if neither function is a constant multiple of each other, and THAT is usually obvious upon inspection of the two functions.
version: 1/3/2013 Review of Elementary ODEs 6 Second-Order Linear Homogeneous Equations with Constant Coefficients§ Consider a differential equation of the form ay′′ + by′ + cy = 0 where a , b , and c are (real) constants. To solve such an equation, assume a solution of the form y(x) = erx (where r is a constant to be determined), and then plug this formula for y into the differential equation. You will then get the corresponding characteristic equation for the de, ar 2 + br + c = 0 .
Solve the characteristic equation. You’ll get two values for r , r = r± = −b ± √ b2 −4ac 2a (with the possibility that r+ = r−). Then: 1.
If r+ and r−are two distinct real values, then the general solution to the differential equation is y(x) = c1er+x + c2er−x where c1 and c2 are arbitrary constants.
2.
If r+ = r−, then r+ is real and the general solution to the differential equation is y(x) = c1er+x + c2xer+x where c1 and c2 are arbitrary constants. (Note: The c2xer+x part of the solution can be derived via the method of “reduction of order”.) 3.
If r+ or r−is complex valued, then they are complex conjugates of each other, r+ = α + iβ and r−= α −iβ for some real constants α and β . The general solution to the differential equation can then be written as y(x) = c1e(α + iβ)x + c2e(α −iβ)x where c1 and c2 are arbitrary constants. However, because e(α ± iβ)x = eαx [cos(βx) ± i sin(βx)] , the general solution to the differential equation can also be written as y(x) = C1eαx cos(βx) + C2eαx sin(βx) where C1 and C2 are arbitrary constants. In practice, the later formula for y is usually preferred because it involves just real-valued functions.
§ see, also, chapter 16 of the online text version: 1/3/2013 Review of Elementary ODEs 7 !◮Example 1.3: Consider y′′ −4y′ + 13y = 0 .
Plugging in y = erx , we get d2 dx2 erx −d dx erx + 13 erx = 0 H ⇒ r 2erx −4rerx + 13erx = 0 H ⇒ r 2 −4r + 13 = 0 .
Thus, r = −(−4) ± p (−4)2 −4 · 13 2 = 4 ± √ −36 2 = 2 ± 3i .
So the general solution to the differential equation can be written as y(x) = c1e(2+3i)x + c2e(2−3i)x or as y(x) = C1e2x cos(3x) + C2e2x sin(3x) , with the later formula usually being preferred.
Second-Order Euler Equations¶ A second-order Euler equation2 is a differential equation that can be written as ax2y′′ + bxy′ + cy = 0 where a , b , and c are (real) constants. To solve such an equation, assume a solution of the form y(x) = xr (where r is a constant to be determined), and then plug this formula for y into the differential equation, and solve for r .
With luck, you will get two distinct real values for r , r1 and r2 , in which case, the general solution to the differential equation is y(x) = c1xr1 + c2xr2 where c1 and c2 are arbitrary constants.
With less luck, you only complex values for r , or only one value for r . See chapter 16 of the online text to see what to do in these cases.
!◮Example 1.4: Consider x2y′′ + xy′ −9y = 0 .
¶ see, also, chapter 19 of the online text 2 also called a Cauchy-Euler equation version: 1/3/2013 Review of Elementary ODEs 8 Plugging in y = xr , we get x2 d2 dx2 xr + x d dx xr −9 xr = 0 H ⇒ x2 r(r −1)xr−2 + x rxr−1 −9 xr = 0 H ⇒ r 2xr −rxr + rxr −9xr = 0 H ⇒ r 2 −9 = 0 H ⇒ r = ±3 .
So the general solution to the differential equation is y(x) = c1x3 + c2x−3 .
Other Methods For solving more involved homogeneous second-order odes, there is still the method of Frobenius (which we will later discuss in some detail). You may also want to look up the method of reduction of order in your old differential equation text or chapter 13 of the online text (or see the last problem in the next homework list).
For solving nonhomogeneous second-order odes, you may want to recall the methods of “undetermined coefficients” (aka the “method of guess”) and “variation of parameters” , described in chapters 21 and 23 of the online text.
version: 1/3/2013 Review of Elementary ODEs 9 Additional Exercises 1.1. In this set, all the differential equations are first-order and separable.
a. Find the general solution for each of the following: i.
dy dx = xy −4x ii.
dy dx = 3y2 −y2 sin(x) iii.
dy dx = xy −3x −2y + 6 iv.
dy dx = y x b. Solve each of the following initial-value problems.
i.
dy dx −2y = −10 with y(0) = 8 ii. y dy dx = sin(x) with y(0) = −4 iii. x dy dx = y2 −y with y(1) = 2 1.2. In this set, all the differential equations are linear first-order equations.
a. Find the general solution for each of the following: i.
dy dx + 2y = 6 ii.
dy dx + 2y = 20e3x iii.
dy dx = 4y + 16x iv.
dy dx −2xy = x b. Solve each of the following initial-value problems: i.
dy dx + 5y = e−3x with y(0) = 0 ii. x dy dx + 3y = 20x2 with y(1) = 10 iii. x dy dx = y + x2 cos(x) with y π 2 = 0 1.3. Find the general solution to each of the following second-order linear equations with constant coefficients. Express your solution in terms of real-valued functions only.
a. y′′ −9y = 0 b. y′′ + 9y = 0 c. y′′ + 6y′ + 9y = 0 d. y′′ + 6y′ −9y = 0 e. y′′ −6y′ + 9y = 0 f. y′′ + 6y′ + 10y = 0 g. y′′ −4y′ + 40y = 0 h. 2y′′ −5y′ + 2y = 0 1.4. Solve the following initial-value problems: a. y′′ −7y′ + 10y = 0 with y(0) = 5 and y′(0) = 16 b. y′′ −10y′ + 25y = 0 with y(0) = 1 and y′(0) = 0 c. y′′ + 25y = 0 with y(0) = 4 and y′(0) = −15 version: 1/3/2013 Review of Elementary ODEs 10 1.5. Find the general solution to each of the following Euler equations on (0, ∞) : a. x2y′′ −5xy′ + 8y = 0 b. x2y′′ −2y = 0 c. x2y′′ −2xy′ = 0 d. 2x2y′′ −xy′ + y = 0 1.6. Solve the following initial-value problems involving Euler equations: a. x2y′′ −6xy′ + 10y = 0 with y(1) = −1 and y′(1) = 7 b. 4x2y′′ + 4xy′ −y = 0 with y(4) = 0 and y′(4) = 2 1.7. Find the general solution to each of the following: a. y′ + 2y = 3 b. xy′ + 2y = 8 c. y′ + 1 x y = 2ex2 d. y′′ + a2y = 0 where a is a positive constant e. y′′ −a2y = 0 where a is a positive constant f. y′′ + 4y′ −5y = 0 g. y′′ −6y′ + 9y = 0 h. x2y′′ −6xy′ + 10y = 0 i. x2y′′ −9xy′ + 25y = 0 (See following note) Note: For the last one, start by assuming y = xr as described for Cauchy-Euler equations. This will lead to one solution. To find the full solution, assume y(x) = xrv(x) where r is the exponent just found and v(x) is a function to be determined. Plug this into the differential equation, simplify, and you should get a relatively easy differential equation to solve for v(x) (it may help to let u(x) = v′(x) at one point). Solve for v (don’t forget any arbitrary constants) and plug the resulting formula into the above formula for y . There, you’ve just done “reduction of order” .
version: 1/3/2013 Review of Elementary ODEs 11 Some Answers to Some of the Exercises WARNING! Most of the following answers were prepared hastily and late at night. They have not been properly proofread! Errors are likely!
1a i. y = 4 + A exp 1 2x2 1a ii. y = c −3x −cos(x) −1 and y = 0 1a iii. y = 3 + A exp 1 2x2 −2x 1a iv. y = Ax 1b i. y = 5 + 3e2x 1b ii. y = − p 18 −2 cos(x) 1b iii. y = 2(2 −x)−1 2a i. y = 3 + ce−2x 2a ii. y = 4e3x + ce−2x 2a iii. y = ce4x −4x −1 2a iv. y = cex2 −1 2 2b i.
1 2 e−3x −e−5x 2b ii. 4x2 + 6x−3 2b iii. x[sin(x) −1] 3a. y(x) = c1e3x + c2e−3x 3b. y(x) = c1 cos(3x) + c2 sin(3x) 3c. y(x) = c1e−3x + c2xe−3x 3d. y(x) = c1e(−3+3 √ 2)x + c2e(−3−3 √ 2)x 3e. y(x) = c1e3x + c2xe3x 3f. y(x) = c1e−3x cos(x) + c2e−3x sin(x) 3g. y(x) = c1e2x cos(6x) + c2e2x sin(6x) 3h. y(x) = c1e2x + c2ex/2 4a. 3e2x + 2e5x 4b. e5x −5xe5x 4c. 4 cos(5x) −3 sin(5x) 5a. y = c1x2 + c2x4 5b. y = c1x2 + c2x−1 5c. y = c1 + c2x3 5d. y = c1x + c2 √x 6a. y = 3x5 −4x2 6b. y = 4x1/2 −16x−1/2 7a.
3 2 + ce−2x 7b. 4 + cx−2 7c.
ex2 + c /x 7d. c1 cos(ax) + c2 sin(ax) 7e. c1eax + c2e−ax 7f. c1ex + c2e−5x 7g. c1e3x + c2xe3x 7h. c1x2 + c2x5 7i. c1x5 + c2x5 ln |x| version: 1/3/2013 |
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5930 | https://www.khanacademy.org/science/ap-college-physics-1/xf557a762645cccc5:ap-physics-1-review/xf557a762645cccc5:ap-physics-1-concept-review/v/ap-physics-1-review-of-energy-and-work | Review of energy and work (video) | Khan Academy
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AP®︎/College Physics 1
Course: AP®︎/College Physics 1>Unit 9
Lesson 1: AP Physics 1 concept review
Review of 1D motion
Review of 2D motion and vectors
Review of forces and Newton's laws
Review of circular motion and orbits
Review of energy and work
Review of momentum and impulse
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Review of energy and work.Created by David SantoPietro.
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jinsungpark 8 years ago Posted 8 years ago. Direct link to jinsungpark's post “At 11:56, wouldn't the ti...” more At 11:56 , wouldn't the time it takes for the box to fall the same because gravity is the only acceleration that makes the box fall. And the system is frictionless.
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Mark Geary 8 years ago Posted 8 years ago. Direct link to Mark Geary's post “No, because the distance ...” more No, because the distance traveled is not the same. The length of the ramp with the angle θ is longer than the ramp with angle 2θ, so if the boxes are accelerated at the same rate with no friction, the box on the right has farther to travel.
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Taha 5 years ago Posted 5 years ago. Direct link to Taha's post “at 4:46 what if the objec...” more at 4:46
what if the object didn't get displaced because of friction would there be work?
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Lawrence 5 years ago Posted 5 years ago. Direct link to Lawrence's post “Great question Taha! No,...” more Great question Taha!
No, if there is enough friction to stop any displacement from occurring, no work would be done. For work to be done on an object, it must be displaced.
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Ridwan Afwan Karim Fauzi 8 years ago Posted 8 years ago. Direct link to Ridwan Afwan Karim Fauzi's post “at 7:57 Does that mean th...” more at 7:57 Does that mean that, if the final velocity of that object were -4 m/s (still going in the same direction) as opposed to 4 m/s as was seen in the video, the net work would still be the same? That doesn't really make sense to me, but mathematically it works out for whatever reason.
i mean, doesn't it take a lot of effort(energy) to slow down the object and then accelerate it again to 4m/s on the opposite direction rather than slow down it a little bit so that it moves 4m/s to the left?
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Shawn Tan 9 years ago Posted 9 years ago. Direct link to Shawn Tan's post “When a force is perpendic...” more When a force is perpendicular to the direction of motion, why is there 0 work being done on the object?
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2002rawlife 5 months ago Posted 5 months ago. Direct link to 2002rawlife's post “Because cos(90) is 0” more Because cos(90) is 0
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Shawn Tan 9 years ago Posted 9 years ago. Direct link to Shawn Tan's post “When a force is perpendic...” more When a force is perpendicular to the direction of motion, why is there 0 work being done on the object?
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Gayle 8 years ago Posted 8 years ago. Direct link to Gayle's post “At 3:43, wouldn't there a...” more At 3:43 , wouldn't there also be potential energy acting on the box?
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Grace 8 years ago Posted 8 years ago. Direct link to Grace's post “why is the work done on t...” more why is the work done on the box by Earth equal to positive mgh?
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Andrew M 8 years ago Posted 8 years ago. Direct link to Andrew M's post “because work is forcedis...” more because work is forcedisplacement, and it's positive if they are in the same direction.
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[Teacher] Not only are there many different kinds of energies, but both objects, and systems of objects, can have energy. Once they have that energy, they can transfer it to another system or object or that energy could transform to a different type of energy inside that system. When energy gets transferred, we call that work, and the amount of work that's done is the amount of energy that was transferred. You often hear people say, "Energy is conserved," which really just means that you can't create or destroy energy. You can simply transfer it between objects or systems. So, what are all the different types of energy? There's kinetic energy which is the energy due to something moving, and the formula's one-half the mass times the speed squared. There's gravitational potential energy which is the energy something has due to its height, and the formula's the mass times the magnitude of acceleration due to gravity times the height of the object. Height above what? Height above whatever you're choosing as the H equals zero reference line. Is that cheating? No, because all that really matters is the change in gravitational potential energy, not the actual value itself. There's also spring potential energy which has to do with a compressed or stretched spring, and the formula's one-half times the spring constant times x, which is not the length of the spring. X is the amount that the spring has been compressed or stretched. These three types of kinetic and potential energy constitute what we call mechanical energy. Mechanical energy's another word for the kinetic energy plus gravitational potential energy plus spring potential energy in a system, and it's important to know that mechanical energy does not include thermal energy. Thermal energy's the heat energy generated by dissipative forces like friction and air resistance, and you can find the amount of thermal energy generated by taking the size of the dissipated force times the distance through which that force was acting. The unit of energy is Joules and energy is not a vector. But maybe the most important thing to remember about energy is if there's no external work done on a system, then there's no change in the energy of that system. In other words, if there's no external work done on a system, the initial energy of that system will equal the final energy of that system, which is the way you solve many conservation of energy problems. So, what's an example problem involving energy look like? Let's say a box started with an initial speed and slides from one platform up to another platform. We'll assume that frictional forces and air resistance are negligible. And for the system that's consisting of the mass and the Earth, what's happening to the total mechanical energy in this system? So, you've got to pay special attention to what is in your system. Since my system includes the mass, which is going to be moving, my system's gonna have kinetic energy, and since my system has two objects that are interacting gravitationally, the mass and the Earth, my system's also going to have gravitational potential energy. So, when I asked about the total mechanical energy of the system, that's really just code for the total kinetic and potential energy of the system. So, as this mass slides up to a higher point on the ramp, the gravitational potential energy increases, but the mass is gonna slow down, so the kinetic energy's gonna decrease. However, since the Earth and the mass are in our system, and there's no dissipative forces, there's no external work done on our system. Yes, the Earth is doing work on the box, but the Earth is part of our system so it can't do external work, and that means energy just gets transferred from one form to another within our system, and the total mechanical energy, here, is gonna remain the same for the entire trip. Now, what if we asked this same question but we consider a system that consists only of the box. In that case, our system has a box that's moving, so it'll have kinetic energy. But, our system no longer includes two objects interacting gravitationally so our system will have no gravitational potential energy. What happens to the total mechanical energy in this case? Well, the only energy that I've got in my system, now, is kinetic energy and since that kinetic energy decreased, the total mechanical energy of the box, as a system, decreases. How does it decrease? It decreases because now the Earth is outside of our system and the work that it is doing on the box is external work and it's taking away energy from the box. What does work mean? In physics, work is the amount of energy transferred from one system, or object, to another. In other words, if a person lifted a box and gave it 10 Joules of gravitational potential energy, we'd say that person did positive 10 Joules of work on the box since that person gave the box 10 Joules of energy. But since the box took 10 Joules of energy from that person, we'd say that the box did negative 10 Joules of work on the person since the box took 10 Joules of energy. So, you can find the work done if you can determine the amount of energy that was transferred. But, there's an alternative formula to find the work done. If something's having work done to it, there's got to be a force on that object, and that object has to be displaced. So, if you take the force on the object times the displacement of the object, and multiply by the cosine of the angle between the force and the displacement, you'll also get the work done. In other words, one way to find the work done is by finding the amount of energy that was transferred. But, another way to find the work done is by taking the magnitude of force exerted on an object times the displacement of the object and then times cosine of the angle between the displacement and the force. Since work is a transfer of energy, it also has units of Joules. And even though work is not a vector, it can be positive or negative. If the force on an object has a component in the direction of motion, that force will do positive work on the object and give the object energy. If the force on the object has a component in the opposite direction of the motion, the work done by that force would be negative and it would take away the object's energy. And, if the force on an object is perpendicular to the motion of the object, that force does zero work on the object. It neither gives the object energy nor takes away the object's energy. So, what's an example problem involving work look like? Let's say a box of mass M slides down a frictionless ramp of height, H, and angle two-theta, as seen in this diagram, here, and a separate box of mass two-M slides down another frictionless ramp of height, H, and angle theta, as seen in this diagram, here. And, we want to know how work done on the object by the Earth compares for each case? The easiest way to find the work done here is by finding the change in energy. The box will gain an amount of kinetic energy equal to the amount of potential energy that it loses. So, the work done by the Earth is just gonna equal positive m-g-h. Both heights are the same. So, the H's are equivalent, but one box has twice the mass. So, the work done by gravity on the mass, two-M, is gonna be twice as great as the work done on the mass one-M. What's the work energy principle mean? The work energy principle states that the total work, or the net work done on an object, is gonna be equal to the change in kinetic energy of that object. So, if you add up all the work done by all forces on an object, that's got to be equal to the change in the kinetic energy of that object. In other words, one-half m-v-final, squared, minus one-half m-v-initial, squared. So, this is a really handy way to find how the speed of an object changes if you can determine the net work on an object. In other words, if there's multiple forces on an object, and you can find the work done by each of those forces, you can determine how much kinetic energy that object gained or lost. So, what's an example of the work energy principle? Let's say a four kilogram box started with a velocity of six meters per second to the left. Some net amount of work is done on that box, and it's now moving with a velocity of four meters per second to the right. We want to know what was the amount of new work done on the box? Without even solving it, we can say since this object's slowed down, energy was taken from it, so the amount of net work had to be negative which means it's either B or D. To figure out which one exactly, we could use the work energy principle which says that the net work done is equal to the change in kinetic energy. So, if we take the final kinetic energy, which is one-half times four kilograms times the final speed squared, and we subtract the initial kinetic energy, one-half times four kilograms times the initial speed, squared, six meters per second, we get negative 40 Joules of net work. If you get a force versus position graph, the area under that graph will represent the work done. So, when you see F versus x, you should think area equals work. But, be careful, area above the x-axis is gonna count as positive work done, and area underneath the x-axis is gonna count as negative work done, and make sure the x-axis really is position. If you get a force versus time graph, the area's impulse, not work. So, what would an example of work as area look like? Let's say a box started at x equals zero with a velocity of five meters per second to the right and a net horizontal force on the box is given by the graph below. We want to know, at what position other than x equals zero, will the box, again, have a velocity of five meters per second to the right? Well, since the box will end with the same speed that it began with, the change in kinetic energy is gonna equal zero. But, that means the net work would also equal zero, since the net work is equal to the change in kinetic energy. So, if the box starts at x equals zero, how far do we have to go in order for us to have no net work done. Between zero and three meters, the work done is gonna be negative, and the area of this triangle is gonna be one-half the base times the height which is one-half times three meters time negative six Newtons which is negative nine Joules of work done, and the area under this triangle, between three and five seconds, would again be one-half base times height, which is one-half times two meters times height of four Newtons which is positive four Joules of work done. So, by the time that the box has made it to five meters, there's been a total amount of work done of negative nine plus four, which is negative five Joules of work. But, we want no net work done. So, we're gonna have to keep going until this positive area contribution is gonna equal the negative area. In other words, if I can make it so that all of this negative area is equal to all of the positive area, my net work's gonna equal zero. My negative area is negative nine. My positive area, so far, is positive four. If I continue on to the six meter mark, I've pick up another positive four Joules of work since the height of this rectangle is four and the width is one meter, which means we're almost there. Four plus four is eight. I'd only need to pick up one more Joule, so I can't go all the way to seven meters. I'd only need to go one more fourth of a meter to pick up one more Joule so that one plus four plus four is equal to negative nine. So, the net work would equal zero somewhere between x equals six and x equals seven which would ensure that the change in kinetic energy is zero and we would end with the same speed that we began with. What does power mean? In physics power is the amount of work done per time, which can also be thought of as the amount of energy transferred per time. In other words, the amount of Joules per second that are transferred, and the name given to a Joule per second is a Watt. So, you can solve for the power by finding the work divided by the time or the change in energy divided by the time. And you can increase the amount of power by increasing the work done or decreasing the amount of time it takes for that work to be done. And just like energy and work, power is not a vector. So, what's an example problem involving power look like? Let's say a box of mass M slid all the way down a frictionless ramp of height H and angle two-theta as seen in this diagram, and a separate mass M slides all the way down a frictionless ramp of height H and angle theta as seen in that diagram, and we want to know how the average power developed by the force of gravity on the boxes compares for each incline? So, we use the formula for power, power's the work done per time. The work done on these boxes is gonna equal the change in kinetic energy of these boxes which would equal the change in potential energy of the boxes, but the mass of the boxes are the same, the gravitational acceleration is the same, and the height they fall from is the same. So, the work done on the boxes are equal, but the time it takes for these boxes to slide down the ramp is not equal. The mass on the steeper ramp will reach the bottom faster which means it has a higher rate of power being done compared to the mass on the less steep ramp. So, even though the same amount of work is being done, the rate at which that work is being done is greater for the steeper ramp compared to the more shallow ramp.
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5931 | https://www.youtube.com/watch?v=kj-rDDZPpxs | Subtracting Decimals with Regrouping: Grade 4
UHouston Math
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Posted: 23 Mar 2020
TEKS: 4.4(A) add and subtract whole numbers and decimals to the
hundredths place using the standard algorithm
CCSS.MATH.CONTENT.5.NBT.B.7
Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.
Transcript:
in this video we will be using the strategy regrouping to subtract and solve the problem Charlie found six dollars and 54 cents while cleaning his bedroom with the money he found he decided to get a party size bag of chips from the store to eat with his friends the bag of chips at the store costs four dollars and 78 cents how much money does Charlie have left after its purchase the number sentence for this problem is 6 and 5,400 minus 4 and 78 hundredths we will be modeling the regrouping strategy through the concrete pictorial and abstract models first we will build on number 6 and 54 hundreds by using the base 10 blocks for our concrete model as shown on the right side of the screen of flat rubbers and swans rods represent tents and cubes represent hundreds we have 6 1 so we will grab six flats 1 2 3 4 5 and 6 then we will grab five rods to represent five tents 1 2 3 4 and 5 and lastly to represent the 400s we will grab 4 cubes 1 2 3 & 4 next we will subtract 8 from the hundreds plate but because there are only 4 in the hundreds place right now we will need to regroup 1/10 into 10 hundreds so 1 2 3 4 5 6 7 8 9 10 now that we have regrouped one comes into hundreds there are now 14 hundredths and from the 14 we'll subtract 8 two three four five six seven eight and there are one two three four five six remaining in the hundreds place next we need to subtract seven tens from six and fifty four hundred to do so we'll have to regroup a one into ten tens so we will take one once and regroup it into ten ten one two three four five six seven eight nine and ten now that we have our group we will subtract seven from the Pens place one two three four five six seven there are one two three four five six seven times remaining in the tens place finally we will subtract four ones for this step we do not need three groups so we will just go ahead and subtract four from the ones place one two three and four this leaves us with one in the ones place this shows us that after his purchased charlie has one dollars and seventy six cents left now we will solve the problem using pictorial abstract models when you read you're using pictorial models it will look very similar to the concrete model however instead of locks will be using squares to represent ones lines to represent tons and dots to represent hundreds as a quick reminder the number sentences six and fifty four hundredths minus four and 78 hundredths to help understand the abstract algorithm will show this alongside the pictorial model to begin we will write out the problem under the abstract so we will write six and fifty four hundredths minus 4 and 78 hundredths then we will draw the number six and fifty four hundredths under the pictorial model to do this we will draw six squares under the ones to show six one two three four five six and then we will draw five lines to represent five tenths one two three four and five and lastly we will draw four dots one two three four to represent the four hundreds now we need to subtract eight from the hundreds place to do so we will need to regroup one times into ten hundreds one two three four five six seven eight nine ten then we'll subtract eight one two three four five six seven eight there are one two three four five six remaining in the hundreds place in the abstract model we can show this up by reroofing one tenth from five tenths making it four tenths and breaking it into ten 100's making the four hundreds into 1400 we will then subtract eight from the 1400s which leaves us with six and the hundreds place then we will subtract seven tens from six and fifty four hundreds to do so we will need to remove one one into ten one two three four five six seven eight nine and ten this gives us 14 tonnes from here we will subtract seven one two three four five six seven the Cecil's with one two three four five six seven seven in the tenths place in the abstract model we can show this stuff by regrouping one one's which makes that five ones into ten tenths which makes that 14 tonnes then we can subtract the seven tonnes from 14 tonnes which leaves us with 7/10 in the tenths place lastly we can subtract 4 ones from the ones place and for this step we do not need to regroup so we can just subtract so 1 2 3 4 that leaves us with 1 in the ones place to show this in the abstract model you will do the same thing there is no regrouping involved so you will just put product 4 from 5 which gives you 1 in the ones place this makes our answer 1 and 76 hundredths for $1 and 76 cents as you saw in the video through the use of concrete pictorial and abstract models we were able to successfully use regrouping to find the answer to the question how much money did trolley have left after his purchase to which we got the answer of 1 dollar and 76 cents |
5932 | https://books.google.com/books/about/Quantitative_Chemical_Analysis.html?id=kIgLJ1De_jwC | Quantitative Chemical Analysis - Daniel C. Harris - Google Books
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Quantitative Chemical Analysis ============================== Daniel C. Harris Macmillan, Apr 30, 2010 - Science - 719 pages The most widely used analytical chemistry textbook in the world, Dan Harris's Quantitative Chemical Analysi s provides a sound physical understanding of the principles of analytical chemistry, showing how these principles are applied in chemistry and related disciplines—especially in life sciences and environmental science. As always, the new edition incorporates real data, spreadsheets, and a wealth of applications, in a witty, personable presentation that engages students without compromising the depth necessary for a thorough and practical understanding of analytical chemistry. More » Preview this book »
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Title Page
Table of Contents
Index
Contents
The Analytical Process 1
Chemical Measurements 13
Tools of the Trade 29
Experimental Error 51
Statistics 68
Quality Assurance and Calibration Methods 96
Chemical Equilibrium 117
Activity and the SystematicTreatment of Equilibrium 142
Applications of Spectrophotometry 419
Spectrophotometers 445
Atomic Spectroscopy 479
Mass Spectrometry 502
Introduction to Analytical Separations 537
Gas Chromatography 565
HighPerformance Liquid Chromatography 595
Chromatographic Methods and Capillary Electrophoresis 634
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Monoprotic AcidBase Equilibria 162
Polyprotic AcidBase Equilibria 185
AcidBase Titrations 205
EDTA Titrations 236
Advanced Topics in Equilibrium 258
Fundamentals of Electrochemistry 279
Electrodes and Potentiometry 308
Redox Titrations 340
Electroanalytical Techniques 361
Fundamentals of Spectrophotometry 393
Gravimetric Analysis Precipitation Titrations and Combustion Analysis 673
Sample Preparation 699
Notes and references NF-1
Glossary NF-25
Appendix AQ-1
Solutions to Exerices 501
Answers to Problems AO-1
Index 101
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Daniel C. Harris
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Daniel C. Harris
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Daniel C. Harris
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absorbanceabsorptionacid dissociation constantactivity coefficientsAnalanalysisanalyteanionaqueousatomsbasebufferCa2+CalculatecalibrationcapillarycarboncathodecationscellchargeChemchemicalchromatogramchromatographycolorcolumncompoundsconcentrationcoulometryCu2+detection limitdetectordilutedissociationdissolvedEDTAelectriceluentelutedemissionend pointenergyEquationequilibrium constantequivalence pointextractionFe2+Fe3+Figurefluorescencefractiongas chromatographygraphhalf-reactionincreasesinjectionion-selective electrodeionic strengthionizationK₁liquidmass balancemass spectrometermeasuredmixturemobile phasemolaritymolecularmoleculesmolesNaOHNernst equationoxidationparticlesPb2+peakpotentialprecipitateproteinprotonatedreactionreagentredoxretentionsampleseparationsignalsolubilitysolution containingsolventspeciesspectrumspreadsheetstandard deviationstationary phaseTabletemperaturetitration curveuncertaintyunknownvoltagevolumewavelengthweak acid
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Title Quantitative Chemical Analysis
AuthorDaniel C. Harris
Edition illustrated
Publisher Macmillan, 2010
ISBN 1429218150, 9781429218153
Length 719 pages
SubjectsScience
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Chemistry
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General
Science / Chemistry / Analytic
Science / Chemistry / General
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5933 | https://math.stackexchange.com/questions/2613051/proof-of-fundamental-theorem-of-arithmetic-uniqueness-part-of-proof | induction - Proof of Fundamental Theorem of Arithmetic: Uniqueness Part of Proof - Mathematics Stack Exchange
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Proof of Fundamental Theorem of Arithmetic: Uniqueness Part of Proof
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I am having trouble trying to understand the uniqueness part of the proof. It uses induction on n (n is claimed to have a unique factorization of primes).
Here is my understanding so far:
Let n n be an integer such that n>1 n>1. Then n=p 1 p 2⋯p k n=p 1 p 2⋯p k, with p 1,p 2,...,p k p 1,p 2,...,p k being primes, is a unique factorization of n n.
Proof: (Uniqueness) (proof by induction) The base case states that for n=2 n=2, n n is a product of primes, which is true (2 2 is prime).
Now assume this is true for integers m m with 1≤m<n 1≤m<n.
This is the part I don't understand. Why is m≥1 m≥1 when 1 1 cannot be written as a product of primes?
and n=p 1 p 2⋯p k=q 1 q 2⋯q l n=p 1 p 2⋯p k=q 1 q 2⋯q l with p 1≤p 2≤⋯≤p k p 1≤p 2≤⋯≤p k and q 1≤q 2≤⋯≤q l q 1≤q 2≤⋯≤q l
Again, I am lost. Why is n n written in such a way when the inductive hypothesis uses m m? Is it because the inductive step uses n n still and the fact for m m is used implicitly?
Then p 1|q i p 1|q i for some 1≤i≤l 1≤i≤l and q 1|p j q 1|p j for some 1≤j≤k 1≤j≤k. Since all p j p j and q i q i are prime, their only divisors are 1 1 and itself. Therefore, p 1=q i p 1=q i and q 1=p j q 1=p j. Since p 1≤p j=q 1≤q i=p 1 p 1≤p j=q 1≤q i=p 1, p 1=q 1 p 1=q 1.
I understand this part of the proof.
Then by inductive hypothesis, n′=p 2⋯p k=q 2⋯q l n′=p 2⋯p k=q 2⋯q l.
Is this because, since p 1=q 1 p 1=q 1, they are cancelled out creating a new integer n′n′? And so does this induction continue until p k=q l p k=q l, which will ultimately lead to 1=1 1=1?
This proof by induction is very brief for me to understand and digest right away.
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edited Jun 12, 2020 at 10:38
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This is the part I don't understand. Why is m≥1 m≥1 when 1 1 cannot be written as a product of primes?
In a technical sense, 1 1is a product of primes - it's the empty product (and the empty set ∅∅ is vacuously a set of primes, in the sense that it does not contain an element which is not prime), but you might as well just assume the author instead wrote 2≤m<n 2≤m<n (except for the small caveat of how the proof functions when n n itself is prime).
Again, I am lost. Why is n n written in such a way when the inductive hypothesis uses m m? Is it because the inductive step uses n n still and the fact for m m is used implicitly?
This is how induction always works: after you assume your base case (2 2) and your inductive hypothesis (all m<n m<n), you then explore the claim for the value of n n itself. Here, we've assumed the claim (unique factorization) is true for all m<n m<n, and now we're going to see if the claim holds as well for the number n n.
So we write two factorizations for n n, and show they must in fact be the same factorization (in such a way that will use the inductive hypothesis).
Is this because, since p 1=q 1 p 1=q 1, they are cancelled out creating a new integer n′n′?
Yes.
And so does this induction continue until p k=q l p k=q l, which will ultimately lead to 1=1 1=1?
Essentially, yes. Logically, this is a strong induction proof (which is why our inductive hypothesis involved all integers m m less than n n, not just one), so we seem to be applying our inductive hypothesis to the value m=n′m=n′ in order to conclude the two factorizations of n′n′ are the same.
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answered Jan 20, 2018 at 6:59
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Oh, I see now about there being an empty product. I never think about the vacuous cases! Thank you for explaining about the inductive hypothesis part, you made it very clear to understand! Now I know why they did what they did. Thank you for taking the time to let me know.user482939 –user482939 2018-01-20 17:56:10 +00:00 Commented Jan 20, 2018 at 17:56
Where does this proof break down for The set 2 Z 2 Z:=:={2a|a∈∈Z Z}?Taylor Rendon –Taylor Rendon 2020-01-27 18:33:27 +00:00 Commented Jan 27, 2020 at 18:33
1 @TaylorRendon Technically it doesn't break down for 2 Z 2 Z. For this rng, the "prime" elements are of the form 2 p 2 p where p=1 p=1 or p p is prime, and any factorization into primes is unique. The catch is that factorizations into primes may not exist in the first place, e.g. 30 30 is neither prime nor a product of "primes" in this setting. "Primes" are characterized by p∣a b⇒p∣a or p∣b p∣a b⇒p∣a or p∣b, and "irreducible" elements are characterized by not being the product of two nonunits. Primes are irreducible in an integral domain but not necessarily vice-versa.anon –anon 2022-04-02 01:29:37 +00:00 Commented Apr 2, 2022 at 1:29
1 For Euclidean domains like Z Z, though, the primes and the irreducible elements are one and the same. In general, factorization into irreducibles is guaranteed to exist but not guaranteed to be unique, and factorization into primes is not guaranteed to exist but guaranteed to be unique when it does.anon –anon 2022-04-02 01:29:40 +00:00 Commented Apr 2, 2022 at 1:29
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5934 | https://testbook.com/question-answer/find-all-the-incongruent-solutions-of-the-congruen--610782bebfaefef2d0529e52 | [Solved] Find all the incongruent solutions of the congruence 18x&nbs
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Find all the incongruent solutions of the congruence 18x≡ 30(mod 42).
3, 12, 19, 26, 33, 40
4, 11, 18, 25, 32, 39
2, 15, 6, 8, 4, 2
no solution exists
Answer (Detailed Solution Below)
Option 2 : 4, 11, 18, 25, 32, 39
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Concept:
The linear congruence ax≡ b (mod n) has a solution if and only if d divides b , where d = gcd(a, n). If d divides b then it has d mutually incongruent solution modulo n.---(1)
If x 0 is one of the solutions of the congruence ax≡ b (mod n) then its incongruent solutions are of the form x 0,x 0+n d,x 0+2 n d, x 0+3 n d, ... and so on.---(2)
Calculations:
Given congruence 18x≡ 30(mod 42) is clearly of the form ax≡ b (mod n).
where a = 18, b = 30, n = 42
And d = gcd(a, n) = gcd(18, 42) = 6
Clearly, 6 divides 42 i.e., d divides b.
∴ By using (1) we have, the congruence 18x≡ 30(mod 42) has d = 6 incongruent solutions modulo 42.
By inspection ,we get x = 4 = x 0 as one solution of the congruence18x≡ 30(mod 42)
Now using (2) we get all the incongruent solutions modulo 42 are
x 0,x 0+n d,x 0+2 n d,x 0+3 n d,x 0+4 n d,x 0+5 n d
= 4, 11, 18, 25, 32, 39 are the 6 incongruent solution mod 42.
Hence, the correct answer is option 2).
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5935 | https://brainly.com/question/41021057 | [FREE] Find the relative extrema of the function f(x) = x^4 - 8x^3 + 25. A. No relative maximum; relative minimum - brainly.com
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Find the relative extrema of the function f(x)=x 4−8 x 3+25.
A. No relative maximum; relative minimum at x=0 and x=6.
B. Relative maximum at x=0; relative minimum at x=6.
C. Relative maximum at x=6; relative minimum at x=0.
D. No relative maximum; relative minimum at x=6.
E. Relative maximum at x=0; no relative minimum.
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The mathematical function f(x)=x⁴-8x³+25 has a relative maximum at x=0, found using the first and second derivative tests. There is also a relative minimum at x=6. The correct option is c.relative maximum at x=6; relative minimum at x=0
Explanation
To find the relative extrema of the mathematical function f(x)=x⁴-8x³+25, we first need to compute its derivative, which will set out the locations of the maximal and minimal points.
The derivative of the function is f'(x)=4x³-24x². Setting f'(x) equal to zero and solving will yield the critical points x = 0 and x = 6.
We then carry out the second derivative test. If the value of the second derivative at a particular point is negative, the function has a relative maximum at that point. If the value is positive, the function has a relative minimum there. Therefore, by calculating the second derivative f''(x), we can perform this test.
f''(x) = 12x² - 48x. Using this, we find that f''(0) is negative, indicating a relative maximum at x=0, and f''(6) is positive, indicating a relative minimum at x=6. So, the correct answer is option C.
Learn more about Relative Extrema here:
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The function f(x)=x 4−8 x 3+25 has a relative maximum at x=0 and a relative minimum at x=6. The correct answer is option B. Relative maximum at x=0; relative minimum at x=6.
Explanation
To find the relative extrema of the function f(x)=x 4−8 x 3+25, we follow these steps:
Find the first derivative:
We calculate the first derivative of the function:
f′(x)=4 x 3−24 x 2.
Set the first derivative to zero:
To find critical points, we set f′(x)=0: 4 x 3−24 x 2=0
Factoring out common terms, we get:
4 x 2(x−6)=0
This gives us critical points at x=0 and x=6.
Find the second derivative:
Next, we compute the second derivative to determine the nature of the critical points:
f′′(x)=12 x 2−48 x.
Evaluate the second derivative at critical points:
For x=0: f′′(0)=12(0)2−48(0)=0.
For x=6: f′′(6)=12(6)2−48(6)=432−288=144.
Interpret the results:
The second derivative test at x=0 is inconclusive since it equals zero. We can use the first derivative test around this point:
For x slightly less than 0, say -1, f′(−1)>0 (function increasing).
For x slightly more than 0, say 1, f′(1)<0 (function decreasing).
Thus, there is a relative maximum at x=0.
The second derivative at x=6 is positive (144), indicating a relative minimum at x=6.
Therefore, the function has a relative maximum at x=0 and a relative minimum at x=6. The correct option is B. Relative maximum at x=0; relative minimum at x=6.
Examples & Evidence
For example, consider a simpler polynomial function like g(x)=x 2−4. Its only critical point occurs at x=0, where it has a minimum since the parabola opens upwards. This illustrates concepts of maxima and minima in simple polynomials.
The analysis of the first and second derivatives, which consistently shows increasing or decreasing behavior around the critical points, verifies the locations of the extrema, confirming that calculations are correct.
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Community Answer Determine the locations ( x-values only) of any relative extrema of f(x)= (1/7) x⁷ − (1/5) x⁵ +10 A) Relative minimum at x=0 and relative maximums at x=±1. B) Relative minimum at x=−1 and relative maximum at x=1. C) Relative minimum at x=1 and relative maximum at x=−1. D) Relative minimums at x=±1 and relative maximum at x=0.
Community Answer 3 Which of the following most accurately identifies the relative maximum and minimum of the polynomial function h(x)? There is a relative maximum at x=1.5 and a relative minimum at x=−2. There is a relative maximum at x=−1 and no relative minimum. There is a relative maximum at x=−1 and a relative minimum at x=0.555. There is a relative minimum at x=−1 and a relative maximum at x=0.555.
Community Answer 2 Estimate the x-coordinates at which the relative maxima and relative minima occur for the function. f(x) = 4x3 + 11x2 + 5 Question 17 options: A. The relative maximum is at x = 1.83, and the relative minimum is at x = 0. B. The relative maximum is at x = –1.83, and the relative minimum is at x = 1. C. The relative maximum is at x = –1.83, and the relative minimum is at x = 0. D. The relative maximum is at x = 1.83, and the relative minimum is at x = 1.
Community Answer Find the values of x that give relative extrema for the function f(x) = (x + D')(x - 2). a. relative maximum: -1; relative minimum: 1 b. relative maximum: -1 and x = 3; relative minimum: -1 c. relative maximum: x = -1; relative minimum: x² d. relative maximum: x = -1; relative minimum: x²
Community Answer Question 20 The derivative f′(x) of an unknown function f(x) has been determined to be f′(x)=(x+4)2(3−x). Use interval testing to find the x-values that correspond to any relative (local) maximums or relative (local) minimums of the original unknown function f(x). None of these: relative maximum at x=4; relative minimum at x=3. no relative maximum: relative minimum at x=3 no relative maximum; relative minimum at x=−4 relative maximum at x−3; no relative minimum relative maximum at x=3; relative minimum at x=−4
Community Answer 4.4 16 Based on the graph of F(x) below, which of the following statements is true about F(x)? A. F(x) has a relative minimum at x = 0 and a relative maximum at x = -1.5 B. Fx) has a relative maximum at x = 0 and a relative minimum at x= -1.5 C. Fx) has a relative maximum at x = 0 and a relative minimum at x= 1.1 D. Fx) has a relative minimum at x = 0 and a relative maximum at x= 1.1
Community Answer Find all relative maxima or minima. f(x)= 2lnx x 6(e1/6 ,3e), relative minimum (e−1/6,−3e −1 ), relative minimum (0,0), relative minimum (0,0), relative maximum; (e1/6 ,3e), relative minimum
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A 3.8-Å X-ray crystal structure of mouse P-gp in an inward-open conformation was originally solved through multi-wavelength anomalous dispersion (MAD) phasing (PDB 3G5U) (Aller et al., 2009) and this was later refined through single-wavelength anomalous dispersion (SAD) phasing (PDB 4M1M) (Li et al., 2014) (Figure 2A). Mouse P-gp shares 87% sequence identity with human P-gp. The overall structure of mouse P-gp is arranged as two “halves” with pseudo two-fold molecular symmetry with two cytoplasmic nucleotide-binding domains (NBDs) separated by ∼30 Å. The structure contains two bundles of six helices (TMs 1-3, 6, 10, 11 and TMs 4, 5, 7-9, 12) and has a large internal cavity open to both the cytoplasm and the inner leaflet of the membrane. Two portals formed by TMs 4/6 and 10/12 provide access routes for hydrophobic molecules directly from the membrane. The putative drug-binding pocket comprises mostly hydrophobic and aromatic residues and the volume of the internal cavity within the lipid bilayer is ∼6000 Å3, large enough for the simultaneous accommodation of at least two compounds. Two additional structures with cyclic hexapeptide inhibitors [cyclic-tris-(R)-valineselenazole (QZ59-RRR) and cyclic-tris-(S)-valineselenazole (QZ59-SSS)] bound to the internal cavity were also solved to 4.4 and 4.35 Å, respectively (PDB 3G60 and 3G61) revealing specific amino acid residues involved in drug recognition (Aller et al., 2009). The 46 residues in the drug translocation pathway of P-gp are conserved with 96% identity in those from mouse and human. The composition of these residues suggests that a significant proportion of drug-protein interactions are electrostatic, including cation–π, CH–π or π–π recognition (Li et al., 2014). The putative mechanism of substrate and drug efflux by P-gp (Figure 2B) involves partitioning of substrate into the bilayer from outside of the cell to the inner leaflet from where it enters the internal drug-binding pocket through an open portal. Residues in the drug-binding pocket interact with substrate in the inward facing conformation. ATP binds to the NBDs and undergoes hydrolysis to produce ADP and energy. There is a concomitant large conformational change that presents the substrate and drug-binding site(s) to the outer leaflet/extracellular space for expulsion to the aqueous phase. It appears that the NBDs hydrolyse ATP in an alternating manner, but it is still not clear whether transport is driven by ATP hydrolysis or ATP binding. Nonetheless, the efflux action of P-gp follows a carrier-mediated primary active transport mechanism that unidirectionally transfers only one molecule at a time. A better understanding of the structure, substrate and inhibitor interactions and molecular mechanism of P-gp would be provided by higher resolution crystal structures of P-gp solved in different conformations (inward-open, occluded, outward-open) and in complex with a range of different ligands.
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5937 | https://www.youtube.com/watch?v=c71IrNXWSqc | How to calculate the Gamma Function Values
sumchief
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Posted: 21 Jun 2022
This video demonstrates how to calculate the values of the Gamma Function including negative and positive values and also the fractions of half etc .
The Gamma integral is a non elementary Integral meaning it cannot be integrated by standard rules such as u sub or integration by parts when it is in indefinite form . But when it is in definite form such as zero to infinity the values can be calculated using u sub integration and or integration by parts .
There are various properties of the Gamma function which make it closely related the the factorial function.
The Gamma Integral from 0 to infinity is by a matter of coincidence equal to the gaussian integral from negative to positive infinity too .
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Transcript:
so in this video we're going to look at the gamma function and the gamma function is defined by this integral 0 To infinity x the aus1 e- X DX a is our parameter which you can see is here for the value of the gamma function now if you take this integration parameters hour here you can't integrate this function by nor methods it's a non-elementary function but by putting in our values here we can integrate sometimes by U sub and sometimes by integration by parts so let's just have a look at that now so gamma a so let's have a look at simplest one of all which should be Gamma One so gamma one obviously one is the a value so we've got x to the 1 - one so integral 0 To infinity x 1 -1 e- x DX 1 - one is zero anything to zero is just one so this term just vanishes away and then we're left with 0 To infinity e- x DX so now just a straightforward integral now so we get minus E to - x 0 to Infinity okay minus E to the minus infinity well anything to the negative Infinity for the exponential function it's just going to approach zero so therefore we just put zero for the first one and then for zero e to the minus 0 is just the same as e to the 0 and as we said here before anything to^ Z is just one so it's minus minus one so this answer is obviously one so we write this value up here now Gamma One equals 1 okay right let's try another value let's try a half so now let's put that in now Gamma 12 so that 0 To infinity x half - 1 e- X DX okay half - 1 is minus a half so now we've got basically 1/ < TK of X so basically we got Infinity to Zer x - 12 so we got 1 over < TK x e- x DX okay how we going to integrate this now well one way we can do we could do a Uub here so with a Uub we can take away this line and just go straight in with u = X2 okay soor the other way around u^2 = X sorry that's the one way around U2 equal x so our X going to sub for u^2 same for this one here it's going to be a little bit tricky let's just go for it near zero to Infinity so X is u^2 u^2 and then multiply that by minus a half and then e- X just becomes eus u^ 2 just to check our parameters of integration when X is0 u^2 will still become zero and infinity still be infinity infinity squ is still Infinity but now we need to sort out our DX so X is U ^2 and DX = 2 U D okay so exchange out for that 2 U du okay let's bring the two out front 0 to Infinity U we can bring out here now what have we got here 2 Min - half so basically we got U to Theus one so we've got 1 over U well that works out very nicely time eus u^ 2 the U and the two have been dealt with so now we're just left with the DU okay that's very satisfying so now we've got 2 times the integral from 0 to Infinity e to Theus u^ 2 du now anybody who does statistics will be very familiar with this function here e to Theus u^2 so this one is the bow distribution graph it just goes along like that and never actually touches the x-axis it starts off here value one it could be any value here but here you're going to use value one and then what we do know is this integral here which is the error function integral um this here value is sare Pi / 2 so the area of this here is square root of pi/ two so what we can say is that this here is root pi/ 2 so we've got 2un piun Pi / 2 Bas equals otk Pi therefore after all that what we can say is gamma of a half is square root Pi let put that in here okay BL this out quite nicely right let's move on to some gamma function calculations so properties of the gamma function so gamma a is basically a - one factorial okay so now what we can do we could do gamma a + one so gamma a + 1 so if you do an A + 1 a + 1 minus one just gives us a so that it becomes a factorial now one property with the factorial is a gamma a is basically a let's just write this down here a factorial here equals a times gamma function a okay right so what does that mean so if we're trying to work out for example the gamma function value for 3 over two so let's try and work this one out 3 over two basically that basically means we got 3 over2 here so 3 over2 + 1 that basically means the a is a half so we got a half and then the gamma a which would be the gamma of a half we've got here so square root of Pi so that equals root pi over two just by using properties of the gamma function so no need to integrate so let's put that one in here okay so if you know 3 over two let's quickly have it Go 5 over two so gamma function 5 over two so for 5 over two using this one here A + 1 if a + 1 is 5 over2 our a is 3 over2 so if we use this property here 3 / 2 gamma 3 /2 so that's times otk Pi / 2 basically we'll get now we'll get 3 pi over 4 3 root pi over four sorry I meant to say so now we got gamma 5 over 2 3 S < TK pi over 4 let's just try just try one more of these fractions so actually we just go here we don't need to cross it out just yet so gamma 7 over two what does that give us so again using the a + one is 7 over2 we can now say a is 5 over two so 5 over two time gamma 5/ 2 which is 3un piun 4 so working that out 15 Pi 15 pi over 8 so now you can see the pattern form in here so 7 over 2 equals 15 squ < TK Pi 8 okay so that's how we work those out now for the whole integers it's a bit more straightforward so let's say we're working out example gamma five so gamma five so a + 1 is five that means here our a is four so four factorial so four factorial 1 2 3 4 so that's just 24 so let's just plug that in there gamma of five equals 24 and then that's the same for all of the integers let's say we plug in here gamma of let's go eight so gamma of eight is 8+ 1 is eight then we want seven factorial so seven factorial is 1 2 3 4 5 6 7 1 2 6 24 120 720 720 7 540 so gamma 8 equals 5,040 okay what about negative numbers negative numbers is a little bit different so let's say we're trying to work out gamma of minus 12 it's not quite the same principle of this so what I do to this nice little shortcut instead of using this what I use is one/ a Time gamma of a + 1 let's just make that looks like a nine let's just make that a little bit looking better looking a so gamma minus a half how we going to do that one over that so it's one over - half times gamma of a + 1 if our a in this case is minus a half a + one is positive a half so now we're going to multiply this by square root of Pi okay so quickly working this out flip the two on the top get minus 2un pi and that's how to get gam of minus half let's just write let's write the negative on here - 2 < TK Pi okay just take this one out let's do another couple so let's just write this down now so gamma of a when a is less than zero that's our little formula for the negative ones so now let's try gamma of - 3/2 so that one equals so - 3 over 2 so it's 1 over -3 / 2 and it's - 3 /2 we need gamma of minus a half gamma minus a half is - 2un P okay so 1 / -3 / 2 that becomes -2 over 3 -2 sare root of Pi Pi okay -2 -2 is + 4 / 3 so that's 4/3 pi so you see here now the value is flipped from negative positive so let's see what happens now gamma- 5/ 2 okay again use this formula here 1 / a so 1 - 5/ 2 and then gamma a + 1 so - 5/ 2 + 1 is - 3 / 2 which is 4/3 pi or Ro Pi I should say okay going carefully with this one - 2 over 5 4 over 3 < tkk - 2 4 is - 8 53 is 15 so it's - 8 over 15 square root Pi so you can see now we're alternating between positive and minus and it would be almost the same values all the way down as to the positive of plus one so here the 7 over2 is a negative version of the minus 5 over2 the 5 over2 is the same as the -3 over2 the 3 over2 is a negative version but reciprocal of minus a half hence why we got the reciprocal in here okay that's all that so what does the graph of this function look like well am I going to take the graph of the positive values for now so this is our X and this is our y this is zero 1 2 3 four and so on okay so the value if you can never plug in a zero for the gamma function is undefined so what happens is the gamma of one is one so that will come down here what we'll also find is that the gamma of two is also two that's equal there so it's G two is also one sorry so this one will come here so basically what it will happen is it will be in a a SYM totic all the way over there hit down to the one here now instead of going straight across it does get a little bit below here but doesn't touch this axis here and then it shoots off to Infinity all the way up here and that's graph of the camma function on the negative you'll see what happens here at minus a half let's just put that in there it becomes minus two so that's going to be down here and at minus 3 over2 it becomes 4 over three it's going to keep alternating here but then it does also lots of other funny things here as well so I'm not going to draw a straight line on it wouldn't do it justice At This stage that'll be for another video another time there you go that's the G the gamma function |
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Use Strategies to Add and Subtract Decimal
Use Strategies to Add and Subtract Decimal
Grade 5
Sep 27, 2022
Key Concepts
Addition and subtraction of decimals by lining up the decimals
Adding decimals using properties of numbers and also Number Line
Subtracting decimals using number line and also using partial difference method
Introduction
In this chapter, we will learn about adding and subtracting decimals just like adding and subtracting whole numbers and using commutative and associative properties of numbers, partial addition, and subtraction, and also using number lines.
Addition of Decimals Using strategies
Decimals can be added like the way we add whole numbers. There are different ways of adding decimals.
Lining them using place values
Using properties of numbers (Commutative property and Associative property)
Using number line
Using partial sums
Let us learn one by one
Lining them using place value
In this method, we follow the following steps:
Write down the numbers one under the other, with the decimal points lined up
Add zeros to the right of the number so that the number to be added are of same digits
Then add using column addition and remember to put the decimal point in the answer
For example,
Add 3.456 to 2.4
Step 1: Line up the decimals
Step 2: Add zeros to the right side of the number if needed
Step 3: Add using column addition
Using properties of numbers
The two properties of numbers are:
Commutative property of addition
Associative property of addition
Commutative property of addition means that you can switch the order of any of the numbers in an addition, the answer remains the same.
For example:
Sum of 4.2 + 3.5 = 7.7
By changing the order of the addends, 3.5 + 4.2 = 7.7
That is, 4.2 + 3.5 = 3.5 + 4.2 = 7.7
Associative property of addition means that you can change the groupings of numbers being added and it does not change the result.
For example, (2.3 + 4.6) + 7.4 = 2.3 + (4.6 + 7.4)
Let us check
LHS = (2.3 + 4.6) + 7.4 = 14.3
RHS = 2.3 + (4.6 + 7.4) = 14.3
⸫ (2.3 + 4.6) + 7.4 = 2.3 + (4.6 + 7.4) = 14.3
Using number line
To add decimals using number line, labelling the number line with decimals is very important. We know how to label a number line using whole numbers.
The number line with decimals will look like this
Here, we started labelling the number line with zero and increase by 0.1. We can label the number line increased by 0.25 or 0.5 or 0.75 etc. We can also start the number line with the numbers given in the question.
For example:
Add 3.6 + 0.8
We can start labelling the number line starting with 3.6 and increase by 0.1
Then start adding 0.8 to 3.6, we get 4.4
⸫ 3.6 + 0.8 = 4.4
Using Partial sums
In this we use what we already know about adding decimals. But in partial addition, we
break the numbers up in the individual places and add.
For example:
Add 4.65 + 2.76 using partial addition
⸫ 4.65 + 2.76 = 7.41
Subtraction of Decimals Using strategies
Decimals can be subtracted just like the subtraction of whole numbers. The different ways of subtracting decimals are:
Lining them using place values
Using number line
Using partial differences
Let us learn one by one
Lining them using place values
This is the same as addition. But instead of adding, we subtract the decimals.
To subtract decimals by lining them using place values, we follow the following steps:
Step 1: Line up the decimal points in a column. When needed add a zero to the left of the number to match the number of digits.
Step 2: Start on the right, and subtract each column in turn. Remember, we are subtracting digits in the same place value position.
Step 3: If the digit you are subtracting is bigger than the digit you are subtracting from, you have to borrow a group of ten from the column to the left.
For example:
Subtract 4.65 – 2. 49
Using number line
To subtract decimal numbers using a number line. Start on the far-right side on the number line and label it backwards by tenths. This is nothing but counting back.
For example:
Subtract 4.8 – 0.9
Draw the number line labelling backward starting from 4.8
Then count backwards by tenths, 9 times
⸫ 4.8 – 0.9 = 3.9
Using Partial differences
Using partial differences helps you to subtract numbers that are difficult to subtract in one step in your head.
The steps to be followed while subtracting numbers using partial differences are as follows:
When subtracting using partial differences, we write the numbers one below the other and start subtracting from left to right
Then we start subtracting the whole number part by place values, then subtract the tenths digit and hundredths digit vice versa.
If the number to be subtracted is greater than the number to be subtracted from, then swap the numbers in head, subtract the smaller one from the bigger one and put the negative sign in the answer.
When writing these results in the answer column subtract the negative numbers from the number with positive sign.
For example:
Subtract 43.85 – 21.63
Now, subtracting by place values
40 – 20 = 20
3 – 1 =
0.8 – 0.6 = 0.2
0.05 – 0.03 = 0.02
22.22
Subtract 54.85 – 31.56
Now, subtracting by place values
50 – 30 = 20
4 – 1 = 3
0.8 – 0.5 = 0.3
0.05 – 0.06 = – 0.01 (⸪0.06 > 0.05)
Since there is a negative number, subtract that from the positive number
0.3 – 0.01 è 0.30 – 0.01 = 0.29
⸫ 54.85 – 31.56 = 20 + 3 + 0.29 = 23.29
Exercise
Add 6. 5+ 3.3 using number line
Casey runs 9.5 miles, 13.2 miles then 11.5 miles the first week and 11.5 miles, 13.2 miles and 9.5 miles the next week. Which property is represented by Casey’s two weeks of running?
Name the property illustrated below.
3.2+(a+5.6)=(3.2+a)+5.6
Add 15.67 + 11.74 using partial sums
Subtract 9.8 – 7.6 using number line
Ms. Gracie is an electrician and has a length of wire that is 54.7m long . She has another length of wire that is 16.7 m long . How much longer is one wire than the other. Use any method to solve this problem
Subtract 25.32 – 13.26 using partial differences.
Concept Map
What have we learned
Adding Decimals using Number line, Lining up the decimal point, Partial sums and using properties of numbers.
Subtracting decimals using Number line, lining up the decimals and partial differences
Comments:
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How Changing Interest Rates Affect Bond Prices
July 31, 2025
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Managing Volatility
Client Conversations
Understanding the impact of interest-rate changes on bond prices is crucial for effectively managing a fixed-income portfolio.
Rising and falling interest rates can have a significant impact on US Treasuries, which are issued by the US government. Other bond types, such as corporate bonds and mortgage-backed bonds, could be impacted differently due to credit risk (i.e., concerns about the issuer’s ability to make timely bond payments) and other factors.
The Hypothetical Impact of Rising and Falling Rates on Treasuries (%, as of 6/30/25)
A basis point (bps) is a unit that is equal to 1/100th of 1%, and is used to denote the change in a financial instrument. The basis point is commonly used for calculating changes in interest rates, equity indexes and the yield of a fixed-income security. For example, +100 bps is the equivalent of a 1% increase in interest rates. Changes to hypothetical return based on a security’s duration and convexity affect return. Duration is a measure of the sensitivity of an investment’s price to changes in interest rates. Convexity is a measure of how a bond’s duration can change based on the magnitude of an interest-rate change. Data Sources: Bloomberg and Hartford Funds, 7/25.
The Interest-Rate See Saw
Bond prices and interest rates have an inverse relationship: When interest rates rise, bond prices fall and vice versa—just like a see saw.
Higher interest rates allow bond investors to collect more interest on new bond purchases, but the principal value of their existing bonds will drop in value. When interest rates increase at a slow and steady pace over several years, bond investors may not feel the impact too much because the higher interest payments help offset the decline in bond principal. When rates rise rapidly, however, it can be painful because the drop in a bond’s principal value is greater than the additional interest income.
Bonds and Interest Rates Have an Inverse Relationship
For illustrative purposes only. Assumes a bond with a fixed semi-annual coupon and 10-year maturity. Source: Hartford Funds.
Purchasing bonds when interest rates are at or close to their peak could be a prudent strategy for capital appreciation since the principal value of bonds will likely increase as interest rates fall. You could also consider purchasing an actively managed bond fund or ETF to let professional money managers decide how to navigate changing interest rates.
Talk to your financial professional for help with managing interest-rate risk in your fixed-income portfolio.
Important Risks: Investing involves risk, including the possible loss of principal. • Fixed-income security risks include credit, liquidity, call, duration, and interest-rate risk. As interest rates rise, bond prices generally fall.
All information provided is for informational and educational purposes only and is not intended to provide investment, tax, accounting, or legal advice. As with all matters of an investment, tax, or legal nature, you should consult with a qualified tax or legal professional regarding your specific legal or tax situation, as applicable. The preceding is not intended to be a recommendation or advice. Tax laws and regulations are complex and subject to change.
CCWP142 4713564
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5941 | https://www.youtube.com/watch?v=rxe1XoXIg1o | AP Biology Exam Prep: Error Bars and Standard Error of the Mean
Gabe Poser - PoseKnows Biology
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Posted: 3 Jan 2023
In this video, I explain and demonstrate how error bars are calculated using standard deviation and standard error of the mean, then use error bars on a graph from an AP Biology exam question.
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Transcript:
Intro hi everybody welcome back it is Mr poser your AP biology teacher today we are continuing based on our last one on graphs um we're studying science practice four a little bit for the AP biology exam um and this is kind of an extension video on like graphs and representing data here because there's going to be some stuff that you're going to see um statistics wise that are going to be on the AP exam and they're just you know not limited to biology this is just you know good to know as far as statistics and data understanding data is a huge deal for uh today's modern world so so I hope this is uh helpful for a number of reasons um but today we're going to be looking at these uh these numbers over here all right so this is where I left you uh with last video all right you're doing your AP exam you come across a question and you gotta make a graph here based on a data table and then you have these numbers over here what do they mean and why are they significant and you'll see why that's funny in just a second all right um but check it out we have had this plus or minus symbol all the way down here on our data points and then we have capital S capital E and then this little X that with a bar on top of it so what does that exactly mean and what are these numbers um indicating over here all right well these numbers are what we call standard error of the mean and it's a measure that indicates how much the sample differs from the mean or how confident you are in your mean value representing your data points all right and that might not mean a whole lot right now um but when we run through an example and we're going to calculate this on our own in a little bit it'll start to make a lot more sense so basically the larger the standard error of the mean value that you have for a data point the less confidence you have in the mean actually representing the average all right so say for example um I am trying to collect the average shoe size of everybody in my AP Biology class right let's say we have average shoe size but I only have four kids in my class um and two of them have size 14 and then the other two have size six okay and then I average those and I get an average of about like I think that would be eight no that wouldn't be eight that would be like ten right so 10 is at very average shoe size but does anybody have that shoe size in my class no right so I would have a very very large standard error of the mean how well does the mean actually you know how different are the values actually representing the mean that's what standard error is okay um so the larger the number as I said the larger the number the greater the standard error of the mean and the less confidence we have in the mean actually representing um the that data point or that uh that group or that sample right so check it out this plus or minus means here indicates a range all right so the number of flies the average number of flies with the ebony body and long Wings is 98 but there's a standard error of the mean about 10 above and below 98 so most of your flies that you're going to find are going to be um in 108 or it's going to be 108 and 88 all right because that's 10 above 98 and 10 below 98 so it's kind of representing a range all right and we're going to Standard Error of the Mean walk through how to do that in just a second but first in order to find standard error of the mean over here we have to find standard deviation and what that is is the value that shows how much variation there is from the average for a set of data points okay um so here's uh here's our standard deviation equation and you will be given this on the AP exam I'm not sure if you're going to actually have to calculate it I'm going to say probably not um but just this is good to know where it comes from um for a lot of uh well not just for AP biology for every other class right or for any other class that involves data collection so any other science class right so here's standard deviation um and it looks like a big scary formula here but it's not that bad all right and then standard error of the mean is just s standard deviation divided by the square root of n um and that's pretty much it all right so uh we're going to be walking through an example here because that is going to give us the clearest indication of how this all works all right so check it out um I have uh Birds on Islands right so it says the number of birds on each island in an island chain or as it follows that's 96 on one 88 and on another 86 84 80 and 70. all right and we're going to calculate the standard deviation of this data set all right so uh how much does the average excuse me how much do these values vary from the average and in order to find that we have to find out what the average is all right an X bar I haven't figured out how to make an X with a little bar on top of it in my uh program here so I wrote X bar there um that is representing what we call our mean or our average right and we've been doing this since grade school like at them all up divided by the number the number of terms right so uh what I did here is that we added up 96 88 86 84 80 and 70 and divided by 6 because that's how many islands there are and we get our average as being 84. yes we're doing good okay so X bar is equal to 84. that is step one of uh calculating standard deviation step two is determine the deviation from the mean of each value and then add them all up okay so how much do R values um vary or differ from 84. all right and there's a mathematical way to do this we have X is representing each one of our values okay and then X bar is representing our means so for example well I did all of them already um check it out we have um our first Island at 96. all right so we have calculating 96 minus 84 squared okay we go 88 minus 84 squared okay because that's our next value basically we're calculating the difference in our values from the mean and squaring them all right um so if we do that I encourage you to try and do this on your calculator yourself okay 84 is our average um and these are each of our values and if you put them in your calculator okay we get these all right because you know think about it 96 minus 84 is 12. square that it's 144. um we add all these numbers up and we get 376 okay so again I'm encouraging you to kind of follow along with me here um as we uh calculate standard deviation all right um so basically again what we did you see the sigma here this Sigma symbol means add them all up it means sum or summation that's what it is okay um and all I'm doing once again here's each of my values is representing x minus X bar which is the average and square that and you add them all up and this is what we get for our data point all right or for our data set 376. step three of this is calculate the degrees of freedom and this part is super easy all degrees of freedom is is basically how many uh how many data points do you have minus one all right so uh basically we have six different Islands which means our data point is or our degrees of freedom is five just because it's six minus one easy right so n this number here is uh representing how many data points that we have and we have six of them all right so six minus one is five all right and then well um next step is put together to put it all together to find s all right this uh Top Value we already calculated as being 376 this bottom value is five and then we take the square root of that I encourage you it to punch that into your calculator right now if you haven't already and check it out 375 or 6 divided by 5 75.2 if you take the square root of that we get a value of 8.67 that is representing our standard deviation how much does our um do our data points differ or how much do they vary from our average okay so uh in order to find standard error of the mean which is what we're going to be graphing here in a second you divide your standard deviation by the square root of n or number of your values and this is this is the easy part right so we are in easy part we already calculated standard deviation 8.67 divided by the square root of 6 and we got 3.54 um and two standard deviations or excuse me two standard error of the mean um like what we saw in that data table okay is just two times this uh standard deviation divided by the square root of n all right so two standard error of the means for us for this data point would be uh 7.08 okay um so this number right here what this is our golden ticket here this number indicates the size of the error bars on a graph okay error bars that's what this Error Bars is all about all right we talked about error bars a little bit um in a previous video okay um I don't remember off the top my head which one it is okay but it's in this playlist I promise you um but error bars represent okay how much does that value uh vary okay and this is uh my interpretation of this uh this this is my graph here that I made all right so if I'm um calculating or if I'm measuring average population size I'm counting up by 10 so you've got my label here um and here's my bar representing my mean 84 all right but this little eye shape over here these are error bars all right and that means from this value I'm going about seven above the mean and I'm going about seven below the mean and this is how much variability um there is in my average my data point that I collected here all right so that would be the size of my error bar and you will be expected to put error bars on your bar graphs and perhaps even on your line graphs as well and that's what they look like you got to know the stand two times the standard error of the mean it's usually going to uh tell you for you all right and you put your error bars just like that okay we're where the top reach the top horizontal section here is representing your average plus okay two standard error of the means and then the bottom is two below or excuse me two standard error that means below your mean okay um so here's the here's the data table from before we're going to graph this now okay where I brought this back from the beginning of the video we got 98 plus or minus 10 ebony body long Wing flies 28 plus or minus seven so on and so forth I'd like you to try and graph this on your own I'm going to show you mine in just a second um pause if you want to try it yourself but if not I'm going to go ahead and move on this is oh hang on there we go this is my graph all right here it is I got the number of flies over here I counted up by 20s as my scale remember scaling in units is still important you know we're still talking about graphs there's my uh data table here's my labels down here and most importantly check it out here are my error bars alright so ebony body long wings are a standard error of the mean or two times standard error the mean was 10 all right so I went 10 below the mean and 10 above the mean to represent that error bar um I think this one was 25 so I went 25 above and 25 below all right and uh yeah this is this is how you do it all right and if this were a line graph you'd do the same thing except for a data point you'd put some uh um error bars on each one of those points okay um and Y as I put over here why bother to put error bars why does error bars matter okay overlapping error bars indicates that there is no statistically significant difference between groups of variables okay and this is going back to testing independent versus dependent variable accepting or rejecting the null hypothesis or the alternative hypothesis right so if this were my data here or these are my data here and check out these gigantic error bars um these are overlapping okay that means that the standard error of the mean is large enough that I cannot actually say statistically that there is a a difference in the values between these three these three data points there's no difference between calcium sensitivity and phosphate oxalate or uric acid okay so this would be a scenario where I accept the null hypothesis because these error bars are so big and check it out they're overlapping one another alright so check it out here's this gigantic range for uric acid okay the other ranges of these other uh error bars fall into that okay you don't have statistically significant data they're the independent variable does not affect the dependent variable so that means the Matrix does not affect calcium sensitivity there okay so check it out on our graph here is the data statistically significant do our error bars overlap well yes it is statistically significant there's no overlap in error bars maybe uh maybe a little bit between ebony body long wings and gray body but best Digital Data excuse me vestigial wings um but the rest of these do not uh overlap at all okay and we can indicate that yes the dependent variable is affected by the independent variable the uh there is a significant difference in the number of flies of each phenotype um over here so there's something going on genetically if you want to know what that this is all about I believe this is topic 5.6 um in my other videos all right um but that will be it for today please let me know if you have any questions and we'll see you next time |
5942 | https://people.tamu.edu/~terdelyi//papers-online/cyclotomic.pdf | ON THE Lq NORM OF CYCLOTOMIC LITTLEWOOD POLYNOMIALS ON THE UNIT CIRCLE Tam´ as Erd´ elyi Abstract. Let Ln be the collection of all (Littlewood) polynomials of degree n with coeffi-cients in {−1, 1}. In this paper we prove that if (P2ν) is a sequence of cyclotomic polynomials P2ν ∈L2ν, then Mq(P2ν) > (2ν + 1)a for every q > 2 with some a = a(q) > 1/2 depending only on q, where Mq(P ) := „ 1 2π Z 2π 0 |P (eit)|q dt «1/q , q > 0 .
The case q = 4 of the above result is due to P. Borwein, Choi, and Ferguson.
A similar result is conjectured for Littlewood polynomials of odd degree. Our main tool here is the Borwein-Choi Factorization Theorem.
1. Introduction Let D be the open unit disk of the complex plane. Its boundary, the unit circle of the complex plane, is denoted by ∂D. Let Kn be the set of all polynomials of degree n with complex coefficients of modulus 1. Elements of Kn are often called (complex) unimodular polynomials of degree n. Let Ln be the set of all polynomials of degree n with coefficients in {−1, 1}. Elements of Ln are often called real unimodular polynomials or Littlewood polynomials of degree n. The Parseval formula yields Z 2π 0 |Pn(eit)|2 dt = 2π(n + 1) for all Pn ∈Kn. Therefore min z∈∂D |Pn(z)| ≤ √ n + 1 ≤max z∈∂D |Pn(z)| .
An old problem (or rather an old theme) is the following.
2000 Mathematics Subject Classifications. 11C08, 41A17 Typeset by A MS-T EX 1 Problem 1.1 (Littlewood’s Flatness Problem). How close can a Pn ∈Kn or Pn ∈Ln come to satisfying (1.1) |Pn(z)| = √ n + 1 , z ∈∂D?
Obviously (1.1) is impossible if n ≥1. So one must look for less than (1.1), but then there are various ways of seeking such an “approximate situation”. One way is the following. In his paper [Li1] Littlewood had suggested that, conceivably, there might exist a sequence (Pn) of polynomials Pn ∈Kn (possibly even Pn ∈Ln) such that (n + 1)−1/2|Pn(eit)| converge to 1 uniformly in t ∈R. We shall call such sequences of unimodular polynomials “ultraflat”. More precisely, we give the following definition.
Definition 1.2. Given a positive number ε, we say that a polynomial Pn ∈Kn is ε-flat if (1 −ε) √ n + 1 ≤|Pn(z)| ≤(1 + ε) √ n + 1 , z ∈∂D .
Definition 1.3. Given a sequence (εnk) of positive numbers tending to 0, we say that a sequence (Pnk) of polynomials Pnk ∈Knk is (εnk)-ultraflat if each Pnk is (εnk)-flat. We simply say that a sequence (Pnk) of polynomials Pnk ∈Knk is ultraflat if it is (εnk)-ultraflat with a suitable sequence (εnk) of positive numbers tending to 0.
The existence of an ultraflat sequence of unimodular polynomials seemed very unlikely, in view of a 1957 conjecture of P. Erd˝ os (Problem 22 in [Er]) asserting that, for all Pn ∈Kn with n ≥1, (1.2) max z∈∂D |Pn(z)| ≥(1 + ε) √ n + 1 , where ε > 0 is an absolute constant (independent of n). Yet, refining a method of K¨ orner [K¨ o], Kahane [Ka] proved that there exists a sequence (Pn) with Pn ∈Kn which is (εn)-ultraflat, where εn = O n−1/17p log n .
See also [QS]. A recent paper of Bombieri and Bourgain [BB] is devoted to the construc-tion of ultraflat sequences of unimodular polynomials. In particular, one obtains a much improved estimate for the error term.
A major part of this paper deals also with the long-standing problem of the effective construction of ultraflat sequences of unimodular polynomials.
Thus the Erd˝ os conjecture (1.2) was disproved for the classes Kn. For the more restricted class Ln the analogous Erd˝ os conjecture is unsettled to this date. It is a common belief that the analogous Erd˝ os conjecture for Ln is true, and consequently there is no ultraflat sequence of polynomials Pn ∈Ln. An interesting result related to Kahane’s breakthrough is given in [Be]. For an account of some of the work done till the mid 1960’s, see Littlewood’s book [Li2] and [QS]. The structure of ultraflat sequences of unimodular polynomials is studied in [Er1], [Er2], [Er3], and [Er4], where several conjectures of Saffari are proved.
2 The Rudin-Shapiro polynomials appear in Harold Shapiro’s 1951 thesis at MIT and are sometimes called just Shapiro polynomials. See Chapter 4 of [Bo] for the construc-tion(s). Cyclotomic properties of the Rudin-Shapiro polynomials are discussed in [BLM].
A sequence (Pn) of Rudin-Shapiro polynomials satisfies Pn ∈Ln and |Pn(z)| ≤C √ n + 1 , z ∈∂D , with an absolute constant C. In this paper we prove that a sequence of cyclotomic Lit-tlewood polynomials of even degree is far from having the above “flatness” property of a sequence of Rudin-Shapiro polynomials. Note that (see page 271 of [BE], for instance) a Littlewood polynomial has Mahler measure one if and only if it is cyclotomic, that is, it has all its zeros on the unit circle ∂D.
For a polynomial P let Mq(P) := 1 2π Z 2π 0 |P(eit)|q dt 1/q , q > 0 , and M∞(P) := max t∈[0,2π]|P(eit)| .
2. Preliminary Results An unpublished observation of the author is the following.
Theorem 2.1. If (P2ν) is a sequence of cyclotomic polynomials P2ν ∈L2ν, then M∞(P2ν) > (2ν + 1)a , where a := 1 −log3 π 2 = 0.5889 . . . > 1 2.
The stronger result below is due to P. Borwein, Choi, and Ferguson [BCF].
Theorem 2.2. If (P2ν) is a sequence of cyclotomic polynomials P2ν ∈L2ν, then M4(P2ν) > (2ν + 1)a with a = 1 4 log2(1 + √ 17) = 0.5892 . . . > 1/2.
In the proof of both theorems above the result of Borwein and Choi [BC] stated below has been a key.
Theorem 2.3. Every cyclotomic polynomial P ∈Ln of even degree can be factorized as P(z) = ±Φp1(±z)Φp2(±zp1) · · ·Φpr(±zp1p2···pr−1) , where n + 1 = p1p2 · · · pr, the numbers pj are primes, not necessarily distinct, and Φp(z) = p−1 X j=0 zj = zp −1 z −1 is the p-th cyclotomic polynomial.
It is conjectured that this characterization also holds for polynomials P ∈Ln of odd degree. This conjecture is based on substantial computation together with a number of special cases.
3 3. New Results Theorem 3.1. If (P2ν) is a sequence of cyclotomic polynomials P2ν ∈L2ν, then Mq(P2ν) > (2ν + 1)a for every q > 2 with some a = a(q) > 1/2 depending only on q.
Theorem 3.2. If (P2ν) is a sequence of cyclotomic polynomials P2ν ∈L2ν, then M1(P2ν) < (2ν + 1)b with some absolute constant 0 < b < 1/2.
It is conjectured that similar results hold for cyclotomic Littlewood polynomials of odd degree.
4. Proofs Although Theorem 2.2 beats Theorem 2.1, we present the short proof of Theorem 2.1 that is simpler than and quite different from that of Theorem 2.2.
Proof of Theorem 2.1. We use the factorization theorem of Borwein and Choi. We prove the theorem by induction on the number of factors. The theorem is obviously true when P2ν has only one factor. The proof of the inductive step goes as follows.
Suppose the theorem is true for f, where f has k −1 factors. We have to prove that theorem is true for g(z) := Φp(±z)f(zp) .
Let M(f) be the maximum modulus of f on the unit circle. The key observation is that M(f) is achieved by |f(zp)| at a system of p equidistant points on the unit circle. Denote these by z1 , z2 , . . . , zp. Then there is at least one zj such that the angular distance between 1 and zj is at most 2π/(2p). Similarly there is at least one zj such that the angular distance between −1 and zj is at most 2π/(2p). Now the proof can be finished by Lemma 4.1 below the proof of which is a straightforward geometric argument. Using Lemma 4.1 the proof of the inductive step is obvious, since a := 1 −log3 π 2 ensures (2/π)p ≥pa for every p ≥3.
In fact, using the prime factorization of 2ν + 1, where 2ν is the degree of P2ν, one can get a larger value of the exponent a in the theorem if the primes in the factorization of 2ν + 1 are large.
□ Lemma 4.1. If z is a point on the unit circle such that the angular distance of z from 1 is at most 2π/(2p), then |Φp(z)| ≥(2/π)p. If z is a point on the unit circle such that the angular distance of z from −1 is at most 2π/(2p). Then |Φp(−z)| ≥(2/π)p.
Proof of Lemma 4.1. Recall that Φp(z) = zp −1 z −1 and | sin t| ≤|t| for every t ∈R.
□ 4 To prove Theorem 3.2 we proceed as follows. First we introduce some notation. As-sociated with a positive integer p and a function f defined on the unit circle ∂D of the complex plane let g(t) := f(eit) , t ∈R , fp(z) := f(zp)Φp(z) , Φp(z) := p−1 X j=0 zj , gp(t) := fp(eit) , that is, gp(t) := f(eipt)hp(t) , hp(t) := Φp(eit) .
Let I(f) := Z ∂D |f(z)| |dz| = Z 2π 0 |f(eit)| dt , I(g) := Z 2π 0 |g(t)| dt, so I(g) = I(f) . The key to the proof of Theorem 3.2 is the following lemma that allows an induction on the number of factors in the decomposition of the cyclotomic polynomial P2ν given by Theorem 2.3.
Lemma 4.2. If f is a continuous function on ∂D such that |f(z)| = |f(z)| for every z ∈∂D, then I(fp) ≤pαI(f) for every odd prime p with an absolute constant 0 < α < 1/2.
Proof of Lemma 4.2. Let k > 0 be an integer and we define Lj,k,p := Z jπ/k (j−1)π/k |gp(t)| dt , j = 1, 2, . . . .
Then Lj,k,p = Lj+2k,k,p, j = 1, 2, . . . , and I(gp) = Z 2π 0 |gp(t)| dt = 2 Z π 0 |gp(t)| dt = 2 kp X j=1 Z jπ/(kp) (j−1)π/(kp) |gp(t)| dt =2 kp X j=1 Z jπ/(kp) (j−1)π/(kp) |f(eipt)| |hp(t)| dt = 2 kp X j=1 Z jπ/(kp) (j−1)π/(kp) |f(eipt)| dt !
Mj,k,p =2 kp X j=1 Z jπ/k (j−1)π/k |f(eiu)| du !
1 p Mj,k,p =2 k X µ=1 Lµ,k,p 1 p (p−1)/2 X ν=0 Mµ+2νk,k,p + (p−3)/2 X ν=0 M2k+1−µ+2νk,k,p =2 k X µ=1 Aµ,k,pLµ,k,p , 5 where Mj,k,p := max Ij,k,p |hp(t)| = max Ij,k,p |Φp(eit)| with Ij,k,p := (j −1)π kp , jπ kp , j = 1, 2, . . . , kp , and (4.1) Aµ,k,p := 1 p (p−1)/2 X ν=0 Mµ+2νk,k,p + (p−3)/2 X ν=0 M2k+1−µ+2νk,k,p , µ = 1, 2, . . . , k .
Here we used that the assumptions on f imply that the value of Z µπ/k (µ−1)π/k |f(eiu)| du remains the same when µ is replaced with µ + 2νk or 2k + 1 −µ + 2νk. The proof of Lemma 4.2 now follows from Lemma 4.3.
□ Lemma 4.3. Let the numbers Aµ,k,p, µ = 1, 2, . . . , k, be defined by (4.1). There is an absolute constant 0 < b < 1/2 such that for every odd prime p there is a positive integer k such that (4.2) Aµ,k,p ≤pb , µ = 1, 2, . . . , k .
Proof of Lemma 4.3. It turns out that for large primes even Aµ,1,p ≤c log p is true, while for smaller primes we choose larger values of k to establish (4.2). Observe that Φp(eit) = sin(pt/2) sin(t/2) , hence |Φp(eit)| ≤ 1 sin(t/2) ≤ 1 2 π t 2 = π t , t ∈(0, π] .
This implies (4.3) |Φp(eit)| ≤min n p, π t o , t ∈(0, π] .
Observe also that |Φp(eit)| ≤ 1 sin(t/2) ≤ 1 sin(π/4) π/4 t 2 ≤ π √ 2 t , t ∈(0, π/2] , 6 hence (4.4) |Φp(eit)| ≤min p, π √ 2 t , t ∈(0, π/2] .
If µ = k = 1, then using (4.3) and (4.4) we easily obtain A1,1,p := 1 p (p−1)/2 X ν=0 M1+2ν,1,p + (p−1)/2 X ν=0 M2+2ν,1,p = 1 p p X j=1 Mj,1,p ≤1 p p + (p−1)/2−1 X j=1 p √ 2j + p−1 X j=(p−1)/2 p j ≤1 + (p−1)/2−1 X j=1 1 √ 2j + p−1 X j=(p−1)/2 1 j ≤1 + 1 √ 2 + 1 √ 2 Z (p−1)/2 1 dx x + 2 p −1 + Z p−1 (p−1)/2 dx x = 1 + 1 √ 2 + 1 √ 2 (ln(p −1) −ln 2) + 2 p −1 + ln 2 ≤1 + 1 √ 2 + 1 −1 √ 2 ln 2 + 2 p −1 + 1 √ 2 ln p ≤p0.48 for every prime p ≥23. Here we used the fact that ln p/p0.48 is decreasing for p ≥23.
Hence the lemma holds for all primes p ≥23 (we choose k = 1). Further, the estimates A1,1,19 ≤1 + 1 √ 2 8 X j=1 1 j + 18 X j=9 1 j ≤3.7 < √ 19 , A1,1,17 ≤1 + 1 √ 2 7 X j=1 1 j + 16 X j=8 1 j ≤3.63 < √ 17 , A1,1,13 ≤1 + 1 √ 2 5 X j=1 1 j + 12 X j=6 1 j ≤3.42 < √ 13 , and A1,1,11 ≤1 + 1 11 sin(π/22) + 1 √ 2 4 X j=1 1 j + 10 X j=5 1 j ≤3.25 < √ 11 , show that lemma holds for all primes 11 ≤p ≤19 (we choose k = 1).
7 Now we study the case p = 7. We have A1,4,7 ≤1 7(7 + 1 + 2.31 + 1.35 + 1.38 + 1.05 + 1.03) ≤2.16 < √ 7 , A2,4,7 ≤1 7(6.83 + 2.15 + 2.07 + 1.42 + 1.23 + 1.06 + 1.02) ≤2.26 < √ 7 , A3,4,7 ≤1 7(6.32 + 3.34 + 1.88 + 1.51 + 1.19 + 1.09 + 1.01) ≤2.34 < √ 7 , and A4,4,7 ≤1 7(5.52 + 4.50 + 1.73 + 1.61 + 1.15 + 1.11 + 1.01) ≤2.38 < √ 7 .
In the above four estimates we used that M1,4,7 = 7 , M2,4,7 = sin(π/8) sin(π/56) ≤6.83 , M3,4,7 = sin(2π/8) sin(2π/56) ≤6.32 , M4,4,7 = sin(3π/8) sin(3π/56) ≤5.52 , M5,4,7 = sin(4π/8) sin(4π/56) ≤4.50 , M6,4,7 = sin(5π/8) sin(5π/56) ≤3.34 , M7,4,7 = sin(6π/8) sin(6π/56) ≤2.15 , M8,4,7 = sin(7π/8) sin(7π/56) ≤1 , and Mj,4,7 = 1 sin((j −1)π/56), j = 9, 10, . . . , 28 .
Hence the lemma is proved for p = 7 (we choose k = 4).
Now we study the case p = 5. We have A1,4,5 ≤1 5(5 + 0.74 + 0.59 + 0.42 + 0.40) ≤1.43 < √ 5 , A2,4,5 ≤1 5(4.88 + 1.56 + 1 + 0.80 + 0.72) ≤1.80 < √ 5 , A3,4,5 ≤1 5(4.53 + 2.42 + 1.25 + 1.09 + 0.93) ≤2.05 < √ 5 , and A4,4,5 ≤1 7(3.96 + 3.24 + 1.25 + 1.25 + 1) ≤2.15 < √ 5 , In the above four estimates we used that M1,4,5 = 5 , M2,4,5 = sin(π/8) sin(π/40) ≤4.88 , 8 M3,4,5 = sin(2π/8) sin(2π/40) ≤4.53 , M4,4,5 = sin(3π/8) sin(3π/40) ≤3.96 , M5,4,5 = sin(4π/8) sin(4π/40) ≤3.24 , M6,4,5 = sin(5π/8) sin(5π/40) ≤2.42 , M7,4,5 = sin(6π/8) sin(6π/40) ≤1.56 , M8,4,5 = sin(7π/8) sin(7π/40) ≤0.74 , M9,4,5 = −sin(9π/8) sin(9π/40) ≤0.59 , M10,4,5 = sin(10π/8) sin(10π/40) = 1 , M11,4,5 ≤1.25 , M12,4,5 ≤1.25 , M13,4,5 ≤1.25 , M14,4,5 = −sin(13π/8) sin(13π/40) ≤1.09 , M15,4,5 = −sin(14π/8) sin(14π/40) ≤0.80 , M16,4,5 = sin(15π/8) sin(15π/40) ≤0.42 , M17,4,5 = sin(17π/8) sin(17π/40) ≤0.40 , M18,4,5 = sin(18π/8) sin(18π/40) ≤0.72 , and M19,4,5 = sin(19π/8) sin(19π/40) ≤0.93 , M20,4,5 = sin(20π/8) sin(20π/40) ≤1 .
Hence the lemma holds for p = 5 (we choose k = 4).
Now we study the case p = 3. We have M1,k,3 = 3 , Mj,k,3 = 1 + 2 cos (j −1)π 3k , j = 2, 3, . . . , 2k , and Mj,k,3 = − 1 + 2 cos jπ 3k , j = 2k + 1, 2k + 2, . . . , 3k , hence with y := (µ −1)π 3k ∈[0, π/3] , µ = 1, 2, . . . , k , we have 3Aµ,k,3 =Mµ,k,3 + M2k+1−µ,k,3 + M2k+µ,k,3 = 1 + 2 cos (µ −1)π 3k + 1 + 2 cos (2k −µ)π 3k − 1 + 2 cos (2k + µ)π 3k =1 + 2 cos y + 2 cos 2π 3 −2π 3k −y −2 cos 2π 3 + 2π 3k + y =1 + 2(cos y + √ 3 sin y) + c(k) = 1 + 4 sin y + π 6 + c(k) ≤5 + c(k) 9 with c(k) →0 as k →∞. Therefore Aµ,k,3 ≤5 3 + c(k) 3 < √ 3 , k = 1, 2, . . . , µ , for all sufficiently large k. Thus the lemma is proved for p = 3 (we choose a sufficiently large k).
□ Proof of Theorem 3.2. The polynomials P2ν can be factorized as it is given in Theorem 2.3. The theorem follows by induction on the number of factors in P2ν. We use Lemma 4.2 in the inductive step.
□ Proof of Theorem 3.1. Let f be a continuous function on ∂D and let Iq(f) := Mq(f)q = 1 2π Z 2π 0 |f(eit|q dt .
Then h(q) := log(Iq(f)) = q log(Mq(f)) is a convex function of q on (0, ∞). This is a simple consequence of H¨ older’s inequality. For the sake of completeness we present the short proof of it. We need to see that Ir(f) ≤Ip(f) r−q p−q Iq(f) p−r p−q , that is (4.5) 1 2π Z 2π 0 |f(eit|r dt p−q ≤ 1 2π Z 2π 0 |f(eit|p dt r−q 1 2π Z 2π 0 |f(eit|q dt q−r .
To see this let α := p −q r −q , β := p −q p −r , γ := p α , δ := q β , hence 1/α + 1/β = 1 and γ + δ = r. Let F(t) := |f(eit)|γ = |f(eit)| p(r−q) p−q , and G(t) := |f(eit)|δ = |f(eit)| q(r−q) p−q , Then by H¨ older’s inequality we conclude 1 2π Z 2π 0 F(t)G(t) dt ≤ Z 2π 0 F(t)α dt 1/α Z 2π 0 G(t)β dt 1/β , and (4.5) follows. Using the convexity of log(Iq(P2ν)) on (0, ∞), for q > 2 we have I2(P2ν) ≤(I1(P2ν)) q−2 q−1 (Iq(P2ν)) 1 q−1 and from Theorem 3.2 we obtain 2ν + 1 ≤((2ν + 1)b) q−2 q−1 (Iq(P2ν)) 1 q−1 , that is (2ν + 1)(q−1)−(q−2)b ≤Iq(P2ν) = Mq(P2ν)q , with an absolute constant 0 < b < 1/2. Hence with a = a(q) := (q −1 −(q −2)b)/q we have (2ν + 1)a = (2ν + 1)(q−1−(q−2)b)/q ≤Mq(P2ν) .
Here a = a(q) := (q −1 −(q −2)b)/q > 1/2, since (1 −2b)(q/2 −1) > 0.
□ 10 References [Be] J. Beck, “Flat” polynomials on the unit circle – note on a problem of Littlewood, Bull. London Math. Soc. (1991), 269–277.
[BN] E. Bombieri and J. Bourgain, On Kahane’s ultraflat polynomials, J. Eur. Math. Soc. 11 (2009, 3), 627–703.
[Bo] P. Borwein, Computational Excursions in Analysis and Number Theory, Springer-Verlag, New York, 2002.
[BC] P. Borwein and K.S. Choi, On cyclotomic polynomials with ±1 coefficients, Experiment. Math.
8 (1995), 399–407.
[BCF] P. Borwein, K.S. Choi, and R. Ferguson, Norm of Littlewood cyclotomic polynomials, Math.
Proc. Cambridge Philos. Soc. 138 (2005), 315-326.
[BE] P. Borwein and T. Erd´ elyi, Polynomials and Polynomial Inequalities, Springer-Verlag, New York, 1995.
[BLM] J. Brillhart, J.S. Lomont, and P. Morton, Cyclotomic properties of the Rudin-Shapiro polyno-mials, J. Reine Angew. Math. 288 (1976), 37–75.
[Er1] T. Erd´ elyi, The phase problem of ultraflat unimodular polynomials: the resolution of the con-jecture of Saffari, Math. Ann. 321 (2001), 905–924.
[Er2] T. Erd´ elyi, How far is a sequence of ultraflat unimodular polynomials from being conjugate reciprocal, Michigan J. Math. 49 (2001), 259–264.
[Er3] T. Erd´ elyi, A proof of Saffari’s “near-orthogonality” conjecture for ultraflat sequences of uni-modular polynomials, C. R. Acad. Sci. Paris S´ er. I Math. 333 (2001), 623–628.
[Er4] T. Erd´ elyi, On the real part of ultraflat sequences of unimodular polynomials: consequences implied by the resolution of the Phase Problem, Math. Ann. 326 (2003), 489–498.
[Er] P. Erd˝ os, Some unsolved problems, Michigan Math. J. 4 (1957), 291–300.
[Ka] J.P. Kahane, Sur les polynomes a coefficient unimodulaires, Bull. London Math. Soc. 12 (1980), 321–342.
[K¨ o] T. K¨ orner, On a polynomial of J.S. Byrnes, Bull. London Math. Soc. 12 (1980), 219–224.
[Li1] J.E. Littlewood, On polynomials P ±zm, P exp(αmi)zm, z = eiθ, J. London Math. Soc. 41 (1966), 367–376.
[Li2] J.E. Littlewood, Some Problems in Real and Complex Analysis, Heath Mathematical Mono-graphs, Lexington, Massachusetts, 1968.
[QS] H. Queffelec and B. Saffari, On Bernstein’s inequality and Kahane’s ultraflat polynomials, J.
Fourier Anal. Appl. 2 (1996, 6), 519–582.
[Sa] B. Saffari, The phase behavior of ultraflat unimodular polynomials, in Probabilistic and Sto-chastic Methods in Analysis, with Applications (1992), Kluwer Academic Publishers, Printed in the Netherlands, 555–572.
Department of Mathematics, Texas A&M University, College Station, Texas 77843 E-mail address: terdelyi@math.tamu.edu 11 |
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PK-11195
1988
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Journal Article•10.1016/0091-3057(88)90202-X•
##### Imidazopyridines as a tool for the characterization of benzodiazepine receptors: a proposal for a pharmacological classification as omega receptor subtypes.
[x] - [x]
S.Z. Langer, S. Arbilla
01 Apr1988-Pharmacology, Biochemistry and Behavior
[x]
TL;DR: The nomenclature of a greek letter omega is proposed, as omega 1, omega 2 and omega 3 to designate respectively the central BZ1, BZ2 and the peripheral BZ receptor.
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[x]
Abstract: At present, the nomenclature of benzodiazepine (BZ) receptors is based on its historical association with the BZ structure. However, it is mainly through the new compounds chemically unrelated to BZs that the central and peripheral subtypes of BZ receptors have been characterized. We therefore propose the nomenclature of a greek letter omega, as ω 1 , ω 2 and ω 3 to designate respectively the central BZ 1 , BZ 2 and the peripheral BZ receptor. Among the several classes of non-BZ drugs with affinity for different receptors, the imidazopyridines provide a valuable tool for the characterization of omega receptor subtypes. Most BZs are non selective ligands for the central ω 1 and ω 2 receptors while the selectivity for ω 1 receptor subtypes is present in several non BZ chemical series: imidazopyridines (zolpidem), triazolopyridines (CL 218872), betacarbolines (β-CCE) and pyrazoloquinolines (CGS 8216). Selective ligands for the ω 2 subtype are not available so far. The so called peripheral BZ receptor is also present in the central nervous system, therefore the proposed nomenclature of ω 3 receptors resolves this paradox because it does not designate location and it is defined in terms of pharmacological specificity. Selective ligands for ω 3 receptors include the BZ Ro 5-4864, and the isoquinolinecarboxamide PK 11195, while the imidazopyridine alpidem is the ligand with the highest affinity for this receptor subtype.
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Journal Article•10.1016/0014-2999(88)90834-5•
##### Presence of peripheral-type benzodiazepine binding sites on human erythrocyte membranes
[x] - [x]
James M. Olson1, Brian J. Ciliax1, William R. Mancini1, Anne B. Young1•Institutions (1)
University of Michigan1
26 Jul1988-European Journal of Pharmacology
[x]
TL;DR: Data provide evidence that a peripheral-type benzodiazepine binding site, pharmacologically similar to the intracellular binding site described in other tissues, is present in the plasma membrane of human erythrocytes.
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Journal Article•
##### Molecular characterization and mitochondrial density of a recognition site for peripheral-type benzodiazepine ligands.
[x] - [x]
L. Antkiewicz-Michaluk1, Alessandro Guidotti, Karl E. Krueger1•Institutions (1)
Georgetown University1
01 Sep1988-Molecular Pharmacology
[x]
TL;DR: The results suggest that the density of peripheral-type benzodiazepine receptors in mitochondria is tissue dependent and apparently regulated independently of the mechanisms by which these two mitochondrial enzymes are expressed or function.
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[x]
Abstract: In a previous report, mitochondria were proposed as a subcellular structure where recognition sites for peripheral benzodiazepine ligands are located in adrenal glands. The present study examines the subcellular distribution of specific binding sites for PK 11195 in eight tissues and compares the relative densities of these binding sites in mitochondrial-enriched fractions with the relative activities of two mitochondrial marker enzymes. In all eight tissues examined, PK 11195 binding sites were found to subfractionate in a manner nearly identical to that of the mitochondrial enzyme succinate dehydrogenase. The subcellular distribution patterns of specific PK 11195 binding sites were unrelated to the distribution patterns of marker enzymes for plasma membranes, lysosomes, or endoplasmic reticulum. Scatchard analyses of mitochondrial fractions from all eight tissues demonstrated a greater than 100-fold difference in the densities of PK 11195 binding sites, the extremes being 140 and 1 pmol/mg of protein in adrenal and brain tissues, respectively. There was no correlation between the relative density of PK 11195 binding sites and the specific activities of succinate dehydrogenase and cytochrome c oxidase. These results suggest that the density of peripheral-type benzodiazepine receptors in mitochondria is tissue dependent and apparently regulated independently of the mechanisms by which these two mitochondrial enzymes are expressed or function. The photoaffinity probe PK 14105 was used to photolabel the peripheral-type benzodiazepine binding sites of mitochondrial fractions prepared from the eight tissues. In all preparations, a 17,000-Da polypeptide is specifically labeled as determined by electrophoresis in sodium dodecyl sulfate-polyacrylamide gels. Thus, it appears that the protein recognition site for isoquinoline carboxamides of peripheral-type benzodiazepine receptor complexes is similar in all mitochondrial preparations.
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Journal Article•10.1111/J.1472-8206.1988.TB00629.X•
##### Limitations of the benzodiazepine receptor nomenclature: a proposal for a pharmacological classification as omega receptor subtypes.
[x] - [x]
S.Z. Langer, S. Arbilla
06 May1988-Fundamental & Clinical Pharmacology
[x]
TL;DR: Among the several classes of non‐BZD drugs with affinity for different receptors, the imidazopyridines provide a valuable tool for the characterization of omega receptor subtypes, and the nomenclature of a Greek letter omega is proposed to designate the central, peripheral and peripheral BZ receptors.
...read more read less
[x]
Abstract: At present, the nomenclature of benzodiazepine (BZ) receptors is based on historical association with the BZ structure. However, it is mainly through the new compounds chemically unrelated to BZ that the central and peripheral subtypes of BZ receptors have been characterized. We therefore propose the nomenclature of a Greek letter omega, as omega 1, omega 2, and omega 3 to designate the central BZ1, BZ2, and peripheral BZ receptors, respectively. Among the several classes of non-BZD drugs with affinity for different receptors, the imidazopyridines provide a valuable tool for the characterization of omega receptor subtypes. Most BZ are nonselective ligands for the central omega 1 and omega 2 receptors, while selectivity for omega 1 receptor subtypes is present in several non BZ chemical series: imidazopyridines (zolpidem), triazolopyridazines (CL 218872), betacarbolines (beta-CCE), and pyrazoloquinolines (CGS 8216). Selective ligands for the omega 2 subtype are not available so far. The so-called peripheral BZ receptor is also present in the central nervous system; therefore, the proposed nomenclature of omega 3 receptors resolves this paradox because it does not designate location and is defined in terms of pharmacological specificity. Selective ligands for omega 3 receptors include the BZ Ro 5-4864, and the isoquinolinecarboxamide PK 11195, while the imidazopyridine alpidem is the ligand with the highest affinity for this receptor subtype.
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Journal Article•
##### Isoquinoline and Peripheral-type Benzodiazepine Binding in Gliomas: Implications for Diagnostic Imaging
[x] - [x]
James M. Olson1, Larry Junck1, Anne B. Young1, John B. Penney1, William R. Mancini1 - Show less +1 more •Institutions (1)
University of Michigan1
15 Oct1988-Cancer Research
[x]
TL;DR: Autoradiograms of postmortem human brain sections containing glioma revealed that [3H]PK 11195 bound specifically to intact tumor cells and not to cells of normal cerebral cortex or necrotic areas of the tumor, supporting the use of radiolabeled PK 11195 for clinical trials of imaging human gliomas by positron emission tomography.
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[x]
Abstract: Binding of the isoquinoline PK 11195 and of the benzodiazepines Ro5-4864 and flunitrazepam was compared in glioma cells and tissues. In human and rat glioma cell cultures [3H]PK 11195 bound with higher affinity ( K d = 14.01 and 15.76 nm, respectively) than either Ro5-4864 ( K i = 1200 and 84.9 nm, respectively) or flunitrazepam ( K i > 10,000 and = 848 nm, respectively). Autoradiograms of postmortem human brain sections containing glioma revealed that [3H]PK 11195 bound specifically to intact tumor cells and not to cells of normal cerebral cortex or necrotic areas of the tumor. Total [3H]Ro5-4864 or [3H]flunitrazepam binding to these sections was indistinguishable from nonspecific binding, and regions of tumor and normal brain could not be delineated. These results support the use of radiolabeled PK 11195 for clinical trials of imaging human gliomas by positron emission tomography.
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Journal Article•
##### Regional distribution of a Ro5 4864 binding site that is functionally coupled to the gamma-aminobutyric acid/benzodiazepine receptor complex in rat brain.
[x] - [x]
Kelvin W. Gee1, Roberta E. Brinton, Bruce S. McEwen•Institutions (1)
University of Southern California1
01 Jan1988-Journal of Pharmacology and Experimental Therapeutics
[x]
TL;DR: These studies lend additional support to the postulate that this drug binding site represents an additional locus for the regulation of GABAergic neurotransmission in the central nervous system.
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[x]
Abstract: The hypothesis that a novel drug binding site linked to a gamma-aminobutyric acid (GABA)-regulated chloride ionophore mediates the excitatory effects of the atypical benzodiazepine (BZ) Ro5 4864 is further evaluated in the present study. Dose-dependent inhibition of [3H]flunitrazepam to the central BZ receptor in rat cerebral cortex by the cage convulsant t-butylbicyclophosphorotionate (TBPS) is modulated by Ro5 4864 and the isoquinoline PK 11195 in a manner consistent with their reported pro/anticonvulsant effects. The ability of Ro5 4864 to enhance the binding of [35S]TBPS to a GABA-regulated chloride ionophore in rat cortex is unchanged after the irreversible labeling of the central BZ receptor by the photoaffinity label Ro15 4513. Together, these observations further suggest that 1) the effect of Ro5 4864 on [35S]TBPS is not mediated by the central BZ receptor and 2) the Ro5 4864 binding site is allosterically coupled to the GABA/BZ receptor-chloride ionophore complex in rat cerebral cortex. Anatomical localization of Ro5 4864-stimulated [35S]TBPS binding in rat brain by autoradiography reveals a distribution of chloride ionophore-coupled Ro5 4864 sites which is in many instances similar to that of the GABA/BZ receptor-chloride ionophore complex. These studies lend additional support to the postulate that this drug binding site represents an additional locus for the regulation of GABAergic neurotransmission in the central nervous system.
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Journal Article•10.1016/0024-3205(88)90464-X•
##### Regulation of renal peripheral benzodiazepine receptors by anion transport inhibitors
[x] - [x]
Anthony S. Basile1, Hartmut W.M. Lueddens1, Phil Skolnick1•Institutions (1)
National Institutes of Health1
01 Jan1988-Life Sciences
[x]
TL;DR: The findings suggest that renal PBR may be selectively modulated in vivo and in vitro by administration of ion transport/exchange inhibitors.
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Journal Article•10.1016/0006-8993(88)90492-1•
##### Increase in central and peripheral benzodiazepine receptors following surgery
[x] - [x]
Faina Okun1, Ronit Weizman2, Yeshayahu Katz1, Arieh Bomzon1, Moussa B.H. Youdim1, Moshe Gavish1 - Show less +2 more •Institutions (2)
Technion – Israel Institute of Technology1, Tel Aviv University2
16 Aug1988-Brain Research
[x]
TL;DR: The surgery resulted in the up-regulation of central benzodiazepine receptors in cerebral cortex and of peripheral BZ binding sites in brain and kidney on the first and third days after operation.
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Journal Article•10.1016/0014-2999(88)90638-3•
##### Chronic ethanol exposure increases peripheral-type benzodiazepine receptors in brain.
[x] - [x]
P.J. Syapin1, Ronald L. Alkana1•Institutions (1)
University of Southern California1
16 Feb1988-European Journal of Pharmacology
[x]
TL;DR: It is suggested that ethanol exposure causes time-dependent changes in brain PBR that may be linked to the development of physical dependence, and further studies are necessary to determine whether the increase inbrain PBR sites of alcohol-dependent mice is causally related to theDevelopment of alcohol dependence.
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Journal Article•10.1016/0006-2952(88)90502-3•
##### Interactions of lipids with peripheral-type benzodiazepine receptors
[x] - [x]
Kevin Beaumont1, Roman Skowroński1, Duke A. Vaughn1, Darrell D. Fanestil1•Institutions (1)
University of California, San Diego1
15 Mar1988-Biochemical Pharmacology
[x]
TL;DR: PBR binding was inhibited by specific lipids and that binding of proposed agonist (RO 5-4864) and antagonist (PK 11195) ligands was differentially affected by unsaturated fatty acids.
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Journal Article•
##### Identification of a high-affinity peripheral-type benzodiazepine binding site in rat aortic smooth muscle membranes.
[x] - [x]
John F. French1, Mohammed A. Matlib•Institutions (1)
University of Cincinnati Academic Health Center1
01 Oct1988-Journal of Pharmacology and Experimental Therapeutics
[x]
TL;DR: The data indicate an abundant high affinity peripheral-type benzodiazepine binding site of unknown function in rat aortic smooth muscle cells.
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[x]
Abstract: The existence of a benzodiazepine binding site in rat aortic smooth muscle membranes was explored employing [3H]Ro5-4864 as radioligand. The binding site was concentrated in the mitochondrial fraction enriched with cytochrome c oxidase and semicarbazide-insensitive monoamine oxidase. [3H]Ro5-4864 binds to the membranes in the mitochondrial fraction with high affinity. The dissociation constant (KD) determined by saturation binding was 2.8 +/- 0.7 nM (n = 5). The association rate constant (k1) was 4.7 +/- 0.8 x 10(6) M1 min-1, and the dissociation rate constant (k-1) was 0.028 +/- 0.005 min-1 (n = 3). The kinetically determined KD was 6.0 +/- 0.8 nM (n = 3) at 0.5 nM [3H]Ro5-4864. The density of binding determined from saturation binding experiments was 14.0 +/- 1.2 pmol/mg protein (n = 5). The Hill coefficient of binding was 0.94 +/- 0.02 (n = 5) indicating that [3H] Ro5-4864 binds to a single site. The [3H]Ro5-4864 binding was inhibited by Ro5-4864 (Ki = 6.1 +/- 1.9 nM), PK 11195 (Ki = 8.9 +/- 1.8 nM), diazepam (Ki = 87.3 +/- 3.4 nM), flunitrazepam (Ki = 94.6 +/- 1.8 nM), clonazepam (Ki = 6.3 +/- 1.3 microM) and Ro15-1788 (Ki = 16.8 +/- 1.5 microM). The rank order of potency of the competitive inhibition of [3H]Ro5-4864 binding (Ro5-4864 = PK 11195 greater than diazepam = flunitrazepam much greater than clonazepam greater than Ro15-1788) is characteristic of the peripheral-type benzodiazepine binding site. The data indicate an abundant high affinity peripheral-type benzodiazepine binding site of unknown function in rat aortic smooth muscle cells.
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Journal Article•10.1021/JM00119A005•
##### Radioiodinated Benzodiazepines: Agents for Mapping Glial Tumors
[x] - [x]
M. E. Van Dort1, Brian J. Ciliax1, David L. Gildersleeve1, Philip S. Sherman1, Karen C. Rosenspire1, Anne B. Young1, Larry Junck, Donald M. Wieland1 - Show less +4 more •Institutions (1)
University of Michigan1
01 Nov1988-Journal of Medicinal Chemistry
[x]
TL;DR: Two isomeric iodinated analogues of the peripheral benzodiazepine binding site (PBS) ligand Ro5-4864 have been synthesized and labeled in high specific activity with iodine-125, indicating high affinity for PBS.
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[x]
Abstract: Two isomeric iodinated analogues of the peripheral benzodiazepine binding site (PBS) ligand Ro5-4864 have been synthesized and labeled in high specific activity with iodine-125. Competitive binding assays conducted with the unlabeled analogues indicate high affinity for PBS. Tissue biodistribution studies in rats with these /sup 125/I-labeled ligands indicate high uptake of radioactivity in the adrenals, heart, and kidney--tissues known to have high concentrations of PBS. Preadministration of the potent PBS antagonist PK 11195 blocked in vivo uptake in adrenal tissue by over 75%, but to a lesser degree in other normal tissues. In vivo binding autoradiography in brain conducted in C6 glioma bearing rats showed dense, PBS-mediated accumulation of radioactivity in the tumor. Ligand 6 labeled with /sup 123/I may have potential for scintigraphic localization of intracranial glioma.
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Journal Article•10.1016/0006-2952(88)90738-1•
##### Purification and characterization of an endogenous protein modulator of radioligand binding to “peripheral-type” benzodiazepine receptors and dihydropyridine ca2+-channel antagonist binding sites
[...]
[x] - [x]
Charles R. Mantione1, Mark E. Goldman1, Brian Martin1, Gordon T. Bolger1, Hartmut W.M. Lueddens1, Steven M. Paul1, Phil Skolnick1 - Show less +3 more •Institutions (1)
National Institutes of Health1
15 Jan1988-Biochemical Pharmacology
[x]
TL;DR: Findings suggest that this endogenous protein may be a PLA2 isoenzyme which may modify both "peripheral-type" benzodiazepine receptors and dihydropyridine Ca2+-channel antagonist binding sites respectively.
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Journal Article•10.1016/0014-2999(88)90563-8•
##### Photoaffinity labeling of peripheral-type benzodiazepine receptors in rat kidney mitochondria with [3H]PK 14105
[x] - [x]
Roman Skowroński1, Darrell D. Fanestil1, Kevin Beaumont1•Institutions (1)
University of California, San Diego1
29 Mar1988-European Journal of Pharmacology
[x]
TL;DR: Results indicate that [3H]PK 14105 identifies the ligand binding domain of the peripheral-type benzodiazepine receptor, which is a peptide with Mr = 18,500, that is of similar size in kidney, heart, brain and adrenals.
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Journal Article•10.1016/0024-3205(88)90294-9•
##### Differential effect of detergents on [3H]Ro 5-4864 and [3H]PK 11195 binding to peripheral-type benzodiazepine-binding sites.
[x] - [x]
M. Awad, Moshe Gavish1•Institutions (1)
Technion – Israel Institute of Technology1
01 Jan1988-Life Sciences
[x]
TL;DR: The results may further support the assumption that Ro 5-4864 and PK 11195 are agonist and antagonist, respectively, of PBS and interact with two different conformations or domains in the peripheral-type benzodiazepine binding site molecule.
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Journal Article•10.1016/0028-3908(88)90155-4•
##### Decreased density of peripheral benzodiazepine binding sites on platelets of currently drinking, but not abstinent alcoholics.
[x] - [x]
Barbara E. Suranyi-Cadotte, F. Lafaille1, M. Dongier1, M. Dumas1, Rémi Quirion1 - Show less +1 more •Institutions (1)
McGill University1
01 Apr1988-Neuropharmacology
[x]
TL;DR: It is suggested that a reduction in the density of peripheral benzodiazepine binding sites on platelets may be a biochemical index of prolonged ethanol use and indicate a possible role for these sites in mediating the chronic effects of alcohol.
...read more read less
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Journal Article•10.1016/0091-3057(88)90210-9•
##### Sleep pharmacology of typical and atypical ligands of benzodiazepine receptors.
[x] - [x]
J.-M. Gaillard, R. Blois
01 Apr1988-Pharmacology, Biochemistry and Behavior
[x]
TL;DR: The results of these experiments indicate a heterogeneity in the mechanism of action of benzodiazepines and non-benzodiazepine ligands of benzidiazepine receptors, because they affect differently the various components of sleep.
...read more read less
[x]
Abstract: The effects of several benzodiazepine and non-benzodiazepine ligands of benzodiazepine receptors have been investigated in sleep of normal young adults. The spectrum of activity of each compound has been characterized using a number of sleep variables in addition to the standard sleep stages. These substances affect all the principal components of sleep, that is the sleep-wake balance, paradoxical sleep, orthodox sleep and the EEG waveforms in the different sleep stages. Some, but not all, modifications induced by flunitrazepam are antagonized by flumazenil and they recover with various time constants after a single administration of the drug. The results of these experiments indicate a heterogeneity in the mechanism of action of benzodiazepine and non-benzodiazepine ligands of benzodiazepine receptors, because they affect differently the various components of sleep. It is not necessary to invoke a heterogeneity of the central benzodiazepine receptors (the BZ1-BZ2 theory) in order to account for these differences, but they can be explained by the concept of spare receptors.
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Journal Article•10.1016/0024-3205(88)90074-4•
##### Selective pharmacological modulation of renal peripheral-type benzodiazepine binding by treatment with diuretic drugs.
[x] - [x]
D.S. Lukeman1, Duke A. Vaughn1, Darrell D. Fanestil1•Institutions (1)
University of California, San Diego1
01 Jan1988-Life Sciences
[x]
TL;DR: It is demonstrated that the PBBS can be selectively "up-regulated" in different regions of the kidney by diuretic drugs with different modes/sites of action.
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Journal Article•10.1007/BF00540960•
##### Reduced affinity of peripheral benzodiazepine binding sites in elderly insomniac patients
[x] - [x]
J. C. Gilbert, D. Valtier, R. Huguet, C. Hulin, J. P. Aquino, Philippe Meyer - Show less +2 more
01 Jan1988-European Journal of Clinical Pharmacology
[x]
TL;DR: There was a twofold reduction in the affinity of these sites in untreated and treated insomniac patients compared to controls, raising the possibility that peripheral-type benzodiazepine sites are involved in abnormal sleep.
...read more read less
[x]
Abstract: Peripheral-type benzodiazepine binding sites on intact platelets from untreated chronic insomniac patients and those chronically treated with benzodiazepine hypnotics were investigated to evaluate their putative involvement in sleep pathology and the influence of treatment. There were 34 elderly subjects in the study, 14 controls (80.7 years) and 20 insomniac patients, of whom 7 were untreated (61.1 years) and 13 were treated (84.4 years). There was an equivalent number of peripheral-type benzodiazepine 3H-PK 11195 binding sites on platelets from untreated (7.61 pmol/mg protein) and treated insomniacs (6.39 pmol/mg protein) and on platelets from the controls (6.21 pmol/mg protein). However, there was a twofold reduction in the affinity of these sites in untreated (Kd=8.02 nM) and treated (Kd=7.40 nM) insomniacs compared to controls (3.79 nM). This difference raises the possibility that peripheral-type benzodiazepine sites are involved in abnormal sleep.
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Journal Article•10.1016/0306-3623(88)90166-8•
##### Effect of PK 11195, an antagonist of benzodiazepine receptors, on platelet aggregation in a model of anxiety in the rat.
[x] - [x]
J.S. Serrano, A. Hevia, A. Fernández-Alonso, J.R. Castillo
01 Jan1988-General Pharmacology-the Vascular System
[x]
TL;DR: Changes induced by anxiety on PAG may be mediated by peripheral-benzodiazepine receptors.
read less
[x]
Abstract: 1. 1. Anxiety induced by forced swimming increases maximal intensity ( h ) of platelet aggregation (PAG) and time to reach it ( t ). 2. 2. PK 11195 pretreatment (12.5 and 25 mg/kg) reverses anxiety-induced PAG changes. At 6.25 mg/kg it inhibits PAG. 3. 3. Changes induced by anxiety on PAG may be mediated by peripheral-benzodiazepine receptors.
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Journal Article•
##### In vivo immunomodulating activity of PK11195, a structurally unrelated ligand for peripheral benzodiazepine binding sites. The possible involvement of central cervous system receptors
[...]
[x] - [x]
M. Lenfant, J. Haumont, P. Horak, L. Sebestova, K. Masek - Show less +1 more
01 Jan1988-International Journal of Immunotherapy
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Algebra 1
Course: Algebra 1>Unit 8
Lesson 9: Intervals where a function is positive, negative, increasing, or decreasing
Increasing, decreasing, positive or negative intervals
Worked example: positive & negative intervals
Positive and negative intervals
Increasing and decreasing intervals
Math>
Algebra 1>
Functions>
Intervals where a function is positive, negative, increasing, or decreasing
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Increasing, decreasing, positive or negative intervals
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About About this video Transcript
Function values can be positive or negative, and they can increase or decrease as the input increases. Here we introduce these basic properties of functions.
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menasir22 10 years ago Posted 10 years ago. Direct link to menasir22's post “At 2:16 the sign is littl...” more At 2:16 the sign is little bit confusing. More explanation. Thanks
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ＫＥＶＩＮ 10 years ago Posted 10 years ago. Direct link to ＫＥＶＩＮ's post “Sal wrote b < x < c. Betw...” more Sal wrote b < x < c. Between the points b and c on the x-axis, but not including those points, the function is negative. Notice, as Sal mentions, that this portion of the graph is below the x-axis. That is your first clue that the function is negative at that spot. Hope this helps.
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josh 10 years ago Posted 10 years ago. Direct link to josh's post “Wouldn't point a - the y ...” more Wouldn't point a - the y line be negative because in the x term it is negative?
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doctorfoxphd 10 years ago Posted 10 years ago. Direct link to doctorfoxphd's post “No, the question is wheth...” more No, the question is whether the function f(x) is positive or negative for this part of the video. That means, according to the vertical axis, or "y" axis, is the value of f(a) positive --is f(x) positive at the point a? In other words, what counts is whether y itself is positive or negative (or zero).
At point a, the function f(x) is equal to zero, which is neither positive nor negative. It makes no difference whether the x value is positive or negative.
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Mothi Ghimire 8 years ago Posted 8 years ago. Direct link to Mothi Ghimire's post “So zero is not a positive...” more So zero is not a positive number?
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Matt Ahlberg 8 years ago Posted 8 years ago. Direct link to Matt Ahlberg's post “Correct. Zero is the div...” more Correct. Zero is the dividing point between positive and negative numbers but it is neither positive or negative.
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Kiki :) 2 years ago Posted 2 years ago. Direct link to Kiki :)'s post “So...How do we know if th...” more So...How do we know if the interval is increasing or decreasing?? I STILL don't get it...Can anyone explain this to me in a simpler and shorter way?
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Kim Seidel 2 years ago Posted 2 years ago. Direct link to Kim Seidel's post “Think in terms of slopes ...” more Think in terms of slopes like with linear equations. When a line has a negative slope, it moves downward as the line moves left to right. If a line has a positive slope, then it moves upwards as the line move left to right.
Now, apply these same ideas to other types of graphs. If the graph is moving downward, then that is a decreasing interval. If the graph is moving upward, then it is a increasing interval.
Hope this helps.
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Robert Zak 2 years ago Posted 2 years ago. Direct link to Robert Zak's post “What's a tangent line?” more What's a tangent line?
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LeafWing 2 years ago Posted 2 years ago. Direct link to LeafWing's post “A tangent line is a line ...” more A tangent line is a line where only one point on the line touches a curve. Hope that helped!
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sbyeon1126 2 years ago Posted 2 years ago. Direct link to sbyeon1126's post “Can somebody pls explain ...” more Can somebody pls explain the difference between the positive and the negative?
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Kim Seidel 2 years ago Posted 2 years ago. Direct link to Kim Seidel's post “A positive interval is th...” more A positive interval is the set of input values where the output value is >0 (so the points sit above the x-axis).
A negative interval is the set of input values where the output value is <0 (so the points sit below the x-axis).
Hope this helps.
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louisaandgreta 4 years ago Posted 4 years ago. Direct link to louisaandgreta's post “1:39 1:58 4:33 Why OR? S...” more 1:39
1:58
4:33
Why OR? Shouldn’t it be AND?
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Kim Seidel 4 years ago Posted 4 years ago. Direct link to Kim Seidel's post “OR means one of the 2 c...” more OR means one of the 2 conditions must apply
AND means both conditions must apply for any value of "x"
For example, in the 1st example in the video, a value of "x" can't both be in the range a<xc. This is why OR is being used.
Hope this helps.
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Veronica Perez 2 years ago Posted 2 years ago. Direct link to Veronica Perez's post “Can a function be increas...” more Can a function be increasing and negative on the same interval?
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Tanner P 2 years ago Posted 2 years ago. Direct link to Tanner P's post “Of course. It happens all...” more Of course. It happens all the time. For example, the function f(x)=x is always increasing (because the slope is positive) and its negative when x<0.
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EnchantressQueen 7 years ago Posted 7 years ago. Direct link to EnchantressQueen's post “What does the variable f ...” more What does the variable f stand for?
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Kim Seidel 7 years ago Posted 7 years ago. Direct link to Kim Seidel's post “If you are referring to t...” more If you are referring to the use of "f" in the video, it tells you the graph represents y = f(x).
This means the variable "y" equals the function called "f" which is defined with the values of "x" as the inputs to the function. The actual function is depicted in the graph.
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pineapple282 2 years ago Posted 2 years ago. Direct link to pineapple282's post “At 0:20 on, Sal refers to...” more At 0:20 on, Sal refers to the point on the graph as a function (f). However, shouldn’t it be f(x) as f is the process it takes to get to f(x)? Can you use f(x) and f interchangeably?
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Video transcript
[Voiceover] What I hope to do in this video is look at this graph y is equal to f of x and think about the intervals where this graph is positive or negative and then think about the intervals when this graph is increasing or decreasing. So first let's just think about when is this function, when is this function positive? Well positive means that the value of the function is greater than zero. It means that the value of the function this means that the function is sitting above the x-axis. So it's sitting above the x-axis in this place right over here that I am highlighting in yellow and it is also sitting above the x-axis over here. And if we wanted to, if we wanted to write those intervals mathematically. Well let's see, let's say that this point, let's say that this point right over here is x equals a. Let's say that this right over here is x equals b and this right over here is x equals c. Then it's positive, it's positive as long as x is between a and b. At x equals a or at x equals b the value of our function is zero but it's positive when x is between a and b, a and b or if x is greater than c. X is, we could write it there, c is less than x or we could write that x is greater than c. These are the intervals when our function is positive. Let me write this, f of x, f of x positive when x is in this interval or this interval or that interval. So when is f of x negative? Let me do this in another color. F of x is going to be negative. Well, it's gonna be negative if x is less than a. So this is if x is less than a or if x is between b and c then we see that f of x is below the x-axis. F of x is down here so this is where it's negative. So here or, or x is between b or c, x is between b and c. And I'm not saying less than or equal to because at b or c the value of the function f of b is zero, f of c is zero. That's where we are actually intersecting the x-axis. So that was reasonably straightforward. Now let's ask ourselves a different question. When is the function increasing or decreasing? So when is f of x, f of x increasing? Well increasing, one way to think about it is every time that x is increasing then y should be increasing or another way to think about it, you have a, you have a positive rate of change of y with respect to x. We could even think about it as imagine if you had a tangent line at any of these points. If you had a tangent line at any of these points the slope of that tangent line is going to be positive. But the easiest way for me to think about it is as you increase x you're going to be increasing y. So where is the function increasing? Well I'm doing it in blue. So it's increasing right until we get to this point right over here, right until we get to that point over there then it starts decreasing until we get to this point right over here and then it starts increasing again. It starts, it starts increasing again. So let me make some more labels here. So let's say that this, this is x equals d and that this right over here, actually let me do that in green color, so let's say this is x equals d. Now it's not a, d, b but you get the picture and let's say that this is x is equal to, x is equal to, let me redo it a little bit, x is equal to e. X is equal to e. So when is this function increasing? Well it's increasing if x is less than d, x is less than d and I'm not gonna say less than or equal to 'cause right at x equals d it looks like just for that moment the slope of the tangent line looks like it would be, it would be constant. We're going from increasing to decreasing so right at d we're neither increasing or decreasing. But then we're also increasing, so if x is less than d or x is greater than e, or x is greater than e. And where is f of x decreasing? So f of x, let me do this in a different color. When is, let me pick a mauve, so f of x decreasing, decreasing well it's going to be right over here. It's gonna be right between d and e. Between x equals d and x equals e but not exactly at those points 'cause at both of those points you're neither increasing nor decreasing but you see right over here as x increases, as you increase your x what's happening to your y? If you go from this point and you increase your x what happened to your y? Your y has decreased. You increase your x, your y has decreased, you increase your x, y has decreased, increase x, y has decreased all the way until this point over here. So f of x is decreasing for x between d and e. So hopefully that gives you a sense of things. Notice, these aren't the same intervals. That we are, the intervals where we're positive or negative don't perfectly coincide with when we are increasing or decreasing. So it's very important to think about these separately even though they kinda sound the same.
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An Important Characteristic of Phase Shifters
An Important Shunt Switch Consideration
Anechoic Chambers
Antenna Design
Antenna measurements
Anti-Virus and Anti-Malware
Antimonide-based compound semiconductors (abcs)
April 2005 Press Release
ARMMS Conference
Artificial transmission lines
Association of Old Crows
Atmospheric Attenuation
Atmospheric Breakdown
Atmospheric Pressure
Attenuator Dissipation
Attenuator Math
Attenuators
Automotive Radar
Automotive radar applications
Averaging example
Avoiding oscillations in microwave packaging
B
BAA
Bad RF connector!
Bagley power divider
Bakelite
Balanced amplifier VSWR
Balanced Amplifiers
Balanced filter
Baluns
Barium strontium titanate (BST)
Barker codes
Basic Concepts
Basic network theory
Basics of computational electromagnetics
Basics of Good EMI/EMC Design
Batteries
BCICTS
BCICTS History
Beam Forming Networks
Beryllium Oxide
Bias Tee
Bias voltage versus supply voltage
Bicononical antenna
Biological Effects of EM Radiation
Bipolar junction transistors (BJTs)
Book: Microwave Engineering by Kaul and Wolff
Books as Free Downloads
Books on Engineering History
Books on Microwave Engineering
Brainwaves and Death (a novel)
Branchline coupler port definition
Branchline Couplers
Breadboard RF Modulator
Bridged Tee Fractional Dissipation
Broadband Four-way and Eight-way Wilkinson Example
BTOP
Butler Matrix
C
Cable Care
Cable Length Rule of Thumb
Calibrating S-parameter Measurements
Capacitor ESR Effects
Capacitor Fabrication
Capacitor Materials
Capacitor Mathematics
Capacitor temperature effects
Capacitor voltage effects
Capacitors
Carbon Nanotubes
Carrottenuator
CASA Center
Cascade Analysis
Cascade Calculator
Cascaded Rat-Race Coupler
Cascaded Splitters and Couplers
Celebrating Black Engineers
Cell Phones
Ceramic Capacitors
Chalcogenide switches
Chalmers University of Technology
Characteristic Impedance
Charge storage capacitor dissipation
Charge Storage Capacitors
Cheapline™
Chicken dots
Chip & Wire Construction
Chromium
Circuit card burn-in
Circular patch antennas
Circular Waveguide
Circularly Polarized Antenna Feed
Circulators
Clemson University
Co-fired Ceramics
Coax
Coax absolute maximum power handling
Coax Cables
Coax Calculator
Coax cutoff frequency
Coax Loss Calculations
Coax Loss due to Dielectric Conduction
Coax Loss due to Loss Tangent
Coax loss versus temperature
Coax Power Handling
Coffee Can Radar
Cold S-parameters
Colorado State University
Comb Generator
Combiner Loss Figure of Merit
Common-Leg Circuits
Compact Wilkinson Example 1: the Webb power divider
Compact Wilkinson Example 2: the Scardelletti power divider
Compact Wilkinson Example 3: the Kang power divider
Compact Wilkinsons
Compression point
Computational Electromagnetics
Computer-Aided Design
Computers
Computing the S-parameters of two cascaded circuits
Conducting Materials
Conductivity
Connector Care
Connector Color Code
Connector Sex
Connector Torque
Controlling Curve Tracer Oscillations
Coplanar Waveguide
Copper
Copper pour on RF PCBs
Corner Reflectors
Corporate Power Dividers
Could your cell phone erase a hotel key?
Coupled Line Coupler Advice
Coupled Line Couplers
Coupled Oscillators
Coupler Fundamentals
Coupler Loss
Couplers and Splitters
Cripps' analysis
Cryogenics in microwaves
Crystal Oscillators
Curve Tracer Example 1
Curve Tracer Example 2
Curve Tracer Example 3
Curve Tracer Measurements
Curve Tracer Modification
Curve tracing without a curve tracer
Cylindrical cavity resonators
Cylindrical phased arrays
D
Danmarks Tekniske Universitet (DTU)
DARPA Microwave Programs
Darwish offset power dividers
DC Block
De-embedding Load-Pull Data
De-embedding S-Parameters
De-embedding using negation elements
Decibels
Deep Space Network
Design of experiments
Designing for High Peak Power
Designing Wilkinsons in Excel
Detectors
Dielectric-Loaded Waveguide
Digital Logic for Microwave Circuits
Digital modulation
Diodes
Diplexers
Direct digital receivers
Directed energy weapons
Directional Couplers
Discrete Devices
Dispersion
Dispersion in microstrip
Distortion 101
Distributed Amplifiers
Dk and Df Extraction Spreadsheet
Doherty Amplifiers
Doppler Radar Teardown
Doppler Shift
Dot Termination
Double-Balanced Mixers
Double-Box Branchlines
Double-Ridged Waveguide
Drill Sizes
Drones
Drop, Droop and Roll-Off
Dual Dielectric Coax Calculator
Dual-Channel Receivers
Dual junction circulators
Duplexers
Dynamic Range
E
eCalibration
Efficiency of Microwave Devices
Eight-Way Wilkinson Example
Eisenhart connector
Eisenhart Section 1 Intro
Eisenhart Section 2 Transmission Lines
Eisenhart Section 3 Planar Circuit Issues
Eisenhart Section 4 Waveguides and Circuits
Eisenhart Section 5 Antenna Element Design
Eisenhart Section 6 Antenna Array Design
Eisenhart Section 7 Extraneous Items
Eisenhart Section 8 Dual Band Common Aperture Array Design
Electro-Static Discharge
Electromagnetic Analysis Software
Electronic article surveillance
Electronic Bandgap Materials
Electronic Decoys
Electronic Warfare
Electronic Warfare and Radar Systems Engineering Handbook
Eli Brookner lectures on phased arrays
EM analysis in MMIC design
EM Analysis Using Sonnet Software
EM Drive
Embedded resistors
Engineering Jokes
Envelope tracking
Epoxy for electronics
Equal ripple transformer example
Equivalent circuit model example 1: gain equalizer
Equivalent circuit model fitter
ESD Protection Circuits
Etch Factor
Even and Odd Mode Impedances
Evolution of the Lange Coupler
Excel S-Parameter Mixer
Experiment
Extrapolating data using Microwave Office
F
Faking SPDT three-port S-parameters
Feedback Amplifiers
Feedthroughs
Ferrite Beads
Ferrite Devices
Ferrite Switches
Ferroelectric Phase Shifters
FET Bias Networks
Fiducials
File Sharing
Filling Factor
Filter Response Types
Filter Schematic Symbols
Filters
Finite Element Analysis
Finite Integration Technique
Finline
Flexible Waveguide
Flip Chip Technology
Fmax and Ft
Fourier transforms
Fourier's Law
FPGAs for a high frequency wizard
FR-4
FraudoCAD
Free book on microwave radiometry
Free EDA software
Free Space Path Loss
Frequency Letter Bands
Frequency Meters
Frequency Multipliers
Frequency selective limiters
Frequency Translators
Frequently Asked Questions
Frey Effect
Full Duplex Transmit/Receive
Fundamentals of EM Waves
Fundamentals of Radio Course
G
Gain
Gain Equalizers
Gallium Arsenide
Gallium Arsenide Semiconductors
Gallium Nitride
Gallium Nitride Semiconductors
Garage MOSFETS
Georgia Tech
Getting organized
Glass Materials
Global Positioning System
Gold
Graceful degradation
Graceful degradation in SSPAs
Grammar and Punctuation
Grating Lobes
Greek letters in microwave engineering
Ground
Ground loops
Ground Penetrating Radar
Group Delay
Group Delay in Filters
Group delay in waveguide
Group Delay Measurements
Group Delay Simulation
Growing Semiconductor Boules
Growing Starting Material
Gunn diode oscillators
Gysel even/odd mode analysis
Gysel N-way Splitter
Gysel Power Splitter
H
H-tree antenna feed
H-tree antenna feed spreadsheet
Hard Substrate Materials
Hardline Balanced Input Technique
Harmonic balance
Hazardous Materials
HBTs
Heat and temperature effects
Helix antennas
Hermeticity
High Permeability Materials
High Temperature Co-fired Ceramics
High-Pass Low-Pass Phase Shifters
Historical Test Equipment
History of microstrip
History of Microwave Diodes
History of Microwave Engineering
History of Microwave Filters
History of Microwave Software
History of MMICs
History of Stripline
History of the Microwave Oven
Homodyne Receivers
Horn Antennas
Hot Vias
Housings
How many sections do you need in a Wilkinson?
How to (not) Trash a Calibration Kit!
How to power up a remote LNA
How to Properly Clean and Care for Connectors
How to size isolation resistors in SSPAs
How We Finally Got it Right (and all the secrets too!)
Hybrid (3 dB) couplers
Hybrid coupler isolation limitation
Hybrid Modules
I
I-Q Mixers
Ideal resistors for broad-band multi-section Wilkinson power dividers
IEEE Journal of Microwaves
IEEE Microwave Theory and Technology Society
IEEE STD 287 Precision RF Connectors
Image Rejection Mixers
IMPATT diodes
IMPATT transmitters
Indian Institutes of Technology
Indium
Indium Phosphide
Indium Phosphide Semiconductors
Induction cooking
Inductor Mathematics
Inductors
Insertion Loss
International Microwave Symposium
International traffic in arms regulations (ITAR)
Internet Connectivity
Interpolating data using Microwave Office
Iridium
Iron
ISIS Diodes
Isolation
Isolation load resistor dissipation in Wilkinson power dividers
Isolators
Isotropy and Anisotropy
J
John Shive's Wave Machines
Josephson junctions - career killer?
Joystick Gamma Amplifier
K
K-Factor Derivation
Katharine Franck Huettner Award
Keffective
Klopfenstein Taper
Know your audience
Kouzoujian Splitter
KU Leuven
L
L-Pads
Laboratory Safety
Laird
Lange Couplers
Laws of Thermodynamics
Layout trick for S-parameter blocks
LDMOS
Lead
Learning to use scikit-rf
LEO Satcom
Light, Phase and Group Velocities
Lim Eom Power Combiner Example #2
Lim-Eom 3-way Power Splitters
Lim-Eom Power Combiner Example #1
Limiters
Line Stretchers
Linear CAD Software
Linear Interpolation for Excel
Linear voltage regulators
Liquid metal switches
Load Pull for Power Devices
Load-modulated balanced amplifier
Load-Pull Effects on Power Combiners
Loaded-Line Phase Shifters
Lockheed Flyby
Log Amplifiers
Long Wavelength Array
Loss or gain?
Low Expansion Alloys
Low Frequency Dispersion in TEM Lines
Low noise amplifiers (LNAs)
Low noise blocks
Low Temperature Co-fired Ceramics
Lumped Element Bias Tee
Lumped Element Filter Calculator
Lumped Element Filters
Lumped element matching calculator
Lumped element Wilkinson power divider example 1
Lumped Element Wilkinson Splitters
Lumped Elements
Lumped-element branchline coupler
Lumped-element two-section branchline coupler
Lumped-element two-section branchline coupler MMIC
M
Magic Tees
Magical Lambda
Magnetic Materials
Magnetron
Manufacturing semi-rigid cables
March hybrid combiner
Marchand balun
Mark 53 VT Fuze
Masers
Massachusetts Institute of Technology
Matching Networks
Materials for Microwave Engineering
Maximally-flat diplexer
Maximally-flat hexaplexer
Maximally-Flat Impedance Transformers
Maximally-flat unequal Wilkinson
Maximum Power Transfer Theorem
Maxwell's Equations
Measuring Characteristic Impedance
Measuring Dielectric Constant
Measuring dielectric constant from group delay
Mechanical Switches
Mechanical Universe from Caltech
Mechanically Adjustable Attenuator
MEMS for microwaves
MEMS Phase Shifters
MEMS Tree of Woe
Men of Science, Santa Barbara News-Press, 1958
Mesa Resistors
Meta-material research at NIT Trichy
Metal-insulator-transition switches
Metamaterials
Metrology
Microstrip
Microstrip "3 dB" Coupler
Microstrip Loss Calculations
Microstrip Loss due to Substrate Conduction
Microstrip Patch Antennas
Microstrip Short Circuit
Microstrip Via Hole Inductance
Microstrip via hole simple model
Microwave Amateur Radio
Microwave Auditory Effects
Microwave Career Killers
Microwave Circuit Card Assemblies
Microwave Coaxial Connectors
Microwave Colleges
Microwave Events Calendar
Microwave FET Tutorial
Microwave Figures of Merit
Microwave Hall of Fame Part I
Microwave Hall of Fame Part III
Microwave Heating
Microwave Imaging
Microwave Impedance Tuners
Microwave Integrated Circuits
Microwave Lab
Microwave Link Budget
Microwave Man
Microwave Market Research
Microwave Measurements
Microwave Medical Applications
Microwave Mortuary
Microwave Nomographs
Microwave ovens
Microwave PCBs - cost versus performance
Microwave Plating 101
Microwave Poetry
Microwave Printed Circuit Boards
Microwave Receivers
Microwave Rules of Thumb
Microwave Semiconductor Processing
Microwave Semiconductor Tradeoffs
Microwave Slang
Microwave Smartphone Apps
Microwave spectroscopy
Microwave Startups
Microwave Switches
Microwave terminations
Microwave Toilets
Microwave Trade Journals
Microwave Tubes
Microwave Hall of Fame Part II
Microwaves101 180 degree phase shifter
Microwaves101 Gift Shop
Microwaves101 Line Stretcher!
Microwaves101 switch FET model
MIL-Specs for Microwaves
Miller effect
Millimeterwave packaging
MIMIC Program
MIMO - an Historical Tutorial
Minimizing Attenuation
Minimum detectable signal
Miscellaneous Dielectric Constants
Mismatch Loss, Etc.
Mismatched termination power handling
Mismatched terminations
Mission Statement
MIT Radiation Laboratory (Rad Lab)
Mitered Bends
Mixed-Mode S-Parameters
Mixer Mess!
Mixer Noise Figure
Mixer Spur Chart
Mixer waveforms
Mixers
MMIC Design
MMIC four bit phase shifter preliminary design
MMIC phase shifter 22 degree bit design
MMIC phase shifter 45 degree bit design
MMIC phase shifter 90 degree bit design
MMIC Phase Shifter Example 1
MMIC Phase Shifter Example 2
MMIC shifter 180 degree bit
MMIC shifter 180 degree bit SPDT switch
MMIC Suppliers
MMIC Switch Design Example 2
MMIC Switch Design Example 3
MMICs
Monopulse Antennas
Monopulse Comparator Networks
Monopulse comparator using branchline couplers
Monopulse comparator using rat-race couplers
MTT IMS 3-Minute Thesis Competition
Multi-bit phase shifter design - how NOT to calculate RMS phase error
Multi-bit phase shifter design using Microwave Office
Multi-Dielectric Coax
Multi-section symmetric couplers
Multi-State Simulations, Part II
Multipaction
Multiplexers
Multistage Wilkinsons
MW101Stuff the Official Microwaves101 Newsletter!
N
N-way Power Splitters
N-way Unequal-Split Wilkinsons
N-way Wilkinson splitters
National Taiwan University
National Electronics Museum
National Institute of Technology, Trichy
Navy Electricity and Electronics Series
Negative resistance devices
Network analyzer calibration standards
Network Analyzer Measurements
New York University Tandon School of Engineering
Nickel
No Link Provided
Noise Conversion
Noise Figure
Noise figure measurement on a VNA!
Noise Figure of Passives
Noise Figure One and Two, Friis and IEEE
Noise Notes
Noise Parameter Equations
Noise Parameter Extraction using Source Pull
Noise Parameters
Noise parameters, a practical example
Noise Temperature
Non-Linear Devices
Non-linear Passive Reciprocal Networks
Nonlinear Transmission Lines
Normalized Determinant Function (NDF)
O
Ode to GREENTAPE
Off-Center Coax
Old School Tools
Optimization
Optimizing circuits using EM analysis
Oscillators
Oscilloscope Measurements
Owen Splitter
Owen Splitter Example #1
P
Packaging
Packaging, assembly, and interconnects
Palladium
Parabolic gain equalizer example 2
Parabolic Gain Equalizers
Parabolic Reflector Antennas
Parallel Plate Waveguide
Parametric amplifiers
Parasitics
Particle Impact Noise Detection
Paschen's Law
Patch Coupler
Percent bandwidth
Periodic Chart
Peripheral Devices
Permalloy
Permeability
Permittivity
PESAs
Phase change switches
Phase delay
Phase delay formula derivation
Phase noise
Phase noise videos
Phase Shifter Multi-State Simulation
Phase Shifter RMS Amplitude Error
Phase Shifter RMS Phase Error
Phase Shifters
Phase-variable mismatched termination
Phased Array Antennas
Phased array tip
Photolithography 101
Physical Constants
Pill terminations
PIN Diodes
PIN Diode Switches
PIN Diode 180 Degree Phase Shifter
Planar RF baluns
Platinum
Pocket Comb Mixer
Pocket Knife Keychain
Pocket Protector
Polarization
Polarization mismatch between antennas
PolyStrata(R) Process
Porcelain
Portmanteaux in Engineering
Power Amp Designer101!
Power Amplifiers
Power combiner loss calculation
Power Combining
Power conditioning circuits
Power density calculation
Power Divider Calculator
Power Handling
Power Handling in Waveguide
Power Meter Measurements
Power splitter or power divider?
Poynting Vector
Precision Connectors
Printed circuit board milling
Problems with VSWR in Medical Applications
Project Connect
Propagation Constant
PTFE
Publish on Microwaves101!
Pulsed versus CW
Pulsed RF Sources
Pulsed RF spectrum spreadsheet
Purdue University
Push-On Connectors
Q
Quadrature (90 degree) property of symmetric coupled lines
Quadrature couplers
Quadrature Reflection Phase Shifters
Quality factor
Quan Attenuator
Quarter-wave Transformers
Quarter-wave Tricks
Quartz
Quasi-true time delay phase shifter
R
Radar Absorbers
Radar Cross-Section Physics
Radar Detector Breakdown
Radar Range Equation
Radar-based Vital Sign Detection
Radar101
Radio Astronomy
Radio direction finding
Radio songs
Radio101
Radiometric Receivers
Radomes
Rat-race couplers
Rat-Race versus Gysel Splitters
RCA
Reactance and Admittance Calculator
Reactance Calculator
Receiver Blocking
Recovering a coaxial cal kit from PFD
Rectangular Waveguide Dimensions
Rectax
Rectax cut-off frequency
Rectennas
Reference Planes
Reflection Attenuators
Reflection Coefficient Sniffer
Reflection Phase Shifters
Reflectionless filters
Reflector Analysis and Accuracy
Reliability
Repairing corrupt S2P files
Resistance Temperature Devices (RTDs)
Resistive Power Splitters
Resistive Taps
Resistor Mathematics
Resistor Trimming
Resistors and Terminations
Resonance of RLC Circuits
Resonant Cavities
Resonant Frequency Calculator
Resonating switch-FET off-capacitance
Resonator Parameters
RETMA Values
RF Ablation of Spinal Nerves
RF Ablation of Varicose Veins
RF cable model
RF Lighting
RF printed wiring board hints
RF Probe Calibration
RF Probing
RF Sheet Resistance
RF sheet resistance in multilayer media
RF test cables need more respect
RF wirebond compensation
RF wirebonds
RF Ablation
RFID
Rhodium
Right angle resistor
Ring Or Crossover?
RMS Phase and Amplitude Errors
RoHS
Rotary Joints
Rotman Lens
Rotodomes
Rubylith
S
S-parameter interpolator
S-parameter Utilities Spreadsheet
S-parameters
Saleh Power Divider
Sample documents
Sapphire
SatCom
SBIR and STTR Programs
Schiffman Phase Shifters
Schiffman phase shifters - a closer look
Schottky Diodes
Scoop-proof connectors
Seam Sealer
Self Resonant Frequency
Semi-rigid coax
Seoul National University
Serial coupled combiner
Sheet Resistance
Short-Slot Waveguide Hybrid
SIGABA
Silicon
Silicon Carbide
Silicon Dioxide
Silicon Semiconductors
Silver
Simultaneous Transmit and Receive (STAR)
Single Sideband Transmission
Single-sideband mixers
Sintered silver paste
Six Sigma
Skin Depth
Skin Depth Calculator
SLCFET
Sliding load
Slotline
Slotted Line Measurements
Slow-Wave Structures
SMA and SSMA Connectors
SMB Connectors
SMC Connectors
Smith Chart Basics
Smith Chart in Excel
Smith Chart Tributes
Smoothing Group Delay Data
Smoothing is Cheating!
SMP Connectors
SMPM Connectors
SNP Format
Soft Substrate Materials
Software Defined Radio (SDR)
Software for Circuit Layout
SOI RF switches
Solder for microwave assemblies
Soldering Basics
Solid state power amplifiers
Solid-state microwave ovens
Solid-State Switches
Solving Equations with Linear Interpolation
Some (not so good) Traces
Songs about Radar
Songs about SatCom
Songs about telecommunications
South Dakota School of Mines and Technology
Special Mixer Mess
Spectral inversion
Spectrum Analyser Mixer Dissection - Page 2
Spectrum Analyser Mixer Dissection - Page 3
Spectrum Analyser Mixer Dissection - Page 4
Spectrum Analyser Mixer Dissection - Page 5
Spectrum analysis
Spectrum Analyzer
Spice Analysis
Split Tee Power Divider
Sponsorship Information
Spur Search
SSMC connectors
SSPAs - a special consideration
Stability factor
Starlink
Stiction
Stripline
Structural Materials
Stuffies of IMS
Sub-Harmonic Mixers
Substrate Integrated Waveguide
SUNY New Paltz
Superconductors
Superheterodyne Receivers
Surface Roughness
Surface roughness effect on propagation delay
Survey Results 2015
Suspended Air Stripline
Suspended Substrate Stripline Couplers (and how we got there from hardline!)
Switch Design
Switch Drivers
Switch FET geometric modeling
Switch FET simple physical model
Switch FETs and FET Switches
Switch Matrices
Switch Power Handling
Switchable Attenuators
Switched filter banks
Switched Filter Phase Shifters
Switched Line Phase Shifters
Switched mode power supplies
T
Tantalum
Tapered Transformers
TDUs or phase shifters?
Technical Writing
Telegrapher's Equations
TEM Wavelength Calculator
Temperature Expansion
Temperature Measurement
Temperature modeling of S-parameters
Temperature variable attenuators
TeraHertz Systems
The History of Electronic Warfare - Part I
The Infrared and Electro-Optical Systems Handbook
The Result - Performance Traces
The Smith Chart - who owns it?
Thermal Analysis
Thermal Conductivity
Thermistors
Thermocouples
Thermoelectric Coolers
Thick-Film Resistors
Thin-Film Networks
Thin-Film Resistors
Three Dimensional Smith Chart
Three-way Branchline Coupler
Three-Way Planar Wilkinsons
Three-way Rat Race Coupler
Time domain analysis using frequency domain data
Time Domain Reflectometry
Time Delay Unit(TDU)
Tin
Tips and Techniques for Microwave Circuit Design
Titanium
TK connectors
Tools for Working Remotely
Toroidal inductors
Torque Wrench
Transfer S-parameters
Transient analysis using EDA software
Transmission Line Loss
Transmission line temperature effects
Transmission line tool
Transmission Lines
Transmission Line Model
Transmit/Receive Modules
Transverse Electro-Magnetic (TEM)
Traveling Wave Divider Secrets Revealed
Traveling Wave Splitter
Traveling wave tubes (TWTs)
Triax
TRL calibration
True Time Delay
Tsai Balun
Tunable filters
Tungsten
Tunnel Diodes
Twelfth-wave Transformer
Twinax
Two-section branchline coupler
U
Unequal Gysel splitters
Unequal split power dividers
Unequal, N-way Gysel power divider
Unequal-split branchline coupler
Unequal-split Wilkinson power divider simplified
Unequal-Split Wilkinsons
Unequal-split Wilkinsons - the rest of the story
Units and Conversion Factors
University of Califonia San Diego
University of Oklahoma
University of Texas El Paso
University of Virginia
Unstable Amplifier Examples
V
V-bonds
Van der Pauw Measurements
Varactor Phase Shifters
Varactors
Variable Attenuators
Variable frequency microwave
Varian Radar Ads-Historical
Vector Modulator
Video Conferencing
Virtual Lobby
Visualizing VSWR
Vivaldi Antenna
Vlad's Lumped Element Filter Designer
VNA Help!
Vocal Acoustics
Voltage regulator modules
Voltage standing wave ratio (VSWR)
W
Waugh Attenuator
Waveguide Cavity Filters
Waveguide Construction
Waveguide Loss
Waveguide Mode Visualization
Waveguide Primer
Waveguide to coax transitions
Waveguide TRL Calibration
Waveguide wave impedance
Waveguide Mathematics
Wavenumber
What makes a good low-noise device?
What's a Neper?
What's that antenna, Daddy (or Mommy?)
What's the frequency?
Where are They Now?
Why Fifty Ohms?
Why there is no way to make a 45 degree hybrid coupler
Wideband 180 Degree Coupler
Wideband receiver harmonic spur suppression
Wilkinson Isolation
Wilkinson Microwave Anisotropy Probe: Mapping the Early Universe
Wilkinson Power Splitters
Wire over ground transmission line
Wire Transmission Lines
Wirebond Impedance and Attenuation
Wirebonding
Wireless
Wireless power transfer
WLAN Standards
Women in Microwaves
Writing process and style
Y
Yagi-Uda antenna
Yield Analysis
YIG Technology
Z
Zener Diodes
Zinc
Zirconium
Reset
About Microwaves101
Advertise with Microwaves101.com!
Frequently Asked Questions
Mission Statement
MW101Stuff the Official Microwaves101 Newsletter!
Pocket Knife Keychain
Sponsorship Information
Survey Results 2015
The Smith Chart - who owns it?
Active Devices
Antimonide-based compound semiconductors (abcs)
Bipolar junction transistors (BJTs)
Carbon Nanotubes
Discrete Devices
Fmax and Ft
Gallium Arsenide Semiconductors
Gallium Nitride Semiconductors
Garage MOSFETS
HBTs
Indium Phosphide Semiconductors
LDMOS
Microwave FET Tutorial
Microwave Semiconductor Tradeoffs
Miller effect
Silicon Semiconductors
Amplifiers
ABCs of Amplifiers
Active Directivity of Amplifiers
Amplifier Classes
Balanced amplifier VSWR
Balanced Amplifiers
Bias Tee
Bias voltage versus supply voltage
Distributed Amplifiers
Doherty Amplifiers
Efficiency of Microwave Devices
Envelope tracking
Feedback Amplifiers
FET Bias Networks
Graceful degradation
Graceful degradation in SSPAs
How to size isolation resistors in SSPAs
Joystick Gamma Amplifier
Load-modulated balanced amplifier
Log Amplifiers
Low noise amplifiers (LNAs)
Low noise blocks
Normalized Determinant Function (NDF)
Parametric amplifiers
Power Amp Designer101!
Power Amplifiers
Power Combining
Power density calculation
Solid state power amplifiers
SSPAs - a special consideration
Unstable Amplifier Examples
Analog Circuits
Active Bias Networks for FETs
Breadboard RF Modulator
Charge Storage Capacitors
Linear voltage regulators
Power conditioning circuits
Switch Drivers
Switched mode power supplies
Voltage regulator modules
Analysis and Theory
A More Exact Coax Attenuation Solution
ABCD Parameters
Characteristic Impedance
Chip & Wire Construction
Coax Loss Calculations
Coax Loss due to Dielectric Conduction
Coax Loss due to Loss Tangent
Computational Electromagnetics
Even and Odd Mode Impedances
Finite Element Analysis
Fourier transforms
Fundamentals of EM Waves
Group Delay Simulation
Gysel even/odd mode analysis
Insertion Loss
K-Factor Derivation
Linear Interpolation for Excel
Maximum Power Transfer Theorem
Minimizing Attenuation
Mismatch Loss, Etc.
Mitered Bends
Phase delay
Poynting Vector
Reflection Coefficient Sniffer
RF Sheet Resistance
RF sheet resistance in multilayer media
Sheet Resistance
Smoothing Group Delay Data
Smoothing is Cheating!
Solving Equations with Linear Interpolation
Stability factor
Surface Roughness
Telegrapher's Equations
Visualizing VSWR
Wavenumber
Why Fifty Ohms?
Wirebond Impedance and Attenuation
Antennas and Feeds
Anechoic Chambers
Antenna Design
Antenna measurements
Beam Forming Networks
Bicononical antenna
Butler Matrix
Circular patch antennas
Circularly Polarized Antenna Feed
H-tree antenna feed
H-tree antenna feed spreadsheet
Horn Antennas
Long Wavelength Array
Microstrip Patch Antennas
Monopulse Antennas
Monopulse comparator using branchline couplers
Monopulse comparator using rat-race couplers
Parabolic Reflector Antennas
Polarization
Polarization mismatch between antennas
Radomes
Reflector Analysis and Accuracy
Rotary Joints
Rotman Lens
Rotodomes
Vivaldi Antenna
What's that antenna, Daddy (or Mommy?)
Yagi-Uda antenna
Applications
Automotive radar applications
Biological Effects of EM Radiation
Drones
Electronic article surveillance
Electronic Decoys
Electronic Warfare
EM Drive
Global Positioning System
Microwave Imaging
Microwave Medical Applications
Microwave spectroscopy
Microwave Toilets
Radio Astronomy
Rectennas
RF Lighting
TeraHertz Systems
Attenuators
3-Bit Attenuator Example
Attenuator Dissipation
Attenuator Math
Attenuators
Carrottenuator
L-Pads
Mechanically Adjustable Attenuator
Quan Attenuator
Reflection Attenuators
Switchable Attenuators
Variable Attenuators
Waugh Attenuator
Basic Concepts
Atmospheric Attenuation
Basic Concepts
Basic network theory
Decibels
Gain
Greek letters in microwave engineering
Ground loops
John Shive's Wave Machines
Loss or gain?
Magical Lambda
Percent bandwidth
Permeability
Permittivity
Phase delay formula derivation
Physical Constants
Quality factor
Reference Planes
Skin Depth
Units and Conversion Factors
Voltage standing wave ratio (VSWR)
What's a Neper?
What's the frequency?
Books and Resources
Book: Microwave Engineering by Kaul and Wolff
Books as Free Downloads
Books on Engineering History
Books on Microwave Engineering
Brainwaves and Death (a novel)
Eisenhart Section 1 Intro
Eisenhart Section 2 Transmission Lines
Eisenhart Section 3 Planar Circuit Issues
Eisenhart Section 4 Waveguides and Circuits
Eisenhart Section 5 Antenna Element Design
Eisenhart Section 6 Antenna Array Design
Eisenhart Section 7 Extraneous Items
Eisenhart Section 8 Dual Band Common Aperture Array Design
Electronic Warfare and Radar Systems Engineering Handbook
Free book on microwave radiometry
IEEE Journal of Microwaves
Microwave Amateur Radio
Microwave Trade Journals
Navy Electricity and Electronics Series
Project Connect
Tips and Techniques for Microwave Circuit Design
Business Development
BTOP
DARPA Microwave Programs
Microwave Market Research
SBIR and STTR Programs
Cables
Cable Care
Cable Length Rule of Thumb
Coax Cables
Manufacturing semi-rigid cables
RF cable model
RF test cables need more respect
Semi-rigid coax
Calculators
Cascade Calculator
Coax Calculator
Computing the S-parameters of two cascaded circuits
Dual Dielectric Coax Calculator
Lumped Element Filter Calculator
Microwave Link Budget
Power Divider Calculator
Reactance and Admittance Calculator
Reactance Calculator
Resonant Frequency Calculator
Skin Depth Calculator
TEM Wavelength Calculator
Career
Active Denial - Career Killer?
Josephson junctions - career killer?
Microwave Career Killers
Microwave Startups
Circuit card assemblies
Etch Factor
Layout trick for S-parameter blocks
Microwave PCBs - cost versus performance
Microwave Printed Circuit Boards
Printed circuit board milling
RF printed wiring board hints
Circulators/Isolators
Dual junction circulators
Connectors
A short history of microwave connectors
Bad RF connector!
Connector Care
Connector Color Code
Connector Sex
Connector Torque
Eisenhart connector
How to Properly Clean and Care for Connectors
IEEE STD 287 Precision RF Connectors
Microwave Coaxial Connectors
Precision Connectors
Push-On Connectors
Scoop-proof connectors
SMA and SSMA Connectors
SMB Connectors
SMC Connectors
SMP Connectors
SMPM Connectors
SNP Format
SSMC connectors
TK connectors
Torque Wrench
Couplers
Branchline coupler port definition
Branchline Couplers
Cascaded Rat-Race Coupler
Cascaded Splitters and Couplers
Coupled Line Coupler Advice
Coupled Line Couplers
Coupler Fundamentals
Coupler Loss
Couplers and Splitters
Directional Couplers
Double-Box Branchlines
Evolution of the Lange Coupler
Hybrid (3 dB) couplers
Hybrid coupler isolation limitation
Lange Couplers
March hybrid combiner
Microstrip "3 dB" Coupler
Multi-section symmetric couplers
Patch Coupler
Power combiner loss calculation
Quadrature (90 degree) property of symmetric coupled lines
Quadrature couplers
Rat-race couplers
Three-way Branchline Coupler
Traveling Wave Divider Secrets Revealed
Why there is no way to make a 45 degree hybrid coupler
Wideband 180 Degree Coupler
Detectors
Detectors
Radar Detector Breakdown
Dividers/Combiners
Bagley power divider
Baluns
Broadband Four-way and Eight-way Wilkinson Example
Combiner Loss Figure of Merit
Compact Wilkinson Example 1: the Webb power divider
Compact Wilkinson Example 2: the Scardelletti power divider
Compact Wilkinson Example 3: the Kang power divider
Compact Wilkinsons
Corporate Power Dividers
Darwish offset power dividers
Designing Wilkinsons in Excel
Eight-Way Wilkinson Example
Gysel N-way Splitter
Gysel Power Splitter
How many sections do you need in a Wilkinson?
Ideal resistors for broad-band multi-section Wilkinson power dividers
Isolation load resistor dissipation in Wilkinson power dividers
Kouzoujian Splitter
Lim-Eom Power Combiner Example #1
Load-Pull Effects on Power Combiners
Lumped-element branchline coupler
Lumped-element two-section branchline coupler
Lumped-element two-section branchline coupler MMIC
Marchand balun
Maximally-flat unequal Wilkinson
Multistage Wilkinsons
N-way Power Splitters
N-way Unequal-Split Wilkinsons
N-way Wilkinson splitters
Planar RF baluns
Power splitter or power divider?
Rat-Race versus Gysel Splitters
Resistive Power Splitters
Saleh Power Divider
Serial coupled combiner
Split Tee Power Divider
Three-Way Planar Wilkinsons
Three-way Rat Race Coupler
Traveling Wave Splitter
Tsai Balun
Two-section branchline coupler
Unequal Gysel splitters
Unequal split power dividers
Unequal, N-way Gysel power divider
Unequal-split branchline coupler
Unequal-split Wilkinson power divider simplified
Unequal-Split Wilkinsons
Unequal-split Wilkinsons - the rest of the story
Wilkinson Isolation
Wilkinson Power Splitters
Electromagnetic EDA Analysis
Basics of computational electromagnetics
EM analysis in MMIC design
EM Analysis Using Sonnet Software
Optimizing circuits using EM analysis
Right angle resistor
Electronically Steerable Antennas
AESAs
Common-Leg Circuits
Cylindrical phased arrays
Grating Lobes
PESAs
Phased Array Antennas
Phased array tip
TDUs or phase shifters?
Transmit/Receive Modules
Electro-Optical Systems
The Infrared and Electro-Optical Systems Handbook
Ferrite Devices
Circulators
Duplexers
Ferrite Beads
Ferrite Devices
Isolation
Isolators
Permalloy
Filters
Balanced filter
Cylindrical cavity resonators
DC Block
Diplexers
Drop, Droop and Roll-Off
Filter Response Types
Filter Schematic Symbols
Filters
Group Delay
Group Delay in Filters
Lumped Element Filters
Maximally-flat diplexer
Maximally-flat hexaplexer
Multiplexers
Reflectionless filters
Resonant Cavities
Resonator Parameters
Switched filter banks
Tunable filters
Vlad's Lumped Element Filter Designer
Waveguide Cavity Filters
YIG Technology
Fun Stuff
A GaAs Poem
Akin's Laws of Spacecraft Design
April 2005 Press Release
Could your cell phone erase a hotel key?
Engineering Jokes
Lockheed Flyby
MEMS Tree of Woe
Microwave Man
Microwave Mortuary
Microwave Poetry
Microwave Slang
Microwaves101 Gift Shop
Ode to GREENTAPE
Portmanteaux in Engineering
Radio songs
Smith Chart Tributes
Songs about Radar
Songs about SatCom
Songs about telecommunications
Stuffies of IMS
Heat and Temperature
Amplifier temperature compensation example
Capacitor temperature effects
Coax loss versus temperature
Cryogenics in microwaves
Directed energy weapons
Finite Integration Technique
Fourier's Law
Heat and temperature effects
Induction cooking
Laws of Thermodynamics
Microwave Auditory Effects
Microwave ovens
Resistance Temperature Devices (RTDs)
Solid-state microwave ovens
Temperature Expansion
Temperature Measurement
Temperature modeling of S-parameters
Temperature variable attenuators
Thermal Analysis
Thermal Conductivity
Thermistors
Thermocouples
Thermoelectric Coolers
Transmission line temperature effects
Variable frequency microwave
History
A Rough Justice
BAA
Celebrating Black Engineers
Historical Test Equipment
History of Microwave Diodes
History of Microwave Engineering
History of Microwave Filters
History of Microwave Software
History of MMICs
History of Stripline
History of the Microwave Oven
Mark 53 VT Fuze
Maxwell's Equations
Mechanical Universe from Caltech
Men of Science, Santa Barbara News-Press, 1958
Microwave Hall of Fame Part I
Microwave Hall of Fame Part III
Microwave Nomographs
Microwave Hall of Fame Part II
MIMO - an Historical Tutorial
MIT Radiation Laboratory (Rad Lab)
National Electronics Museum
Old School Tools
Pocket Protector
RCA
Rubylith
SIGABA
The History of Electronic Warfare - Part I
Varian Radar Ads-Historical
Where are They Now?
Wilkinson Microwave Anisotropy Probe: Mapping the Early Universe
Linear EDA Analysis
ADS Example 1. Two-state devices
Multi-State Simulations, Part II
Optimization
Repairing corrupt S2P files
Transient analysis using EDA software
Linear Network Parameters
Mixed-Mode S-Parameters
S-parameter Utilities Spreadsheet
S-parameters
Transfer S-parameters
Lumped Elements
Adams' resistive splitter
Capacitor ESR Effects
Capacitor Fabrication
Capacitor Mathematics
Capacitor voltage effects
Capacitors
Ceramic Capacitors
Charge storage capacitor dissipation
Embedded resistors
Inductor Mathematics
Inductors
Lumped Element Bias Tee
Lumped element matching calculator
Lumped element Wilkinson power divider example 1
Lumped Element Wilkinson Splitters
Lumped Elements
Mesa Resistors
Parasitics
Resistor Mathematics
Resonance of RLC Circuits
RETMA Values
Self Resonant Frequency
Toroidal inductors
Manufacturing
Additive manufacturing
Design of experiments
Fiducials
Microwave Semiconductor Processing
Photolithography 101
Resistor Trimming
Six Sigma
Soldering Basics
Thick-Film Resistors
Thin-Film Networks
Thin-Film Resistors
Van der Pauw Measurements
Yield Analysis
Material Properties
Absorbing Materials
Air
Alumina 96%
Alumina 99.5%
Aluminum
Aluminum Nitride
Bakelite
Barium strontium titanate (BST)
Beryllium Oxide
Capacitor Materials
Chromium
Co-fired Ceramics
Conducting Materials
Conductivity
Copper
FR-4
Gallium Arsenide
Gallium Nitride
Glass Materials
Gold
Growing Semiconductor Boules
Growing Starting Material
Hard Substrate Materials
Hazardous Materials
High Permeability Materials
High Temperature Co-fired Ceramics
Indium
Indium Phosphide
Iridium
Iron
Isotropy and Anisotropy
Lead
Low Expansion Alloys
Magnetic Materials
Materials for Microwave Engineering
Metamaterials
Miscellaneous Dielectric Constants
Nickel
Palladium
Periodic Chart
Platinum
Porcelain
PTFE
Quartz
Rhodium
Sapphire
Silicon
Silicon Carbide
Silicon Dioxide
Silver
Soft Substrate Materials
Solder for microwave assemblies
Structural Materials
Superconductors
Tantalum
Tin
Titanium
Tungsten
Zinc
Zirconium
Measurement and Characterization
3D load-pull plot example
A Prescription For THz Transistor Characterization
Airline calibration standards
Averaging example
Calibrating S-parameter Measurements
Cold S-parameters
Corner Reflectors
De-embedding Load-Pull Data
De-embedding S-Parameters
De-embedding using negation elements
Extrapolating data using Microwave Office
Group Delay Measurements
How to (not) Trash a Calibration Kit!
Interpolating data using Microwave Office
Laboratory Safety
Measuring Characteristic Impedance
Measuring Dielectric Constant
Measuring dielectric constant from group delay
Metrology
Network analyzer calibration standards
Network Analyzer Measurements
Noise Parameter Equations
Noise Parameter Extraction using Source Pull
Noise parameters, a practical example
Oscilloscope Measurements
Power Meter Measurements
Recovering a coaxial cal kit from PFD
RF Probing
Slotted Line Measurements
Time domain analysis using frequency domain data
Time Domain Reflectometry
TRL calibration
VNA Help!
Medical Applications
Problems with VSWR in Medical Applications
RF Ablation of Spinal Nerves
RF Ablation of Varicose Veins
RF Ablation
Microwave Tricks
Electronic Bandgap Materials
Equal ripple transformer example
Gain Equalizers
Parabolic gain equalizer example 2
Parabolic Gain Equalizers
Quarter-wave Transformers
Quarter-wave Tricks
Slow-Wave Structures
Military
International traffic in arms regulations (ITAR)
MIL-Specs for Microwaves
Mixers
Double-Balanced Mixers
Excel S-Parameter Mixer
Hardline Balanced Input Technique
How We Finally Got it Right (and all the secrets too!)
I-Q Mixers
Image Rejection Mixers
Mixer Mess!
Mixer Noise Figure
Mixer Spur Chart
Mixer waveforms
Mixers
Pocket Comb Mixer
Ring Or Crossover?
Single-sideband mixers
Some (not so good) Traces
Special Mixer Mess
Spectral inversion
Spectrum Analyser Mixer Dissection - Page 2
Spectrum Analyser Mixer Dissection - Page 3
Spectrum Analyser Mixer Dissection - Page 4
Spectrum Analyser Mixer Dissection - Page 5
Spur Search
Sub-Harmonic Mixers
Suspended Substrate Stripline Couplers (and how we got there from hardline!)
The Result - Performance Traces
Vector Modulator
MMICs
Air bridge inductance
MMIC Design
MMIC Suppliers
MMICs
RFID
Modeling
Equivalent circuit model example 1: gain equalizer
Equivalent circuit model fitter
Microstrip via hole simple model
S-parameter interpolator
Switch FET simple physical model
Nonlinear Devices
Compression point
Cripps' analysis
Diodes
Distortion 101
Frequency Multipliers
Harmonic balance
IMPATT diodes
Limiters
Load Pull for Power Devices
Negative resistance devices
Non-Linear Devices
Non-linear Passive Reciprocal Networks
PIN Diodes
Schottky Diodes
Tunnel Diodes
Varactors
Zener Diodes
Oscillators and Sources
Coupled Oscillators
Crystal Oscillators
Gunn diode oscillators
IMPATT transmitters
ISIS Diodes
Masers
Oscillators
Phase noise
Phase noise videos
Pulsed RF Sources
Packaging
American Wire Gauge
Atmospheric Pressure
Avoiding oscillations in microwave packaging
Basics of Good EMI/EMC Design
Batteries
Chicken dots
Drill Sizes
Electro-Static Discharge
Epoxy for electronics
ESD Protection Circuits
Feedthroughs
Flip Chip Technology
Hermeticity
Hot Vias
Housings
Hybrid Modules
Laird
Low Temperature Co-fired Ceramics
Microwave Circuit Card Assemblies
Microwave Integrated Circuits
Microwave Plating 101
Millimeterwave packaging
Packaging
Packaging, assembly, and interconnects
Particle Impact Noise Detection
RF wirebond compensation
RF wirebonds
RoHS
Seam Sealer
Sintered silver paste
V-bonds
Wirebonding
Phase Shifters/Time Delay Units
180 Degree Hybrid Phase Shifters
An Important Characteristic of Phase Shifters
Ferroelectric Phase Shifters
Frequency Translators
High-Pass Low-Pass Phase Shifters
Line Stretchers
Loaded-Line Phase Shifters
MEMS Phase Shifters
Microwaves101 180 degree phase shifter
Microwaves101 Line Stretcher!
MMIC four bit phase shifter preliminary design
MMIC phase shifter 22 degree bit design
MMIC phase shifter 45 degree bit design
MMIC phase shifter 90 degree bit design
MMIC Phase Shifter Example 1
MMIC Phase Shifter Example 2
MMIC shifter 180 degree bit
MMIC shifter 180 degree bit SPDT switch
Multi-bit phase shifter design - how NOT to calculate RMS phase error
Multi-bit phase shifter design using Microwave Office
Phase Shifter Multi-State Simulation
Phase Shifter RMS Amplitude Error
Phase Shifter RMS Phase Error
Phase Shifters
PIN Diode 180 Degree Phase Shifter
Quadrature Reflection Phase Shifters
Quasi-true time delay phase shifter
Reflection Phase Shifters
RMS Phase and Amplitude Errors
Schiffman Phase Shifters
Schiffman phase shifters - a closer look
Switched Filter Phase Shifters
Switched Line Phase Shifters
Time Delay Unit(TDU)
True Time Delay
Varactor Phase Shifters
Power Handling
Atmospheric Breakdown
Coax absolute maximum power handling
Coax Power Handling
Designing for High Peak Power
Multipaction
Paschen's Law
Power Handling
Power Handling in Waveguide
Wireless power transfer
Practical Advice
Faking SPDT three-port S-parameters
Ground
Microwave Figures of Merit
Microwave Rules of Thumb
Printed Circuit Boards
Copper pour on RF PCBs
Professional Societies
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Association of Old Crows
BCICTS
BCICTS History
IEEE Microwave Theory and Technology Society
International Microwave Symposium
Katharine Franck Huettner Award
Microwave Events Calendar
Publish on Microwaves101!
Women in Microwaves
Radar
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Barker codes
CASA Center
Coffee Can Radar
Doppler Radar Teardown
Doppler Shift
Eli Brookner lectures on phased arrays
Ground Penetrating Radar
Radar Absorbers
Radar Cross-Section Physics
Radar Range Equation
Radar-based Vital Sign Detection
Radar101
Radio and Communications
5G
Cell Phones
Deep Space Network
Digital modulation
Free Space Path Loss
Full Duplex Transmit/Receive
Fundamentals of Radio Course
How to power up a remote LNA
LEO Satcom
Radio direction finding
Radio101
SatCom
Simultaneous Transmit and Receive (STAR)
Single Sideband Transmission
Software Defined Radio (SDR)
Starlink
Wireless
WLAN Standards
Receivers
A Note on Noise
Direct digital receivers
Dual-Channel Receivers
Dynamic Range
Homodyne Receivers
Microwave Receivers
Minimum detectable signal
Noise Conversion
Noise Figure
Noise Figure of Passives
Noise Figure One and Two, Friis and IEEE
Noise Notes
Noise Parameters
Noise Temperature
Radiometric Receivers
Receiver Blocking
Superheterodyne Receivers
What makes a good low-noise device?
Wideband receiver harmonic spur suppression
Reliability
Circuit card burn-in
Reliability
Smith Chart
Smith Chart Basics
Smith Chart in Excel
Three Dimensional Smith Chart
Software
Computer-Aided Design
Electromagnetic Analysis Software
FraudoCAD
Free EDA software
Linear CAD Software
Microwave Smartphone Apps
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Spectrum analysis
Pulsed RF spectrum spreadsheet
Spectrum analysis
Splitters
Lim Eom Power Combiner Example #2
Lim-Eom 3-way Power Splitters
Owen Splitter
Owen Splitter Example #1
Resistive Taps
Switches
An Important Shunt Switch Consideration
Chalcogenide switches
Ferrite Switches
Liquid metal switches
Mechanical Switches
MEMS for microwaves
Metal-insulator-transition switches
Microwave Switches
Microwaves101 switch FET model
MMIC Switch Design Example 3
Phase change switches
PIN Diode Switches
Resonating switch-FET off-capacitance
SLCFET
SOI RF switches
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Stiction
Switch Design
Switch FET geometric modeling
Switch FETs and FET Switches
Switch Matrices
Switch Power Handling
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Cascade Analysis
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Technical Writing and Presentations
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Know your audience
Sample documents
Technical Writing
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Terminations
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Microstrip Short Circuit
Microwave terminations
Mismatched termination power handling
Mismatched terminations
Phase-variable mismatched termination
Pill terminations
Test Equipment
Controlling Curve Tracer Oscillations
Curve Tracer Example 1
Curve Tracer Example 2
Curve Tracer Example 3
Curve Tracer Measurements
Curve Tracer Modification
Curve tracing without a curve tracer
eCalibration
Frequency Meters
Microwave Impedance Tuners
Microwave Lab
Microwave Measurements
Noise figure measurement on a VNA!
RF Probe Calibration
Sliding load
Spectrum Analyzer
Transformers and Matching Networks
Matching Networks
Maximally-Flat Impedance Transformers
Tapered Transformers
Twelfth-wave Transformer
Transmission Lines
A more accurate Tline model
Artificial transmission lines
Cheapline™
Coax
Coax cutoff frequency
Dispersion
Dispersion in microstrip
Dk and Df Extraction Spreadsheet
Filling Factor
Finline
History of microstrip
Keffective
Klopfenstein Taper
Light, Phase and Group Velocities
Low Frequency Dispersion in TEM Lines
Microstrip
Microstrip Loss Calculations
Microstrip Loss due to Substrate Conduction
Multi-Dielectric Coax
Nonlinear Transmission Lines
Off-Center Coax
PolyStrata(R) Process
Propagation Constant
Rectax
Rectax cut-off frequency
Slotline
Stripline
Surface roughness effect on propagation delay
Suspended Air Stripline
Transmission Line Loss
Transmission line tool
Transmission Lines
Transmission Line Model
Transverse Electro-Magnetic (TEM)
Triax
Twinax
Wire over ground transmission line
Wire Transmission Lines
Universities
Chalmers University of Technology
Clemson University
Colorado State University
Danmarks Tekniske Universitet (DTU)
Georgia Tech
Indian Institutes of Technology
KU Leuven
Massachusetts Institute of Technology
Meta-material research at NIT Trichy
Microwave Colleges
MTT IMS 3-Minute Thesis Competition
National Taiwan University
National Institute of Technology, Trichy
New York University Tandon School of Engineering
Purdue University
Seoul National University
SUNY New Paltz
University of Califonia San Diego
University of Oklahoma
University of Texas El Paso
University of Virginia
Vacuum Electronics
Magnetron
Microwave Tubes
Traveling wave tubes (TWTs)
Waveguides
Circular Waveguide
Coplanar Waveguide
Dielectric-Loaded Waveguide
Double-Ridged Waveguide
Flexible Waveguide
Frequency Letter Bands
Group delay in waveguide
Magic Tees
Parallel Plate Waveguide
Rectangular Waveguide Dimensions
Short-Slot Waveguide Hybrid
Substrate Integrated Waveguide
Waveguide Construction
Waveguide Loss
Waveguide Mode Visualization
Waveguide Primer
Waveguide to coax transitions
Waveguide TRL Calibration
Waveguide wave impedance
Waveguide Mathematics
IEEE Microwave Theory and Technology Society
Click here to go to our microwave events page
Who is the IEEE? From their web site... The IEEE name was originally an acronym for the Institute of Electrical and Electronics Engineers, Inc. Today, the organization's scope of interest has expanded into so many related fields, that it is simply referred to by the letters I-E-E-E (pronounced Eye-triple-E).
For microwave engineers, you won't find a better source of information than the IEEE. However, some of the articles and presentations can be quite tedious as authors compete to show how smart they are. So there will always be a place for web sites such as Microwaves101 to dumb down the subject, spice it up and make it less painful than stepping on a rusty nail...
IEEE Explore and copyright restrictions on IEEE papers
The IEEE has a great web site where most of their articles are archived, it's called IEEExplore. You (or your company) need to join the IEEE before you can use it. Click here to go to IEEE Explore.
The terms of use are very specific, you should read them. Many engineers think it's OK to download articles to their hard drives, then email them to each other or put them in electronic collections on a server. Guess what? You are not supposed to do that! Microwaves101 tries to respect copywrited material, so you won't see us offering any of their articles as downloads. But we might quote them from time to time....
One exception to this is theJournal of Microwaves, which is published "open access". That means you don't need to pay to read it, and you don't need to be an IEEE member.
IEEE MTT-S
The Microwave Theory and Technology Society (MTT-S) is a subgroup within the Institute of Electrical and Electronics Engineers (IEEE). Although we might target the IEEE with so-called humor here at Microwaves101, we have to give them respect for holding the world's repository of all electronics knowledge, including all things microwave. The collection of papers on the IEEE site go back at least to 1960. So why not just go to IEEE instead of Microwaves101? Two reasons: first, all of the IEEE papers are written so that you need a solid background in microwave theory to understand them. Some of the papers will give you not much more than a headache. And second, their collection is only accessible if you are a member (which costs money!) Sometimes your employer will pay so that you will have access to this great collection.
The main IEEE MTT-S web site can be accessed by clicking here. Here is the purpose of this organization (we copied it from the MTT-S web site, hope they don't mind):
"The IEEE Microwave Theory and Technology Society (MTT-S) is a transnational society with more than 9,000 members and 80 chapters worldwide. Our society promotes the advancement of microwave theory and its applications, usually at frequencies from 200 MHz to 1 THz and beyond."
If you live in a big U.S. city, chances are there is a local IEEE chapter that organizes seminars and short courses from time to time. U.S. chapters include Los Angeles, San Fernando Valley, Santa Clara Valley, San Diego, Denver, Washington DC/Northern VA, Springfield, MA, Melbourne, Florida, Long Island, NY. There are local chapters in many other countries, comrades!
The IEEE Long Island section of MTT has some great lecture notes that you can access for free on their web site Some especially good notes on noise figure are available for example.
MTT Webinars
If you're an MTT-S member, you've probably seen the emails promoting their webinar series. Good news for everyone: MTT-S now has the webinar library on their site! They've got videos on all levels, from intro to in-depth, and they even have a handy filter at the top of the page that helps you find the exact video you need. Bonus: they'll be adding content regularly, so there will always be something new to learn. These are free to watchi if you're an MTT member, with small charges if you're not.
The IMS Symposium
The "big show" of microwaves is put on by MTT-S every year, it's called the International Microwave Symposium (IMS). Typically 20,000 engineers from all over the world attend for up to a week. As many as 600 microwave vendors advertise their wares in the exhibition hall, and there is a student paper contest, great for recruiting smart college kids and scoring free drinks. For the latest on IMS and other microwave-related events, check outour calendar!
See you there!
IEEE MTT Scholarships and Graduate Fellowships
In their own words...
The purpose of the IEEE Microwave Theory & Technology (MTT) graduate fellowship is to recognize and provide financial assistance to graduate students who show promise and interest in pursuing a graduate degree in microwave engineering.
Up to six $6000 awards may be granted each year. The awards are presented at the International Microwave Symposium (IMS). Limited travel support is available to enable the winners to attend the IMS.
Update!! MTT now offers up to a dozen fellowships, and several undergraduation scholarships. For more information please visit:
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5946 | https://youglish.com/pronounce/excoriation/english/us | Excoriation | 25 pronunciations of Excoriation in American English
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How to pronounce excoriation in American English (1 out of 25):
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and how much that notion of preemptive self-excoriation
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Having trouble pronouncing 'excoriation' ? Learn how to pronounce one of the nearby words below:
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Phonetic:
When you begin to speak English, it's essential to get used to the common sounds of the language, and the best way to do this is to check out the phonetics. Below is the UK transcription for 'excoriation':
Modern IPA: ɪksgóːrɪjɛ́jʃən
Traditional IPA: ɪkˌskɔːriːˈeɪʃən
5 syllables: "ik" + "SKAW" + "ree" + "AY" + "shuhn"
Test your pronunciation on words that have sound similarities with 'excoriation':
excavation
expatriation
exportation
excretion
excoriated
exclamation
exhortation
exploration
expropriation
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Tips to improve your English pronunciation:
Here are a few tips that should help you perfect your pronunciation of 'excoriation':
Sound it Out: Break down the word 'excoriation' into its individual sounds "ik" + "skaw" + "ree" + "ay" + "shuhn". Say these sounds out loud, exaggerating them at first. Practice until you can consistently produce them clearly.
Self-Record & Review: Record yourself saying 'excoriation' in sentences. Listen back to identify areas for improvement.
YouTube Pronunciation Guides: Search YouTube for how to pronounce 'excoriation' in English.
Pick Your Accent: Mixing multiple accents can be confusing, so pick one accent (US or UK) and stick to it for smoother learning.
Here are a few tips to level up your english pronunciation:
Mimic the Experts: Immerse yourself in English by listening to audiobooks, podcasts, or movies with subtitles. Try shadowing—listen to a short sentence and repeat it immediately, mimicking the intonation and pronunciation.
Become Your Own Pronunciation Coach: Record yourself speaking English and listen back. Identify areas for improvement, focusing on clarity, word stress, and intonation.
Train Your Ear with Minimal Pairs: Practice minimal pairs (words that differ by only one sound, like ship vs. sheep) to improve your ability to distinguish between similar sounds.
Explore Online Resources: Websites & apps offer targeted pronunciation exercises. Explore YouTube channels dedicated to pronunciation, like Rachel's English and English with James for additional pronunciation practice and learning.
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5947 | https://www.coursera.org/specializations/chip-based-vlsi-design-for-industrial-applications | Chip based VLSI design for Industrial Applications Specialization
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Chip based VLSI design for Industrial Applications Specialization
Master FPGA Design for Industrial Application. Master FPGA architecture, VHDL programming, and IoT integration for industrial applications, unlocking career opportunities in digital design
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Explore digital design basics, Boolean algebra, logic gates, circuits, memory types, and PLDs, gearing you for diverse VLSI challenges.
Master CMOS VLSI design, analog/digital circuits and their implementation with Electric VLSI EDA Tool & LTspice, ensuring industry proficiency.
Master FPGA architecture, sensor interfacing, digital protocols and real-time applications with Xilinx Vivado for industrial automation & IoT roles.
Skills you'll gain
Programmable Logic Controllers
Serial Peripheral Interface
Electronics Engineering
Field-Programmable Gate Array (FPGA)
Hardware Design
Electronic Components
Application Specific Integrated Circuits
Electronics
Embedded Systems
Schematic Diagrams
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Specialization - 4 course series
Embark on a transformative exploration into the dynamic field of Very Large-Scale Integration (VLSI) Design. Unravel the intricacies of semiconductor technology and chip design, delving into the multifaceted world of VLSI with real-time facets of designing integrated circuits. Our comprehensive course structure covers essential topics such as digital design fundamentals including Boolean algebra and logic gates, combinational circuits and arithmetic logic for binary operations, sequential circuits and state machines for designing complex systems, memory and programmable logic for advanced functionalities, VLSI chip design and simulation using Electric VLSI EDA Tool with a focus on CMOS technology and IC design principles, VHDL programming using Xilinx ISE for digital circuit design and analysis, and FPGA architecture for industrial applications using Vivado with hands-on experiences in designing digital logic circuits, interfacing sensors and communication protocols (RS232, SPI, and I2C) and implementing IoT solutions. This ensures a holistic understanding and practical skills in VLSI, chip design, VHDL programming, and FPGA-based system design for industrial innovations.
Applied Learning Project
Through comprehensive training, learners will develop proficiency in VLSI chip design, VHDL programming, FPGA architecture, and industrial automation. This knowledge equips them for diverse roles in semiconductor design and FPGA-based applications across various industries. They will possess the skills needed to tackle challenges in digital system design, embedded systems, and IoT integration, enabling them to contribute effectively to technological advancements and innovation in the field.
Fundamentals of Digital Design for VLSI Chip Design
Course 1
•
17 hours
What you'll learn
This comprehensive learning module delves into Boolean algebra and its applications in digital circuit design, covering fundamental concepts like Boolean variables, logic gates, and their relationship with digital logic circuits. Participants explore Boolean expressions, simplification techniques, and consensus theorems, including the advanced Quine McCluskey method.
The module also addresses combinational circuits, detailing the design and functionality of adders, subtractors, parity circuits, and multipliers. Encoding complexities are navigated with insights into encoders, decoders, multiplexers, and demultiplexers. Binary shifting operations, emphasizing logical and arithmetic shifting with multiplexers for efficient design, are covered.
Moving forward, the module provides an in-depth exploration of sequential circuits, including latch and flip-flop circuits like SR latch, JK flip-flop, and more. Hazards in digital circuits, along with registers, bidirectional shift registers, and various counters, are thoroughly explained. The exploration concludes with Mealy and Moore state sequential circuits.
Additionally, participants gain a comprehensive understanding of memory systems, programmable logic devices, and VLSI physical design considerations. The module covers SRAM and DRAM, tri-state digital buffers, Read-Only Memory (ROM), and Programmable Logic Devices (PLD) such as PROM, PLA, and PAL. Architecture and implementation of Complex Programmable Logic Devices (CPLD) and Field-Programmable Gate Arrays (FPGA) are discussed, along with the VLSI design cycle and design styles for CPLD, SPLD, and FPGA.
By the end of this course, you will be able to:
Understand the distinctions between analog and digital signals and the transformative benefits of digitization.
Comprehend various number systems, Boolean algebra, and its application to logic gates.
Master Boolean expression manipulation, canonical forms, and simplification techniques.
Proficiently handle SOP and POS expressions, recognizing relationships between minterms and maxterms.
Recognize the universality of NAND and NOR gates, implementing functions using De Morgan's Law.
Master Karnaugh map techniques, including advanced methods and handling don't care conditions.
Gain a comprehensive understanding of combinational circuits, covering principles and applications.
Understand binary addition principles and design various adder circuits, including 4-bit ripple carry adders.
Explore advanced adder designs for arithmetic operations.
Proficiently design binary subtractors, analyze overflow/underflow scenarios, and understand signed number representation.
Understand parity generation, detection, and various methods of binary multiplication.
Master the design and application of various multipliers, incorporating the Booth algorithm.
Understand applications of comparators, encoders, and decoders in digital systems.
Proficiently use multiplexers and demultiplexers in digital circuit design, recognizing their role as function generators.
Understand binary shifting operations, designing logical shifters, and principles of arithmetic and barrel shifting.
Grasp foundational principles of sequential circuits, focusing on storage elements and designing an SR latch.
Understand the operation of JK flip-flops, addressing race around conditions, and design master-slave JK flip-flops and Gated SR latches.
Gain proficiency in designing and analyzing various types of counters in sequential circuits.
Understand principles and design techniques for Mealy and Moore state sequential circuits.
Grasp fundamental principles of memory, differentiating internal structures between SRAM and DRAM, and gain practical skills in addressing memory, controlling tri-state digital buffers, and understanding ROM, PLD, and various PLDs.
Skills you'll gain
Category: Computational Logic
Computational Logic
Category: Hardware Design
Hardware Design
Category: Field-Programmable Gate Array (FPGA)
Field-Programmable Gate Array (FPGA)
Category: Electronic Components
Electronic Components
Category: Electronics Engineering
Electronics Engineering
Category: Electronics
Electronics
Category: Data Storage Technologies
Data Storage Technologies
Category: Programmable Logic Controllers
Programmable Logic Controllers
Category: Data Storage
Data Storage
Category: Application Specific Integrated Circuits
Application Specific Integrated Circuits
Category: Electronic Systems
Electronic Systems
Category: Semiconductors
Semiconductors
Category: Computer Architecture
Computer Architecture
VLSI Chip Design and Simulation with Electric VLSI EDA Tool
Course 2
•
13 hours
What you'll learn
This course provides a comprehensive exploration of CMOS VLSI design and simulation, covering IC technology, CMOS structures, historical timelines, processor intricacies, MOS transistor design, non-ideal characteristics, power dissipation, low-power design techniques, and practical insights into CMOS logic gates. Participants will delve into fundamental components and circuit design in the "Analog Circuit CMOS Chip Design and Simulation" module, using the Electric VLSI EDA tool. This includes stick diagrams, tool installation and usage, and hands-on experience in schematic/layout representations, enhancing electronic circuit design proficiency. In the "Digital Circuit CMOS Chip Design and Simulation" module, participants create systematic workflows for schematic/layout designs using the Electric VLSI EDA tool. The curriculum covers logic gates, and half adder circuits, providing a holistic understanding of CMOS logic circuit design. Throughout the course, participants acquire a robust skill set, combining theoretical knowledge with practical expertise in CMOS VLSI design and simulation.
By the end of this course, you will be able to:
Develop a profound understanding of Integrated Circuit (IC) technology, exploring its historical timeline and key inventions.
Discuss Moore’s Law and technology scaling, recognizing the importance of processors in Very Large-Scale Integration (VLSI).
Gain proficiency in MOS transistors, explaining their types and comprehending their working process, including operational modes of both PMOS and NMOS transistors.
Describe ideal transistor I-V characteristics and delve into non-ideal transistor characteristics, including leakage currents and their impact on device performance.
Understand the workings of the CMOS inverter, covering both its static behavior and power dissipation characteristics.
Explain components and mechanisms involved in CMOS power dissipation, addressing both static and dynamic aspects.
Explore benefits of low-power design techniques, analyzing factors influencing power consumption, and learning various power reduction techniques.
Understand the purpose of power gating in reducing overall power consumption and learn techniques to minimize short-circuit power consumption.
Explain the fundamentals of CMOS logic gates, including the series and parallel connections of NMOS and PMOS transistors.
Acquire skills in designing basic logic gates using Complementary Metal-Oxide-Semiconductor (CMOS) technology.
Develop skills in designing CMOS circuits using stick diagrams, creating blueprints for physical layouts adhering to semiconductor manufacturing process design rules.
Install and set up Electric VLSI EDA tool for VLSI circuit design, exploring components, schematic and layout editors, and conducting essential checks.
Understand PMOS and NMOS transistor concepts, design schematic and layout representations, perform various checks, and conduct simulations for current-voltage characteristics.
Grasp the CMOS inverter concept, create schematic and layout designs, and simulate the inverter to analyze behavior and characteristics.
Explore common-source and common-drain amplifiers in analog circuit design, designing schematics, layouts, and performing simulations to analyze performance.
Investigate the three-stage oscillator concept, design schematics and layout representations with CMOS inverters, and analyze performance through waveform simulations.
Comprehend CMOS NAND gate concepts, design schematics, validate layouts, and simulate for logical behavior analysis with diverse input scenarios.
Explore various digital circuit elements such as AND, NOR, and OR gates, XOR gate, and half adder, designing schematics, layouts, and performing simulations.
Skills you'll gain
Category: Simulation and Simulation Software
Simulation and Simulation Software
Category: Schematic Diagrams
Schematic Diagrams
Category: Electronics
Electronics
Category: Electronic Components
Electronic Components
Category: Electronics Engineering
Electronics Engineering
Category: Semiconductors
Semiconductors
Category: Computer Architecture
Computer Architecture
Category: Hardware Design
Hardware Design
Category: Verification And Validation
Verification And Validation
Category: Software Installation
Software Installation
Category: Data Validation
Data Validation
Category: Electrical Engineering
Electrical Engineering
Category: Low Voltage
Low Voltage
Design of Digital Circuits with VHDL Programming
Course 3
•
18 hours
What you'll learn
This course is designed to provide a comprehensive understanding of digital circuit design using VHDL programming with Xilinx ISE. Participants will learn the fundamentals of VHDL, simulation modeling, and design methodologies for digital circuits, including combinational and sequential circuits. Practical exercises using Xilinx ISE will enhance hands-on skills in circuit implementation, simulation, and analysis.
By the end of this course, you will be able to:
Understand the structure and behavior of digital circuits using VHDL.
Design and simulate digital circuits using Xilinx ISE.
Implement combinational and sequential logic circuits in VHDL.
Analyze and verify the functionality of digital circuits through simulation.
Skills you'll gain
Category: Computer Engineering
Computer Engineering
Category: Hardware Design
Hardware Design
Category: Field-Programmable Gate Array (FPGA)
Field-Programmable Gate Array (FPGA)
Category: Simulations
Simulations
Category: Embedded Systems
Embedded Systems
Category: Integrated Development Environments
Integrated Development Environments
Category: Electronics Engineering
Electronics Engineering
Category: Electronic Hardware
Electronic Hardware
Category: Simulation and Simulation Software
Simulation and Simulation Software
Category: Verification And Validation
Verification And Validation
FPGA Architecture Based System for Industrial Application
Course 4
•
12 hours
What you'll learn
The course "FPGA Architecture Based System for Industrial Application Using Vivado" is a comprehensive program that focuses on the design and implementation of FPGA-based VLSI systems for industrial applications. Participants will gain practical knowledge and hands-on experience in utilizing Xilinx Vivado software with Artix 7 FPGA boards to develop digital arithmetics, integrate sensors and motors, implement communication protocols, and create IoT applications.
By the end of this course, you will be able to:
• Understand the architecture and features of Artix 7 FPGA boards.
• Install and utilize Xilinx Vivado software for FPGA projects.
• Design and implement digital arithmetics including LEDs, adders, buzzer, and pushbuttons using VHDL on FPGA boards.
• Integrate sensors such as accelerometers, gesture recognition sensors, and ultrasonic sensors with FPGAs.
• Interface motors like stepper motors and DC motors with FPGA kits.
• Implement communication protocols including RS232, I2C, and SPI for data exchange.
• Develop IoT applications for remote monitoring and control using FPGA technology.
• Analyze RTL schematics and configure constraint files for FPGA-based designs.
• Validate hardware logic and functionality through simulation and real-time implementation.
• Demonstrate proficiency in designing complex VLSI systems for industrial use cases.
Skills you'll gain
Category: Field-Programmable Gate Array (FPGA)
Field-Programmable Gate Array (FPGA)
Category: Embedded Systems
Embedded Systems
Category: Internet Of Things
Internet Of Things
Category: Serial Peripheral Interface
Serial Peripheral Interface
Category: Software Development Tools
Software Development Tools
Category: Embedded Software
Embedded Software
Category: Software Installation
Software Installation
Category: Computer Architecture
Computer Architecture
Category: Hardware Architecture
Hardware Architecture
Category: Hardware Design
Hardware Design
Category: System Design and Implementation
System Design and Implementation
Category: Digital Communications
Digital Communications
Category: Network Protocols
Network Protocols
Category: Electronic Hardware
Electronic Hardware
Category: Verification And Validation
Verification And Validation
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Research Article
1 December 1989
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Rotavirus gene structure and function
Authors: M K Estes, J CohenAuthors Info & Affiliations
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Abstract
Knowledge of the structure and function of the genes and proteins of the rotaviruses has expanded rapidly. Information obtained in the last 5 years has revealed unexpected and unique molecular properties of rotavirus proteins of general interest to virologists, biochemists, and cell biologists. Rotaviruses share some features of replication with reoviruses, yet antigenic and molecular properties of the outer capsid proteins, VP4 (a protein whose cleavage is required for infectivity, possibly by mediating fusion with the cell membrane) and VP7 (a glycoprotein), show more similarities with those of other viruses such as the orthomyxoviruses, paramyxoviruses, and alphaviruses. Rotavirus morphogenesis is a unique process, during which immature subviral particles bud through the membrane of the endoplasmic reticulum (ER). During this process, transiently enveloped particles form, the outer capsid proteins are assembled onto particles, and mature particles accumulate in the lumen of the ER. Two ER-specific viral glycoproteins are involved in virus maturation, and these glycoproteins have been shown to be useful models for studying protein targeting and retention in the ER and for studying mechanisms of virus budding. New ideas and approaches to understanding how each gene functions to replicate and assemble the segmented viral genome have emerged from knowledge of the primary structure of rotavirus genes and their proteins and from knowledge of the properties of domains on individual proteins. Localization of type-specific and cross-reactive neutralizing epitopes on the outer capsid proteins is becoming increasingly useful in dissecting the protective immune response, including evaluation of vaccine trials, with the practical possibility of enhancing the production of new, more effective vaccines. Finally, future analyses with recently characterized immunologic and gene probes and new animal models can be expected to provide a basic understanding of what regulates the primary interactions of these viruses with the gastrointestinal tract and the subsequent responses of infected hosts.
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Microbiological Reviews
Volume 53 • Number 4 • December 1989
Pages: 410 - 449
PubMed: 2556635
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Published online: 1 December 1989
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Rotavirus gene structure and function. Microbiol Rev
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5949 | https://fiveable.me/key-terms/physical-chemistry-i/geometric-isomerism | Geometric isomerism - (Physical Chemistry I) - Vocab, Definition, Explanations | Fiveable | Fiveable
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Physical Chemistry I
Geometric isomerism
🧤physical chemistry i review
key term - Geometric isomerism
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MLA
Definition
Geometric isomerism refers to a type of stereoisomerism where molecules with the same molecular formula have different spatial arrangements of atoms or groups around a double bond or a ring structure. This phenomenon plays a crucial role in determining the physical and chemical properties of compounds, as the different arrangements can lead to distinct behaviors in chemical reactions and interactions with other molecules.
5 Must Know Facts For Your Next Test
Geometric isomers can exhibit significantly different physical properties, such as boiling points and solubility, due to their distinct arrangements.
The presence of double bonds or rings in a molecule is essential for geometric isomerism to occur since these features restrict rotation and create fixed geometries.
Geometric isomerism has important implications in biological systems, such as in the activity of pharmaceuticals, where one isomer may be therapeutically active while another may be inactive or harmful.
In addition to cis-trans configurations, geometric isomerism can also include more complex forms in larger molecules, often leading to multiple possible arrangements.
Understanding geometric isomerism is crucial for predicting reaction mechanisms, as the orientation of substituents can influence the pathway and outcome of chemical reactions.
Review Questions
How does geometric isomerism influence the physical properties of compounds?
Geometric isomerism impacts physical properties like boiling points and solubility because the different spatial arrangements of atoms lead to variations in intermolecular interactions. For instance, cis isomers may have higher boiling points due to stronger dipole-dipole interactions compared to their trans counterparts, which can be more linear and less polar. This difference can affect how substances behave in different environments, making it critical to understand these variations in practical applications.
Discuss how geometric isomerism can affect chemical reactivity in organic compounds.
Geometric isomerism can significantly influence chemical reactivity by altering how molecules interact during reactions. For example, the spatial arrangement of groups around a double bond may affect steric hindrance and electronic interactions, leading to differences in reaction rates and products. Certain geometric isomers might be more reactive than others due to their specific conformations that align better with reactants in a reaction mechanism.
Evaluate the role of geometric isomerism in drug design and its implications for pharmacology.
Geometric isomerism plays a pivotal role in drug design as different isomers can have drastically different biological activities. For instance, one geometric isomer may bind effectively to a target receptor, producing a desired therapeutic effect, while its counterpart may not bind at all or could even cause adverse effects. This necessitates careful consideration during drug development, as ensuring the correct isomer is utilized can greatly influence efficacy and safety profiles for medications.
Related terms
cis-trans isomerism:A specific form of geometric isomerism where two substituents on either side of a double bond or ring structure can be positioned on the same side (cis) or opposite sides (trans).
stereoisomerism:The broad category of isomerism that involves compounds with the same molecular formula but different spatial arrangements of atoms, which includes both geometric and optical isomerism.
conformational isomerism:A type of stereoisomerism where the rotation around single bonds leads to different spatial orientations of the atoms in a molecule.
"Geometric isomerism" also found in:
Subjects (3)
General Chemistry II
Inorganic Chemistry I
Inorganic Chemistry II
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5950 | https://radiopaedia.org/cases/boerhaave-syndrome-7?lang=us | Boerhaave syndrome | Radiology Case | Radiopaedia.org
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Boerhaave syndrome
Case contributed by Domenico Nicoletti
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Citation:
Nicoletti D, Boerhaave syndrome. Case study, Radiopaedia.org (Accessed on 29 Sep 2025)
DOI:
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rID:
27683
Case published:
14 Feb 2014, Domenico Nicoletti
Revisions:
7 times, by 5 contributors - see full revision history and disclosures
Systems:
Chest, Gastrointestinal
Tags:
core condition, pneumomediastinum
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Presentation
No history of previous illness but with an ambiguous reference to recent food and drink overindulgence, was presented to Emergency department complaining of the sudden onset of gradually increasing, lower thorax post emetic pain and subcutaneous emphysema.
Patient Data
Age: 50 years
Gender: Male
Chest
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The suspicion of spontaneous esophageal rupture was confirmed on CT scan and esophagogram by the presentation of bilateral pleural effusion together with right sided contrast extravasation from the lower third of the esophagus and pneumomediastinum.
Case Discussion
Boerhaave syndrome is suspected on a clinical basis and confirmed with radiologic studies. As gastric content passes to mediastinum and usually pleural space, a delay in diagnosis raises morbidity and mortality significantly.
Boerhaave syndrome is a spontaneous longitudinal perforation of the esophagus due to forceful emesis first described by Hermann Boerhaave in the 18th century.
This pathology is best treated with definitive repair and mediastinal and/or pleural drainage procedures.
2 articles feature images from this case
Boerhaave syndrome
Hamman syndrome
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5951 | https://dictionary.cambridge.org/us/dictionary/english/cultivate | Cambridge Dictionary +Plus
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English (US)
Meaning of cultivate in English
cultivate verb [T] (USE LAND)
Add to word list
Add to word list
C1
to prepare land and grow crops on it, or to grow a particular crop:
Most of the land there is too poor to cultivate.
The villagers cultivate mostly corn and beans.
SMART Vocabulary: related words and phrases
Farming - general words
agrarianism
agricultural extension
agriculturally
agrivoltaic
food insecurity
food secure
food security
hand-reared
handpick
harvester
homestead
mixed farming
reap
self-sufficiency
self-sufficient
sharecropping
sugaring
the Agrarian Revolution
thrash
tillable
See more results »
cultivate verb [T] (DEVELOP)
C2
to try to develop and improve something:
cultivate an image She has cultivated an image as a tough negotiator.
If you cultivate a relationship, you make a special effort to establish and develop it, because you think it might be useful to you:
The new prime minister is cultivating relationships with East Asian countries.
SMART Vocabulary: related words and phrases
Making things better
add salt to something idiom
allay
alleviate
alleviation
ameliorate
lighten
liven (something) up phrasal verb
lube
lubricate
make a difference idiom
make a world of difference idiom
rationalize
rectify
remedial
repair
repolish
restructure
revamp
revitalize
sharpen
See more results »
You can also find related words, phrases, and synonyms in the topics:
Encouraging and urging on
(Definition of cultivate from the Cambridge Advanced Learner's Dictionary & Thesaurus © Cambridge University Press)
cultivate | Intermediate English
cultivate
verb [ T ]
us
/ˈkʌl·təˌveɪt/
cultivate verb [T] (GROW)
Add to word list
Add to word list
to prepare land and grow crops on it, or to grow a particular crop:
He cultivated soybeans on most of the land.
cultivate verb [T] (DEVELOP)
to create a new condition by directed effort:
We’re trying to help these kids cultivate an interest in science.
To cultivate is also to try to become friendly with someone because that person may be able to help you:
to cultivate friendships
cultivation
noun [ U ] us
/ˌkʌl·təˈveɪ·ʃən/
Simple changes in rice cultivation can reduce methane emissions.
(Definition of cultivate from the Cambridge Academic Content Dictionary © Cambridge University Press)
cultivate | Business English
cultivate
verb [ T ]
uk
/ˈkʌltɪveɪt/ us
Add to word list
Add to word list
PRODUCTION
to prepare land and grow crops on it, or to grow a particular crop:
Corn, or maize, is indigenous to Mexico and has been cultivated there for some 6,000 years.
to try to develop and improve something:
She has cultivated an image as a shrewd investor.
if you cultivate a relationship, you make a special effort to develop it, because you think it might be useful to you:
He cultivated business contacts in ten major cities.
cultivation
noun [ U ]
the cultivation of crops/the land
This course will help with the cultivation of key marketing skills.
If you wish to retain customers, you must work at the cultivation of the customer relationship.
(Definition of cultivate from the Cambridge Business English Dictionary © Cambridge University Press)
Examples of cultivate
cultivate
Cultivating a strong brand, though, is no simple task.
From OCRegister
In order to succeed you must know how to cultivate your community.
From Huffington Post
It is what you do after college with the experiences you explore, the friendships you cultivate, and the paths you choose that make the difference.
From CNN
But the engineering marvels of a termite mound -- internal temperature control, ventilation, cultivated fungal gardens -- should not be sneezed at, either.
From Phys.Org
She credits him with teaching her how to cook while cultivating her love for food.
From NOLA.com
For as long as humans have been cultivating grains, cooks have been baking flatbreads in the embers or on primitive wood-fired ovens.
From Los Angeles Times
Everyone cultivates a fashionable, skin-deep vulnerability; underneath, they're superheroes with jujitsu skills and heads full of put-downs.
From The New Yorker
Neither made overtly political films and instead ran campaigns based on the heroic personas they had cultivated in their film careers.
From The Atlantic
Being all things to all people, of course, is a skill that good politicians look to cultivate.
From Washington Post
Kindness is a virtue that we need to cultivate and value.
From Huffington Post
Leaders must cultivate relationships in multiple ways and using multiple tools.
From Huffington Post
But facts don't matter to those who cultivate raw and unthinking emotion.
From Washington Times
Yoga is a union of all steps -- cultivating the body, breath, mind and the inner nature.
From Huffington Post
But if not, you need to cultivate your will -- or pick a different resolution -- before you are likely to find a way forward.
From Huffington Post
They need to be expressed and cultivated in order for each of us to thrive.
From Huffington Post
These examples are from corpora and from sources on the web. Any opinions in the examples do not represent the opinion of the Cambridge Dictionary editors or of Cambridge University Press or its licensors.
What is the pronunciation of cultivate?
Translations of cultivate
in Chinese (Traditional)
種植, 耕作, 栽培…
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in Chinese (Simplified)
种植, 耕作, 栽培…
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in Spanish
cultivar, entrenar, desarrollar…
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in Portuguese
cultivar…
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in more languages
in Marathi
in Japanese
in Turkish
in French
in Catalan
in Dutch
in Tamil
in Hindi
in Gujarati
in Danish
in Swedish
in Malay
in German
in Norwegian
in Urdu
in Ukrainian
in Russian
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in Arabic
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in Indonesian
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in Polish
in Korean
in Italian
जमिनीवर लागवड करणे…
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～を栽培する, 耕（たがや）す, 養（やしな）う…
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toprağı işlemek, ekip biçmek, sürüp ekmek…
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cultiver…
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cultivar…
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bebouwen, telen…
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நிலத்தை தயார் செய்து அதில் பயிர்களை வளர்ப்பது அல்லது குறிப்பிட்ட பயிரை வளர்ப்பது…
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(फसल उगाने हेतु) खेती करना…
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પાક ઉગાડવો, ખેતી કરવી…
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kultivere, udvikle, dyrke…
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bruka, bearbeta, odla…
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menggemburkan, membiakkan…
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bebauen, züchten…
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dyrke, kultivere, utvikle…
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جوتنا, بونا, کھیتی کرنا…
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обробляти, культивувати, розводити…
See more
возделывать, выращивать, культивировать…
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పండించడం…
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يَفلَح الأرْض…
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চাষ করা…
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obdělávat půdu, pěstovat…
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mengolah, menanam…
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เตรียมดินสำหรับเพาะปลูก, เพาะปลูก…
See more
cày cấy, trồng trọt…
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uprawiać, wytwarzać, kultywować…
See more
재배하다…
See more
coltivare…
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cultivate (USE LAND)
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5952 | http://mathcentral.uregina.ca/QQ/database/QQ.09.02/amanda3.html | Natural logarithms
Quandaries and Queries
Hi,
my name is Amanda, and I'm going into my senior year of high school. I will be taking AP calculus, and my teacher gave us some homework over the summer. However, there are two things that I do not understand how to do. The first is, she wants us to be able to generate a unit circle by hand using 30, 60 and 90 degree triangles. I have used the unit circle in trigonometry, however I was never taught how to draw it. Secondly, I need to know how to do natural logarithms without a calculator. I was not taught how to do this, and the worksheet I was given only showed me how to complete them using a calculator.
Thank you so much for any help that you might be able to offer
Amanda
Hi Amanda,
This looks like a very tough class. I was taught how to draw a circle using compasses, pennies or glasses, but never with triangles. Also, I know how to compute the natural logarithm using tables, a slide rule or a calculator, but without these tools, the best I can do is approximate them. (And even Napier
the guy who devised logarithms, had no easy way to compute them; he made up tables and eventually invented the slide rule.)
To approximate natural logarithms, you can make a small table as follows: the base e is about 2.7, so that ln(2.7) is approximately1. Then, e e is approximately 7.3, so that ln(7.3) is approximately2. Then, e e e is approximately 19.7, so that ln(19.7) is approximately 3, and so on.
In this way, you make up a list 2.7, 7.3, 19.7, 53.2, ... of numbers whose
natural logarithms are approximately 1, 2, 3, 4, ... For numbers that fall between
these values, you need to interpolate: 10 is between 7.3 and 19.7, so
ln(10) should be between 2 and 3. Perhaps 2.2 is a good estimate;
try it out and see how it works.
There is another way to approximate natural logarithms if you are allowed to use one of these solar powered calculators with a square root button but no ln button:
enter the number whose logarithm you want to calculate (say 19.7)
press the square root button ten times
subtract 1
multiply by 1024.
On my calculator, I get 2.9849... when I perform these operations. This is quite good since it is supposed to be an approximation of ln(19.7) which is about 3. You can try it with 2.7 and 7.3 to convince yourself that it works well enough. Then one day in your second or third calculus course you will learn why this works.
Claude
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5953 | https://en.wikipedia.org/wiki/Staphyloxanthin | Jump to content
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Staphyloxanthin
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From Wikipedia, the free encyclopedia
Staphyloxanthin
| |
| Names |
| IUPAC name [(2S,3R,4S,5S,6R)-3,4, 5-Trihydroxy-6-[[(12S)-12-methyltetradecanoyl]oxymethyl]oxan-2-yl] (2E,4E,6E,8E,10E,12E,14E,16E,18E)-2,6,10,15,19,23-hexamethyltetracosa-2,4,6,8,10,12,14,16,18,22-decaenoate |
| Identifiers |
| CAS Number | 71869-01-7N |
| 3D model (JSmol) | Interactive image |
| ChEBI | CHEBI:71690N |
| ChemSpider | 65323059Y |
| PubChem CID | 56928085 |
| CompTox Dashboard (EPA) | DTXSID20222141 |
| InChI InChI=1S/C51H78O8/c1-10-39(4)25-16-14-12-11-13-15-17-33-44(36-52)51(9,35-23-32-43(8)49(57)59-50-48(56)47(55)46(54)45(37-53)58-50)34-19-18-26-40(5)28-21-30-42(7)31-22-29-41(6)27-20-24-38(2)3/h17-19,21-24,26,28-33,35-36,39,44-48,50,53-56H,10-16,20,25,27,34,37H2,1-9H3/b19-18+,28-21+,31-22+,33-17-,35-23+,40-26+,41-29+,42-30+,43-32+/t39?,44?,45-,46-,47+,48-,50-,51?/m1/s1Y Key: ZGBLADNGFNFPBV-OQMOJWPESA-NY InChI=1/C51H78O8/c1-10-39(4)25-16-14-12-11-13-15-17-33-44(36-52)51(9,35-23-32-43(8)49(57)59-50-48(56)47(55)46(54)45(37-53)58-50)34-19-18-26-40(5)28-21-30-42(7)31-22-29-41(6)27-20-24-38(2)3/h17-19,21-24,26,28-33,35-36,39,44-48,50,53-56H,10-16,20,25,27,34,37H2,1-9H3/b19-18+,28-21+,31-22+,33-17-,35-23+,40-26+,41-29+,42-30+,43-32+/t39?,44?,45-,46-,47+,48-,50-,51?/m1/s1 Key: ZGBLADNGFNFPBV-OQMOJWPEBF |
| SMILES O=C(O[C@H]1OC@@HCO)\C(=C\C=C\C(C)(C\C=C\C=C(\C=C\C=C(\C=C\C=C(/C)CC\C=C(/C)C)C)C)C(\C=C/CCCCCCCC(C)CC)C=O)C |
| Properties |
| Chemical formula | C51H78O8 |
| Molar mass | 819.177 g·mol−1 |
| Except where otherwise noted, data are given for materials in their standard state (at 25 °C [77 °F], 100 kPa). N verify (what is YN ?) Infobox references |
Chemical compound
Staphyloxanthin is a carotenoid pigment that is produced by some strains of Staphylococcus aureus, and is responsible for the characteristic golden color that gives S. aureus its species name. Staphyloxanthin also acts as a virulence factor. It has an antioxidant action that helps the microbe evade death by reactive oxygen species produced by the host immune system.
When comparing a normal strain of S. aureus with a strain modified to lack staphyloxanthin, the wildtype pigmented strain was more likely to survive incubation with an oxidizing chemical such as hydrogen peroxide than the mutant strain was. Colonies of the two strains were also exposed to human neutrophils. The mutant colonies quickly succumbed while many of the pigmented colonies survived. Wounds on mice were inoculated with the two strains. The pigmented strains created lingering abscesses. Wounds with the unpigmented strains healed quickly. These tests suggest that the staphyloxanthin may be key to the ability of S. aureus to survive immune system attacks.
Drugs designed to inhibit the bacterium's production of the staphyloxanthin may weaken it and renew its susceptibility to antibiotics. In fact, because of similarities in the pathways for biosynthesis of staphyloxanthin and human cholesterol, a drug developed in the context of cholesterol-lowering therapy was shown to block S. aureus pigmentation and disease progression in a mouse infection model.
Genomically, the crt operon primarily underlies the biosynthesis of staphyloxanthin; however, another enzyme, AldH, found outside of the operon, was later identified as also needed for the carotenoid’s production. Recent comparative genomics research has further revealed that the crt operon is widespread across the Staphylococcus genus. Non-aureus pigmented staphylococci have long been noted and experimental support for staphyloxanthin production now exists in Staphylococcus xylosus, Staphylococcus warneri, Staphylococcus epidermidis, and Staphylococcus capitis. Further, some lineages of S. aureus have been found to lack crt genes and some non-aureus staphylococcal isolates have been found to feature multiple copies of crt.
References
[edit]
^ a b Clauditz A, Resch A, Wieland KP, Peschel A, Götz F (August 2006). "Staphyloxanthin plays a role in the fitness of Staphylococcus aureus and its ability to cope with oxidative stress". Infection and Immunity. 74 (8): 4950–3. doi:10.1128/IAI.00204-06. PMC 1539600. PMID 16861688.
^ Campbell, Amy E.; McCready-Vangi, Amelia R.; Uberoi, Aayushi; Murga-Garrido, Sofía M.; Lovins, Victoria M.; White, Ellen K.; Pan, Jamie Ting-Chun; Knight, Simon A. B.; Morgenstern, Alexis R.; Bianco, Colleen; Planet, Paul J.; Gardner, Sue E.; Grice, Elizabeth A. (2023-10-31). "Variable staphyloxanthin production by Staphylococcus aureus drives strain-dependent effects on diabetic wound-healing outcomes". Cell Reports. 42 (10): 113281. doi:10.1016/j.celrep.2023.113281. ISSN 2211-1247. PMC 10680119. PMID 37858460.{{cite journal}}: CS1 maint: article number as page number (link)
^ Liu GY, Essex A, Buchanan JT, Datta V, Hoffman HM, Bastian JF, Fierer J, Nizet V (2005). "Staphylococcus aureus golden pigment impairs neutrophil killing and promotes virulence through its antioxidant activity". J Exp Med. 202 (2): 209–15. doi:10.1084/jem.20050846. PMC 2213009. PMID 16009720.
^ Liu CI, Liu GY, Song Y, Yin F, Hensler ME, Jeng WY, Nizet V, Wang AH, Oldfield E (2008). "A cholesterol biosynthesis inhibitor blocks Staphylococcus aureus virulence". Science. 319 (5868): 391–94. Bibcode:2008Sci...319.1391L. doi:10.1126/science.1153018. PMC 2747771. PMID 18276850.
^ Pelz, Alexandra; Wieland, Karsten-Peter; Putzbach, Karsten; Hentschel, Petra; Albert, Klaus; Götz, Friedrich (2005-09-16). "Structure and Biosynthesis of Staphyloxanthin from Staphylococcus aureus". Journal of Biological Chemistry. 280 (37): 32493–32498. doi:10.1074/jbc.M505070200. ISSN 0021-9258. PMID 16020541.
^ Kim, Se Hyeuk; Lee, Pyung Cheon (2012-06-22). "Functional expression and extension of staphylococcal staphyloxanthin biosynthetic pathway in Escherichia coli". The Journal of Biological Chemistry. 287 (26): 21575–21583. doi:10.1074/jbc.M112.343020. ISSN 1083-351X. PMC 3381123. PMID 22535955.
^ a b c d Salamzade, Rauf; Cheong, J.Z. Alex; Sandstrom, Shelby; Swaney, Mary Hannah; Stubbendieck, Reed M.; Starr, Nicole Lane; Currie, Cameron R.; Singh, Anne Marie; Kalan, Lindsay R. (2023-04-28). "Evolutionary investigations of the biosynthetic diversity in the skin microbiome using lsaBGC". Microbial Genomics. 9 (4). doi:10.1099/mgen.0.000988. ISSN 2057-5858. PMC 10210951. PMID 37115189.
^ Kloos, Wesley E.; Schleifer, Karl H. (1975). "Isolation and Characterization of Staphylococci from Human Skin II. Descriptions of Four New Species: Staphylococcus warneri, Staphylococcus capitis, Staphylococcus hominis, and Staphylococcus simulans1". International Journal of Systematic and Evolutionary Microbiology. 25 (1): 62–79. doi:10.1099/00207713-25-1-62. ISSN 1466-5034.
^ Seel, Waldemar; Baust, Denise; Sons, Dominik; Albers, Maren; Etzbach, Lara; Fuss, Janina; Lipski, André (2020-01-15). "Carotenoids are used as regulators for membrane fluidity by Staphylococcus xylosus". Scientific Reports. 10 (1): 330. Bibcode:2020NatSR..10..330S. doi:10.1038/s41598-019-57006-5. ISSN 2045-2322. PMC 6962212. PMID 31941915.
^ Vermassen, Aurore; Dordet-Frisoni, Emilie; de La Foye, Anne; Micheau, Pierre; Laroute, Valérie; Leroy, Sabine; Talon, Régine (2016-02-05). "Adaptation of Staphylococcus xylosus to Nutrients and Osmotic Stress in a Salted Meat Model". Frontiers in Microbiology. 7: 87. doi:10.3389/fmicb.2016.00087. ISSN 1664-302X. PMC 4742526. PMID 26903967.
^ Cosetta, Casey M.; Niccum, Brittany; Kamkari, Nick; Dente, Michael; Podniesinski, Matthew; Wolfe, Benjamin E. (September 2023). "Bacterial-fungal interactions promote parallel evolution of global transcriptional regulators in a widespread Staphylococcus species". The ISME Journal. 17 (9): 1504–1516. Bibcode:2023ISMEJ..17.1504C. doi:10.1038/s41396-023-01462-5. ISSN 1751-7370. PMC 10432416. PMID 37524910.
^ Siems, Katharina; Runzheimer, Katharina; Rebrosova, Katarina; Etzbach, Lara; Auerhammer, Alina; Rehm, Anna; Schwengers, Oliver; Šiler, Martin; Samek, Ota; Růžička, Filip; Moeller, Ralf (2023-09-29). "Identification of staphyloxanthin and derivates in yellow-pigmented Staphylococcus capitis subsp. capitis". Frontiers in Microbiology. 14. doi:10.3389/fmicb.2023.1272734. ISSN 1664-302X. PMC 10570620.
^ Holt, Deborah C.; Holden, Matthew T. G.; Tong, Steven Y. C.; Castillo-Ramirez, Santiago; Clarke, Louise; Quail, Michael A.; Currie, Bart J.; Parkhill, Julian; Bentley, Stephen D.; Feil, Edward J.; Giffard, Philip M. (2011). "A very early-branching Staphylococcus aureus lineage lacking the carotenoid pigment staphyloxanthin". Genome Biology and Evolution. 3: 881–895. doi:10.1093/gbe/evr078. ISSN 1759-6653. PMC 3175761. PMID 21813488.
^ Jeong, Do-Won; Heo, Sojeong; Ryu, Sangryeol; Blom, Jochen; Lee, Jong-Hoon (2017-07-14). "Genomic insights into the virulence and salt tolerance of Staphylococcus equorum". Scientific Reports. 7 (1): 5383. Bibcode:2017NatSR...7.5383J. doi:10.1038/s41598-017-05918-5. ISSN 2045-2322. PMC 5511256. PMID 28710456.
External links
[edit]
Staphyloxanthin on www.genome.jp
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5954 | https://www.statology.org/probability-of-a-and-b/ | Given two events, A and B, to “find the probability of A and B” means to find the probability that event A and event B both occur.
We typically write this probability in one of two ways:
P(A and B) – Written form
P(A∩B) – Notation form
The way we calculate this probability depends on whether or not events A and B are independent or dependent.
If A and B are independent, then the formula we use to calculate P(A∩B) is simply:
```
Independent Events: P(A∩B) = P(A) P(B)
```
If A and B are dependent, then the formula we use to calculate P(A∩B) is:
Dependent Events:P(A∩B) = P(A) P(B|A)
Note that P(B|A) is the conditional probability of event B occurring, givenevent A occurs.
The following examples show how to use these formulas in practice.
Examples of P(A∩B) for Independent Events
The following examples show how to calculate P(A∩B) when A and B are independent events.
Example 1: The probability that your favorite baseball team wins the World Series is 1/30 and the probability that your favorite football team wins the Super Bowl is 1/32. What is the probability that both of your favorite teams win their respective championships?
Solution: In this example, the probability of each event occurring is independent of the other. Thus, the probability that they both occur is calculated as:
P(A∩B) = (1/30) (1/32) = 1/960 = .00104.
Example 2: You roll a dice and flip a coin at the same time. What is the probability that the dice lands on 4 and the coin lands on tails?
Solution: In this example, the probability of each event occurring is independent of the other. Thus, the probability that they both occur is calculated as:
P(A∩B) = (1/6) (1/2) = 1/12 = .083333.
Examples of P(A∩B) for Dependent Events
The following examples show how to calculate P(A∩B) when A and B are dependent events.
Example 1: An urn contains 4 red balls and 4 green balls. You randomly choose one ball from the urn. Then, without replacement, you select another ball. What is the probability that you choose a red ball each time?
Solution: In this example, the color of the ball that we choose the first time affects the probability of choosing a red ball the second time. Thus, the two events are dependent.
Let’s define event A as the probability of selecting a red ball the first time. This probability is P(A) = 4/8. Next, we have to find the probability of selecting a red ball again, given that the first ball was red. In this case, there are only 3 red balls left to choose and only 7 total balls in the urn. Thus, P(B|A) is 3/7.
Thus, the probability that we select a red ball each time would be calculated as:
P(A∩B) = P(A) P(B|A) = (4/8) (3/7) = 0.214.
Example 2: In a certain classroom there are 15 boys and 12 girls. Suppose we place the names of each student in a bag. We randomly choose one name from the bag. Then, without replacement, we choose another name. What is the probability that both names are boys?
Solution: In this example, the name we choose the first time affects the probability of choosing a boy name during the second draw. Thus, the two events are dependent.
Let’s define event A as the probability of selecting a boy first time. This probability is P(A) = 15/27. Next, we have to find the probability of selecting a boy again, given that the first name was a boy. In this case, there are only 14 boys left to choose and only 26 total names in the bag. Thus, P(B|A) is 14/26.
Thus, the probability that we select a boy name each time would be calculated as:
P(A∩B) = P(A) P(B|A) = (15/27) (14/26) = 0.299.
Zach Bobbitt
Hey there. My name is Zach Bobbitt. I have a Masters of Science degree in Applied Statistics and I’ve worked on machine learning algorithms for professional businesses in both healthcare and retail. I’m passionate about statistics, machine learning, and data visualization and I created Statology to be a resource for both students and teachers alike. My goal with this site is to help you learn statistics through using simple terms, plenty of real-world examples, and helpful illustrations.
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5955 | https://jdmitrigallow.com/teaching/logic19/10.pdf | Natural Deduction Proofs for SL Derived Rules 1 A Ass. (LEM) 2 A ∧A ∧I 1 3 ¬A Ass. (LEM) 4 ¬A ∨A ∨I 3 5 (A ∧A) ∨(¬A ∨A) LEM 1–2, 3–4 1 Outline Derived Rules 2 Derived Rules • The plan: use the basic rules to establish additional rules 3 Reiteration Reiteration (R) A ◃ A 4 Reiteration m A 5 Reiteration m A k A∧A ∧I m 5 Reiteration m A k A∧A ∧I m k+1 A ∧E k 5 Disjunctive Syllogism Disjunctive Syllogism (DS) A∨B ¬A ◃ B A∨B ¬B ◃ A 6 Disjunctive Syllogism m A∨B n ¬A 7 Disjunctive Syllogism m A∨B n ¬A k A Ass. (∨E) 7 Disjunctive Syllogism m A∨B n ¬A k A Ass. (∨E) k+1 ⊥ ⊥I n, k 7 Disjunctive Syllogism m A∨B n ¬A k A Ass. (∨E) k+1 ⊥ ⊥I n, k k+2 B ⊥E k+1 7 Disjunctive Syllogism m A∨B n ¬A k A Ass. (∨E) k+1 ⊥ ⊥I n, k k+2 B ⊥E k+1 k+3 B Ass. (∨E) 7 Disjunctive Syllogism m A∨B n ¬A k A Ass. (∨E) k+1 ⊥ ⊥I n, k k+2 B ⊥E k+1 k+3 B Ass. (∨E) k+4 B R k+3 7 Disjunctive Syllogism m A∨B n ¬A k A Ass. (∨E) k+1 ⊥ ⊥I n, k k+2 B ⊥E k+1 k+3 B Ass. (∨E) k+4 B R k+3 k+5 B ∨E m, k–k+2, k+3–k+4 7 Double Negation Elimination Double Negation Elimination (DNE) ¬¬A ◃ A 8 Double Negation Elimination m ¬¬A 9 Double Negation Elimination m ¬¬A k ¬A Ass. (¬E) 9 Double Negation Elimination m ¬¬A k ¬A Ass. (¬E) k+1 ⊥ ⊥I m, k 9 Double Negation Elimination m ¬¬A k ¬A Ass. (¬E) k+1 ⊥ ⊥I m, k k+2 A ¬E k–k+1 9 Modus Tollens Modus Tollens (MT) A →B ¬B ◃ ¬A 10 Modus Tollens m A →B n ¬B 11 Modus Tollens m A →B n ¬B k A Ass. (¬I) 11 Modus Tollens m A →B n ¬B k A Ass. (¬I) k+1 B →E m, k 11 Modus Tollens m A →B n ¬B k A Ass. (¬I) k+1 B →E m, k k+2 ⊥ ⊥I n, k+1 11 Modus Tollens m A →B n ¬B k A Ass. (¬I) k+1 B →E m, k k+2 ⊥ ⊥I n, k+1 k+3 ¬A ¬I k–k+2 11 Law of Excluded Middle Law of Excluded Middle (LEM) A .
.
.
B ¬A .
.
.
B ◃ B 12 Law of Excluded Middle 1 13 Law of Excluded Middle 1 ¬(A∨¬A) Ass. (¬E) 13 Law of Excluded Middle 1 ¬(A∨¬A) Ass. (¬E) 2 A Ass. (¬I) 13 Law of Excluded Middle 1 ¬(A∨¬A) Ass. (¬E) 2 A Ass. (¬I) 3 A∨¬A ∨I 2 13 Law of Excluded Middle 1 ¬(A∨¬A) Ass. (¬E) 2 A Ass. (¬I) 3 A∨¬A ∨I 2 4 ⊥ ⊥I 1, 3 13 Law of Excluded Middle 1 ¬(A∨¬A) Ass. (¬E) 2 A Ass. (¬I) 3 A∨¬A ∨I 2 4 ⊥ ⊥I 1, 3 5 ¬A ¬I 2–4 13 Law of Excluded Middle 1 ¬(A∨¬A) Ass. (¬E) 2 A Ass. (¬I) 3 A∨¬A ∨I 2 4 ⊥ ⊥I 1, 3 5 ¬A ¬I 2–4 6 A∨¬A ∨I 5 13 Law of Excluded Middle 1 ¬(A∨¬A) Ass. (¬E) 2 A Ass. (¬I) 3 A∨¬A ∨I 2 4 ⊥ ⊥I 1, 3 5 ¬A ¬I 2–4 6 A∨¬A ∨I 5 7 ⊥ ⊥I 1, 6 13 Law of Excluded Middle 1 ¬(A∨¬A) Ass. (¬E) 2 A Ass. (¬I) 3 A∨¬A ∨I 2 4 ⊥ ⊥I 1, 3 5 ¬A ¬I 2–4 6 A∨¬A ∨I 5 7 ⊥ ⊥I 1, 6 8 A∨¬A ¬E 1–7 13 Law of Excluded Middle Law of Excluded Middle (LEM) A .
.
.
B ¬A .
.
.
B ◃ B 14 DeMorgan’s Rules DeMorgan’s Rules (DeM) ¬(A∧B) ▹◃ ¬A∨¬B ¬(A∨B) ▹◃ ¬A∧¬B 15 1 ¬(A∧B) 16 1 ¬(A∧B) 2 A Ass. (LEM) 16 1 ¬(A∧B) 2 A Ass. (LEM) 3 B Ass. (¬I) 16 1 ¬(A∧B) 2 A Ass. (LEM) 3 B Ass. (¬I) 4 A∧B ∧I 2, 3 16 1 ¬(A∧B) 2 A Ass. (LEM) 3 B Ass. (¬I) 4 A∧B ∧I 2, 3 5 ⊥ ⊥I 1, 4 16 1 ¬(A∧B) 2 A Ass. (LEM) 3 B Ass. (¬I) 4 A∧B ∧I 2, 3 5 ⊥ ⊥I 1, 4 6 ¬B ¬I 3–5 16 1 ¬(A∧B) 2 A Ass. (LEM) 3 B Ass. (¬I) 4 A∧B ∧I 2, 3 5 ⊥ ⊥I 1, 4 6 ¬B ¬I 3–5 7 ¬A∨¬B ∨I 6 16 1 ¬(A∧B) 2 A Ass. (LEM) 3 B Ass. (¬I) 4 A∧B ∧I 2, 3 5 ⊥ ⊥I 1, 4 6 ¬B ¬I 3–5 7 ¬A∨¬B ∨I 6 8 ¬A Ass. (LEM) 16 1 ¬(A∧B) 2 A Ass. (LEM) 3 B Ass. (¬I) 4 A∧B ∧I 2, 3 5 ⊥ ⊥I 1, 4 6 ¬B ¬I 3–5 7 ¬A∨¬B ∨I 6 8 ¬A Ass. (LEM) 9 ¬A∨¬B ∨I 7 16 1 ¬(A∧B) 2 A Ass. (LEM) 3 B Ass. (¬I) 4 A∧B ∧I 2, 3 5 ⊥ ⊥I 1, 4 6 ¬B ¬I 3–5 7 ¬A∨¬B ∨I 6 8 ¬A Ass. (LEM) 9 ¬A∨¬B ∨I 7 10 ¬A∨¬B LEM 2–7, 8–9 16 1 ¬A∨¬B 17 1 ¬A∨¬B 2 A∧B Ass. (¬I) 17 1 ¬A∨¬B 2 A∧B Ass. (¬I) 3 A ∧E 2 17 1 ¬A∨¬B 2 A∧B Ass. (¬I) 3 A ∧E 2 4 B ∧E 2 17 1 ¬A∨¬B 2 A∧B Ass. (¬I) 3 A ∧E 2 4 B ∧E 2 5 ¬A Ass. (∨E) 17 1 ¬A∨¬B 2 A∧B Ass. (¬I) 3 A ∧E 2 4 B ∧E 2 5 ¬A Ass. (∨E) 6 ⊥ ⊥I 3, 5 17 1 ¬A∨¬B 2 A∧B Ass. (¬I) 3 A ∧E 2 4 B ∧E 2 5 ¬A Ass. (∨E) 6 ⊥ ⊥I 3, 5 7 ¬B Ass. (∨E) 17 1 ¬A∨¬B 2 A∧B Ass. (¬I) 3 A ∧E 2 4 B ∧E 2 5 ¬A Ass. (∨E) 6 ⊥ ⊥I 3, 5 7 ¬B Ass. (∨E) 8 ⊥ ⊥I 4, 7 17 1 ¬A∨¬B 2 A∧B Ass. (¬I) 3 A ∧E 2 4 B ∧E 2 5 ¬A Ass. (∨E) 6 ⊥ ⊥I 3, 5 7 ¬B Ass. (∨E) 8 ⊥ ⊥I 4, 7 9 ⊥ ∨E 1, 5–6, 7–8 17 1 ¬A∨¬B 2 A∧B Ass. (¬I) 3 A ∧E 2 4 B ∧E 2 5 ¬A Ass. (∨E) 6 ⊥ ⊥I 3, 5 7 ¬B Ass. (∨E) 8 ⊥ ⊥I 4, 7 9 ⊥ ∨E 1, 5–6, 7–8 10 ¬(A∧B) ¬I 2–9 17 1 ¬(A∨B) 18 1 ¬(A∨B) 2 A Ass. (LEM) 18 1 ¬(A∨B) 2 A Ass. (LEM) 3 A∨B ∨I 2 18 1 ¬(A∨B) 2 A Ass. (LEM) 3 A∨B ∨I 2 4 ⊥ ⊥I 1, 3 18 1 ¬(A∨B) 2 A Ass. (LEM) 3 A∨B ∨I 2 4 ⊥ ⊥I 1, 3 5 ¬A∧¬B ⊥E 4 18 1 ¬(A∨B) 2 A Ass. (LEM) 3 A∨B ∨I 2 4 ⊥ ⊥I 1, 3 5 ¬A∧¬B ⊥E 4 6 ¬A Ass. (LEM) 18 1 ¬(A∨B) 2 A Ass. (LEM) 3 A∨B ∨I 2 4 ⊥ ⊥I 1, 3 5 ¬A∧¬B ⊥E 4 6 ¬A Ass. (LEM) 7 B Ass. (¬I) 18 1 ¬(A∨B) 2 A Ass. (LEM) 3 A∨B ∨I 2 4 ⊥ ⊥I 1, 3 5 ¬A∧¬B ⊥E 4 6 ¬A Ass. (LEM) 7 B Ass. (¬I) 8 A∨B ∨I 7 18 1 ¬(A∨B) 2 A Ass. (LEM) 3 A∨B ∨I 2 4 ⊥ ⊥I 1, 3 5 ¬A∧¬B ⊥E 4 6 ¬A Ass. (LEM) 7 B Ass. (¬I) 8 A∨B ∨I 7 9 ⊥ ⊥I 1, 8 18 1 ¬(A∨B) 2 A Ass. (LEM) 3 A∨B ∨I 2 4 ⊥ ⊥I 1, 3 5 ¬A∧¬B ⊥E 4 6 ¬A Ass. (LEM) 7 B Ass. (¬I) 8 A∨B ∨I 7 9 ⊥ ⊥I 1, 8 10 ¬B ¬I 7–9 18 1 ¬(A∨B) 2 A Ass. (LEM) 3 A∨B ∨I 2 4 ⊥ ⊥I 1, 3 5 ¬A∧¬B ⊥E 4 6 ¬A Ass. (LEM) 7 B Ass. (¬I) 8 A∨B ∨I 7 9 ⊥ ⊥I 1, 8 10 ¬B ¬I 7–9 11 ¬A∧¬B ∧I 6, 0 18 1 ¬(A∨B) 2 A Ass. (LEM) 3 A∨B ∨I 2 4 ⊥ ⊥I 1, 3 5 ¬A∧¬B ⊥E 4 6 ¬A Ass. (LEM) 7 B Ass. (¬I) 8 A∨B ∨I 7 9 ⊥ ⊥I 1, 8 10 ¬B ¬I 7–9 11 ¬A∧¬B ∧I 6, 0 12 ¬A∧¬B LEM 2–5, 6–11 18 1 ¬A∧¬B 19 1 ¬A∧¬B 2 ¬A ∧E 1 19 1 ¬A∧¬B 2 ¬A ∧E 1 3 ¬B ∧E 1 19 1 ¬A∧¬B 2 ¬A ∧E 1 3 ¬B ∧E 1 4 A Ass. (LEM) 19 1 ¬A∧¬B 2 ¬A ∧E 1 3 ¬B ∧E 1 4 A Ass. (LEM) 5 ⊥ ⊥I 2, 4 19 1 ¬A∧¬B 2 ¬A ∧E 1 3 ¬B ∧E 1 4 A Ass. (LEM) 5 ⊥ ⊥I 2, 4 6 ¬(A∨B) ⊥E 5 19 1 ¬A∧¬B 2 ¬A ∧E 1 3 ¬B ∧E 1 4 A Ass. (LEM) 5 ⊥ ⊥I 2, 4 6 ¬(A∨B) ⊥E 5 7 ¬A Ass. (LEM) 19 1 ¬A∧¬B 2 ¬A ∧E 1 3 ¬B ∧E 1 4 A Ass. (LEM) 5 ⊥ ⊥I 2, 4 6 ¬(A∨B) ⊥E 5 7 ¬A Ass. (LEM) 8 A∨B Ass. (¬I) 19 1 ¬A∧¬B 2 ¬A ∧E 1 3 ¬B ∧E 1 4 A Ass. (LEM) 5 ⊥ ⊥I 2, 4 6 ¬(A∨B) ⊥E 5 7 ¬A Ass. (LEM) 8 A∨B Ass. (¬I) 9 A DS 3, 8 19 1 ¬A∧¬B 2 ¬A ∧E 1 3 ¬B ∧E 1 4 A Ass. (LEM) 5 ⊥ ⊥I 2, 4 6 ¬(A∨B) ⊥E 5 7 ¬A Ass. (LEM) 8 A∨B Ass. (¬I) 9 A DS 3, 8 10 ⊥ ⊥I 7, 9 19 1 ¬A∧¬B 2 ¬A ∧E 1 3 ¬B ∧E 1 4 A Ass. (LEM) 5 ⊥ ⊥I 2, 4 6 ¬(A∨B) ⊥E 5 7 ¬A Ass. (LEM) 8 A∨B Ass. (¬I) 9 A DS 3, 8 10 ⊥ ⊥I 7, 9 11 ¬(A∨B) ¬I 8–10 19 1 ¬A∧¬B 2 ¬A ∧E 1 3 ¬B ∧E 1 4 A Ass. (LEM) 5 ⊥ ⊥I 2, 4 6 ¬(A∨B) ⊥E 5 7 ¬A Ass. (LEM) 8 A∨B Ass. (¬I) 9 A DS 3, 8 10 ⊥ ⊥I 7, 9 11 ¬(A∨B) ¬I 8–10 12 ¬(A∨B) LEM 4–6, 7–11 19 A Sample Proof 1 (P →Q) ∨(Q ∧R) 2 ¬Q 20 A Sample Proof 1 (P →Q) ∨(Q ∧R) 2 ¬Q 3 ¬Q ∨¬R ∨I 2 20 A Sample Proof 1 (P →Q) ∨(Q ∧R) 2 ¬Q 3 ¬Q ∨¬R ∨I 2 4 ¬(Q ∧R) DeM 3 20 A Sample Proof 1 (P →Q) ∨(Q ∧R) 2 ¬Q 3 ¬Q ∨¬R ∨I 2 4 ¬(Q ∧R) DeM 3 5 P →Q DS 1, 4 20 A Sample Proof 1 (P →Q) ∨(Q ∧R) 2 ¬Q 3 ¬Q ∨¬R ∨I 2 4 ¬(Q ∧R) DeM 3 5 P →Q DS 1, 4 6 ¬P MT 2, 5 20 A Sample Proof 1 (P →Q) ∨(Q ∧R) 2 ¬Q 3 ¬Q ∨¬R ∨I 2 4 ¬(Q ∧R) DeM 3 5 P →Q DS 1, 4 6 ¬P MT 2, 5 20 A Sample Proof 1 P →Q 21 A Sample Proof 1 P →Q 2 ¬(¬P ∨Q) Ass. (¬E) 21 A Sample Proof 1 P →Q 2 ¬(¬P ∨Q) Ass. (¬E) 3 ¬¬P ∧¬Q DeM 2 21 A Sample Proof 1 P →Q 2 ¬(¬P ∨Q) Ass. (¬E) 3 ¬¬P ∧¬Q DeM 2 4 ¬¬P ∧E 3 21 A Sample Proof 1 P →Q 2 ¬(¬P ∨Q) Ass. (¬E) 3 ¬¬P ∧¬Q DeM 2 4 ¬¬P ∧E 3 5 P DNE 4 21 A Sample Proof 1 P →Q 2 ¬(¬P ∨Q) Ass. (¬E) 3 ¬¬P ∧¬Q DeM 2 4 ¬¬P ∧E 3 5 P DNE 4 6 Q →E 1, 5 21 A Sample Proof 1 P →Q 2 ¬(¬P ∨Q) Ass. (¬E) 3 ¬¬P ∧¬Q DeM 2 4 ¬¬P ∧E 3 5 P DNE 4 6 Q →E 1, 5 7 ¬Q ∧E 3 21 A Sample Proof 1 P →Q 2 ¬(¬P ∨Q) Ass. (¬E) 3 ¬¬P ∧¬Q DeM 2 4 ¬¬P ∧E 3 5 P DNE 4 6 Q →E 1, 5 7 ¬Q ∧E 3 8 ⊥ ⊥I 6, 7 21 A Sample Proof 1 P →Q 2 ¬(¬P ∨Q) Ass. (¬E) 3 ¬¬P ∧¬Q DeM 2 4 ¬¬P ∧E 3 5 P DNE 4 6 Q →E 1, 5 7 ¬Q ∧E 3 8 ⊥ ⊥I 6, 7 9 ¬P ∨Q ¬E 2–8 21 |
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Your new article (Euclidean Division Algorithm) duplicates the pre-existing article Euclidean algorithm. I suggest you convert your article into a redirect and move any new content to the other article. --JBL 18:05, 19 February 2007 (EST)
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5957 | https://www.101cookingfortwo.com/how-to-roast-a-turkey-breast-the-easy-way/ | 101 Cooking For Two
Easy Oven-Roasted Turkey Breast at 350°F (No Fuss)
Last Updated: Sep 15, 2025 by Dan Mikesell AKA DrDan · 37 Comments
Learn how to oven-roast a turkey breast at 350°F the easy way—crispy skin, juicy meat, and no complicated steps or temperature flipping.
Cook any turkey breast this way. Previously frozen bone-in breasts are the most common, but boneless will cook a little faster, and fresh may benefit from a brine.
🤔 Quick Answer: How Long to Cook a 5 to 10 lb Turkey Breast
✅ 325°F works too, but it takes 1–2 minutes more per pound and the skin won’t brown as well. For the best combination of skin and meat, stick with 350°F.
Featured Comment from Norma :
⭐⭐⭐⭐⭐"This is the best, most straightforward instruction I have seen.
Thank you."
🧡 Why You’ll Love This Recipe
🦃 Ingredients
👨🍳Quick Overview: How to Bake a Bone-In Turkey Breast
1. Prep
1a. Completely thaw and pat the breast dry. Set aside or discard any extra parts, including gravy packets.
1b. Preheat oven to 350°F. Position the oven rack so the center of the turkey breast will sit in the middle of the oven—usually one or two positions below center.
1c. Prep a roasting pan with a rack (if available) and a good coat of cooking spray.
2. Remove the backbone and trim
If present, cut out the backbone using a heavy knife or kitchen shears. Trim off any excess skin (like the neck flap) as needed.
3. Break the rib sections
Use firm pressure to break back the rib sections so the breast sits flat and stable.
✅ Pro Tip: The natural weak spot is about an inch from the breastbone. Fold the ribs back to help support the breast while roasting.
4. Position the turkey
Move the breast to a roasting pan, using the broken-back rib sections to help stabilize it upright.
(See image for positioning—yes, that pop-up timer is just along for the ride.)
5. Season and roast
Brush with oil or melted butter and sprinkle with kosher salt. Add herbs or other seasoning if desired.
Roast at 350°F until the thickest part reaches 165°F—about 2 hours for a 7-pound breast.
✅ Pro Tip: Basting isn’t required, but I like to brush on a little more butter or oil halfway through. Rotate the pan 180° in case your oven has hot spots—most do, even if you don’t think so.
Let rest, tented with foil, for 10–15 minutes before carving. Perfect time to make gravy—see the recipe card for instructions.
👇 Scroll down for the printable recipe card and step-by-step photo instructions—or keep reading for extra tips, veggie sides, and serving ideas.
⏰ How long to cook a turkey breast
The general rule is 16 to 20 minutes per pound in a 350°F oven.
Estimated times at 350°F (planning only):
Cooking time depends on the thickness of the meat, not just the weight. Larger breasts tend to cook faster per pound than smaller ones.
✅ Boneless turkey breasts cook the same way at 350°F, but they’re usually a bit quicker per pound. Start checking with a thermometer 10–15 minutes earlier than you would for bone-in.
🛑 Important: Always cook to an internal temperature of 165°F in the thickest part—not by time alone. Residual heat may raise the temperature slightly after removing it from the oven.
🌡️ The best oven temperature for cooking turkey breast is 350°F
Stick with 350°F. It’s the sweet spot for crispy skin and juicy meat—without fussing with foil or juggling oven settings.
Some recipes tell you to start hot at 425°F, then drop low to 325°F for hours. It looks done early, but you spend the rest of the cook tenting to keep the skin from burning. One steady 350°F does the job without the fuss.
Yes, you can use 325°F, but it takes 1–2 minutes more per pound and the skin won’t brown as well. At 375°F, the skin browns too fast and you’ll likely end up tenting with foil.
A countertop turkey roaster also works well and frees up your main oven. Just use the same temperature and method.
Convection ovens are fine too, but not necessary. They can over-brown the skin, so if you use convection, reduce the temp to 325°F and watch the color.onvection, reduce the temp to 325°F and watch the color.
🌡️ The best turkey temperature is 165°F
The turkey breast is done when the thickest part reaches 165°F. That’s the safe, fully cooked temperature—and the key to juicy, tender meat. For more details, see the USDA turkey safety guidelines.
Don’t go over 165°F or you’ll dry it out. Don’t stop short either. Some recipes say to pull at 160°F and let carryover heat finish the job. It might get there, but I’d rather be sure I’m serving safe food.
Ignore the pop-up timer. They’re unreliable—sometimes too late (dry turkey), sometimes too early (unsafe). Always use an instant-read thermometer.
Don’t judge by color. Fully cooked turkey can still look slightly pink near the bone. That’s from natural compounds like myoglobin, not undercooking. If the internal temp is 165°F, it’s safe regardless of color.
✅ Boneless turkey breasts follow the same rule—finish at 165°F. They may cook a little faster, so start checking early.
😊 Tips for the best results
🧂 Seasoning
All you really need is kosher salt and a little butter—simple and traditional.
Want crispier skin? Use oil instead of butter.
Feel free to add fresh or dried herbs like sage, thyme, or rosemary. Garlic or dry rubs also work well. For extra flavor, you can tuck herb butter under the skin, but it’s not necessary for juicy results.
Save this recipe!
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Also send free occasional recipe updates—see details..
❓ How much turkey to buy per person?
Plan on ½ to 1¼ pounds per person. That sounds like a wide range—and it is—but both ends can be correct.
Remember, number of servings ≠ number of people. Many folks go back for seconds, and most want leftovers (especially if gravy is involved).
❄️ How to thaw a turkey breast
Never thaw turkey at room temperature. It’s not safe. Use one of these two methods:
✅ Refrigerator method (preferred)
Leave the turkey breast unopened in its original packaging and place it on a tray in the refrigerator.
⚡ Cold water method (faster)
🛒Shopping
Like most recipes, the success of cooking a turkey breast depends on obtaining the best final temperature. You must be able to monitor the end point of cooking accurately and avoid overcooking the skin.
Here are some suggestions I like, but you can find many more good products that will work at your local big-box store. All links below are affiliate links, meaning I make a small profit from your purchases. This commission does not affect your price. We are a participant in the Amazon Services LLC Associates Program. As an Amazon associate, I earn from qualifying purchases.
Proctor Silex Turkey Roaster Oven
Maverick XR-50 4 Probe Remote Thermometer
Thermapen™ One from Thermoworks™
ThermoPro TP19 Instant Read Thermometer
🧂 Should you brine a turkey breast?
Almost always, no. Most frozen turkey breasts are already injected or pre-brined, even if it’s not obvious on the front of the package. If you brine one of those, you’ll likely end up with mushy or overly salty turkey.
If you happen to have a completely unbrined, uninjected breast, brining can help—but that’s rare. And not needed here. Just follow the instructions and you’ll be fine.
How to brine a turkey breast if indicated?
If you are sure your turkey has not been injected or brined, you can add a simple brine of 1 gallon of water, 1 cup of salt, and ½ cup of sugar for 12 to 24 hours.
🥣 How to make turkey gravy
Of course, you want gravy—but with a turkey breast, you won’t get many pan drippings, so a classic roux-based gravy doesn’t really work.
Instead, this recipe uses a slurry method, combining the drippings you do get with chicken or turkey broth (or gravy base) for a smooth, flavorful result.
Full instructions are in the recipe card. For more on how to make gravy from scratch, see How To Make Gravy at Home.
🍴 Serving and related recipes
Besides the mandatory mashed potatoes and peas at our table, we also serve:
Cranberry sauce and other sides are always welcome—go with whatever fits your family traditions.
Looking for more turkey? Try these:
❄️What to do with leftovers
Store cooked turkey in an airtight container in the refrigerator for up to four days. It also freezes well for up to four months. Flour-based gravy stores the same way and reheats better than cornstarch-based versions.
We always save extra gravy for leftovers. Warm up chopped turkey in the gravy (add a splash of water if it’s too thick) and serve it over mashed potatoes, sausage dressing, or just by itself.
Want something a little more creative? Try one of these:
Easy Turkey Tetrazzini
Turkey Tetrazzini is the perfect leftover Thanksgiving turkey recipe. It features tender turkey, creamy sauce, pasta, vegetables, and a crispy Parmesan topping. You may want to cook extra turkey this year for this casserole.
❓FAQs
No—it’s not safe. For stuffing to be fully cooked, it would need to hit 165°, but by then the turkey would be dry and overdone. For best results, cook the stuffing separately.
No. Roast uncovered so the skin browns. A 5–7 pound breast takes about 1¾ to 2 hours at 350°F.
If you have a larger breast, the skin may brown before the meat is fully cooked—just tent lightly with foil near the end to prevent over-browning.
Yes. Boneless turkey breasts cook the same way at 350°F—finish at 165°F in the thickest part. They’re usually a little faster per pound, so start checking with a thermometer about 10–15 minutes earlier than you would for bone-in.
The key is to not overcook it—use a thermometer and pull it at 165°.
If your turkey isn’t injected or pre-brined (rare), brining can help, but most don’t need it.
⚕️Food safety
Treat all raw poultry as contaminated—wash your hands thoroughly before and after handling it.
Do not rinse raw poultry. It spreads bacteria through splatter, contaminating the surrounding area. For more, see Chicken… To Rinse or Not To Rinse?
Turkey must be cooked to an internal temperature of 165° in the thickest part. Use an instant-read thermometer—no guessing.
Cooking times are estimates only (including this recipe). They’re useful for planning, but you can’t cook safely by time alone.
📖The Recipe Card
Oven-Roasted Turkey Breast at 350°F (Easy & Juicy)
Video Slideshow
Ingredients
Step-by-Step Instructions Step-by-Step Photos ONOFF
Prep
Remove the backbone and trim
Break the rib sections
Season and roast
Optional Gravy
Recipe Notes
Pro Tips:
Your Own Private Notes
To adjust the recipe size:
You can adjust the number of servings above; however, only the amount in the ingredient list is adjusted, not the instructions.
Nutrition Estimate (may vary)
Editor's Note: First Published on March 25, 2018. Updated with expanded options, refreshed photos, and a table of contents to help navigation.
More Turkey Recipes
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Mama K says
December 25, 2024 at 11:34 pm
This was my first time to cook a turkey breast (without the rest of the bird), and I was actually just trying to figure out which side to place up in the pan when I stumbled upon this recipe. I had never even considered splitting anything apart, but I decided to give it a go. I am certainly glad that I did!! This was super easy, faster than the original packaging predicted, and the finished product was tender & delicious. I'm bookmarking this one for next time I need to be reminded of exactly "what did I do last time that worked out so well? 🤔" Thank you!!
David says
November 22, 2023 at 8:14 pm
I didn't want to rate this recipe, just put a comment that I had a 7lb turkey breast and it took 3.5 hours to get to 160 (in some places). it was 170 in other places and 155 in others. This seems to always happen to me when taking meat temps. Do you think my thermometer is broken? I made sure it was deep but not touching bone for readings. I can tell it seems very tough already. I'm afraid it will be too dry but I was just following the temp instructions.
Dan Mikesell AKA DrDan says
November 22, 2023 at 9:08 pm
Hi David,
Welcome to the blog.
About your issues. Something is wrong with the thermometer or the oven. Or the breast is not fully thawed. My suspicion is the thermometer. You can test it by boiling water on your stovetop and check the temp which will be 212°.
Since the meat is dry, it usually means overcooked. The oven is less likely to be far that far off, although prolonged cooking at a lower temp would dry out the meat.
Hope that helps.. my money is on the thermometer.
Dan
PS. forgot to add to fully preheat the oven before starting.
Dana Stieferman says
October 30, 2023 at 11:41 am
I need to do a whole turkey for a gathering. All I find on your site are for turkey breasts - can you get us a whole-turkey recipe for next year? I would prefer a smoked recipe, would like to try a grilled turkey, but a roasted recipe will do.
Thanks a bunch!@ Happy Thanksgiving.
Dan Mikesell AKA DrDan says
October 30, 2023 at 1:00 pm
Hi Dana
Welcome to the blog.
Email with attached PDF sent.
Dan
Jim Coggs says
April 14, 2022 at 8:24 am
Hi Dr. Dan
Can you put stuffing under the breast?
Maybe cook a little longer?
Dan Mikesell AKA DrDan says
April 14, 2022 at 11:36 am
Hi Jim,
Welcome to the blog and good question.
It is all about safety which means temperature. So yes, as long as you get the stuffing to 165 in all areas. But you do not want to overcook the turkey. So, remove the breast when it is 165 and you can leave the dress in to reach the required 165 while the breast rests to absorb the fluid back into the cells.
I see two issues, first, the pan is tied up with the dressing so my gravy is either skipped or another method is needed. And the second is the dressing will be cooked under the weight of the breast, leading a compressed texture I probably wouldn't prefer.
Hope that helps.
Dan
Anita says
November 25, 2021 at 6:20 pm
I found this recipe last year and forgot to save it and Thank Goodness I came across it again this year. It's just me and my husband right now and even if we do get invited over to family I always like to make turkey that weekend and have our own dressing and gravy. Yes, it's Thanksgiving Day right now and the breast is going in the oven in about 30 minutes. This time I will try and cut the backbone properly. And thank you for the Penzley's tip. I have family that is in a city right now where they have a location. I am going to ask them to pick it up for me. Thanks Dr. Dan!
Hi, I'm DrDan.
Welcome to 101 Cooking for Two, the home of great everyday recipes with easy step-by-step photo instructions.
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5958 | http://files.pucp.edu.pe/departamento/economia/DDD123.pdf | 123 TEORIA DE LA PRODUCCION Y COSTOS: UNA EXPOSICION DIDACTICA Leopoldo Vilcapoma DOCUMENTO DE TRABAJO 123 TEORIA DE LA PRODUCCION Y COSTOS: UNA EXPOSICION DIDACTICA Leopoldo Vilcapoma RESUMEN En este trabajo se examinan los diferentes conceptos derivados de la función de producción total y de la función de costos totales. Desde el punto de vista tecnológico el análisis se realiza considerando el sistema tecnológico de Leontief y el sistema tecnológico de von Neumann. Desde el punto de vista económico se asume que la unidad productiva es precio-aceptante en el mercado de factores. Para ello se muestra la "parametrización" de la tecnología, a partir de los procesos de producción, obteniendo la función de producción en el caso de la existencia de un número finito de procesos técnicos. Debido a que esta deducción no he podido encontrar en ninguna referencia bibliográfica, se presenta un breve apéndice matemático. Toda la exposición se realiza en términos muy simples, empleando la matemática elemental. ABSTRACT This paper presents different concepts related to total production functions and total cost functions. From the technological point of view, the analysis is made considering the Leontief technological system and the von Neumann technological system. From the economical point of view, it is assumed that the firm is price-taker in the factors market. This paper shows the parametrization of technology from the production processes, deducing a production function for a finite number of technological processes. Because this is a new result, it is explained in a brief mathematical appendix. This is a very simple exposition, using only the college mathematics. 2 TEORIA DE LA PRODUCCION Y COSTOS: UNA EXPOSICION DIDACTICA Leopoldo Vilcapoma Consideremos el caso de una unidad productiva, tal como se concibe en la teoría microeconómica tradicional. En el análisis de la relación entre el proceso de producción y los costos para una unidad simple de producción emplearemos los siguientes supuestos: (1) Existen solamente dos factores de producción, la mano de obra (L) y la tierra (T).1 Además, estos medios de producción son factores primarios, esto es, no son producidos ni producibles en el proceso considerado. (2) La duración de la jornada de trabajo (δ) es constante, además δ=1. Esto nos permite asociar de manera directa, los factores de producción L y T, con sus servicios.2 (3) La producción es obtenida directamente empleando los factores L y T. Esto es, no existen relaciones intersectoriales.3 (4) La producción es disyunta. El proceso de producción permite obtener un sólo bien. Obviamente, esto implica que no estamos considerando los otros elementos que salen del proceso de producción4 (mano de obra y tierra usadas, por ejemplo). Profesor del Departamento de Economía de la Pontificia Universidad Católica del Perú. El autor desea agradecer los valiosos comentarios de Adolfo Figueroa, Oscar Rodríguez y Cecilia Garavito. 1 No hay ninguna pérdida de generalidades si consideramos otros factores de producción. 2 En adelante nos referiremos a L (T) como los servicios de la mano de obra (servicios de la tierra), o simplemente como mano de obra (tierra). 3 En el análisis de una unidad simple de producción, en nuestros términos, no se requiere la consideración de las relaciones intersectoriales. Si el propósito es examinar las interrelaciones entre distintos procesos de producción al interior de una gran unidad de producción, podría ser conveniente tomar en cuenta las relaciones intersectoriales. Si el propósito es analizar una economía que es más o menos integrada con un modelo de equilibrio general, si sería crucial la considerar de tales interrelaciones (véase Leontief (1986)). 4 Existen modelos de producción, sobre todo en la tradición clásica tratan la producción conjunta. Véase por ejemplo, entre otros, von Neumann (1938), Roemer (1982). 3 Las condiciones de producción que asumimos son muy simples, tanto desde el punto de vista tecnológico como económico. Todos los aspectos tecnológicos del proceso de producción se resumen en la relación que existe entre la producción y los factores de producción empleados, dada cierta duración de la jornada de trabajo.5 Todos los aspectos económicos de la producción se expresan en la relación de los precios de los factores con los diferentes niveles de producción. Circunscribimos nuestro examen, como ya indicamos, al caso de una unidad simple de producción.6 I. PRODUCCION EN EL CASO DE UNA TECNOLOGIA DE RENDIMIENTOS CONSTANTES A ESCALA Desde el punto de vista tecnológico, asumiremos que la producción es de rendimientos constantes a escala. Esto significa que ante un variación proporcional en el empleo de todos los factores de producción relevantes, se obtiene una variación, en la misma proporción, en el máximo volumen de producción obtenible. Así, una duplicación de los factores permite obtener el doble del nivel de producción. Una reducción de todos los factores en 50% permite obtener, a lo más, la mitad de producción. En esta parte analizaremos el caso en que los procesos de producción relevantes sean de proporciones factoriales constante, luego consideramos el caso de proporciones factoriales variables. 1.1 Producción con el sistema tecnológico de Leontief Supongamos que los factores de producción, en el proceso considerado, son limitacionales.7 Este es un supuesto que hace Leontief para describir la estructura productiva y analizar el funcionamiento de la economía norteamericana.8 5 Un análisis detallado de la naturaleza de la función de producción se encuentra en Georgescu-Roegen (1973, 1976). Véase también Figueroa (1993). 6 Existe otras dimensiones del proceso de producción que no se considera en el presente análisis. Un aspecto importante es, por ejemplo, la organización y los aspectos sociales del proceso de producción (Coase (1992), Kreps (1990), entre otros). 7 Un factor se define como limitacional, si un incremento en su disponibilidad y uso es un condición necesaria, pero no suficiente para obtener incremento en los niveles de producción. Se dice que un factor es limitativo, si un incremento en su disponibilidad es condición suficiente para obtener incrementos en los niveles de producción. Véase Frisch (1963), Leontief (1951), Georgescu Roegen (1955). 4 Si la producción de cierto bien se describe a través del vector P1, tal vector muestra la combinación de insumos (L1 y T1) que permite obtener la producción de q1. Podríamos considerar el vector P1 como un vector observado empíricamente, o como un vector que proviene de los conocimientos tecnológicos. Si además asumimos que: 1) el método de producción empleado en P es el único existente; 2) existen rendimientos constantes a escala; 3) existe aditividad y divisibilidad en la producción; entonces, tal método de producción se puede representar en una escala unitaria Pu, donde se muestre los requerimientos de L y T para producir una unidad del bien; y además, todas las posibilidades de producción podría ser mostrada empleando el vector Pu.9 Con estos supuestos P1 y Pu describen el mismo método productivo. La diferencia entre Pu y P1 es únicamente de escala de operación. En el gráfico 1.1, se presente de manera completa los procesos de producción tal como han sido especificados. Tanto P1 como Pu, se encuentran en un espacio de 3 dimensiones. Las combinaciones de trabajo (L) y tierra (T) se encuentran en el subespacio de factores (L,T). Cualquier cambio en Pu implica un cambio tecnológico. Podemos definir en tal caso, aL = L / q, aT = T/ q, que denotan la cantidad de mano de obra (aL) y la cantidad de tierra (aT) para producir 1 unidad del bien q, que se puede emplear para definir los procesos: a T a = P L 1 q = P T 1 1 u 1 1 1 El significado de estos vectores es directo: dados todos los supuestos antes mencionados, las cantidades (mínimas) de L1 y T1, permiten obtener un nivel (máximo) de producción q1. O de manera equivalente, las cantidades aL y aT permiten obtener 1 de producto (como máximo). La representación de la tecnología por medio de los procesos P, en verdad, se reducen a un punto (el proceso Pu) o a una expansión lineal de dicho punto (q1, Pu). Cualquier otra combinación o emplea más factores o no es factible con la tecnología disponible. 8 Véase W. Leontief (1951, 1986). 9 Todos estos supuestos son únicamente con el objeto de simplificar el análisis. Por ejemplo, si la producción no fuera continua, entonces desde un punto de vista conceptual el tratamiento, cualitativamente sería similar. 5 A veces es conveniente, por simplicidad, expresar la relación entre insumos y producto en un espacio de 2 dimensiones. Para ello se considera las curvas de nivel del espacio (q, L, T). Entonces, los distintos niveles de producción se pueden expresar por un conjunto de combinaciones de trabajo y tierra tales que permitan obtener un mismo nivel de producción. Así, podemos definir una isocuanta de producción como el conjunto de combinaciones de trabajo y capital tales que proporcionan el mismo nivel de producción.10 Véase el gráfico 1.2, donde se muestra dos curvas de nivel, asociados a los niveles de producción q=1 y q=q1, de la superficie de producción correspondiente al gráfico 1.1. En general, la dependencia de la producción q respecto a los factores, se expresa con el concepto de "función de producción". Así, considerando la información contenida en el vector P, y con los supuestos mencionados, se puede expresar la función de producción: T] a , L a [ = ] a T/ , a L/ [ = ) T (L, f = q T L T L min min que describe máximo nivel de producción "q" que se puede obtener con las cantidades mínimas L y T de mano de obra y tierra, respectivamente. Obviamente, si aL es el requerimiento de factor i para obtener 1 unidad de producto, 1/aL = a es el producto por unidad del factor i. En adelante sólo emplearemos los coeficientes ai. 10 Existen varias maneras, no necesariamente equivalentes de definir una isocuanta de producción. Por ejemplo: 1. Como el máximo nivel de producto que se puede obtener para cada combinación de factores. 2. Como la cantidad mínima de factores necesarias para producir determinado nivel de producción. Ambos conceptos pueden ser esencialmente útiles según sea el caso. Véase por ejemplo Georgescu-Roegen (1955) donde se emplea la primera definición. 6 1.1.1 La producción en el largo plazo11 Si asumimos que la relación entre insumo y producto es tal que en la expresión ] a T/ , a [L/ = ) T (L, f = q T L min L y T son variables, entonces tenemos la función de producción de largo plazo. Tal función de producción se expresa gráficamente en el espacio de factores (y producto) por una superficie de producción (como se muestra en el gráfico 1.1). Además, las curvas de nivel de esta superficie son las isocuantas de producción, que en este caso tienen la forma de escuadra. Obviamente, estas isocuantas satisfacen las condiciones usualmente establecidas de convexidad y densidad12 13. Si examinamos el espacio de factores y producto (ver gráficos 1.1 y 1.2), en el punto C, el factor L es un factor limitativo, en tanto que el factor T es redundante. En el punto B el factor T es un factor limitativo, en tanto que el factor L es redundante. En el punto A tanto L como T son factores limitacionales: un incremento en L o T no es suficiente para aumentar el nivel de producción.14 Así, la identificación de un factor como limitacional, limitativo o redundante depende del punto o región del espacio de producción que se considera para caracterizar la redundancia, limitatividad o limitacional de los factores.15 Dada la tecnología y la disponibilidad de factores, las características de los factores de producción corresponden a cierta vecindad del espacio de producción. 11 "Largo plazo" en este caso significa que no existe ningún factor de producción fijo. "Corto plazo" significa que existe por lo menos algún factor fijo. 12 La importancia de la convexidad y la continuidad se hace evidente cuando consideramos la producción con varios procesos. 13 La explicación de estos conceptos se pueden encontrar en Nikaido(1978), Madden(1986). 14 En la representación gráfica, R1 denota la región del espacio de factores donde la mano de obra es limitada (y la tierra redundante), R3 la región donde la mano de obra es redundante (y la tierra limitativa), en R2 ambos factores son limitacionales. 15 Se podría considerar la dotación total de factores para caracterizar que tipo de factores son los considerados. 7 Como ya se indicó, el proceso de producción unitario resume la tecnología. Además, se asume que tal proceso de producción es reproducible en cualquier escala. En este caso, el presupuesto es que la escala de operaciones no es de importancia.16 Si existe cierto proceso "unitario" que permite obtener un nivel de producción q = 1, es posible replicar dicho proceso q1 veces para obtener cierta producción q = q1.17 Los diferentes niveles de producción en el largo plazo corresponden a los niveles de producción obtenidos a lo largo del rayo P, pues dado que no existen factores fijos en este caso, se puede modificar los factores de producción en la forma más conveniente, sin ninguna restricción en términos de la cantidad de factores, pues no hay ningún factor fijo. Por ejemplo, el nivel de producción deseado es q1, se puede obtener empleando por lo menos L1 y T1 de trabajo y tierra. Si queremos producir q2, la cantidad de insumos que permiten alcanzar dicho nivel de producción son L2 y T2, respectivamente. La producción de largo plazo, está expresada en los niveles de producción a los que se asocian los "niveles" de las isocuantas. 1.1.2 La producción en el corto plazo Para configurar un contexto de corto plazo asumamos que el stock de tierra está fijada en cierta magnitud T1. Dado este factor fijo, podemos obtener ciertos niveles de producción modificando el empleo del factor variable L. Así, obtenemos la curva de producto total de corto plazo (ver gráfico 2.1 y 2.2). Por nuestros supuestos, ante incremento en el empleo, el nivel de producción aumenta linealmente hasta el punto A, después del cual permanece constante, pues las cantidades adicionales de trabajo no son suficientes para que el producto total se incremente. En B, la mano de obra es redundante mientras que la tierra es limitante. En A ambos factores son limitacionales. La inclinación de la curva de producto total está relacionada con la "productividad del trabajo" de manera directa. Por ejemplo, si el requerimiento de trabajo por unidad de producto es 1/2, entonces con 10 unidades de L se tiene la 16 En algunos modelos puede ser importante la consideración explícita de la escala de producción. 17 Algunas precisiones sobre el carácter de tal replica han sido mencionadas en Arrow K. y F. Hahn (1977). 8 capacidad de producir 20 unidades de q. Si dicho requerimiento disminuye a la mitad, entonces con las mismas 10 unidades de L es posible obtener 20 unidades de q. La función de producción total de corto plazo (q (L, T1)) se puede expresar, algebraicamente, mediante la siguiente ecuación: L > L si L ) a (1/ L L 0 si L ) a (1/ = ) T (L, q 1 1 L 1 L 1 ≤ ≤ y su forma se muestra en el gráfico 2.2. 1.1.3 Las funciones de producto medio y marginal La evaluación del producto por unidad de trabajo depende del máximo nivel de producción alcanzable con ciertos niveles de empleo. En general, dado la tecnología, el producto medio de la mano de obra (PMeL) depende del nivel de producción y nivel de empleo de los factores. Formalmente, eso se observa en la siguiente ecuación. L > L si /L L ) a (1/ L L < 0 si ) a (1/ = q/L = ) (L Pme 1 1 L 1 L ≤ Como se observa, la curva de producto marginal del trabajo (PMgL) es constante e igual a 1/aL hasta el punto A" (Ver gráfico 3.1). Después disminuye a cero. Es claro que mientras existan cantidades disponibles de tierra, si el empleo aumenta en una unidad, entonces el producto aumentará en (1/aL). Para expresarlo en términos numéricos, siendo aL=1/2, si se aumenta en 1 la cantidad de mano de obra empleada, el producto total aumentará en el "margen" en 2 unidades. Si se alcanza el punto A, entonces cantidades adicionales de trabajo no pueden incrementar los niveles de producción (pues, la tierra se convierte en un factor limitativo o limitante). L > L si 0 L L < 0 si ) a (1/ = dq/dL = ) (L Pmg 1 1 L ≤ Aquí observamos lo siguiente: 9 (1) La relación general entre producto medio y marginal se mantiene. Mientras el producto medio es constante, el producto marginal es constante y además es igual el producto medio. Si el producto medio es decreciente entonces el producto marginal es constante y menor que el producto medio. Además, si el producto marginal es constante, el producto medio no tiene que ser necesariamente constante. (2) La denominada "ley de rendimientos finalmente decrecientes de un factor" también está presente en este caso particular de tecnología lineal. Tal ley tecnológica dice que el producto marginal de un factor variable disminuye en cierto punto, cuando existen factores fijos. Aquí dicha ley no se manifiesta como una disminución "suave," sino como una variación "abrupta" del producto marginal. 1.1.4 Los costos en el largo plazo Sean los niveles de producción q1, q2 y en general q, que se obtienen con los procesos [q1, L1, T1], [q2, L2, T2], [q, L, T]. Si ningún factor de producción es fijo, ¿cuáles serían los costos totales (mínimos) en que se incurre al usar (las cantidades mínimas necesarias de) factores para obtener tales niveles de producción? Si los precios de la mano de obra y tierra son w y r, respectivamente, entonces los costos totales estarán dados por las siguientes expresiones: rT + wL = CT(q) . . . T r + wL = ) q ( CT T r + wL = ) q ( CT 2 2 2 1 1 1 Dado que las cantidades de L y T que se emplean para obtener q están especificadas por Pu, podemos expresar, de manera equivalente: q ) aT r + aL (w = q aT r + q aL w = ) q ( CT q ) aT r + aL (w = q aT r + q aL w = ) q ( CT 2 2 2 2 1 1 1 1 10 o en general q ) aT r + aL (w = q aT r + q aL w = ) (q CT Si graficamos la función anterior relacionando q y CT(), obtenemos la curva de costos totales. Como estamos considerando las posibilidades de producción manteniendo variables todos los factores de producción, tenemos la función de costos de largo plazo CTLP(q). El resultado de una curva de costos de largo plazo lineal es consecuencia de los supuestos de rendimientos constantes a escala y que la empresa es precio-aceptante en el mercado de factores. Si consideramos la interacción de la empresa con otras empresas o tomando en cuenta la industria, estos supuestos no necesariamente conducen a este tipo de costo total.18 El costo mínimo de producir q1 es CT (q1), de producir q2 es CT (q2) y de q3 es CT (q3). La curva de costos de largo plazo relaciona los costos totales mínimos con diferentes niveles de producción asumiendo que todos los factores son variables y que los precios de los factores y la tecnología están dados (Ver gráficos 4.1 y 4.2). Obviamente, también se puede deducir la curva de costes de corto plazo. 1.1.5 Los costos en el corto plazo Para obtener la curva de costes de corto plazo consideramos la función de producción de corto plazo. Asumiendo la existencia del factor fijo T en el nivel T1, mientras que el factor L es variable. Los costos totales se pueden clasificar en costos fijos y costos variables. La presencia del factor fijo T está relacionado a un costo rT1, como tanto T1 y r están fijos, rT1 es un costo fijo para la empresa. Además, como no existen otros costos fijos asociados a otros factores, rT1 es el costo fijo total de la empresa. Dado que los incrementos en el nivel de producción se obtienen aumentando el nivel de empleo, a este incremento del empleo corresponde un aumento en el costo total. Por lo tanto, el costo total de corto plazo está constituido por el costo fijo (del factor fijo: tierra) y el costo variable: el costo variable (relacionado al factor variable: trabajo L). La siguiente expresión relaciona niveles de producción de corto plazo con los costos totales: 18 Véase por ejemplo, el trabajo clásico Viner (1931). 11 q, a w + T r = ) (q L w + rT = ) (q CTCP L 1 1 donde L(q) = aL q es el nivel de empleo necesario para alcanzar q. Se puede constatar, por lo menos gráficamente, que la curva de corto plazo nunca es menor que la curva de costos de largo plazo para ninguna combinación de insumos. 1.1.6 El costo medio y marginal Es conveniente responder la siguiente pregunta. ¿En cuánto se modifica el costo total cuando se obtiene niveles de producto adicionales? ¿Cuál es el costo promedio de determinado nivel de producto? El concepto de costo marginal (CMg) se refiere al costo adicional debido a cantidades adicionales de producción. El costo medio (CMe) se refiere al costo por unidad de producción. En términos formales: q / ) CT(q = ) CMe(q q / ) CT(q = ) CMg(q ∆ ∆ Si consideramos la función de producción de largo plazo, entonces los conceptos anteriores devienen en la función de costo marginal de largo plazo (CMgLP) y costo medio de largo plazo (MeLP). Las funciones de costo medio y costo marginal de largo plazo consisten en las siguientes expresiones: q / ) (q CTLP = a r + a w = ) (q CMeLP T L Esta ecuación expresa que en el largo plazo, el costo promedio de una unidad de q es w aL + r aT. Del mismo modo: a r + a w = ) (q CMgLP T L 12 Así, en el largo plazo, aumentar en una unidad el nivel de producción implica un costo adicional de w aL + r aT. Por otra parte, tenemos los costos medios y marginales de corto plazo: a w = ) (q CMgCP a w + q / rT = ) (q CMeCP L L 1 Se observa que, antes de la combinación óptima, el costo marginal de corto plazo es menor que el costo marginal de largo plazo. La explicación radica en que para obtener una unidad adicional en el corto plazo es necesario emplear solamente los requerimientos adicionales de L (y no de T, hasta T1, ya que tenemos disponible), mientras que en el largo plazo necesitamos emplear tanto trabajo como tierra. Por lo tanto, para incrementar en 1 el nivel de producción, en el largo plazo es necesario incurrir en un costo de waL + raT, el cual obviamente es mayor que waL. Esto no sucede en la "región" de producción donde el producto marginal del factor es cero. (Nótese que en el tramo no decreciente el CMgCP es indeterminado.) La curva de costo medio de corto plazo es decreciente y luego crece indefinidamente. Esto es así pues, llegado a ciertos niveles de producción cuando el trabajo es redundante, mayores niveles de empleo se traducirán únicamente en aumentos en los costos y no en la producción, pues existen restricciones tecnológicas: la tierra se convierte en un factor limitante. 1.1.7 Un modelo de producción del empresario capitalista Podemos definir una ecuación de describa el ingreso total (IT) que tiene la empresa: p.q = ) (q IT Si el precio de venta no se modifica antes distintos niveles de producción IT será una función lineal. Podemos definir una ecuación que describa los beneficios (B) de la empresa: 13 ) (q CT - ) (q IT = ) (q B 1.1.8 Un modelo de producción del empresario capitalista En el contexto de una economía de mercado, ¿cuál sería el nivel de producción de un empresario capitalista? La respuesta depende de una serie de consideraciones económicas y tecnológicas. Se requiere una serie de supuestos sobre el productor. Hagamos los siguientes supuestos: 1. La racionalidad del empresario es buscar hacer máximo los beneficios (B). 2. El empresario es precio aceptante tanto en el mercado de bienes como de factores. 3. La tecnología es de Leontief y está dada. 4. La empresa tiene un factor fijo de T = T1. 5. La jornada de trabajo esta dada. Entonces, la condición de equilibrio implica que el empresario produce siempre que su ingreso marginal sea mayor que su costo marginal y produce hasta que el factor fijo se convierta en limitativo. Para resumir, las variables exógenas son: 1. Los coeficientes tecnológicos: (aT, aL ) 2. Los precios de los factores (w y r) y del bien (p) 3. El factor fijo T=T1. Como los costos de los factores están dados, también lo está el costo fijo CF = rT1 14 En tanto que las variables endógenas son: 1. La producción (q) y empleo de factor variable (L). 2. Los costos variables (wL) y los costos totales (CT). 3. El ingreso total (IT) y los beneficios totales(B). 4. La distribución del ingreso (dy).19 ¿Cómo están relacionadas las variables endógenas a las exógenas? Esto requiere hacer un ejercicio de estática comparativa.20 Se puede probar que los signos de la relación entre las variables exógenas y endógenas son los como se indica en la siguiente tabla. Tabla 1 T1 p w r aL 21 aT 22 δ23 q + 0 0 0 0 + L + 0 0 0 0 + wL + 0 + 0 - + CT + 0 + + - + IT + + 0 0 0 + B + + - - - + dy 0 - + + - - 19 Existen varias manera de cuantificar la distribución del ingreso. En nuestro caso dy = w.L/B. 20 Desde un punto de vista formal, se puede probar que el equilibrio es estable y único. Sólo en este caso tiene sentido hacer la "estática comparativa". 21 En este caso, se asume que el efecto es una reducción de este coeficiente. 22 En este caso, como en el anterior, se asume que se da una disminución de este coeficiente. 23 En realidad no hay una teoría de la relación entre stocks y servicios dada la jornada laboral. Sería conveniente hacer los supuestos necesarios respecto a esta relación. Por fines de simplificar la exposición, en adelante no consideraremos las modificaciones en la jornada laboral. Para una discusión sobre la importancia de la dimensión temporal en el análisis del proceso de producción. Véase: Georgescu-Roegen (1965, 1976), Figueroa (1993). 15 Tenemos pues, un modelo muy particular que sirve para analizar el comportamiento de una empresa capitalista individual. 1.1.9 La demanda de trabajo y la oferta del productor Tomando en cuenta el contexto del empresario de la sección anterior, si consideramos los niveles de empleo asociados a diferentes niveles de salario, entonces obtendremos la demanda de trabajo. La demanda de trabajo consiste en el conjunto de combinaciones de tasa de salario y mano de obra tal que se maximizan los beneficios, asumiendo dados la tecnología, los precios de (los demás) factores y producto. Del mismo modo, si asociamos diferentes niveles de precios de venta del producto con los niveles de producción, obtenemos la función de oferta del producto, que consiste en las combinaciones de precio y cantidad ofrecida, tales que se maximizan los beneficios, asumiendo dados la tecnología, los precios de los factores. En este caso, dentro de determinados rangos, la gráfica de la demanda de trabajo y de la oferta de producto serán curvas perfectamente verticales en los planos (L, w) y (p, q) respectivamente. La demanda de trabajo para determinados niveles de salarios se muestran en el gráfico 3.2; donde se ilustra que la demanda está definida para cierto rango de la tasa de salarios. La explicación radica en que para salarios mayores a w1, no es rentable el empleo de L, y por lo tanto no es viable la producción desde el punto de vista del productor capitalista. La oferta del productor individual es perfectamente inelástica para ciertos argumentos de los precios de venta. Esta idea se ilustra en el gráfico 5.2. 1.2 Producción con el sistema tecnológico de von Neumann En su famoso artículo sobre crecimiento económico von Neumann24, asumió la existencia de una economía que dispone de un conjunto finito de procesos primarios. En esta parte de nuestra exposición consideramos el caso más simple de 2 procesos primarios. En la próxima sección consideraremos la presencia de "n" 24 J. von Neumann, "A Model of General Economic Equilibrium" (1938). 16 procesos primarios de producción. En ambos casos existe un número infinito de procesos de producción derivados. Las posibilidades tecnológicas de producción se podrían representar por los vectores P(1), P(2) y P(1, 2) = P= α P(1) + (1-α) P(2). Donde P(1) y P(2) son procesos primarios y P(1, 2) es un conjunto de procesos derivados. Una representación explícita de los procesos primarios estaría dado por los siguientes vectores, donde Pu1 y Pu2 permiten producir a una escala unitaria.25 a a T T a = P a = P L = P L = P 1 1 q q T(2) T(1) (2) (1) L(2) u2 L(1) u1 (2) (2) (1) (1) (2) (1) (2) (1) Obviamente, esta información se puede expresar también con el concepto de función de producción. 1.2.1 Producción en el largo plazo La producción en largo plazo, se puede expresar mediante la siguiente ecuación26: R en encuentra se T) (L, si a / T R en encuentra se T) (L, si ] a a - a a [ / T] a - a ( - L a - a [( = q R en encuentra se T) , si(L a / L 3 T(2) 2 T(2) L(1) T(1) L(2) L(2) (1) T(2) T(1) 1 (1) L Los niveles de producción en el largo plazo se pueden representar en una superficie de nivel, tal como se ilustra en los gráficos 6.1 y 6.2. Las isocuantas son las curvas de nivel de superficie de producción y muestran la convexidad usual27. 25 En la representación de cada de estos vectores, se expresa la producción, mano de obra y tierra respectivamente en cada componente. 26 El procedimiento a través del cual se obtiene la parametrización de la tecnología, expresada en principio por un conjunto de procesos de producción, hallando una expresión de la función de producción, se encuentra brevemente explicada en el apéndice matemático. 17 En este caso, desde el punto de vista del ahorro de los recursos, la zona de producción donde no hay desperdicio de recursos está dado por R2 región donde L y T son limitantes. En el subespacio R1 y R3, existe algún factor que es redundante. 1.2.2. Producción en el corto plazo Supongamos que disponemos de T2 como factor fijo (Gráfico 7.1). Dada la tecnología, los niveles de producción de corto plazo que se pueden obtener se asocian al producto alcanzable modificando las cantidades de mano de obra empleada, manteniendo constante la cantidad de tierra. Se observa también como en el caso anterior, el producto total obtenible depende de manera directa de la productividad del trabajo. Si fuera posible emplear tanto P1 y P2 en alguna escala convenientemente elegida y por lo tanto emplear procesos derivados, la curva del producto total sería una curva cóncava (aunque no estrictamente cóncava) en todo su dominio. Estos resultados se muestran en el gráfico 7.2, donde se grafican las curvas de producto total de corto plazo para los tamaños de tierra T1, y T2 respectivamente. Se puede observar que la curva de producto total es cóncava, además que las funciones de producción de corto plazo obtenidas con menores niveles de factor fijo son "envueltas" por las funciones que poseen un mayor nivel de factor fijo. De este modo, la curva de producto total de corto plazo sigue la siguiente regla: R en encuentra se ) T (L, si a / T R en encuentra se ) T (L, si ] a a - a a [ / T] ) a - a ( - L ) a - a [( = q(L) R en encuentra se ) T (L, si L ) a / (1 3 1 T(2) 2 1 T(2) L(1) T(1) L(2) L(2) L(1) T(2) T(1) 1 1 L(1) En R1, el producto total crece linealmente. Sin embargo, considerando R1 y R2 se observa que la producción aumenta a tasa decreciente (por segmentos). 27 Sin el supuesto de divisibilidad no sería posible obtener isocuantas convexas en el espacio de producción. Consecuentemente, las curvas de producto total de corto plazo, tampoco serían funciones cóncavas para todo nivel de mano de obra. 18 1.2.3 Las funciones de producto medio y marginal El producto medio del factor variable tiene la siguiente expresión: R en encuentra se ) T (L, si a / T R en encuentra se ) T (L, si ] a a - a a [ / ] L / T ) a - a ( - ) a - a [( = Pme(L) R en encuentra se ) T (L, si L ) a / (1 3 1 T(2) 1 2 1 T(2) L(1) T(1) L(2) 1 L(2) L(1) T(2) T(1) 1 1 L(1) El producto marginal es: R en encuentra se T) (L, si 0 R en encuentra se ) T (L, i s ] a a - a a [ / ] ) a - a [( = Pmg(L) R en esta en encuentra se ) T (L, si ) a / (1 3 2 T(2) L(1) T(1) L(2) T(2) T(1) 1 L(1) Desde un punto de vista geométrico, estos resultados se muestran en los gráficos 8.1 y 8.2. 1.2.4 Los costes totales Podemos determinar en este caso los costes mínimos asociados a cada nivel de producción. Para esto basta relacionar los costes asociados a las cantidades mínimas de insumos para obtener cierta producción. Si los precios de los factores están dados, si la producción de 1 unidad requiere de aL(1) de trabajo y aT(1) de tierra, la producción empleando solamente el P(1) estaría expresado por: q ) r a + w a ( = r R + w L = ) q ( CRLP (1) T(1) L(1) 1 1 (1) Si P(2) es el proceso empleado el costo está expresado por q r) a + w a ( = r R + w L = ) q ( CRLP (2) T(2) L(2) 2 2 (2) 19 ¿En qué casos emplearemos uno de los procesos? ¿Cuándo será conveniente usar una combinación de los procesos? La respuesta depende de los precios relativos. Para dar una respuesta, supongamos que el proceso P(2) es más intensivo en trabajo que el proceso P(1). Si el valor de m definido como: ] a - a [ / ] a - a [ = m L(2) L(1) T(2) T(1) y se busca producir con el mínimo costo, entonces si ambos. o P el , P el /r w = m P el emplear se /r, w > m P el emplear se entonces /r, w < m (2) (1) (2) (1) Es posible obtener una relación entre los costos y los niveles de producción. Como tales costos están asociados a los niveles de producción de largo plazo, tales funciones son pues funciones de costos de largo plazo (CTLP). (Véase gráficos 9.1 y 9.2) En el corto plazo tenemos cierto factor fijo T1, cuyo costo es rT1. Asociado a tal factor fijo tenemos un costo fijo (CF). Dado T1 la única manera de incrementar la producción es empleando cantidades adicionales de mano de obra. Si queremos producir una unidad adicional del bien q, es necesario emplear adicionalmente aL(1) de L, a un costo unitario de aL(1) w. El costo total de producir q unidades se puede expresar mediante la ecuación: L(q) w + rT = CTCP(q) 1 En particular, considerando el valor explícito de L(q), la mano de obra necesaria para producir q, se tiene: (q) CT > (q) CT para q, ] ) a - a [( / ] T ) a - a ( + q ) a a - a a [( w + rT = CTCP(q) R n encuentrae se ) T (L, si q, a w + rT T(2) T(1) 1 L(2) L(1) T(2) L(1) T(1) L(2) 1 1 1 L 1 20 La forma de la curva de costos totales está estrechamente relacionada con los procesos de producción empleados; por lo tanto, con las proporciones factoriales que se utilizan en cada caso. En el primer tramo se emplea enteramente el primer proceso P(1), luego se pasa a emplear ambos procesos en R2. La curva de costos totales empieza a aumentar a una mayor tasa en el segundo tramo debido a que la cantidad del factor variable por unidad de producción aumenta a medida que se expande la producción. Finalmente, en R3 no es posible incrementar la producción incrementando el empleo del factor variable (consiguientemente elevando los costos). 1.2.5 Los costos medios y marginales Las funciones de costo medio y costo marginal de largo plazo consisten en las siguientes expresiones: ) (q CMe > ) (q CMe para q, ] ) a - a [( / ] /q T ) a - a ( + ) a a - a a [( w + /q rT = CMeCP(q) R en ) T (L, si , a w + q / rt T(2) T(1) 1 L(2) L(1) T(2) L(1) T(1) L(2) 1 1 1 L 1 Esta ecuación expresa que en el largo plazo, el costo promedio de una unidad de q es w aL(1)+r aT(1), y dada la configuración de precios relativos que se asumieron, el proceso empleado es P(1). Los costos marginales de corto plazo se expresan en las siguientes fórmulas: ) (q CT > ) (q CT para q, R en ) T (L, si ] ) a - a [( / ] ) a a - a a [( w = CMgCP(q) R en ) T (L, si , a w 2 1 T(2) T(1) T(2) L(1) T(1) L(2) 1 1 L Se observa que, en R1, el costo marginal de corto plazo es menor que el costo marginal de largo plazo. La explicación radica en que para obtener una unidad adicional en el corto plazo es necesario solamente los requerimientos adicionales de L (y no de T, hasta T1, ya que tenemos disponible), mientras que en el largo plazo 21 necesitamos emplear tanto trabajo como tierra. Por lo tanto, para incrementar en 1 el nivel de producción, en el largo plazo necesitamos waL+raT, el cual obviamente es mayor que waL. Esto no sucede en la "región" de producción donde el producto marginal del factor es cero; aquí el costo marginal deviene en una magnitud indeterminada. (Véase los gráficos 10.1 y 10.2) La curva de costo medio de corto plazo es decreciente y luego crece indefinidamente. Esto es así pues, llegado a ciertos niveles de producción cuando el trabajo es redundante mayores niveles de empleo se traducirán únicamente en aumentos en los costos y no en la producción, pues existen restricciones tecnológicas: la tierra se convierte en un factor limitante. 1.2.6 La demanda de Trabajo y la oferta Si consideramos un empresario capitalista, podemos relacionar los niveles de empleo asociados a diferentes niveles de salario, entonces obtendremos, como antes, la demanda de trabajo. Del mismo modo, si asociamos diferentes niveles de precios de venta del producto con los niveles de producción, obtenemos la oferta de bienes producidos. Con los supuestos indicados, el gráfico de la demanda de trabajo es una curva con tramos perfectamente verticales en el plano (L, w), tal como se muestra en el gráfico 8.2. Para salarios mayores a w1, el nivel de empleo que hace máximo el beneficio es cero, si el salario se encuentra entre W1 y Wo, el empleo es L1. Finalmente, si el salario es menor Wo, el empleo es L2 en el plano (p, q), tal como se ilustra en el gráfico 10.2. Si el precio es superior a p1, el nivel de producción que hace máximo el beneficio es q4 si p se encuentra entre p1 y po la cantidad a producir será q2. Finalmente, si el precio es menor que po, la cantidad ofrecida será cero. 1.2.7 Un modelo de un empresario capitalista Sea un empresario capitalista. Además, supongamos que nuestro productor es precio aceptante en el mercado del bien que produce y en el mercado de los factores que emplea. ¿Cuál será la producción? ¿Cuál el nivel de empleo? La respuesta depende de una serie de consideraciones económicas y tecnológicas. Hagamos los siguientes supuestos: 22 1. La racionalidad del empresario es buscar hacer máximo los beneficios. 2. El productor no puede modificar los precios de q, L y T, esto es, que el productor es precio aceptante en le mercado de factores. 3. Que la tecnología es de von Neumann, con dos procesos primarios y está dada. 4. Que el empresario tiene un factor fijo de T = T1. 5. La jornada de trabajo está determinada exógenamente. El empresario produce eligiendo el nivel de producción que hace máxima la diferencia entre sus ingresos totales y costos totales. En particular, el empresario tiende a producir más siempre que su ingreso marginal sea mayor que su costo marginal, y a la inversa, el empresario tiende a reducir su producción si es que su ingreso marginal es menor que su costo marginal. Para resumir el modelo, las variables exógenas son: 1. La tecnología. (aT(1), aL(1) aT(2), aL(2)). La manera como operan este proceso. 2. Los precios de los factores (w y r) y del producto (p). 3. El factor fijo T=T1. Como los costos de los factores están dados, también lo está el costo fijo CF = rT1. 4. La duración de la jornada de trabajo δ. En tanto que las variables endógenas son: 1. La producción (q) y empleo de factor variable (L). 2. Los costos variables (wL) y los costos totales (CT). 3. El ingreso total(IT) y los beneficios totales (B). 4. El proceso tecnológico empleado (Pi). 5. La distribución del ingreso (dy). ¿Cómo están relacionadas las variables endógenas a las exógenas? En términos intuitivos se puede probar que, bajo determinadas condiciones, los signos de variaciones positivas en algunas variables exógenas sobre las variables endógenas indicadas, son la siguiente tabla: 23 Tabla 2 T1 W r p aL(1) aT(1) aL(1) aT(2) q + 0 0 0 + L + 0 0 0 + wL + + 0 - + CT + + + - + IT + 0 0 0 + B + - - - + Pi 0 P(1) 0 0 0 P(1) P(2) dy 0 + + - 1.3 Producción con el sistema tecnológico de von Neumann con n procesos En esta exposición consideraremos el caso de n procesos primarios, pero manteniendo el supuesto de 2 factores de producción. Supongamos que los procesos están ordenados en términos crecientes respecto a la intensidad de cierto factor, por ejemplo, la mano de obra. Las posibilidades tecnológicas de producción se podría representar por los vectores P ) -(1 + P = P = P , P , P (2) (1) (1,2) (2) (1) α α P ) -(1 + P = P = P , P , P (3) (2) (2,3) (3) (2) α α ......... P ) -(1 + P = P = P , P , P (n) 1) -(n n) 1, -(n (n) 1) -(n α α o en términos generales: P ) -(1 + P = P = P , P , P 1) + (i 1) + i (i, 1) + (i (i) α α con i= 0,1,..n-1. 24 a a T T a = P a = P L = P L = P 1 1 q q T(2) T(1) (2) (1) L(2) 2 L(1) 1 (2) (2) (1) (1) (2) (1) (2) (1) o expresado en una escala unitaria: a a T T a = P a = P L = P L = P 1 1 q q T(2) T(1) (2) (1) L(2) Un L(1) 1 (2) U2 (1) U1 (2) (1) (2) (1) Obviamente, esta información se puede expresar en el concepto de función de producción. 1.3.1 Producción en el largo plazo La producción en largo plazo, se puede expresar mediante la siguiente ecuación28: R en encuentra se ) T (L, si a / T 1 -n .. 1 = i para , R cono el en encuentra se T) (L, si ] a a - a a [ / ] )T a - a ( - )L a - a [( = q R en encuentra se T) (L, Si a L/ 3 T(n) i) 2(i, 1) + T(i L(i) T(i) 1) + L(i 1) + L(i L(i) 1) + T(i T(i) 1 L(1) Los niveles de producción en el largo plazo se pueden representar en una superficie de nivel. Como ya se vio antes, las curvas de nivel son las isocuantas de producción y presentan convexidad usual. Si excluyéramos el supuesto de divisibilidad no sería posible obtener isocuantas convexas en el espacio de producción. 28 Esta ecuación es una representación analítica de un sistema tecnológico con n procesos primarios. Importantes implicancias para el análisis del cambio técnico y crecimiento de países subdesarrollados, se pueden encontrar en Stiglitz-Akerlof (1969), Eckaus (1955). 25 En este caso, desde el punto de vista del ahorro de los recursos, la zona de producción donde no hay desperdicio de recursos está dado por R2 región donde L y T son limitantes. En el subespacio que no es el cono, existe algún factor que es redundante. 1.3.2. Producción en el corto plazo Aquí disponemos de T como factor fijo. Los niveles de producción de corto plazo dependen de la cantidad de mano de obra, dado la cantidad constante de tierra y la tecnología. Se observa también como en el caso anterior, el producto total obtenible depende de manera directa de la productividad del trabajo y esta se encuentra asociada al nivel de empleo. Como es posible emplear tanto P(i) y P(i+1) en alguna escala convenientemente elegida y, por lo tanto, emplear procesos derivados, la curva de producto total es una curva cóncava en todo su dominio. De este modo, la curva de producto total de corto plazo sigue la siguiente regla: R en encuentra se ) T (L, si a / T R en encuentra se T) (L, si ] a a - a a [( / ] )T a - a ( - )L a - a [( = (L) q R en esta ) T (L, (Si L < L si ) L a (1/ 3 1 T(n) 2 1) + T(i L(i) T(i) 1) + L(i 1) + L(i L(i) 1) + T(i T(i) 1 1 1 L(1) En R1, el producto total crece linealmente. Sin embargo, aumenta a tasa decreciente, considerando los diferentes pares de procesos primarios en R2, los aumentos en la producción es con producto marginal disminuye por tramos sucesivos. Cuanto mayor sea el número de procesos y más amplio el ángulo del cono R2, la curva de producto total y las isocuantas serán más parecidas a las isocuantas que se construyen asumiendo la existencia de infinitos procesos29. 29 La relación existente entre procesos primarios representados con isocuantas de "coeficientes fijos" y las isocuantas suaves y continuamente diferenciables ha sido mencionado por Stiglitz-Akerlof (1969). 26 1.3.3 Las funciones de producto medio y marginal El producto medio del factor variable tiene la siguiente fórmula: R en encuentra se ) T (L, si a / T R en encuentra se ) T (L, si ] a a - a a [( / ] L / T ) a - a ( - ) a - a [( = (L) Pme R en esta ) T (L, (Si L < L si L a 1/ 3 1 T(n) 1 2 1 1) + T(i L(i) T(i) 1) + L(i 1 1) + L(i L(i) 1) + T(i T(i) 1 1 1 L(1) El producto marginal es: R en encuentra se ) T (L, i s 0 R en encuentra se T) (L, si ] a a - a a [( / )] a - a [( = (L) Pmg R en esta T) (L, (Si L < L si a 1/ 3 2 1) + T(i L(i) T(i) 1) + L(i 1) + T(i T(i) 1 1 L(1) 1.3.4 Los costes totales Podemos determinar en este caso los costes mínimos asociados a cada nivel de producción. Para esto basta relacionar los costes asociados a las cantidades mínimas de insumos para obtener ciertos niveles de producción. Si los precios de los factores están dados, si la producción de 1 unidad requiere de a(i)L de trabajo y a(i)T de tierra. La producción empleando solamente el P(i) estaría expresado por: r a + w a = 1) = q ( CTPL (i)T (i)L (i) Si P(j) es el proceso empleado el costo esta expresado por r a + w a = 1) = q ( CTPL (j)T (j)L (j) ¿En qué casos emplearemos uno de los procesos, o cierta combinación de los procesos? La respuesta depende de los precios relativos. Para dar una respuesta, 27 supongamos que el proceso P(i) es más "trabajo intensivo" que el P(1). Entonces, se puede probar que si el valor de m ] a - a [ / ] a - a [ = m (j)L (i)L (j)T (i)T ij es tal que ambos. o P el , P el /r w = m P el emplear se /r, w > m P el emplear se entonces /r, w < m (j) (i) ij (j) ij (1) ij Como son tales costos están asociados a niveles de producción de largo plazo, tales funciones de costos de largo plazo (CTLP). En el corto plazo tenemos cierto factor fijo T1, cuyo costo es rT1. Asociado a tal factor fijo tenemos un costo fijo (CF). La única manera de incrementar la producción es empleando cantidades adicionales de mano de obra. Si queremos producir una unidad adicional del bien q, es necesario emplear adicionalmente aL(i) de L, a un costo unitario de aL(i)w. El costo total de producir q unidades se puede expresar mediante la ecuación: L w + rT = CTCP (q) 1 (q) En particular: CT > CT para q, ] a - a [( / ] T ) a - a ( + q ) a a - a + a [( w + rT [ = CTCP R en ) T (L, si q, a w + T r (q) (q) 1) + T(i T(i) 1 1) + L(i L(i) 1) + T(i L(i) T(i) 1) + L(i 1 (1) 1 1 L 1 La forma de la curva de costos totales está estrechamente relacionada con los procesos de producción empleados; por lo tanto, con las proporciones factoriales que se utilizan en cada caso. En el primer tramo se emplea enteramente el primer proceso P1, luego se pasa los procesos P(1) y P(2), ... P(i) y P(i+1). La curva de costos totales empieza a aumentar a una mayor tasa en el segundo tramo debido a que la cantidad del factor variable por unidad de producción aumenta a medida que se 28 expande la producción. El resultado de las curvas de costos de corto plazo convexas es consecuencia de curvas de producto total cóncavas. 1.3.5 Los costos medios y marginales Las funciones de costo medio y costo marginal de largo plazo consisten en las siguientes expresiones: CMe > CMe para q, ] a - a [( / ] /q T ) a - a ( + ) a a - a a [( w + q / rT = CTCP R en ) T (L, si , a w + q / T r (q) (q) 1) + T(i T(i) 1 1) + L(i L(i) 1) + T(i L(i) T(i) 1) + L(i 1 1 1 L(1) 1 Esta ecuación expresa que en el largo plazo, el costo promedio de una unidad de q es w aL(1)+r aT(1). Con la configuración de precios relativos que se asumió el proceso empleado es P(1). Los costos marginales de corto plazo se expresan en las siguientes formulas: CT > CT para q, R en ) T (L, si )] a - a [( / ] ) a a - a a [( w = CMgCP R en ) T (L, si , a w (q) (q) 2 1 T(2) T(1) T(2) L(1) T(1) L(2) (q) 1 1 L Se observa que el costo marginal de corto plazo es menor que el costo marginal de largo plazo, antes de llegar a R3. La explicación es directa: para obtener una unidad adicional en el corto plazo es necesario solamente los requerimientos adicionales de L (y no de T, hasta T1, ya que tenemos disponible), mientras que en el largo plazo necesitamos emplear tanto trabajo como tierra. Por lo tanto, para incrementar en 1 el nivel de producción, en el largo plazo necesitamos waL+raT, el cual obviamente es mayor que waL. Esto no sucede en la "región" de producción donde el producto marginal del factor es cero. La curva de costo medio de corto plazo es decreciente y luego crece indefinidamente. Esto es así pues, llegado a ciertos niveles de producción cuando el 29 trabajo es redundante mayores niveles de empleo se traducirán únicamente en aumentos en los costos y no en la producción, pues existen restricciones tecnológicas para esto. 1.4. Una digresión: El caso de infinitos procesos de producción primarios En esta sección haremos una breve explicación de las consecuencias de incrementar el número de procesos primarios. Si el número de procesos primarios aumenta, intuitivamente se puede observar las isocuantas, las funciones de producción y de costo de corto plazo, se van configurando como funciones suaves tal como se muestra en los libros de texto y en el gráfico 12.1. Sin embargo, conviene advertir las siguientes posibilidades: (1) Aunque el número de procesos considerados sea infinito, las isocuantas pueden no ser "suaves" y diferenciables en todos los puntos del espacio de factores. (2) Para que la isocuanta resulte en una curva "suavemente" diferenciable no es necesario que existan "infinitos procesos" en cierta vecindad finita. Puede no ser el caso, y aún ser la isocuanta una curva "suave". Si asumimos que los precios de los factores y la tecnología están dados, isocuantas, curvas de producción y costos "suaves" satisfacen las siguientes relaciones: (1) Si las isocuantas son convexas y el producto marginal de todos los factores es innegativo, entonces la curva de producción total de corto plazo es cóncava. (2) Si la función de producción es cóncava, entonces la curva de costos tiene que ser convexa. 30 APENDICE: Acerca de la parametrización de la tecnología Es frecuente representar directamente la tecnología por una función apropiada que reproduzca las propiedades teóricas que se asumen sobre la naturaleza de los procesos de producción. Es posible generar a través de un procedimiento constructivo, a partir de los procesos primarios de producción, la función de producción que relacione las cantidades de insumos empleados con el máximo nivel de producción obtenible. Consideremos en principio el caso más simple, donde existen dos procesos primarios de producción: P(1) y P(2). Además asumiendo rendimientos constantes a escala, divisibilidad y aditividad, es posible construir conjunto de procesos derivados P(1,2). Este último proceso genera un cono convexo, que en nuestra exposición se ha denotado por R2. La producción está obviamente determinada si (L,T) pertenece a la región R1 y R3, como ya se señaló en el texto. La determinación de la función producción en R2 es la que merece cierta explicación. Si se quiere lograr la producción asociada al vector P = [q L T], entonces debemos hallar la combinación convexa λ(1) + (1-λ) P(2) tal que multiplicada k veces nos permite obtener el vector P. Esto es: ] [q P = ) P ) -(1 + P ( k (2) (1) λ λ A partir de este sistema de ecuaciones se puede encontrar la expresión del producto total cuando (L,T) se encuentra en R2. De modo que, la función de producción tiene la siguiente expresión algebraica: 31 R en encuentra se ) T (L, si a / T R cono el en encuentra se T) (L, si ] a a - a a [( / ] )T a - a ( - )L a - a [( = T) q(L, R en esta T) (L, si a L/ 3 T(2) 2 T(2) L(1) T(1) L(2) L(2) L(1) T(2) T(i) 1 L(1) La generalización al caso de n procesos primarios es directa. 32 33 34 35 36 37 38 39 40 41 42 BIBLIOGRAFIA ATKINSON, A. Y STIGLITZ, J. (1969) "A New View of Technological Change," The Economic Journal, Vol. LXXXV. Setiembre. AKERLOF, G. Y STIGLITZ, J. (1969) "Capital, Wages and Structural Unemployment," The Economic Journal, Vol. LXXXV. Junio. ARROW K. Y HAHN F. (1977) Análisis General Competitivo, Editorial Fondo de Cultura Económica, México. COASE, RONALD H. (1992) The firm, the Market and Law, The University of Chicago Press, Chicago. ECKAUS, RICHARD (1955) "El Problema de la proporciones en el uso de los factores en los países subdesarrollados," en Breit y Hochman (eds.) (1973) Microeconomía, México, Interamericana. FRISCH, RAGNAR (1963) Las Leyes Técnicas y Económicas de la Producción, Sagitario, Barcelona. GEORGESCU-ROEGEN, N. (1967) Analytical Economics, Harvard University Press, Cambridge. (1955) "Limitationality, Limitativeness and Economic Equilibrium," en Georgescu Roegen (1967). (1976) Energy and Economic Myths, Pergamon Press, New York. FIGUEROA, ADOLFO (1993) "Algunas Notas sobre la Teoría de la Producción," Serie Ensayos Teóricos, No. 1, Marzo de 1993. HAHN, FRANK (1971) Readings in the Theory of Growth, St. Martin Press, London. KREPS, D. (1990) A course in Microeconmic Theory, Princeton University Press, NJ. LEONTIEF, WASSILY (1986) Input-Output Economics, Oxford University Press, New York. 43 (1986) The Structure of American Economy 1919-1939, Oxford University Press, New York, 1951. (Hay edición en español). MADDEN, P. (1986) Concavidad y Optimización en Microeconomía, Alianza Editorial, Madrid. NIKAIDO, H. (1978) Métodos Matemáticos del Análisis Económico Moderno, Ed. Vicens-Vives, Barcelona. ROEMER, JOHN (1982) A General Theory of Explotation and Class, Harvard University Press, Cambridge. STIGLER, GEORGE (1968) Ensayos sobre la teoría de los precios, Ed. Aguilar, Madrid. VON NEUMANN, J. (1938) "A Model of General Economic Equilibrium" (1938), reimpreso en F. Hanhn (1971). VINER, JACOB (1931) "Curvas de Coste y Curvas de Oferta," en Stigler G. y Boulding K. (eds.) (1968) Ensayos sobre la teoría de los Precios, Ed. Aguilar, Madrid. 44 45 PUBLICACIONES Libros Adolfo Figueroa (1993) Crisis Distributiva en el Perú. Pontificia Universidad Católica del Perú - Fondo Editorial. Mario D. Tello (1993) Mecanismos Hacia el Crecimiento Económico. Fondo Editorial. Pontificia Universidad Católica del Perú. Consorcio de Investigación Económica. Máximo Vega-Centeno (1993) Desarrollo Económico y Desarrollo Tecnológico. Pontificia Universidad Católica del Perú - Fondo Editorial. Adolfo Figueroa (1992) Teorías Económicas del Capitalismo. Fondo Editorial. Pontificia Universidad Católica del Perú. Serie Documentos de Trabajo No. 115, Gloria Canales, "Dolarización y Fragilidad Financiera en el Perú". Noviembre, 1993. No. 116, Oscar Dancourt, Jorge Rojas "El Perú desde 1990: El Fin de la Restricción Externa". Noviembre, 1993. No. 117, Oscar Dancourt, "Sobre el Retraso Cambiario y la Repatriación de Capitales en una Economía Dolarizada". Noviembre, 1993. No. 118, Alan Fairlie Reinoso, "Una Lectura Peruana del Plan de Convertibilidad Argentino". Febrero, 1994. No. 119, Felix Jiménez, "El dinero y Relación con los Precios: Del Monetarismo Neoclásico al Tratado del Dinero de Keynes". Setiembre, 1994. No. 120, Felix Jiménez, "Dinero, Inversión y Financiamiento: Apuntes sobre el Discurso Teórico de J.M. Keynes". Setiembre, 1994. No. 121, Cecilia Garavito, "Oferta de Trabajo en Lima Metropolitana: 1989-1992," Mayo 1995. No. 122, Waldo Mendoza, "Dinero, Tipo de Cambio y Expectativas," Setiembre 1995. Serie Informes de Coyuntura Informe de Coyuntura: Perú: 1994. Oscar Dancourt, Waldo Mendoza y Lucía Romero, Marzo 1995. Informe de Coyuntura: Primer Trimestre de 1995. Oscar Dancourt, Waldo Mendoza y Lucía Romero, Mayo 1995. Informe de Coyuntura: Segundo Trimestre de 1995. Oscar Dancourt, Waldo Mendoza y Lucía Romero, Julio 1995. Informe de Coyuntura: Tercer Trimestre de 1995. Oscar Dancourt y Waldo Mendoza, Julio 1995. |
5959 | https://math.stackexchange.com/questions/4391205/solving-xs-at-y-for-quadratic-given-one-point-origin-at-0-0 | geometry - Solving x's at y for quadratic given one point, origin at (0,0) - Mathematics Stack Exchange
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Solving x's at y for quadratic given one point, origin at (0,0)
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I have a parabola where I know the origin is (0,0)(0,0), I know a single point, and I know it's symmetrical on the Y axis. I only need positive or negative x x, I can work with either. It has been a couple decades, I am rusty.
Given any other y y, I want to find x x.
So far, I think I'm getting a a correctly. For my known (x,y)(x,y) point I plug them into:
a=y/x 2 a=y/x 2
Then I'm trying to use that a a value to find x x given y y.
I think, given 0=(b+s q r t(b−4 a c))/2 a 0=(b+s q r t(b−4 a c))/2 a, I'm trying to find an X-intercept given the Y-intercept is at (0,−y)(0,−y), basically an offset of my known y y. Then b b is zero, because the origin X is at 0, but this may be where I'm going wrong?
0=s q r t(4 a c)/2 a 0=s q r t(4 a c)/2 a
Where c c = −y−y. I guarantee that y y is the negative number, so no imaginaries under the sqrt.
So for known c c and a a, that should give me x x?
But the values I'm getting don't line up with visual parabola I have. (Full disclosure, this is a programming thing, I'm trying to make some things line up with a half-parabola-curve where all I really know are width and height (x and y), that it's symmetrical, and starting at 0,0. I know various vertical positions of the things I'm trying to line up horizontally.)
geometry
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edited Feb 25, 2022 at 21:51
KReiser
75.6k 16 16 gold badges 65 65 silver badges 118 118 bronze badges
asked Feb 25, 2022 at 21:49
Randy HallRandy Hall
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1 Is the origin part of the parabola? Then c=0 c=0, and the problem is solvable. You are given (x 0,y 0)(x 0,y 0), and a point (x,y)(x,y), but in the second case you only know y y, and not x x. Then write a a in terms of (x 0,y 0)(x 0,y 0) and in terms of (x,y)(x,y). You will get y 0 x 2 0=y x 2 y 0 x 0 2=y x 2 You have only one unknown, so it should be easy to solve. Don't even need the quadratic equation, just take a square root.Andrei –Andrei 2022-02-25 22:05:40 +00:00 Commented Feb 25, 2022 at 22:05
So using h/w 2=y/x 2 h/w 2=y/x 2 should be like x=s q r t(y/(h/w 2))x=s q r t(y/(h/w 2)) ? (not sure how to do the subscript things)Randy Hall –Randy Hall 2022-02-25 22:14:37 +00:00 Commented Feb 25, 2022 at 22:14
a=y x 2⟹x 2=y a.a=y x 2⟹x 2=y a.user2661923 –user2661923 2022-02-25 22:15:19 +00:00 Commented Feb 25, 2022 at 22:15
1 @RandyHall correct. Use $x_0$ to get x 0 x 0, and $\sqrt{\frac{x_0^2 y}{y_0}}$ to get x 2 0 y y 0−−−√x 0 2 y y 0 Andrei –Andrei 2022-02-26 04:05:41 +00:00 Commented Feb 26, 2022 at 4:05
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5960 | http://www.math.emory.edu/~dcai7/day3.html | MATH 111 Calculus I (\quad\;\;) Instructor: Difeng Cai
1.3 Transformation and Composition of Functions
Transformation of functions
Suppose (f(x)) is a function
with domain ([a,b]) and range ([p,q]).
The domain and range after transformation are shown below.
| transformation | domain | range | action |
--- --- |
| (f(x)) | ([a,b]) | ([p,q]) | debut |
| (f(x)+s) | ([a,b]) | ([p+s,q+s]) | vertical shift by (s) |
| (f(x+s)) | ([a-s,b-s]) | ([p,q]) | horizontal shift by (-s) |
| (s f(x)) | ([a,b]) | ([sp,sq]) | vertical scaling by (s) |
| (f(sx)) | ([\frac{a}{s},\frac{b}{s}]) | ([p,q]) | horizontal scaling by (\frac{1}{s}) |
| (-f(x)) | ([a,b]) | ([-q,-p]) | reflection about (x)-axis |
| (f(-x)) | ([-b,-a]) | ([p,q]) | reflection about (y)-axis |
Trick: You can actually tell what the transformation does to the graph of f(x)
from the change in the domain or range, e.g., a stretch, shift, reflection, etc.
Composite functions and their domains
Definition.
the composite function (f\circ g) is defined by ((f\circ g)(x) = f( g(x) )). Namely, first apply (g) to (x), then apply (f) to the result.
The domain of (f\circ g) is the set of all (x) such that both (g(x)) and (f(g(x))) are well-defined.
Example. (f(x)=\sqrt{x}), (g(x)=\sqrt{2-x}).
Find the domain and expression of (g\circ f).
Solution:
We know that domain of (f) is ([0,\infty)) and
domain of (g) is ((-\infty,2]).
((g\circ f)(x) = g\left(f(x)\right)).
For (g\left(f(x)\right)) to make sense, we require:
(f(x)\in (-\infty,2]) and (x\in [0,\infty)), which implies,
(x\in [0,4]) .
Thus we conclude that ([0,4]) is the domain of
(g\circ f=g(f(x))=\sqrt{2-\sqrt{x}}).
Decompose complicated functions
Example. (F(x)=\sqrt{3+x^2}). find (f,g) such that (F=f\circ g).
Solution:
Given an input, (F) does this:
You can check that (F=f\circ g).
1.4 Exponential Functions
(f(x)=b^x) is an exponential function because the variable, (x) is the exponent. The base (b) is a positive constant.
If (b\neq 1), domain: (\mathbb{R}=(-\infty,\infty)); range: ((0,\infty)).
Examples.
If (n) is a positive integer,
[f(n)=b^n=b \cdots b \quad (n \text{ factors}). ]
Examples. (f(0)=b^0=1) and
[f(-x)=b^{-x}=\frac{1}{b^x}.]
The Number (e)
In addition to the renowned superstar number (\pi\approx 3.14159\dots), there is another number that is special enough to be endowed with its own symbol. That number is [e\approx 2.71828\dots]
The notation (e) was chosen by Euler (a famous Swiss mathematcian) and
the number is often called Euler's number.
We will see why the number (e) is singled out later when we learn derivatives of functions.
Graph
Graphs of (f(x)=a^x) for different values of (a):
Example. Sketch the graph of the function (y=3-2^x) and determine its domain and range.
Domain: (\mathbb{R}); range: ((-\infty,3))
Ex. Find the domain of each function.
((a).\; f(x)=\frac{1+e^x}{1-e^{x}}\quad
(b).\; g(t)=\sqrt{10^t-100}\quad
(c).\; f(t)= \frac{\cos t}{e^t})
Answer: (a). (x\neq 0); (b). (t\geq 2); (c). ((-\infty,\infty))
Law of Exponents
Law of Exponents:
[b^{x+y}=b^x b^y,\quad (b^x)^y=b^{xy},\quad (ab)^x=a^x b^x]
Ex. Simplify the following expressions:
[(a)\; 8^{4/3}\quad (b)\; \frac{1}{\sqrt{x^4}}\quad (c).\; x(3x^2)^3 \quad (d).\; \frac{x^{3}x^{2k}}{x^{k+2}} ]
Answer: ((a). 2^4=16 \quad (b). x^{-4/3}\quad (c). 27x^{7}\quad (d). x^{k+1})
Applications (Modelling with exponential functions)
Example. Assume the volume of ice (F) on an unknown planet decays with respect to time (t)(year) in the following pattern:
every 5 years, the volume shrinks by half.
If the volume today (year (0)) is (1836) degrees. Find the volume (F) at year (t).
Solution:
[
\begin{aligned}
F(0) &=1836 \
F(5) &= \frac{1}{2}(1836) \
F(10) &= \frac{1}{2}\frac{1}{2}(1836)=\frac{1}{2^2}(1836) \
F(15) &= \frac{1}{2}\frac{1}{2^2}(1836)=\frac{1}{2^3}(1836)
\end{aligned}
]
We deduce that
[F(t) = \frac{1}{2^{t/5}}(1836)=1836 (2^{-1/5})^t.] |
5961 | https://www.oxfordlearnersdictionaries.com/us/definition/american_english/incongruous | Definition of incongruous adjective from the Oxford Advanced American Dictionary
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5962 | https://www.youtube.com/watch?v=TGhBafCN6ms | Percentages Multiplicative Reasoning
MrModoniMaths
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Posted: 5 Dec 2020
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percentages again multiplicative reasoning is going to be probably applied here kirsten buys a laptop okay we understand that she gets discount of 20 off the normal price so that means if she gets a discount off 20 off how much are we left with we're going to be left with 80 percent to pay then it goes on to say kirsten pays 480 for the laptop so this is how much he has to pay so we can set up multiplicative reasoning with say 480 pounds is how much we have to pay which is equal to 80 percent okay so that's from that so set up multiplicative reasoning work out the discount the discount is 20 so i need to get to 20 so how do i go from there to there divide by four how do i go from there to there divide by four okay so 480 divided by four that's 120. okay um and there you have it that's the answer so again we saw multiplicative reasoning used with percentages now every single question with percentages if you can find out what the quantity i.e the amount pounds whatever it is is as a percentage you can always start off like that and you should be the only question where you might not use multiplicative reasoning is if it is com um should have been relatively straightforward dual buys a washing machine okay understand that 20 v80 is added on so when v8 is added on you're adding it onto the hundred percent so if it's added on what where do we end up at we end up at one hundred and twenty percent joules then pays a total of six hundred so six hundred pounds is equi what they pay which is the same as 120 because that's v80 is added on to the original okay originals always 100 so from there we can use multiplicative reasoning what is the price of the washing machine with no v80 we want to get to 100 percent so from 120 200 you might not go there directly you might find out first of all what 10 is to get to 10 you divide by 12. so you could divide by 12 um here as well so 600 divided by 12 gets us 50 pounds and then if you times it by 100 times by 10 you'll get to this one which is 500 pounds so the answer is 500 is how much they pay |
5963 | https://sites.millersville.edu/bikenaga/number-theory/periodic-continued-fractions/periodic-continued-fractions.html | Periodic Continued Fractions
Periodic Continued Fractions
A periodic continued fraction is one which "repeats" --- for example,
In general, a periodic continued fraction has the form
If n is the length of the smallest repeating part, we say that the period is n. Thus, in the example above, the period is 2.
The primary result of this section is a theorem of Lagrange which charactertizes periodic continued fractions: They correspond to irrational numbers which are roots of quadratic equations with integer coefficients. Moreover, one part of the proof will require the construction of an algorithm for computing the continued fraction expansion for a quadratic irrational --- it is different from the general continued fraction algorithm, and important in its own right.
Before I begin, I should note that this section is rather long and technical. I've tried to write out the details with care to make them easy to follow, but they are often kind of dry. You've been warned!
Definition. A quadratic irrational is an irrational number which is a root of a quadratic equation
Proposition. A number is a quadratic irrational if and only if it can be written in the form , where , , and q is positive and not a perfect square.
Proof. Suppose x is a quadratic irrational. Then x is a root of
By the quadratic formula,
and and are integers, and , since .
If , then , which is a rational number, contrary to assumption.
If , then x is complex, again contrary to assumption.
Hence, .
Finally, if is a perfect square, then is rational. Hence, is not a perfect square.
For the converse, suppose , where , , and q is positive and not a perfect square. Then
This is a quadratic equation with integer coefficients, and since . Therefore, x is a quadratic irrational.
I'll prove the theorem of Lagrange in two parts. First, I'll show that periodic continued fractions represent quadratic irrationals.
I need a series of lemmas; the lemmas are motivated by the informal procedure of the following example.
Example. Write as a quadratic irrational.
I'll write x in closed form. Let . Then
On the other hand,
After some simplification, I get
y must be positive, so . Therefore,
The idea of the lemmas is simply to emulate the algebra I just did.
Lemma 1. If x is a quadratic irrational and is an integer, then is a quadratic irrational.
Proof. Write , where , , and b is positive and not a perfect square. Then
(I've suppressed the ugly algebra involved in combining the fractions and rationalizing the denominator.) The last expression is a quadratic irrational; note that , because b is not a perfect square.
Lemma 2. If x is a quadratic irrational and are integers, then the following expression is a quadratic irrational:
Proof. I'll use induction. The case was done in Lemma 1.
Suppose , and suppose the result is true for . nsider
By induction, the following subfraction is a quadratic irrational:
But the original fraction is just , so it's a quadratic irrational by Lemma 1. This completes the induction step, so the result is true for all .
Lemma 3. Let . Let
Then y can be written as , where .
Proof. Your experience with algebra should tell you this is obvious, but I'll give the proof by induction anyway.
For , I have
This has the right form.
Take , and assume the result is true for . Consider
By induction, for some ,
The original fraction is therefore
(I've suppressed some easy but ugly algebra again.) The last fraction is in the right form, so this completes the induction step. The result is therefore true for all .
I'm ready to prove that periodic continued fractions are quadratic irrationals. First, I'll consider those that start repeating immediately. These continued fractions are purely periodic, and I'll discuss them in more detail later.
Lemma 4. If , then is a quadratic irrational.
Proof. First, x is irrational, because it is an infinite continued fraction.
By Lemma 3, for some ,
Hence,
Therefore, x is a quadratic irrational.
In the general case, the fraction does not start repeating immediately.
Theorem. (Lagrange) Suppose , and let
Then x is a quadratic irrational.
Proof. is a quadratic irrational by Lemma 4. Therefore, is a quadratic irrational by Lemma 2.
The converse states the quadratic irrationals give rise to periodic continued fractions. Before I give the proof, here's an example which shows how you can go from a quadratic equation to a periodic continued fraction (at least in this case).
Suppose x is a quadratic irrational satisfying . This gives
Now substitute for x in the right side:
Do it again:
It's clear that you can keep going, and so .
The proof that quadratic irrationals give rise to periodic continued fractions will come out of an algorithm for computing the continued fraction for a quadratic irrational, which is useful in its own right. First, I need to be able to write a quadratic irrational in a "standard form".
Recall that a general quadatic irrational is an expression of the form , where , , , and b is not a perfect square. In the next lemma, I'll show that a quadratic irrational can be written in a special form, which I'll need for the algorithm which follows.
Lemma. Every quadratic irrational can be written in the form , where:
(a) .
(b) .
(c) .
(d) , and d is not a perfect square.
Proof. Let be a quadratic irrational, where , , , and b is not a perfect square. Write
First,
Thus, .
Obviously, .
Since , I have .
Since , I have . Since b is not a perfect square, is not a perfect square.
For example, is not in the form specified by the lemma. But I can write
Now which is divisible by 9, so the last fraction is in the correct form.
With a quadratic irrational expressed in this special form, I can construct an algorithm for computing its continued fraction. I'll compare this to the general continued fraction algorithm below.
Theorem. Let be a quadratic irrational, where , , , , and d is not a perfect square.
Then there are infinite sequences of integers , , and and an infinite sequence of irrational numbers defined by:
These sequences satisfy:
(a) and and for .
(b) .
Proof. Step 1. , , and are integers for . Further, and and for .
Clearly, is an integer for , and and are integers by definition. I also have .
Now
Since , it follows that . This means that is an integer, and I have . This shows that .
Thus, the assertions in this step are true for .
Suppose the assertions hold for k. Thus, and are integers and . Then is an integer.
If , then
This contradicts the assumption that d is not a square. Hence, .
Next,
This proves that , so is an integer.
From I get , so . This establishes the assertions in Step 1 by induction.
Step 2. .
Hence, . Since , this shows that the 's are the partial quotients of x.
Example. Use the quadratic irrational algorithm to compute the first 5 terms and convergents of the continued fraction for .
Note that , so the quadratic irrational is in the correct form.
The computation starts with , and
Then
Continuing in this way, I obtain:
It is interesting to compare this algorithm to the general continued fraction algorithm:
If I apply the general algorithm to , I get
I used a typical computer program to do the computation; you may get a slightly different result depending on what software you use. Notice that the partial quotients (which should be periodic) have started to be incorrect.
The quadratic irrational algorithm gives
The general algorithm is unstable because the division magnifies the round-off errors which will occur in the floating-point computations. The quadratic irrational algorithm is more stable, and continues to produce the correct periodic partial quotients.
I will digress mometarily to prove an easy observation about the quadratic irrational algorithm. It's not needed for Lagrange's theorem, but I'll need it later in discussing continued fractions for irrationals of the form .
Proposition. Let be a quadratic irrational, where , , , , and d is not a perfect square. Suppose the quadratic irrational algorithm is applied to x, yielding sequences , , and .
Then if and only if .
Proof. If , then
Conversely, suppose . Then
Equating rational and irrational parts on the two sides, I have
Since , I have . Thus, .
As an example, here are the first 7 values of , , and for . Compare the values of and :
We continue with our discussion of Lagrange's theorem. Next, I need to derive some results on conjugates of quadratic irrationals.
Definition. Let , where . Let , and suppose d is not a perfect square. The conjugate of is
Note that if , then I get the number rational number , whose conjugate is itself.
The properties that follow are sufficiently formal that " " could be replaced with "x". Note that in all the expressions the "radical part" is the same ( ). I've switched from " " to " " for readability, since I need two quadratic irrationals for each property. The proof of this proposition is just tedious basic algebra, so you could skip it, or just work out some of the parts yourself.
Proposition. Let
Assume that , , , and u is not a perfect square.
(a) .
(b) .
(c) .
Proof. (a)
(b)
(c)
The fourth and fifth equalities used (b).
Now I can prove the other half of Lagrange's theorem.
Theorem. (Lagrange) The continued fraction for a quadratic irrational is periodic.
Proof. I will use the notation of the quadratic irrational continued fraction algorithm. Thus, I assume with , , , d is not a perfect square, and . Then the sequences , , , and are defined recursively by the algorithm.
The idea is to show that for large values of n, the 's and 's only assume only finitely many values. This will imply that the same is true for the 's, and hence that the continued fraction for x is periodic.
The first and longest step sets things up by showing that the 's are eventually positive.
Step 1. for sufficiently large n.
Recall that if x is an irrational number, the general continued fraction algorithm is
I showed that
The convergent of this continued fraction is equal to (since this is a finite continued fraction), and the convergents algorithm say that the convergent is . Thus,
In our situation, is a quadratic irrational . But the quadratic irrational continued fraction algorithm shows that is a quadratic irrational of the form --- that is, and are quadratic irrationals with the same radical term . Therefore, I may apply the properties of conjugates I derived:
Solve for :
As the convergents and both approach , and . It follows that
Moreover, for large n I have Image 276: $q_{n - 2}, q_{n - 1}
0$ . So for large n,
Since , I have Image 279: $x_n - \overline{x_n}
0$ for large n.
Now using the notation of the quadratic irrational continued fraction algorithm,
So
Thus, for sufficiently large n,
Step 2. assumes only finitely many values For sufficiently large n.
The quadratic irrational algorithm gives
For large n, I know is a positive integer, so . Hence,
For sufficiently large n I know is positive. Therefore, assumes only finitely many values (between 0 and d) for sufficiently large n.
Step 3. assumes only finitely many values For sufficiently large n.
For sufficiently large n I have . Hence,
Thus, . So assumes only finitely many values for sufficiently large n (and the same is true for ).
Step 4. The continued fraction for x is periodic.
Steps 3 and 4 show that for sufficiently large n the pairs assume only finitely many values. Therefore, there are distinct integers i and j --- say --- such that . Then
Thus,
Moreover, the quadratic irrational algorithm shows that if , then , and successive pairs continue to be equal. So
This proves that the continued fraction for x is periodic.
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5964 | https://web.ma.utexas.edu/users/gordanz/notes/random_walk_color.pdf | Lecture 3: Random Walks 1 of 15 Course: Introduction to Stochastic Processes Term: Fall 2019 Instructor: Gordan Žitkovi´ c Lecture 3 Random Walks 3.1 Stochastic Processes Definition 3.1.1. A stochastic process is a sequence - finite or infinite - of random variables.
We usually write {Xn}n∈N0 of {Xn}0≤n≤T, depending on whether we are talking about an infinite or a finite sequence. The number T ∈N0 is called the (time) horizon, and we sometimes set T = +∞when the sequence is infinite. The index n is often interpreted as time, so that a stochastic process can be thought of as a model of a random process evolving in time. The initial value of the index n is often normalized to 0, even though other values - such as 1 - are also used. This it usually very clear from the context.
It is important that all the random variables X0, X1, . . . “live” on the same sample space.
This way, we can talk about the notion of a trajectory or sample path of a stochastic process: it is, simply, the sequence of numbers X0(ω), X1(ω), . . .
but with ω considered “fixed”. In other words, we can think of a stochastic process as a random variable whose values are not numbers, but sequences of numbers. This will become much clearer once we introduce enough ex-amples.
3.2 The Simple Symmetric Random Walk Definition 3.2.1. A stochastic process {Xn}n∈N0 is said to be a simple symmetric random walk if 1. X0 = 0, 2. the random variables δ1 = X1 −X0, δ2 = X2 −X1, . . . are indepen-dent Last Updated: September 25, 2019 Lecture 3: Random Walks 2 of 15 3. each δn is a coin toss, i.e., its distribution is given by δn ∼ −1 1 1 2 1 2 Remark 3.2.2.
1. Definition 3.2.1 captures the main features of an idealized notion of a particle that gets shoved, randomly, in one of two possible directions.
These “shoves” are modeled by the random variables δ1, δ2, . . . and the position of the particle after n “shoves” is Xn; indeed, Xn = δ1 + δ2 + · · · + δn, for n ∈N.
It is important to assume that any two “shoves” are independent of each other; the most important properties of random walks depend on this in a critical way.
2. Sometimes, we only need a finite number of steps of a random walk, so we only care about the random variables X1, . . . , XT. This stochastic process (now with a finite time horizon T) will also be called a random walk, and it should be clear from the context whether we need a finite or infinite horizon.
3. The starting point X0 = 0 is just a normalization. Sometimes we need more flexibility and allow our process to start at X0 = x for some x ∈N.
To stress that fact, we talk about the random walk started at x. If no starting point is mentioned, you should assume X0 = 0.
4. We will talk about the biased or assymetric random walks a bit later. The only difference will be that the probabilities of each δn taking values 1 or −1 will not be 1 2 (but will also not change from step to step).
We defined a notion of a sample path (or a trajectory) of a stochastic process. For a random walk on a finite horizon T, a trajectory is simply a sequence of natural numbers starting from 0. Different realizations of the coin-tosses δn will lead to different trajectories, but not every sequence of natural numbers corresponds to a trajectory. For example (0, 3, 4, 5) is not a sample path of a random walk because X1 can only take values 1 or −1. In fact, a finite sequence (x0, x1, . . . , xT) is a (possible) sample path of a random walk if and only if x0 = 0 and xk −xk−1 ∈{−1, 1} for each k.
Last Updated: September 25, 2019 Lecture 3: Random Walks 3 of 15 æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ 5 10 15 20 -6 -4 -2 2 4 6 æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ 5 10 15 20 -6 -4 -2 2 4 6 Figure 1. Two sample paths of a random walk Two figures above show two different trajectories of a simple random walk. Each one corresponds to a (different) frozen ω ∈Ω, with n going from 0 to 20.
Unlike in Figure 1. above, Figure 2. below shows two “time slices” of the same random process; in each graph, the time t is fixed (n = 15 vs. n = 25) but the various values random variables X15 and X25 can take are presented through the probability mass functions.
-20 -10 0 10 20 0.00 0.05 0.10 0.15 0.20 0.25 0.30 -20 -10 0 10 20 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Figure 2. probability mass functions of Xn, corresponding to n = 15 and n = 20.
3.3 The canonical probability space Let us build a sample space on which a random walk with a finite horizon T can be constructed. Since the basic building blocks of the random walk are its increments, it makes sense to define the elementary outcome ω as a Last Updated: September 25, 2019 Lecture 3: Random Walks 4 of 15 sequence, of size T, consisting of 1s and −1s. More precisely, we take Ω= {(c1, . . . , cT) : c1, . . . , cT ∈{−1, 1}} = {−1, 1} × · · · × {−1, 1} | {z } T times = {−1, 1}T.
What probability would be appropriate here? That is easy, too, since the individual coin tosses need to be fair and independent. This dictates that any two sequences be equally likely. There are 2T possible sequences of length T of 1s and −1, so P[{ω}] = 2−T, for each ω.
This way, given an ω = (c1, . . . , cn) ∈Ωand n = 1, . . . , T, we define δn(ω) = cn and Xn(ω) = c1 + · · · + cn, with X0(ω) = 0. It is easy to see that {Xn} is, indeed, a random walk.
This setup also makes the computation of various probabilities that have to do with the random walk easy (at least in theory) - we simply count the number of sequences of coin tosses that correspond to the event in question, and then divide by 2T.
Example 3.3.1. Let us compute the probability that Xn = k, for some n ∈0, . . . , T and k ∈N. For this, we first need to count the number of ω ∈Ωfor which Xn(ω) = k. Using the definition above, this is equivalent to counting the number of n-tuples (c1, . . . , cT) of 1s and −1s such that c1 + c2 + · · · + cn = k.
To solve that problem, we note that, in order for n numbers, each of which is either 1 or −1, to add up to k, we must have n1 −n−1 = k, where n1 is the number of 1s and n−1 the number of −1s in the sequence (c1, . . . , cn). On the other hand, it is always the case that n1 + n−1 = n, so we must have n1 = 1 2(n + k) and n−1 = 1 2(n −k).
The first thing we observe that this is not possible unless n + k (and then also n −k) is an even number. If you think about it for a second, it makes perfect sense. The random walk can only take odd values at odd times and even values at even times. Hence, P[Xn = k] = 0 if n and k have different parity (one is odd and the other even).
Suppose now that n + k is even. Then n −k is also even, and so n1 and n−1, defined above, are natural numbers. Our question then becomes How many sequences (c1, c2, . . . , cT) with values in {−1, 1} are there such that there are exactly n1 1s among the first n elements?
We can pick the positions of n1 1s among the first n spots in ( n n1) ways, and fill the remaining n−1 slots by −1s. The values at positions Last Updated: September 25, 2019 Lecture 3: Random Walks 5 of 15 between n + 1 and T do not affect the value of Xn, so they can be chosen arbitrarily, in 2T−n ways. Therefore, the answer to our question is ( n n1)2T−n.
To turn the count into a probability, we need to divide by 2T. All in all, we get the following result: for n ∈{0, . . . , T} and −n ≤k ≤n, we have P[Xn = k] = ( ( n (n+k)/2)2−n, if n + k is even, and 0, otherwise.
Another approach to this questions would be via the fact that 1 2(Xn + n) = 1 2(δ1 + 1) + 1 2(δ2 + 1) + · · · + 1 2(δn + 1).
Since δm takes values ±1 with probability 1 2, each, Ym = 1 2(δm + 1) is a Bernoulli random variable, with parameter p = 1 2. They are indepen-dent (since δs are), and, so, 1 2(Xn + n) has the binomial distribution, with parameters n and 1 2. It follows that for −n ≤k ≤n and n and k of the same parity, we have P[Xn = k] = P[ 1 2(Xn + n) = 1 2(n + k)] = n (n + k)/2 2−n.
3.4 Biased random walks If the steps of the random walk preferred one direction to the other, the definition would need to be tweaked a little bit: Definition 3.4.1. A stochastic process {Xn}n∈N0 is said to be a (sim-ple) biased random walk with parameter p ∈(0, 1) if 1. X0 = 0, 2. the random variables δ1 = X1 −X0, δ2 = X2 −X1, . . . are indepen-dent 3. each δn is a p-biased coin toss, i.e., its distribution is given by δn ∼ −1 1 1 −p p Could we reuse the sample space Ωto build a biased random walk? Yes, we could, but we would need to assign different probabilities to elementary outcomes. Indeed, if p = 0.99, the probability that all the increments δ take the value +1 is larger than the probability that all steps take the value −1.
Last Updated: September 25, 2019 Lecture 3: Random Walks 6 of 15 More generally, the sequence ω = (c1, . . . , cN) consisting of n1 1s and n−1 −1s should be assigned the probability pn1(1 −p)n−1. Only then will δs be independent and distributed as −1 1 1 −p p , as required.
We can still use this sample space to figure out distributions of various random variables, but we cannot always simply “count and divide by the size of Ω” like we could when p = 1 2. Sometimes it still works, as the following example shows: Example 3.4.2. Let us try to compute the same probability as in Ex-ample 3.3.1 above, namely P[Xn = k], but now in the biased case.
To simplify our lives, we can assume without loss of generality that n = N, i.e., that nothing happens after n. We still need to identify those ω = (c1, . . . , cT) for which Xn(ω) = k, and it turns out that the reasoning is the same as in the symmetric case. We need exactly n1 = 1 2(n + k) of the cs to be equal to 1 and exactly n−1 = n −n1 of them to be equal to −1. The lucky break is that each sequence with exactly n1 1s carries the same probability, namely pn1(1 −p)n−1, no matter where these 1s are. In other words, it just happened that all ω in the event {Xn = k} have the same probability. Therefore, the probability of {Xn = k} is simply pn1(1 −p)n−n1 multiplied by the number of ωs that constitute it. We have already computed that in Example 3.3.1 - the answer is ( n n1) - and, so P[Xn = k] = ( ( n (n+k)/2) p(n+k)/2(1 −p)(n−k)/2, if n + k is even, and 0, otherwise.
As in the symmetric case, this also follows from the fact that 1 2(Xn + n) is binomial, with parameters n and p.
3.5 The Reflection Principle Counting trajectories in order to compute probabilities can be quite powerful, as the following example shows. It also reveals a potential weakness of the combinatorial approach: it works best when all ω are equally likely (i.e., when p = 1 2 in the case of the random walk).
We start by asking a simple question; what is the typical record value of the random walk, i.e., how far “up” does it typically get? Clearly, the largest value it can attain is T at time T, provided that all coin tosses came up +1.
This is, however, extremely unlikely - it happens with probability 2−T. On the other hand, this maximal value is at last 0, since X0 = 0, already. A bit of thought reveals that any value between those two extremes is possible, but it is not at all easy to compute their probabilities.
Last Updated: September 25, 2019 Lecture 3: Random Walks 7 of 15 More precisely, if {Xn} is a simple random walk with time horizon T. We define the running maximum process {Mn}n∈N0 by Mn = max(X0, . . . , Xn), for 0 ≤n ≤T.
It turns out that a nice counting trick - known as the reflection principle -can help us compute the distribution of Mn for each n.
Proposition 3.5.1. Let {Xn}0≤n≤T be a simple symmetric random walk.
For 1 ≤n ≤T, the support of the random variable Mn = max(X0, . . . , Xn) is {0, 1, . . . , n} and its probability mass function is given by P[Mn = k] = P[Xn = k] + P[Xn = k + 1] = n ⌊n+k+1 2 ⌋ 2−n, for k = 0, . . . , n, where ⌊x⌋denotes the largest integer smaller than or equal to x.
Proof. As usual, we may assume without loss of generality that n = T since the values of δn+1, . . . , δT do not affect Mn at all.
We start by picking a level l ∈{0, 1, . . . , n} and first compute the proba-bility P[Mn ≥l]. The symmetry assumption ensures that all trajectories are equally likely, so we can do this by counting the number of trajectories whose maximal level reached is at least l, and then multiply by 2−n.
What makes the computation of P[Mn ≥l] a bit easier than that of P[Mn = l] is the following equivalence Mn ≥l if and only if Xk = l for some k.
In words, the set of trajectories whose maximum is at least l is exactly the same as the set of trajectories that hit the level l at some time. Let us denote the set of ω with this property by Al, so that P[Mn ≥l] = P[Al].
We can further split Al into three disjoint events A> l , A= l and A< l , depend-ing on whether Xn < l, Xn = l and Xn > l. The idea behind the reflection principle is that A> l and A< l have exactly the same number of elements. To see that that is, indeed, true, we is take an ω ∈A> l and denote by F(ω) the first time the corresponding trajectory visits the level l. After that, we flip the portion the trajectory between F(ω) + 1 and n around the level l. In terms of ω, this amounts to flipping the signs of its last n −F(ω) + 1 entries (see Figure 3 below). It is easy to see that this establishes a bijection between the sets A> l and A< l , making these two sets equal in size.
Last Updated: September 25, 2019 Lecture 3: Random Walks 8 of 15 æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ 2 4 6 8 10 12 14 -2 2 4 6 Figure 3. A trajectory (blue) and what you get (red) when you flip its “tail” after the first visit to l = 4 which happened at time F(ω) = 8. Note that the red trajectory takes the same values as the blue one up to n = 8.
The punchline is that the trajectories in A> l as well as in A= l are easy to count. For them, the requirement that the level l is hit at a certain point is redundant; if you are at or above l at the very end, you must have hit l at a certain point (if nothing else, at time n). Therefore, A> l is simply the family of those ω whose final positions are somewhere strictly above l, and, therefore, P[A> l ] = P[Xn = l + 1 or Xn = l + 2 or . . . or Xn = n] = n ∑ k=l+1 P[Xn = k], and, therefore, by the reflection principle, P[A< l ] = P[A> l ] = n ∑ k=l+1 P[Xn = k].
We still need to account for A= l , i.e., for the trajectories that end up exactly at the level l. Just like above, P[A= l ] = P[Xn = l].
Putting all of this together, we get P[Al] = P[Xn = l] + 2 n ∑ k=l+1 P[Xn = k], so that P[Mn = l] = P[Mn ≥l] −P[Mn ≥l + 1] = P[Al] −P[Al+1] = P[Xn = l] + P[Xn = l + 1].
To show the versatility of the reflection principle, let us use it to solve a classical problem in combinatorics.
Last Updated: September 25, 2019 Lecture 3: Random Walks 9 of 15 Example 3.5.2 (The Ballot Problem). Suppose that two candidates, Daisy and Oscar, are running for office, and T ∈N voters cast their ballots. Votes are counted the old-fashioned way, namely by the same official, one by one, until all T of them have been processed. After each ballot is opened, the official records the number of votes each candi-date has received so far. At the end, the official announces that Daisy has won by a margin of k > 0 votes, i.e., that Daisy got (T + k)/2 votes and Oscar the remaining (T −k)/2 votes. What is the probability that at no time during the counting has Oscar been in the lead?
We assume that the order in which the official counts the votes is com-pletely independent of the actual votes, and that each voter chooses Daisy with probability p ∈(0, 1) and Oscar with probability q = 1 −p.
For 0 ≤n ≤T, let Xn be the number of votes received by Daisy minus the number of votes received by Oscar in the first n ballots. When the n + 1-st vote is counted, Xn either increases by 1 (if the vote was for Daisy), or decreases by 1 otherwise. The votes are independent of each other and X0 = 0, so Xn, 0 ≤n ≤T is a simple random walk with the time horizon T. The probability of an up-step is p ∈(0, 1), so this random walk is not necessarily symmetric. The ballot problem can now be restated as follows: For a simple random walk {Xn}0≤n≤T, what is the probability that Xn ≥0 for all n ∈{0, . . . , T}, given that XT = k?
The first step towards understanding the solution is the realization that the exact value of p does not matter. Indeed, we are interested in the conditional probability P[F|G] = P[F ∩G]/P[G], where F de-notes the set of ω whose corresponding trajectories always stay non-negative, while the trajectories corresponding to ω ∈G reach k at time n. Each ω ∈G consists of exactly (T + k)/2 up-steps (1s) and (T −k)/2 down steps (−1s), so its probability weight is equal to p(T+k)/2q(T−k)/2.
Therefore, with #A denoting the number of elements in the set A, we get P[F|G] = P[F ∩G] P[G] = #(F ∩G) p(T+k)/2q(T−k)/2 #G p(T+k)/2q(T−k)/2 = #(F ∩G) #G .
This is quite amazing in and of itself. This conditional probability does not depend on p at all!
Since we already know how to count the number of elements in G (there are ( T (T+k)/2)), “all” that remains to be done is to count the num-ber of elements in G ∩F. The elements in G ∩F form a portion of all the elements in G whose trajectories don’t hit the level l = −1; this Last Updated: September 25, 2019 Lecture 3: Random Walks 10 of 15 way, #(G ∩F) = #G −#H, where H is the set of all paths which finish at k, but cross (or, at least, touch) the level l = −1 in the process. Can we use the reflection principle to find #H? Yes, we can. In fact, you can convince yourself that the reflection of any trajectory correspond-ing to ω ∈H around the level l = −1 after its last hitting time of that level produces a trajectory that starts at 0 and ends at −k −2, and vice versa. The number of paths from 0 to −k −2 is easy to count - it is equal to ( T (T+k)/2+1). Putting everything together, we get P[F|G] = ( T n1) −( T n1+1) ( T n1) = k + 1 n1 + 1, where n1 = T + k 2 .
The last equality follows from the definition of binomial coefficients (T i ) = T!
i!(T−i)!.
The Ballot problem has a long history (going back to at least 1887) and has spurred a lot of research in combinatorics and probability. In fact, people still write research papers on some of its generalizations. When posed outside the context of probability, it is often phrased as “in how many ways can the counting be performed . . . ” (the difference being only in the normalizing factor ( T n1) appearing in Example 3.5.2 above). A special case k = 0 seems to be even more popular - the number of 2n-step paths from 0 to 0 never going below zero is called the Catalan number and equals to Cn = 1 n + 1 2n n .
(3.5.1) Here is another nice consequence of the reflection principle, i.e., its ap-plication to the running maximum. Our formula for the distribution of the maximum of the random walk on {0, . . . , T} can be used to answer the fol-lowing question: What is the probability that the random walk will reach the level l in T steps (or fewer)?
Indeed, {Xn}n∈N will reach l during the first T steps if and only if MT ≥l.
Therefore the answer to the above question is P[MT ≥l] = P[XT ≥l] + P[XT ≥l + 1]. A special case is the following: What is the probability that X will stay at or below 0 throughout the interval {0, . . . , T}?
Clearly, {Xn}n∈N will stay non-positive if it never hits the level 1. The probability of that is P[MT = 0] = P[XT = 0] + P[XT = 1]. What happens to this expression as T get larger and larger. In other words, if I give my Last Updated: September 25, 2019 Lecture 3: Random Walks 11 of 15 walk enough time, can I guarantee that it will reach the level 1?
Let us compute. For simplicity, let us consider only even time horizons T = 2N, so that P[MT = 0] = P[X2N = 0]. Using the formula for the distribution of X2N, we get P[X2N = 0] = 2N N 2−2N, so our problem reduces to the investigation of the behavior of (2N N )2−2N, as N gets larger and larger, i.e., lim N→∞ 2N N 2−2N.
(3.5.2) To evaluate this limit, we need to know about the precise asymptotics of N!, as N →∞: Proposition 3.5.3 (Stirling’s formula). We have N! ∼ √ 2πN N e N , where AN ∼BN means limN→∞ AN BN = 1.
Let us use Stirling’s formula in (3.5.2): 2N N 2−2N = (2N)!
N!N! 2−2N ∼ √ 2π2N(2N/e)2N √ 2πN(N/e)N√ 2πN(N/e)N 2−2N = 1 √ πN Therefore, lim N→∞ 2N N 2−2N = 0, and it follows that the answer to our question is positive: Yes, the simple symmetric random walk will reach the level 1, with certainty, given enough time.
By symmetry, the level 1 can be replaced by −1. Also, once we hit 1, the ran-dom walk “renews itself” (this property is called the Strong Markov Property and we will talk about it later), so it will eventually hit the level 2, as well.
Continuing the same way, we get the following remarkable result Last Updated: September 25, 2019 Lecture 3: Random Walks 12 of 15 Theorem 3.5.4. The symple symmetric random walk will visit any point in Z = {. . . , −2, −1, 0, 1, 2, . . . }, eventually.
3.6 Problems Problem 3.6.1. Let {Xn}n∈N0 be a simple symmetric random walk. The dis-tribution of the product X1X2 is (a) 0 2 1 2 1 2 (b) −2 0 2 1 4 1 2 1 4 (c) −2 −1 0 1 2 1 5 1 5 1 5 1 5 1 5 (d) −1 0 −1 1 4 1 2 1 4 (e) none of the above Problem 3.6.2. Let {Xn}n∈N0 be a simple symmetric random walk with the time horizon T = 3. The probability that X will never hit the level 2 or the level −2 is (a) 1 4 (b) 1 3 (c) 1 2 (d) 3 8 (e) none of the above Problem 3.6.3. Let {Xn}n∈N0 be a simple symmetric random walk. Then (a) X1 and X2 are independent (b) X4 −X2 is independent of X3.
(c) X4 −X2 is independent of X6 −X5 (d) X1 + X3 is independent of X2 + X4 (e) none of the above Problem 3.6.4. Let {Xn}n∈N0 be a simple symmetric random walk. Then (a) X1 and X2 are independent Last Updated: September 25, 2019 Lecture 3: Random Walks 13 of 15 (b) X3 −X1 is independent of X2.
(c) P[X32 = 44|X12 = 0] = P[X22 = 44|X2 = 0, X1 = 1] (d) P[X13 = 4] = (13 4 )2−13 (e) none of the above Problem 3.6.5. Let {Xn}n∈N0 be a simple symmetric random walk. Which of the following processes are simple random walks?
1. {2Xn}n∈N0 ?
2. {X2 n}n∈N0 ?
3. {−Xn}n∈N0 ?
4. {Yn}n∈N0, where Yn = X5+n −X5 ?
How about the case p ̸= 1 2?
Problem 3.6.6. Let {Xn}n∈N0 be a biased simple random walk with p = P[X1 = 1] = 1/3. Compute the following: 1. P[X2 = 0] 2. P[X7 = X16] 3. P[X2 = X4 = X8], 4. P[Xn < 10 for all 0 ≤n ≤10].
Problem 3.6.7. Let {Xn}n∈N0 be a symmetric simple random walk. Compute the following 1. P[X2n = 0], n ∈N0, 2. P[Xn = X2n], n ∈N0, 3. P[ |X1X2X3| = 2], 4. P[X7 + X12 = X1 + X16].
Problem 3.6.8. Let {Xn}n∈N0 be a symmetric simple random walk with p = P[X1 = 1] ∈(0, 1). For n ∈N, the probability P[X2n = X4n and X6n = X8n] is given by (a) (2n n )2−4n Last Updated: September 25, 2019 Lecture 3: Random Walks 14 of 15 (b) ( n ⌊n/2⌋)2−4n (c) (4n 2n)2−4n (d) (2n n ) 22−4n (e) none of the above Problem 3.6.9. Let {Xn}n∈N0 be a simple random walk with P[X1 = 1] = p ∈(0, 1). Define Yn = 1 n n ∑ k=1 Xk, for n ∈N.
Compute E[Yn] and Var[Yn], for n ∈N.
Hint: You can use the following formulas: n ∑ j=1 j = n(n + 1) 2 , n ∑ j=1 j2 = n(n + 1)(2n + 1) 6 without proof.
Problem 3.6.10. Let {Xn}n∈N0 be a simple symmetric random walk. Given n ∈N0 and k ∈N, compute Var[Xn], Cov[Xn, Xn+k] and corr[Xn, Xn+k], where Cov stands for the covariance and corr for the correlation. Note: If you forgot what these are, look them up.
Problem 3.6.11. Let {Xn}0≤n≤10 be a simple symmetric random walk with time horizon T = 10. What is the probability it will never reach the level 5?
Problem 3.6.12. Let {Xn}n∈N0 be a simple symmetric random walk. Given n ∈N, what is the probability that X does not visit 0 during the time interval 1, . . . , n.
Problem 3.6.13. Luke starts a random walk, where each step takes him to the left or to the right, with the two alternatives being equally likely and independent of the previous steps. 11 steps to his right is a cookie jar, and Luke gets to take a (single) cookie every time he reaches that position. He performs exactly 15 steps, and then stops.
1. What is the probability that Luke will be exactly by the cookie jar when he stops?
Last Updated: September 25, 2019 Lecture 3: Random Walks 15 of 15 2. What is the probability that Luke stops with with exactly 3 cookies in his hand?
3. What is the probability that Luke stops with at least one cookie in his hand?
4. Suppose now that we place a bowl of broccoli soup one step to the right of the cookie jar. It smells so bad that, if reached, Luke will throw away all the cookies he is currently carrying (if any) and run away pinching his nose. What is the probability that Luke will finish his 15-step walk without ever encountering the yucky bowl of broccoli soup and with at least one cookie in his hand?
Problem 3.6.14. A fair coin is tossed repeatedly and the record of the out-comes is kept. Tossing stops the moment the total number of heads obtained so far exceeds the total number of tails by 3. For example, a possible sequence of tosses could look like HHTTTHHTHHTHH. What is the probability that the length of such a sequence is at most 10?
Problem 3.6.15. () Let Cn denote the n-th Catalan number (defined in (3.5.1)).
1. Use the reflection principle to show that Cn is the number of trajectories (x0, . . . , x2n) of a random walk with time horizon T = 2n such that xk ≥0, for all k ∈{0, 1, . . . , 2n} and x2n = 0.
2. Prove the Segner’s recurrence formula Cn+1 = ∑n i=0 CiCn−i. (Hint: Don’t compute - just think about paths).
3. Show that Cn is the number of ways that the vertices of a regular 2n-gon can be paired so that the line segments joining paired vertices do not intersect.
4. Prove that Cn = 2n n − 2n n + 1 , both algebraically (using the formula for the binomial coefficient) and combinatorially (by counting).
Last Updated: September 25, 2019 |
5965 | https://www.quora.com/Is-there-a-mathematical-equation-model-for-density-of-liquid-e-g-water-with-temperature | Is there a mathematical equation model for density of liquid (e.g. water) with temperature? - Quora
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Is there a mathematical equation model for density of liquid (e.g. water) with temperature?
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Yes, the density of a liquid, such as water, typically varies with temperature. For many liquids, including water, the relationship between density and temperature can be modeled using empirical equations. Here are a couple of common approaches:
Empirical Polynomial Equation:
For water, density can be approximated by a polynomial function of temperature. A commonly used equation for the density of water in kilograms per cubic meter (kg/m³) as a function of temperature in degrees Celsius (°C) is:
[math]\rho(T) = a + bT + cT^2 + dT^3[/math]
Where:
[math]\rho(T)[/math] = density at temperature [math]T[/math]
[math]a, b, c, d[/math] = empirical
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Yes, the density of a liquid, such as water, typically varies with temperature. For many liquids, including water, the relationship between density and temperature can be modeled using empirical equations. Here are a couple of common approaches:
Empirical Polynomial Equation:
For water, density can be approximated by a polynomial function of temperature. A commonly used equation for the density of water in kilograms per cubic meter (kg/m³) as a function of temperature in degrees Celsius (°C) is:
[math]\rho(T) = a + bT + cT^2 + dT^3[/math]
Where:
[math]\rho(T)[/math] = density at temperature [math]T[/math]
[math]a, b, c, d[/math] = empirical coefficients specific to the liquid (for water, these coefficients can be derived from experimental data).
Specific Equation for Water:
A more specific equation for the density of water, based on empirical data, is:
[math]\rho(T) = 999.83952 + 16.945176T - 7.987040 \times 10^{-3} T^2 - 46.170461 \times 10^{-6} T^3 + 105.56302 \times 10^{-9} T^4 - 280.54253 \times 10^{-12} T^5[/math]
This equation is valid in the temperature range of 0°C to 100°C and provides a good approximation of water's density.
General Trends:
The density of water decreases as temperature increases, reaching its maximum density at around 4°C (approximately 1000 kg/m³).
Beyond 4°C, the density continues to decrease until it reaches 0°C, where it starts to form ice, which is less dense than liquid water.
Conclusion:
These models can be useful for various applications in science and engineering, especially when precise measurements of density are required at different temperatures. For practical purposes, tables or charts of water density at various temperatures are also widely available.
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What is the formula for density of liquid?
At which temperature is the density of water at a minimum?
Does the density of water depend on its temperature?
What is the average density of mixture when 1 kg of water is mixed with 2 kg of liquid of density 3g/cm?
What is the relationship between the temperature and the density of liquid?
William Hubbard
Former Nuclear Engineer at United States Navy (USN) (2002–2007) · Author has 116 answers and 361.4K answer views
·8y
Simple answer: Yes.
The relationship of density of water at different temperatures is not something that you would recognize in everyday life. The reason is that water boils at 212F and the density of water doesnt change much from room temperature to boiling point.
So when does it temperature affect water density? At very high pressures like in a pressurized water nuclear reactor. You can heat water to around 600F and it won't boil. Instead of an “equation”, engineers refer to a phase diagram to understand the temperature/density relationship.
At that high temperature and pressure, the density of
Continue Reading
Simple answer: Yes.
The relationship of density of water at different temperatures is not something that you would recognize in everyday life. The reason is that water boils at 212F and the density of water doesnt change much from room temperature to boiling point.
So when does it temperature affect water density? At very high pressures like in a pressurized water nuclear reactor. You can heat water to around 600F and it won't boil. Instead of an “equation”, engineers refer to a phase diagram to understand the temperature/density relationship.
At that high temperature and pressure, the density of water will fluctuate with significance due to small changes in temperature and this density of the water is what determines the “reactor power”. Read more here:
PHYSICS AND KINETICS OF TRIGA REACTORS
This phenomenon can literally be witnessed in real-time by watching the water level in the coolant surge tank (aka “pressurizer”) as the coolant temperature fluctuates. At normal operating pressure, there are thumb-rules about how many inches the water level will change due to a 1 degree change in temperature.
Hope this helps!
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Michael Stevenson
B.A. in Mathematics, Oberlin College · Author has 5.4K answers and 1.6M answer views
·8y
This would stray well away from pure math. My guess is that methods in statistical mechanics could make predictions with some success.
Of course, there are models like the Standard Model of Physics which are mathematical and may even be shown correct to a certain number of decimal places, but solving - maybe even writing - the equation which would give a macro quantity like density could take longer than the lifetime of the universe with today’s technology.
Most of our information about water is experimental data first, with theoretical confirmation, say. It’s pretty well mapped out now (see a p
Continue Reading
This would stray well away from pure math. My guess is that methods in statistical mechanics could make predictions with some success.
Of course, there are models like the Standard Model of Physics which are mathematical and may even be shown correct to a certain number of decimal places, but solving - maybe even writing - the equation which would give a macro quantity like density could take longer than the lifetime of the universe with today’s technology.
Most of our information about water is experimental data first, with theoretical confirmation, say. It’s pretty well mapped out now (see a phase diagram) though every now and then I think I hear about a new type of ice or behavior near a critical point. Mathematical regression models would fit various parts of the curve. The rate of change with respect to temperature could even be all or mostly accounted for due to known molecular effects.
So you see the direction of inquiry here is not proceeding from pure math. An applied mathematician might be able to offer proofs bounding (giving a range for) water’s density at various temperatures. As always, mathematical postulates used for physics are ultimately justified empirically via natural philosophy (physics).
Reading Can the ideal gas law be applied to liquids? | Socratic may help.
Disclaimer / not a doctor
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Carol Van Zoeren
PhD in Chemistry, Massachusetts Institute of Technology (Graduated 1989) · Author has 87 answers and 147.4K answer views
·7y
Originally Answered: How does the density of a liquid vary with temperature? ·
I LOVE this question, because it hits on my favorite molecule, H2O. Most substances — solid, liquid, gas — expand with higher temperature, so the density goes down. But water, H2O, breaks that rule, and more.
The root cause of why water is water is because the molecule is bent. Unlike molecules with similar molecular weights (carbon dioxide — CO2. methane — CH4. Oxygen — O2. Nitrogen — N2). H2O is bent because the Oxygen atom is sorta OK with unfilled molecular orbitals. Methane needs all 4 of its molecular orbitals filled, so CH4 is tetrahedral and “balanced” aka symmetric. CO2 is linear, and
Continue Reading
I LOVE this question, because it hits on my favorite molecule, H2O. Most substances — solid, liquid, gas — expand with higher temperature, so the density goes down. But water, H2O, breaks that rule, and more.
The root cause of why water is water is because the molecule is bent. Unlike molecules with similar molecular weights (carbon dioxide — CO2. methane — CH4. Oxygen — O2. Nitrogen — N2). H2O is bent because the Oxygen atom is sorta OK with unfilled molecular orbitals. Methane needs all 4 of its molecular orbitals filled, so CH4 is tetrahedral and “balanced” aka symmetric. CO2 is linear, and O2 and N2 are diatomic. Bottom line, those with such symmetry can’t have a molecular dipole. In contrast, water is a dipole — one side of the molecule is more negative, and the other part is more positive. Because of this molecular dipole, H2O molecules have an electrostatic attraction to each other that other small molecules don’t have. So water molecules need a lot more energy to break that electrostatic molecular attraction, and become a gas, than non-polar molecules. I anthropomorphize chemistry, so I think of it as water molecules have a sister bond, and the non-polar molecules have a 4th cousin bond.
One real world result is that water is a liquid at much higher temperatures than similar small molecules like CO2. In other words, water is a liquid at “room temperature.” Which is a bit of a chicken and egg thing. Because we think of “room temperature” as something pretty comfortable for beings such as ourselves. And we beings evolved in a “room temperature” environment, so of course that’s what we find comfortable. If water were not a liquid at this “room temperature”, we would not have evolved to fit it. And if water ceased to be a liquid at “room temperature”, we could not survive.
But more than that — and this gets to the OP density question — unlike most every other substance, the density of water does not monotonically increase as the temperature goes down. As has been mentioned, water reaches maximum density at 4 degrees Celsius — just a bit above freezing. Above AND below this temperature, it’s density is lower (i.e. it floats). And this gets to my toss-away comment above that “the Oxygen atom is sorta OK with unfilled molecular orbitals.” Yes, a bent H2O molecule isn’t lookin’ to cause trouble (i.e react with anything) on its own. But when close enough to others of its own kind, the H’s of one H20 molecule senses an even more stable existence by connecting with the unfilled molecular orbitals of the Oxygen in another H2O molecule. The H can’t embrace either the O to which it’s bound or to the “foster” O, but must hold each at arms length. They push away to an equilibrium position.
Like I said, anthropomorphizing. The technical term for this is hydrogen bonding. And it causes water molecules to push away from each other below 4C. And so what? Well, that’s why when you freeze vegetables, they’re all mushy when they thaw — the “push away” busted the structural cellular walls. Same as heating boils the water in your veggies — both bust the cellular walls and make ’em mushy (or, in the case of kale, edible).
It’s also why ice floats. No big deal, right? Um, when life was starting in the primordial ooze, think those bugs coulda tolerated their structural cellular walls being busted by muscular hydrogen bonding water molecules? Bodies of water freeze from the top down rather than the bottom up. If not, life would have had a tough time.
Two take homes: 1) that’s why astrobiologists get so excited at evidence of water. And 2) I’m a recovering atheist, and the miracle of H2O is a big part of my recovery.
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When to use Proof by Contradiction (with an example)
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So I came upon this issue when trying to prove the following inequality:
|xy| $\le$ |x - y|
one of the triangle inequalities.
So one way I thought of proving this is to square both sides. With a few simple steps, the inequality simplifies to |x||y| $\ge$ xy , which we know is true. But is this enough to really prove this inequality?
The other way to prove it would be with contradiction; ie, "Just suppose that |xy| > |x - y| " and then show that this simplifies to a false inequality.
Are these two different approaches comparable in how deeply they prove the inequality?
proof-verification
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edited May 8, 2018 at 22:21
user553645
user553645
asked May 8, 2018 at 22:14
user553645user553645
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$\begingroup$ Unfortunately, your first method of proof is completely wrong: if "squaring both sides gives a true result" were a valid method of proof, then $-1 = 1$. Not to mention that $|x| |y| \le xy$ is also wrong. $\endgroup$
user296602
– user296602
2018-05-08 22:16:44 +00:00
Commented May 8, 2018 at 22:16
$\begingroup$ Also $|-1| \cdot |+1|$ is not less than $-1$ $\endgroup$
AnalysisStudent0414
– AnalysisStudent0414
2018-05-08 22:17:48 +00:00
Commented May 8, 2018 at 22:17
$\begingroup$ The inequality was a typo, thanks for pointing that out. As for the proof in general, now that I look back I realize that I constructed the approach incorrectly. Instead, start with |x-y|. (|x-y|)^2 = (x-y)^2 $\le$ |x|^2 - 2|x||y| +|y|^2 That's the way I meant to do this $\endgroup$
user553645
– user553645
2018-05-08 22:21:21 +00:00
Commented May 8, 2018 at 22:21
$\begingroup$ $x \le y \not \implies x^2 \le y^2$ and $x^2 \le y^2 \not \implies x \le y$ as signs may not be consistent. $\endgroup$
fleablood
– fleablood
2018-05-08 22:23:35 +00:00
Commented May 8, 2018 at 22:23
$\begingroup$ So you're saying that the method I offered in the comments was incorrect? I found a textbook that does it this way; however, it is a book specifically for an introductory course, so it may be assuming less than you are. $\endgroup$
user553645
– user553645
2018-05-08 22:28:19 +00:00
Commented May 8, 2018 at 22:28
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Are these two different approaches comparable in how deeply they prove the inequality?
You basically ask if between two different kind of proofs one is "better" or "more valid" than the other. The answer is No. Proof is proof: As long as the reasoning is logically correct, a proof proves the statement it is a proof of. A proof by contradiction is as good as any other.
On the other hand, different proofs for the same statement (regardless if one of them is by contradiction or not) can provide different insight into a problem or statement, but that is subjective and not measurable.
EDIT: I should mention that there is a philosophical branch of mathematics called "Constructivism", where proof by contradiction and axiom of choice are not allowed. If there are merits in that philosophy, I let for you to decide. However, the majority of the mathematical doesn't feel obliged to follow the constructionistic ways and will accept proofs by contradiction without problem.
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edited May 8, 2018 at 22:33
answered May 8, 2018 at 22:23
SK19SK19
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$\begingroup$ With their comments on my sloppiness I was able to edit the original example. Thank you for addressing the question, despite that! So I guess my intuition was that an algebraic construction of something "felt" more rigorous than contradiction. But based on what I'm reading, a proof by contradiction is still using algebra - just from the other point of view. In general though, are statements considered proven if we only have a proof by contradiction? Or does existence of contradiction imply existence of other proofs? $\endgroup$
user553645
– user553645
2018-05-08 22:26:28 +00:00
Commented May 8, 2018 at 22:26
$\begingroup$ @kasa I added a paragraph regarding the constructive aspect. Also, if there is one proof, there are infinite proofs, because you can add unnecessary arguments to your proof. To ponder how many fundamentally different proofs of a statement there are would mean another question (that I will not be able to answer). $\endgroup$
SK19
– SK19
2018-05-08 22:36:15 +00:00
Commented May 8, 2018 at 22:36
$\begingroup$ @kasa A proof by contradiction (nothing to do with whether you use algebra or not) is just a much a proof as one that works directly from hypothesis to conclusion. But as this answer points out, a direct proof may (and usually does) offer insight into why the theorem is true. Silly example: I know you can get from my daughter's house to mine by car, since if you couldn't she wouldn't have been able to visit me yesterday. But a proof showing the route to take would be much more informative. $\endgroup$
Ethan Bolker
– Ethan Bolker
2018-05-08 22:38:40 +00:00
Commented May 8, 2018 at 22:38
Add a comment |
0
$\begingroup$
Proof by contradiction is best used when you believe a statement is false, and are unable to find an algebraic proof, or when you see an obvious counterexample that can immediately tell you that the statement is false. The most commonplace use of it I've seen is in proof of irrationality.
Is it stronger/weaker than a standard proof? Absolutely not. To prove a statement false is to prove a statement false, whether you find one counterexample or infinitely many, the statement is still false.
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answered May 8, 2018 at 22:32
Rhys HughesRhys Hughes
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$\endgroup$
2
$\begingroup$ What exactly is an "algebraic proof"? $\endgroup$
SK19
– SK19
2018-05-08 22:38:11 +00:00
Commented May 8, 2018 at 22:38
$\begingroup$ Your answer suggests that proof by contradiction is what you do to prove something false. I don't think that's a good way to look at it. For example, to prove that a polynomial of odd degree has a root, you can assume it doesn't and get a contradiction by invoking the intermediate value theorem. Proof by contradiction is a good strategy here because actually showing how to calculate a root would be harder (though possible). $\endgroup$
Ethan Bolker
– Ethan Bolker
2018-05-08 22:45:31 +00:00
Commented May 8, 2018 at 22:45
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Viviani’s theorem states that in an equilateral triangle, the sum of the perpendicular distances from any interior point to the three sides is always equal to the height of the triangle. This means that if a point is chosen anywhere inside the triangle, and perpendicular lines are drawn from this point to each of the three sides, the total length of these three perpendiculars will be the same as the height of the original equilateral triangle. The theorem can be proved by considering the areas of the three smaller triangles formed by these perpendiculars and showing that their total area equals the area of the original triangle, leading to the conclusion that the sum of the perpendicular distances remains constant.
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5968 | https://www.yourdictionary.com/brindle | Brindle Definition & Meaning | YourDictionary
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Brindle Definition
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A brindled color.
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A brindled animal.
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Brindled.
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Having such a colouration; brindled.
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Any color or variety of colors is acceptable, white, black, blue, fawn, brindle or parti-coloured are all OK.
They can also be black, brindle, red, fawn or tricolor.
Variegated yarns can give a brindle pattern to the dog's coat.
Fawn and brindle are the colours preferred.
The coat should be thick, short and very silky, the favourite colours being white and white marked with brindle.
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5969 | https://www.nutritioncluster.net/sites/nutritioncluster.com/files/2024-03/CMAM%20Signposting%20Brief_final%20version.pdf | 1 Community Management of Acute Malnutrition Resources Signposting Document March 2023 Note: This Community Management of Acute Malnutrition (CMAM) signposting document aims to offer some of the key CMAM-related resources (focusing on humanitarian contexts) as of February 2023. We recognise that some resources may become outdated once the World Health Organisation (WHO) publishes its updated global guideline for the prevention and management of wasting and nutritional oedema (acute malnutrition). The Global Action Plan on Child Wasting (GAP) agencies, led by UNICEF and WHO are currently in the process of developing a wider resource library for wasting and nutritional oedema (acute malnutrition) which will be housed on the www.childwasting.org website. We thus advise practitioners to visit the www.childwasting.org website for updates on the resource repository and to use the library once it is published. Background: The GNC Technical Alliance is a technical assistance platform that responds to technical requests1 by leveraging and building on existing nutrition resources, capacities, initiatives, and coordination structures. A regular technical request posed to the Alliance in relation to wasting has centered on which community management of acute malnutrition (CMAM) related resources and tools, including training tools, are available. While a plethora of resources and tools exist on CMAM, including a wide range of agency-specific tools, with several platforms to share such resources, there remains a need for a consolidated list and repository of resources and tools that country-level practitioners can easily access and use when developing country-specific tools and resources. The need for a broader CMAM-related resource repository website is currently being discussed by global actors. Given these discussions, this signposting document aims to serve as an initial step to developing such a repository and highlight some of the key CMAM related resources and tools. This signposting document aims to provide easy access to various global-level CMAM-related resources organised across the following themes: a. Core CMAM documents 1 Anyone working in the humanitarian nutrition space can ask a technical related question to the GNC Technical Alliance here. 2 i. CMAM in Emergencies ii. Generic CMAM resources iii. Moderate Acute Malnutrition/ Moderate Wasting iv. Severe Acute Malnutrition/ Severe Wasting v. Growth Faltering among infants u6m vi. Community Mobilisation b. Examples of Programme documents and adaptations i. Simplified Approaches ii. Resources related to Community Health Workers implementing CMAM iii. The CMAM Surge Approach iv. Additional resources focused on COVID-19 v. CMAM and other health concerns vi. CMAM and IMCI integration c. Program planning, Management, Evaluation, Reporting, and Assessments i. Operational management support, planning, costing ii. Nutrition Assessments and reporting iii. Prevention of acute malnutrition (wasting)- non-exhaustive list iv. Additional relevant resources While important to capture all country-level and global-level resources to assist practitioners in identifying and using relevant materials, given the scope of work and capacity, such a broad repository was not possible. The scope was thus narrowed through the following inclusion and exclusion criteria. Inclusion Criteria: - CMAM-related materials include policy guidance documents, tools, training, and orientation documents that support the implementation of CMAM-related programming. - Global-level tools, although if no global tool existed on a particular topic, country examples were sought. - Resources in English- although the priority of the Alliance is to provide resources in as many languages as possible, this signposting document was limited to English resources at this stage. Exclusion Criteria: - Documents from specific countries or regions with little broader reference - Country specific CMAM guidelines - Resources that relate to the other Global Thematic Working Groups such as resources related to infant feeding in emergencies, nutrition information systems, etc. 3 - Nutrition assessment-related resources aside from those that relate to coverage and monitoring. - Research articles and reviews that relate to CMAM. - The search for relevant CMAM resources was finalized on the 28th of February 2023 and thus, resources published after this date were not included. - To ensure the relevance of resources, newer resources were highlighted in this review- resources older than 2008 were not typically included unless deemed relevant and thus included in the column entitled ‘Underpinning guidance, outdated and other resources. These documents aim to provide contextual understanding as well as offer practitioners additional resources on a specific area should they wish to explore a topic further. While other audiences may benefit from this signposting brief, the primary audience for this work are practitioners working in CMAM programmes. 4 List of Resources Themes Resources Brief overview Underpinning guidance, Outdated and additional resources Core CMAM documents CMAM in Emergencies Agora Training Modules: Nutrition in Emergencies (NiE) Learning Channel (Link) Basic training on Nutrition in Emergencies (Link) Acute Malnutrition in Emergencies Preparedness and Response (Link) Harmonized Training Package Version 2 (ENN & Nutrition Works, 2011) Module 1: Introduction to nutrition in emergencies (Fact Sheet, resource list, technical notes, trainer’s guide) HTP v 2 Module 2: The humanitarian system: roles, responsibilities, and coordination (Fact Sheet, resource list, technical notes, trainer’s guide) HTP v 2 Module 15: Priority health interventions that impact nutrition status in emergencies (Fact Sheet, UNICEF offers free online courses through its Agora platform. This platform is easy to access, offers digital certificates and is free of charge. A host of nutrition related training resources are housed on this platform, including those related to Nutrition in Emergencies and CMAM. The Harmonised Training Package (HTP) is a resource for trainers in the NiE sector and it can be used by individuals to increase their technical knowledge of the sector. It is designed to provide trainers from any implementing agency or academic institution with information from which to design and implement a training course according to the specific needs of the target audience, the length of time available for training, and according to the training objectives. It can be used as stand-alone modules or as combined modules depending on the training needs. The Agora modules cover much of the content of the HTP but the Community therapeutic care (CTC): a new approach to managing acute malnutrition in emergencies and beyond. (Technical notes, FANTA, 2004) Community-based therapeutic care A new paradigm for selective feeding in nutritional crises (Guidance Brief, Humanitarian Practice Network, 2004) Acute Malnutrition in Emergencies Preparedness and Response (UNICEF) ?id=28662 5 resource list, technical notes, trainer’s guide) HTP v 2 Module 19: Working with communities in emergencies (Fact Sheet, resource list, technical notes, trainer’s guide) HTP v 2 Module 22: Gender-responsive nutrition in emergencies (Fact Sheet, resource list, technical notes, trainer’s guide) CMAM Toolkit A Rapid Start-up Resource for Emergency Nutrition Link (Save the Children, 2017) The Sphere Handbook Link (Policy Guidelines, Sphere Association, 2018) HTP offers an offline version of training resources. The CMAM Toolkit is a collection of tools needed by program managers to begin the implementation of CMAM programs. This Toolkit is ‘not’ a recreation of existing tools/resources but an easy-to-use compilation of these tools and resources. The Sphere Handbook aims to improve the quality of humanitarian response in situations of disaster and conflict and to enhance the accountability of 6 Sphere Training toolkit (facilitator guide) (Sphere Association, 2019) A Toolkit for Addressing Nutrition in Emergency Situations Link (Inter-Agency Standing Committee, Nutrition Cluster, 2008) humanitarian action to crisis-affected people. The pack of 20 training modules was updated based on the feedback of trainers and learners who have been using the previous version of the tool since 2012. The pack contains five days’ worth of materials. Its’ modular nature means it can be used in its entirety, broken up into shorter workshops, or injected into broader initiatives such as humanitarian degree programs. The toolkit is intended as an easy-to-use field guide that outlines the key basic interventions for nutritional support to individuals and groups during an emergency. The toolkit offers guidance and support for nutritionists and humanitarian workers to ensure that basic guidelines are followed and the basic nutritional needs of populations in emergencies are met. It is not intended to be an exhaustive resource for each intervention presented, but rather an overview of interventions to be 7 Guidelines for Selective Feeding: The Management of Malnutrition in Emergencies Link (UNHCR & WFP. 2011) considered with references and links to more detailed technical guidance for each issue. This revised version is intended as a practical guide to design, implement, monitor, and evaluate selective feeding programs in emergencies. Generic Project Model: Community-Based Management of Acute Malnutrition (CMAM) Link (WVI, 2017) Training Guide for Community-Based Management of Acute Malnutrition (CMAM): Guide for Trainers Link, PowerPoint (FANTA III- 2018) Severe acute malnutrition | MSF Medical Guidelines Link (2022). This resource describes the CMAM approach, evidence for the approach, and how to contextualise the approach to different contexts. This CMAM training guide is designed for healthcare managers and providers who manage, supervise, and implement services for the management of acute malnutrition, including those involved in community outreach activities. The MSF protocols are focused on the diagnosis and management of inpatient SAM in children 6 to 59 months with a Position Paper on Evolution of CMAM Implementation (AAH, 2021). Updating National CMAM Guidelines Lessons from Previous Experiences (Global Nutrition Cluster Technical Alliance, UNICEF, World Vision, Action Against Hunger, ENN, 2022) WHO child growth standards and the identification of severe acute malnutrition in infants and children: a joint statement by the World Health Organization and the United Nations Children's Fund (2009) Community-Based Therapeutic Care (CTC): Field Manual (Valid International and Concern Worldwide, 2006) 8 CMAM Report: A comprehensive monitoring and reporting package for Community-based Management of Acute Malnutrition Link (Save the Children, 2015) Treating Acute Malnutrition with Ready-to-Use Therapeutic Food: A Community Outpatient Approach April 2013 Link (Technical Guidelines, EDESIA, 2013) Managing the Supply Chain of Specialized Nutritious Foods Link (Guidelines, WFP, 2019) particular focus on children with medical complications. The CMAM Report e-learning is a distance learning tool for nutrition and/or M&E staff at field, country, and head office levels with a specific focus on monitoring CMAM programs. This manual describes the use of these approaches for small-to-medium size organizations or programs that wish to add the treatment of acute malnutrition to the services they are currently providing in their communities. This manual has been produced to provide comprehensive guidance on the supply chain management of SNF. Global action plan on child wasting: a framework for action to accelerate progress in preventing and managing child wasting and the achievement of the Sustainable Development Goals (Joint statement by the Principles of FAO, WHO, UNHCR, UNICEF, WFP, and UN OCHA, 2020). Essential nutrition actions: improving maternal, new-born, infant, and young child health and nutrition (WHO, 2013) Moderate Acute Malnutrition / Moderate Wasting Moderate Acute Malnutrition: A Decision Tool for Emergencies Link (Global Nutrition Cluster, 2017) This decision-making tool aims to guide program managers to identify the most appropriate and feasible program strategy to address MAM in a particular emergency setting. It explicitly incorporates a range of contextual factors Module 6: Supplementary Feeding for the Management of Moderate Acute Malnutrition (MAM) in the Context of CMAM from Community-Based management of acute malnutrition training guide 2008 9 HTP v 2 Module 12: Management of moderate acute malnutrition (2011) Fact sheet Resource List, Technical Note. Trainer’s guide Module 8: Supplementary Feeding Program Link (UNHCR, 2008) Management of wasting during a shortage or absence of specialised nutritious food products Link (GNC-TA, 2022). into the decision-making process, in addition to population-level nutrition status before and during the emergency. The Harmonised Training Package (HTP) is primarily a resource for trainers in the Nutrition in Emergencies (NiE) sector and can be used by individuals to increase their technical knowledge of the sector. It can be used as stand-alone modules or as combined modules depending on the training needs. This module outlines the monitoring requirements for the following nutritionally at-risk groups, most targeted for supplementary feeding in refugee operations. This note applies predominantly to situations where specialised nutritious foods (SNF) have been used and which are facing an absence or shortage of products. It does not replace guidance already existing in national guidelines for the (USAID, Valid International, Concern Worldwide, UNICEF) 10 Technical note: supplementary foods for the management of moderate acute malnutrition in infants and children 6– 59 months of age Link (WHO, 2012) treatment of wasting or where the use of SNF is not indicated, accepted, or established. This technical note summarises the available evidence and presents some principles underlying the dietary management of children with moderate acute malnutrition with a proposed nutrient composition profile for supplementary foods relevant to situations in which their use may be warranted. Severe Acute Malnutrition / Severe Wasting Guideline: updates on the management of severe acute malnutrition in infants and children Link (WHO, 2013) HTP v 2 Module 13: Management of Severe acute malnutrition (2011) Fact sheet, Resource List, Technical note, Trainers guide This guideline provides global, evidence-informed recommendations on several specific issues related to the management of severe acute malnutrition in infants and children. The Harmonised Training Package (HTP) is primarily a resource for trainers in the Nutrition in Emergencies (NiE) sector and it can be used by individuals to increase their technical knowledge of the sector. It can be used as stand-alone modules or as combined modules depending on the training needs. Module 04- Outpatient Care for the Management of SAM Without Medical Complications from Community-Based management of acute malnutrition training guide 2008 (USAID, Valid International, Concern Worldwide, UNICEF) Scale-up of severe wasting management within the health system: A stakeholder perspective on current progress (ENN, ECF & Irish Aid, 2021) 11 Module 8: Outpatient management of severe acute malnutrition Link (WHO, 2021) Guidelines for the Integrated Management of Severe Acute Malnutrition: In- and Out-Patient Treatment Link (ACF International, 2011) WHO guideline on the dairy protein content in ready-to-use therapeutic foods for the treatment of uncomplicated severe acute malnutrition Link (2021) The training package is based on the 2002 WHO Training course on the management of severe malnutrition, which was updated in 2009. In 2013, WHO issued the Guideline: updates on the management of severe acute malnutrition in infants and children, which provided updated recommendations. This guideline outlines ACF’s guidelines on SAM Treatment, including both inpatient and outpatient management. This guideline provides global, evidence-informed recommendations focusing on whether reduced dairy or non-dairy RUTF should be used for treating uncomplicated severe acute malnutrition. This training course on the inpatient management of severe acute Management of severe malnutrition: a manual for physicians and other senior health workers (who.int) (WHO, 1999) 12 Training course on the inpatient management of severe acute malnutrition: (WHO, 2022) Facilitator Guide, Clinical Instructor Guide Course Director Guide Introducing Updated Packaging and Reconstitution Guidance for Therapeutic Milk Link (WHO & UNICEF, 2017). malnutrition includes training modules, training guides, and supporting materials. UNICEF and WHO updated their guidance on Therapeutic milk (TM) with updated packaging and reconstitution guidance. Growth Faltering among infants u6m WHO recommendations for care of the preterm or low-birth-weight infant Link (WHO, 2022) Management of small and nutritionally At-risk infants under six months and their mothers (MAMI) Care Pathway Package, Version 3 (Irish Aid, ECF & ENN, 2021) MAMI Care Pathway Package The recommendations in this guideline are intended to inform the development of national and subnational health policies, clinical protocols, and programmatic guides. The MAMI Care Pathway Package provides practitioners with a resource to screen, assess, and manage small and nutritionally at-risk infants under six months and their mothers (MAMI). It has been updated from the WHO recommendations on interventions to improve preterm birth outcomes (WHO, 2015) Community Management of at-risk Mothers and Infants under six months of age (C-MAMI) (Version 2.0. 2018) [Irish Aid, Save The Children, GOAL, LSHTM & ENN] 13 MAMI Counselling Cards and Support Actions Booklet MAMI health Workers Users Booklet MAMI mid-upper arm circumference (MUAC) tape Link (GOAL) 2018 version through stakeholder consultation, literature reviews, and based on learnings of previous implementation experiences. MAMI-MUAC tapes using reduced thresholds have been developed by GOAL to help identify at-risk infants under 6 months. Community Mobilisation Technical Brief: Community Engagement for CMAM Link (Coverage Monitoring Network, 2015) How to Conduct Community Assessment Link (Guide, Coverage Monitoring Network, 2015) This resource explores community engagement. Community engagement for CMAM aims to engage community actors and obtain their commitments for actions they will take to support caretakers of children with acute malnutrition to access and use CMAM services, and their support to community engagement activities, including volunteers’ and outreach workers’ work. This document articulates how to conduct community assessments and also provides guidance on essential tool development for community assessments. Community Outreach for Community-Based Management of Acute Malnutrition in Sudan: A Review of Experiences and the Development of a Strategy (Technical report, USAID, FANTA.2, 2010) 14 Care Groups: A Reference Guide for Practitioners Link (USAID, FSN Network, FHI, Core Group, World Relief, 2016) Mobilising Communities for Improved Nutrition: A Manual and Guide for Training Community Leaders (USAID, PATH & IYCN, 2011) Manual for Training Community Leaders Guide for Training Community Leaders This brief guide is meant to serve as a companion to Care Groups: A Training Manual for Program Design and Implementation, and additional details on all topics covered in this guide are provided in the Training Manual. This guide was developed in response to practitioner requests and assumes the reader already has a general understanding of the Care Group methodology. This resource contains instructions for facilitating a one-day workshop with community leaders (e.g., community, religious, and business leaders and government officials) to support and improve children’s health and well-being through improved infant and young child feeding and maternal dietary practices. It also provides information aimed directly at community leaders that will help them support women and families to follow recommended practices concerning the issues above. Examples of Programmes and Adaptations Simplified Approaches Simplified Approaches for The Provision of Care to Children with This document describes simplified approaches for improving care for Simplified approaches to the treatment of wasting. ENN Technical Brief (2020) 15 Wasting and other forms of Acute Malnutrition Link (Technical guidance, UNICEF, 2020) Global Training Toolkit on Simplified Approaches for the Detection and Treatment of Child Wasting Link (UNICEF, ACF, STC, IRC, HLA, 2022) Family MUAC Approach in the Time of COVID-19: Implementation Guidance for Program Managers Link (Save The Children, 2020) children with wasting, including adaptations to national and global protocols that aim to remove barriers to treatment, improve coverage and reduce the costs of care. This training manual focuses on simplified approaches to detect and treat childhood wasting. It comprises modules on Family MUAC, Community Health Worker-led Treatment, Reduced Frequency of Follow-Up Visits, and Simplified Treatment Protocols. This document provides step-by-step guidance for implementing the Family MUAC Approach, in both Covid-19 and non-Covid-19 contexts, to increase CMAM coverage. This guidance can also be used in line with simplified treatment protocols during a Covid-19 pandemic where routine CMAM approaches may not be feasible. A guide-in-brief to the Family MUAC (or Mother MUAC as it is also sometimes Rapid Review: Treatment of Wasting Using Simplified Approaches by UNICEF (2020) IRC Simplified protocol for Acute Malnutrition (2020) A Simplified, Combined Protocol: Evidence Overview (IRC, 2022) Simplified Approaches Community of Practice (State of Acute malnutrition, 2019) CMAM expanded admissions guidance. (Interagency Nutrition Meeting, 2014) 16 Family MUAC: Supporting entire communities to screen for acute malnutrition Link. (Brief guide, Action Against Hunger, 2020) Family MUAC training toolkit (GOAL, 2019) Training Guide for Family MUAC approach Presentation: Family MUAC approach Family MUAC M&E toolkit Family MUAC: Care Group Lesson Reference guide for practitioners Link (Action Against Hunger, 2020) called) community screening approach which empowers mothers, caregivers, and other family members to screen their children for acute malnutrition using color-coded MUAC tapes. The GOAL HQ program quality technical team developed a standardised Family MUAC training module with an associated monitoring and evaluation framework for multi-country use. This lesson was developed by Action Against Hunger’s Nutrition teams in Nigeria and Uganda, with the support of the Social and Behaviour Change Adviser. The lesson is part of a curriculum for the care group approach and was developed using the facilitation cue’ illustrations and methods for meeting facilitation from Care Groups: A Reference Guide for Practitioners Excel-based M&E tools for recording data that include screening forms, admission 17 M&E tools for the Family MUAC approach Link (Toolkit, IMC, 2019) Mother-MUAC Teaching Mothers To Screen For Malnutrition- Guidelines for the training of trainers Link (ALIMA, 2016). Video: w1oRXBoo forms, training records, and monthly training registration. This guideline was developed by ALIMA to support the training of mothers on the Family MUAC approach. Community Health Workers implementin g CMAM Toolkit for CHW Community-Based Treatment of Uncomplicated Wasting for Children 6-59 Months in the Context of Covid-19 (SIMPLIFIED REGISTERS FOR CHWs) Combined, SAM only, MAM only (Toolkit, International Rescue Committee, 2020) Toolkit for CHW Community-Based Treatment of Uncomplicated Wasting for Children 6-59 Months in the Context of Covid-19 Link (Toolkit, International Rescue Committee/UNICEF, 2020) Simplified treatment registers (SAM register, MAM register, or combined register) to enable CHWs to document caregiver and child information, admit and track treatment progress, and record the treatment/discharge outcomes of each child they treat. The Toolkit for CHW Community-Based Treatment of Uncomplicated Wasting for Children 6-59 Months in the Context of Covid-19 brings together existing evidence and operational experience to provide implementers key recommendations and considerations for Community-based Management of Acute Malnutrition Using Community Health Activists in Angola (Report, World Vision, 2013) Treating Malnutrition in The Community (Brief report, IRC, 2017) Developing and Strengthening Community Health worker program at scale: A Reference Guide for Program Managers and Policy Makers (USAID & MCHIP, 2013) 18 rolling out the approach as well as step-by-step guidance on an implementation protocol. Prioritizing Community Health Worker Data for Informed Decision-Making (Report, Frontline Health Workers Coalition, 2008) CMAM Surge CMAM Surge Toolkit: Implementation guides and tools (Concern Worldwide, 2016) Operational Guide Link Facilitator guide Link, slides Introduction to CMAM Surge Link Value for money Framework Link Concern developed CMAM Surge Global Guidance to support the implementation of CMAM Surge which aims to help government health teams respond to relative changes in treatment capacity and caseloads of wasted children. The Operational Guide provides an overview of the approach and describes eight steps covering the implementation and monitoring processes. Currently, the approach focuses on the management of SAM but can also include MAM if this is part of routine health services. The ‘CMAM Surge’ approach: setting the scene CMAM Surge: the way forward (FEX 64, ENN, 2021). New Design Framework for CMAM Programming (by Peter Hailey and Daniel Tewoldeberha) (2010) ested Additional resources focused on COVID-19 Prevention, Early Detection and Treatment of Wasting in Children 0-59 Months through National Health Systems in the Context of COVID-19: Implementation Guidance Link (WHO & UNICEF, 2020) This document serves as a tool for implementing the recommendations reflected in existing WHO and UNICEF guidance on the delivery of services through national health systems for the prevention, early detection, and treatment of child wasting in the context of COVID-19. This note reflects broad guidance for all levels of the health system, including community health services that offer prevention, early detection, and treatment services for child wasting. Adaptations to the Management of Acute Malnutrition in the Context of COVID-19 (USAID, Action Against Hunger, UNICEF, 2022) Adaptations to the Management of Acute Malnutrition in the Context of COVID-19 (Report, Action Against Hunger, USAID & UNICEF, 2020). 19 Technical brief on Management of Child Wasting in the context of Covid-19 Link (UNICEF, GNC-TA, GNC, 2020) Adapting Community-based Management of Acute Malnutrition in the context of COVID-19 Link (Interim guidance, Concern Worldwide, 2020) Operational guidelines to contain and reduce transmission of novel coronavirus (covid-19) for nutrition service providers and clients of WFP nutrition-specific programs Link (World Food Program -Uganda, 2020) This Brief is meant to provide information specific to services and programs for the management of child wasting in the context of COVID-19, and it contains information that is not already available elsewhere. This Brief does not cover wider mitigation and response measures available in other guidance. The guide attempts to provide practical steps based on international guidelines and recommendations to ensure essential treatment services for acute malnutrition continue as much as possible while minimising the risk of COVID-19 transmission. The primary focus of this guide is the management of the outpatient component of CMAM. This operational guideline is intended to support primary care teams in the practical implementation of nutrition-specific programming during the COVID-19 pandemic. WFP’s additional recommendations for the management of maternal and child malnutrition prevention and treatment in the context of COVID-19 (2020) ALIMA Suggested Actions for Consideration in the Management of Child Wasting in the Context of COVID-19 (2022). Continuing care during COVID-19 Adopting Life-Saving Approaches to Treat Acute Malnutrition (Policy Brief, IRC, 2020) 20 CMAM and other Health concerns Checklist for Determining HIV Status for Children 0-59 Months with Severe Acute Malnutrition Link (Save the Children, 2015) Cholera and acute malnutrition Link (Medical Guidelines, MSF) Nutritional care and support for patients with tuberculosis Link (WHO, 2013). Guidelines for an Integrated Approach to the Nutritional care of HIV-infected children (6 months-14 years) Link Chart Booklet Guide for local adaptation A checklist was created to help health workers determine the HIV status of children 0-59 months with severe acute malnutrition that are enrolled in the management of acute malnutrition program. This resource is a part of the management of the cholera epidemic in general and cholera case management in which acute malnutrition is one of the possible conditions. This guideline explores nutrition care for patients with tuberculosis. Its primary audience is health workers providing care to people with tuberculosis. However, the guideline is also intended for a wider audience including policymakers, their expert advisers, and technical and program staff at organizations involved in the design, implementation, and scaling-up of nutrition actions for public health. The Guidelines provide a framework for integrating nutrition support into the routine care of HIV-infected children. Manual for the integration of childcare practices and mental health within nutrition programs (Manual, Action Against Hunger, 2006) 21 (Handbook, WHO, 2009) Technical Brief: Severe Acute Malnutrition and Infection Link (Technical brief, 2013) This brief outlined the evidence underpinning important questions relating to the management of infectious diseases in children with SAM and highlighted research gaps. CMAM and IMCI integration Integrating Early Detection and Treatment of Child Wasting into Routine Primary Health Care Services: A resource guide to support National planning Link (UNICEF, 2021) Community Case Management (CCM) In Humanitarian Settings: Guidelines for Humanitarian Workers Link (Save the Children & USAID, 2019) This new resource offers an easy-to-follow process for governments to identify integration actions that can help achieve wasting program goals within routine primary health care services. This guide includes tools and resources to support national decision-makers in developing their integration plans. This guide provides emergency responders with basic information on iCCM, support for making key decisions to implement iCCM during a spike in humanitarian needs or throughout protracted crisis settings, and choices for transition after the acute phase has ended. Child health in the community - "Community IMCI": Briefing package for facilitators (Training material, WHO, 2004) 22 Pocket book of hospital care for children: guidelines for the management of common childhood illnesses, 2nd ed. Link (WHO, 2013) Community Case Management Essentials Treating Common Childhood Illnesses in the Community: A guide for program Managers Link (USAID, Core group, Save the Children, MCHIP, & BASICS, 2010) Guidelines for the management of common childhood illnesses. This is the second edition of the Pocket book of hospital care for children. It is for use by doctors, nurses, and other health workers who are responsible for the care of young children at first-level referral hospitals. The Pocket Book is one of a series of documents and tools that support the Integrated Management of Childhood Illness (IMCI). This guide documents what is known about CCM and how to make it work. First, health program managers are introduced to the basics. Then, CCM Essentials walks its readers through the process of designing and managing a high-quality CCM program. Program planning, Management, Evaluation, Reporting, and Assessments Operational managemen t support, planning, costing CMAM Costing tools (USAID, FANTA-2, fhi360, 2012) CMAM Costing Workbook version 1.1 Link CMAM Costing tool user guide, version 1.1 Link CMAM costing tool exercise, version 1.1 Link This Microsoft Excel-based application estimates the costs of establishing, maintaining, and/or expanding services for CMAM at the national, sub-national, and district levels. This information helps program managers determine whether their plans for CMAM are financially feasible, identify the resources needed, Coverage matters: a collation of content on coverage monitoring of CMAM programs. (ENN, Coverage Monitoring Network, 2014) Measuring Coverage in Community-based Therapeutic Care Programs 23 CMAM Costing tool case study, version 1.1 Link Caseload, targets, and supplies calculation tool (nutrition cluster) Link (GNC, 2016) Guidance Note: New Design for the Mid-Upper Arm Circumference (MUAC) Tape Link, Design (UNICEF, 2020) Nutrition Program Design Assistant: A Tool for Program Planners (NPDA) Workbook Version 2, Revised 2015 (Link, Excel) (Save the Children, USAID, FANTA-III, FHI360, Core Group- Advancing Community Health Worldwide) and formulate an effective implementation plan. (Note that this tool deals only with the management of SAM.) An Excel workbook is designed to aid the calculation of caseloads, targets, and supplies for Nutrition Cluster interventions as per available guidance. In response to the need of scaling up the early detection and treatment of children who are wasted, UNICEF has updated the design of the MUAC tape so it can be easily utilized by caregivers. Further, the modified design adheres to the safety and hygiene precaution measures in the context of COVID-19 The Nutrition Program Design Assistant is a tool to help organisations design the nutrition component of their community-based maternal and child health, food security, or another development program. The tool has two components: through Semi-Quantitative Evaluation of Access and Coverage (SQUEAC): An Overview (Coverage Monitoring Network) 24 Module 7 Planning CMAM Services at the District Level, from Community-Based management of acute malnutrition training guide 2008 Link (USAID, Valid International, Concern Worldwide, UNICEF). How do we estimate the caseload for SAM and/or MAM in children 6-59 months in each period? Link (2012) (1) a reference guide for understanding the nutrition situation and identifying and selecting program approaches, and (2) a workbook to record information, decisions, and decision-making rationale. This module introduces participants to the issues and considerations in the design and planning of a community-based management of acute malnutrition (CMAM) service or program. This module focuses on the different steps used to plan a CMAM service or program. It aims to provide participants with the tools and conceptual frameworks for thinking through the planning stages according to the context. Brief on estimating caseload for SAM and/or MAM in children 6-59 months in each period? Nutrition Assessment s and reporting Emergency Nutrition Assessment Guidelines for field workers (Save the Children UK, 2004) Link The ENA Manual is intended to provide straightforward and comprehensive guidance to nutritionists and other fieldworkers responsible for assessments Minimum reporting package (MRP) for emergency supplementary and therapeutic feeding programs. User guidelines. 25 Module 8: Monitoring and Reporting on CMAM from Community-Based management of acute malnutrition training guide 2008 Link (USAID, Valid International, Concern Worldwide, UNICEF) SQUEAC: Low resource method to evaluate access and coverage of programs Link. (Technical Guidelines, CMN, 2016) in emergency settings. It is divided into 6 parts including a how to assess the causes of malnutrition and how to interpret assessment data alongside mortality and malnutrition information. This module introduces participants to the basic principles of monitoring, reporting on and supervising community-based management of acute malnutrition (CMAM) services, with a focus on outpatient care. The module describes how individual children are tracked and monitored in CMAM and how monitoring information and data are collected and reported for the service/program. The purpose and function of support and supervisory visits are discussed. The SQUEAC (Semi- Quantitative Evaluation of Access and Coverage) coverage methodology uses a two-stage screening test model to assess the coverage of programs. (Save the Children, USAID, ENN & European Commission, 2012 26 Semi-Quantitative Evaluation of Access and Coverage (SQUEAC)/ Simplified Lot Quality Assurance Sampling Evaluation of Access and Coverage (SLEAC) Technical Reference Link (FANTA III & USAID, 2012) Open review of coverage methodologies. Questions, comments & ways forward. SQUEAC, SLEAC, S3M Link (Technical guideline, Epicentre, 2015) Standardized Monitoring and Assessment of Relief and Transitions The Technical Reference comprises dedicated sections on SQUEAC and SLEAC methods and a series of case studies that address: Assessing evidence and coverage in very high coverage programs, assessing evidence and coverage in moderate coverage programs, sampling without maps or lists, using satellite imagery to assist sampling in urban settings, active and adaptive case-finding in rural settings, within-community sampling in an IDP camp, within-community sampling in urban settings and the case of the ‘hidden defaulters’. This review was commissioned by the CMN with the explicit objective of improving existing coverage methodologies through participatory review with users of the methodologies. The following package is the set of materials for Survey Manager level 27 (SMART) Survey planning tool Kit For survey Manager (AAH- Canada, 2014) Strengthening Nutrition Emergency Response Preparedness (ERP) Interim ERP step-by-step guide (GNC, USAID, 2022) ERP Step-by-step guide | Global Nutrition Cluster ERP plan template | Global Nutrition Cluster The Preparedness actions work plan | Global Nutrition Cluster training (individuals who will be planning, supervising, analysing, and report writing of the SMART survey data). The ERP guide outlines a step-by-step process describing how to undertake ERP planning. It is the central element of the GNC ERP toolkit that, in addition to this guide, includes templates, tools, and e-learning modules to support Nutrition ERP planning. Prevention of acute malnutrition (wasting)- non-exhaustive list Best Practice in Preventing Child Wasting within the Wider Context of Undernutrition Link (ENN, Irish Aid, & ECF, 2021) The brief is based on work since 2014 by the Wasting Stunting Technical Interest Group (WaSt TIG) and the Emergency Nutrition Network (ENN). It updates previous reports on wasting prevention by ENN and builds on ENN’s recent position paper on wasting. This technical brief examines current evidence, knowledge, and practice Wasting in the wider context of undernutrition: An ENN position paper June 2020 28 Preventing Moderate Acute Malnutrition (MAM) Through Nutrition-Sensitive Interventions Link (Technical guidelines, CMAM Forum, 2014) Practical pathways to integrate nutrition and water, sanitation and hygiene: Global Brief Link (Water Aid & Action Against Hunger, 2019) WASH’ Nutrition: A practical guidebook on increasing nutritional impact through the integration of WASH and Nutrition programs Link (Action Against Hunger, UNICEF & EU, 2017) relating to the prevention of moderate acute malnutrition (MAM) through nutrition-sensitive interventions in various sectors. This brief was developed based on a review of many types of evidence including ‘grey’ research, published research, systematic literature reviews, ongoing impact evaluations, and other plausible research systematically collected. This global brief brings together commonalities and key enablers, or pathways, for progress to collaborate and integrate WASH and nutrition policies and programs in countries with high undernutrition burdens. This operational guidebook has been developed to provide practitioners with usable information and tools so that they can design and implement effective WASH and nutrition programs. 29 Integrating WASH and nutrition Link (Learning brief, FHI 360, 2015) Livestock and Nutrition LEGS: A Discussion Paper for the Livestock Emergency Guidelines and Standards (LEGS, 2020) WASH-plus recognizes the importance of integrating water, sanitation, and hygiene into other development priorities, such as nutrition, to achieve its objective of healthy households and communities. This discussion paper was commissioned by LEGS to review the key issues relating to nutrition in the context of livestock-based emergency interventions. The paper presents the outcome of a literature review that summarises key issues for livestock and nutrition. It also presents three short case studies illustrating the impacts of livestock emergency responses on nutrition. Additional relevant resources Improving Pre-Service Nutrition Education and Training of Frontline Health Care Providers Link (Technical brief, FANTA-III, USAID, fhi360, 2018) The purpose of this technical brief is to describe the process and methods the Food and Nutrition Technical Assistance III Project (FANTA) used to improve pre-service nutrition education and training of frontline healthcare providers, including nurses and midwives, in two countries. MULTI-SECTORAL NUTRITION STRATEGY 2014–2025 Technical Guidance Brief (USAID, 2016) Social protection and nutrition (Australian Department of Foreign Affairs and Trade, 2015) 30 Guide to Anthropometry: A Practical Tool for Program Planners, Managers, and Implementers Link (FANTA-III, USAID, fhi360, 2018) T he Nutrition Counselling Visit for Young Children (Video) et/document/the-nutrition-counselling-visit-for-young-children/ (Global Health Media & UNICEF, 2018) Integrated Food Security Phase Classification manual IPC version 3.0, Link (IPC -GSU, 2019) This user-friendly reference offers information and step-by-step instructions on using anthropometry to assess the nutritional status of individuals and communities. The guide, which replaces the 2003 Anthropometric Indicators Measurement Guide, explains anthropometric measurements/indices and the nutrition conditions they assess in different demographic groups, discusses how to interpret anthropometric data, and offers guidance on selecting equipment for taking measurements in low-resource settings. This video shows health workers how to assess and guide a caregiver during a nutrition counselling visit. The IPC Technical Manual 3.1 is an update of version 3.0 (published in 2019) to improve the understanding of ‘Famine’ and ‘Famine Likely’ as well as strengthen the methods used to guide the convergence of evidence by using 31 A practical toolkit for People in Need's integrated programming for improved nutrition. Link (People in need, 2014). Defining Nutrition Assessment, Counselling, and Support (NACS) Link (Technical notes, FANTA, 2012) international standards and cut-offs for Famine classifications. The IPC Manual is a set of internationally accepted guidelines for the analysis of food insecurity and malnutrition, developed through consultations with food security and nutrition experts from 15 organisations. This toolkit recommended e-learning courses, dozens of carefully selected guidelines, experience from PIN missions, support of PIN’s Senior Advisors, and much more is available to help you to integrate IPIN into your programming and effectively contribute to reducing undernutrition. The nutrition assessment, counselling, and support (NACS) approach aims to improve the nutritional status of individuals and populations by integrating nutrition into policies, programs, and the health service delivery infrastructure. Most of the experience with this approach has come from working with people living with HIV (PLHIV), but lessons from this experience 32 A Gender-Transformative Framework for Nutrition Advancing Nutrition and Gender Equality Together Link (WVI, AAH, Care, STC, NI, Bruyere, Mother Food International, 2020) are being adapted and extended to standardised case management for malnourished people with other infectious diseases and non-communicable diseases. The Gender-Transformative Framework for Nutrition (GTFN) is a conceptual model supported by research and practice that enables improved gender analysis, solutions design, and monitoring and evaluation of nutrition approaches, as well as interventions promoting women and girls’ empowerment. The GTFN applies systems thinking that enables users to critically examine the multi-sectoral drivers of malnutrition using a gender equality and empowerment lens 33 List of relevant websites on CMAM - - - - - - Malnutrition (who.int) - - - - - - - - - - - 5.pdf - - - - - - - - - Nutrition | Results for Development (r4d.org) 34 List of Abbreviations Acronyms Abbreviations AAH/ACF Action Against Hunger/ Action Contre la Faim CMAM Community Management of Acute Malnutrition CMN Coverage Monitoring Network CTC Community-Based Therapeutic Care ECF Eleanor Crook Foundation ENA Emergency Nutrition Assessment ENN Emergency Nutrition Network FANTA Food And Nutrition Technical Assistance FAO Food and Agriculture Organization FHI Family Health International FSN Network Food Security and Nutrition Network GNC Global Nutrition Cluster GNC-TA Global Nutrition Cluster – Technical Alliance GTAM Global Technical Assistance Mechanism for Nutrition GTWG Global Technical Working Group HLA Humanitarian Leadership Academy HTP Harmonized Training Package iCCM Integrated Community Case Management IMC International Medical Corps IMCI Integrated Management of Childhood Illness IPC-GSU Integrated food security Phase Classification- Global Support Unit IRC International Rescue Committee (IRC) IYCFE Infant and Young Child Feeding in Emergencies IYCN Infant and Young Child Nutrition MAM Moderate Acute Malnutrition MCHIP Maternal and Child Health Integrated Program MSF Médecins Sans Frontières MUAC Mid Upper Arm Circumference NACS nutrition assessment, counselling, and support NI Nutrition International SAM Severe Acute Malnutrition SLEAC Simplified LQAS 1 Evaluation of Access and Coverage SMART Standardized Monitoring and Assessment of Relief and Transitions SQUEAC Semi-Quantitative Evaluation of Access and Coverage STC Save The Children UNHCR United Nations High Commissioner for Refugees UNICEF United Nations Children’s Fund UNOCHA United Nations Office for the Coordination of Humanitarian Affairs WFP World Food Programme WHO World Health Organisation WVI World Vision International 35 |
5970 | http://master.get.com.tw/file/Paper/Transfer/MDU/102/Logic.pdf | 第 1 頁 共 3 頁 邏輯推理考古題 答題注意事項: 1. 本試題共45 題，皆為單選題。1~10 題，每題2 分；11~20 題，每題3 分；21~45 題，每題6 分；合計滿分200 分。答錯不倒扣。 2. 請考生將答案以2B 鉛筆劃記於答案卡上，使用非2B 鉛筆劃記答案卡，導致讀卡機不能判讀答案者，考生自行負 責；或考生只將答案書寫於試題紙上而未劃記於答案卡者不予計分。 3. 請考生自行核對准考證號碼，並將號碼書寫於試題紙上。考試完畢，請將試題紙與答案卡一併繳回，未繳回者， 不予計分。 測驗試題 請仔細閱讀下列1~5 題，每題是一語句，不過其中的字沒有依照順序排列，並有一個多餘的字。請你把那個 多餘的字找出來。 1. 前門拒虎後門狼進鬥 (A) 虎 (B) 狼 (C) 拒 (D) 進 (E) 鬥。 2. 不做明人作暗事(A)做 (B)作 (C)明 (D)人 (E)不。 3. 無不知言以也知禮人(A)知 (B)言 (C)禮 (D)也 (E)無。 4. 利下上征交而國矣之危(A) 上 (B) 交 (C) 矣 (D) 之 (E) 危。 5. 智義里為仁美不處仁擇得焉 (A) 義 (B) 仁 (C) 擇 (D) 焉 (E) 智。 請仔細閱讀下列6~15 題，並找出相對應之語句： 6. 宋….._，好像…..曲。(A)詩…..民 (B)小說…..明 (C)詞…..元 (D)詞…..唐。 7. 元曲…..關漢卿、白樸，好像唐詩…..、_。(A)王安石、杜甫 (B)岳飛、文天祥 (C)李白、王維 (D) 賈誼、司馬相如。 8. 之於人體，好像燃料之於(A)食物…..飛機 (B)血液…..電車 (C)手足…..機車 (D)食物…..冰箱。 9. 紅樓夢之於_，好像之於後漢書(A)明、曹雪芹，漢、班固 (B)元、施耐庵，明、歸有光 (C)清、 曹雪芹，晉、范曄 (D)清、羅貫中，漢、范曄。 10. 聲望：名望=肆力：(A)放縱 (B)蠻力 (C)盡力 (D)學業。 11. 宜蘭之於﹔好比之於風 (A)海…台中 (B)北部…南部 (C)雨…新竹 (D)水…高雄 12. 父母….._ ， 好像…..孝 、 敬 (A)責難…..姊妹 (B)教訓…..兄弟 (C)慈 、 愛…..子 、 女 (D) 家庭…..朋友 13. 鉛筆之於 ，好比之於板擦 (A)黑板…老師 (B)工人…校長 (C)橡皮擦…粉筆 (D)樹木…河 流 14. 之於主義，好像個人之於 (A)政黨…思想 (B)政黨…行動 (C)信仰…行動 (D)國家…行 動 15. 之於止渴，好像畫餅之於(A)飲水…觀賞 (B)水果…評鑑 (C) 望梅…充飢 (D)芒果…食物 背面尚有試題 PDF created with pdfFactory Pro trial version www.pdffactory.com 第 2 頁 共 3 頁 請仔細閱讀下列16~20 題，並依字序分別為(A)(B)(C)(D)，請選出錯字： 16. 明謀皓齒。 17. 杯盤狼藉。 18. 拍按叫絶。 19. 班門弄釜。 20. 萬賴無聲。 請仔細閱讀下列各題，並選出一個合適答案： 21. 2，8，18，32，50， (A)76 (B)70 (C)72 (D)64 22. 3，1，6，2， ，3，12，4 (A) 9 (B)12 (C)15 (D)18 23. 2，6，12，20，30， (A)38 (B)39 (C)41 (D)42 24. 0，2，1，5，2，8，3， (A)9 (B)10 (C)11 (D)12 25. 1，4，16，36，64， (A)81 (B)100 (C)121 (D)144 26. BA, DC, , HG, JI, LK (A)BC (B)ED (C)FE (D)EF (E)CB 27. 3, 1 ,6 ,3 ,9, 5 ,12 ,7 , (A)8 (B)16 (C)15 (D)10。 28. 5, 12, 22, 35, 51 ,70, (A)92 (B)93 (C)82 (D)88。 29. 1, 2, 3, 4, 5 ,8, 7, (A)9 (B)12 (C)6 (D)16。 30. 0 ,2, 0 ,4 ,0, 6, 0, (A)3 (B)5 (C)8 (D)9。 31. 1, 0 ,3, 0, 5 ,0 ,7, (A) 8 (B)0 (C) 9 (D)6。 請計算下列各題，並選出一個合適答案： 32. 面積16 平方公分的正方形，其周長為幾公分？(A)16 (B)20 (C)24 (D) 64。 33. 周長a，半徑0.5 的圓，圓周率為何？(A)2a (B)3a (C) 2 a (D)a。 34. 某數的十位數為x，個位數為y，某數是多少？(A)x+y (B)10y+x (C)x-y (D)10x+y。 35. x=0.5，求 x 1 1 1 1 1 1 + + + ？(A)2 (B)1/2 (C)7/4 (D)4/7。 36. 13 個人住一間有7 個人無處住，15 個人住一間有1 人無處住，問共有多少人？(A)46 (B)47 (C)48 (D)50 。 37. 鉛筆6 支13.5 元，則一打需多少錢？(A)40.5 元 (B)13.5 元 (C)27 元 (D)54 元。 38. 某物品價格1000 元，購買1 個加稅後為1006 元，則稅率為多少？(A)6% (B)0.4% (C)0.6% (D)0.3%。 39. 503.05 右邊的5 是左邊5 的幾倍？(A)10000 (B)1000 (C)1/1000 (D)1/10000。 40. 矩形，一邊長為x，其周長為y，請問面積？(A)xy (B)x(y-2x) (C)x(y-2x)/2 (D)x(y-x)/2。 次頁尚有試題 PDF created with pdfFactory Pro trial version www.pdffactory.com 第 3 頁 共 3 頁 41. 投籃時(如下圖)，請問那一點之速度最小？(A)A (B)B (C)C (D)D。 A B C D 42. 若下圖中正方形XYZW 之面積等於△RST 之面積，求邊長RT=? (A)2 (B)24/9 (C)4 (D)48/9 (E)8 43. 如下圖，若每個圓的面積是4π，則灰色面積是多少？(A)36-4π (B)36-16π (C)64-4π (D)64-16π 。 44. 45. 參考答案 1.E 2.B 3.C 4.D 5.D 6.C 7.C 8A 9.C 10.C 11.C 12.A 13.C 14.C 15.C 16.B 17.D 18.B 19.D 20.B 21.C 22.A 23.D 24.C 25.B 26.C 27.C 28.A 29.D 30.C 31.B 32.A 33.D 34.D 35.C 36.A 37.C 38.C 39.D 40.C 41.C 42.E 43.D 44.D 45.C X Y W Z 6 R T S 9 PDF created with pdfFactory Pro trial version www.pdffactory.com 注意： ◆ 本試題共計100 分，每題2 分。 ◆ 請將正確答案填寫在「答案卷」或「答案卡」上，並請依題號作答。 ◆ 請考生自行填上准考證號碼，考完後， 「試題」 、 「答案卷」或「答案卡」一併繳回。 ※ 語詞分辨，請仔細閱讀下列1~15 題，並選出一個答案。 例如: (A)人類 (B)猪 (C)狗 (D)鳥……… 答案: D 1.(A)美麗 (B)高大 (C)精神 (D)矮小。 2.(A)房屋 (B)樓房 (C)平房 (D)三合院。 3.(A)這對夫婦 (B)真是郎才女貌 (C)而住屋之豪 (D)彷彿仙島樂園。 4.(A)小健很愛她 (B)所以每次她來找小健 (C)他都覺得 (D)不厭其煩。 5.(A)桌子 (B)電燈 (C)電視機 (D)跑步。 6.(A)清明節 (B)端午節 (C)元旦 (D)中秋節 (E)重陽節 7.(A)菊花 (B)野花 (C)蓮花 (D)牡丹花 (E)茶花 8.(A)巴哈 (B)蕭邦 (C)莫札特 (D)安徒生 (E)貝多芬 9.(A)孔子 (B)老子 (C)子游 (D)子貢 (E)子禽 10.(A)李政道 (B)丁肇中 (C)楊振寧 (D)李遠哲 (E) 朱棣文 11.__之於喜怳，好像失敗之於_（a）成功/雀躍（b）成功/沮喪（c）沮喪/錯愕（d）收穫/播種 12.之於耳朵，好像影像之於（a）聲音/眼睛（b）聲音/雙手（c）味道/眼睛（d）味道/聲音 13._之於電風扇，好像火之於（a）風/瓦斯爐（b）風/電子鍋（c）火/瓦斯爐（d）火/電子鍋 14.鉛球之於__，好像_之於徑賽（a）球賽/撐竿跳（b）田賽/馬拉松（c）球賽/羽毛球（d）田賽/撐竿跳 15.黑板之於，好像__之於毛筆（a）板擦/橡皮擦（b）粉筆/宣紙（c）粉筆/麥克筆（d）板擦/宣紙 ※ 請仔細閱讀下列16~20 題，並依字序分別為(A)(B)(C)(D)，請選出錯字： 16.明鏡高旋。 17.明目彰膽。 18.燕而新婚。 19.犖犖大則。 20.帥爾操觚。 ※請仔細閱讀下列各題，並選出一個合適答案： 21. 2，8，18，32，50， (A)76 (B)70 (C)72 (D)64 22. 3，1，6，2， ，3，12，4 (A) 9 (B)12 (C)15 (D)18 23. 2，6，12，20，30， (A)38 (B)39 (C)41 (D)42 24. 0，2，1，5，2，8，3， (A)9 (B)10 (C)11 (D)12 25. 1，4，16，36，64， (A)81 (B)100 (C)121 (D)144 26. BA, DC, , HG, JI, LK (A)BC (B)ED (C)FE (D)EF (E)CB 27. 3, 1 ,6 ,3 ,9, 5 ,12 ,7 , (A)8 (B)16 (C)15 (D)10。 28. 5, 12, 22, 35, 51 ,70, (A)92 (B)93 (C)82 (D)88。 29. 1, 2, 3, 4, 5 ,8, 7, (A)9 (B)12 (C)6 (D)16。 30. 0 ,2, 0 ,4 ,0, 6, 0, (A)3 (B)5 (C)8 (D)9。 背面還有試題 下面是甲、乙、丙、丁、戊五個學生高度關係： 乙矮於丙，高於戊。丁高於甲。丙高於甲，矮於丁。 五人在課室內同坐於一直行，學校規定，矮的坐於前，次矮的坐於第二，以此類推。 31.誰坐第三位置？（a）甲（b）乙（c）丙（d）丁（e）戊 32.最後兩個位置（第四、第五）順序是誰？（a）乙戊（b）丙甲（c）丙戊（d）甲丁（e）丙丁 在一條短短的馬路上，有趙、錢、孫、李四家房屋。趙家不在李家隔壁，孫家不與錢家相鄰，錢家亦不與李家相鄰 33.李家的位置是：（a）在中間（b）距趙家最遠（c）距趙家最近（d）與孫家相鄰 （e）距錢家最近 34.錢家的位置是：（a）與趙家相鄰（b）距孫家最遠（c）距孫家最近（d）距李家最近 （e）在中間 甲、乙、丙、丁和戊五人吃水果的愛好如下： 甲：香蕉、蘋果。乙、梨子、蘋果。 丙：榴槤、橙、水蜜桃。丁、香蕉、橙。 戊：水蜜桃、蘋果。 有一次，朋友買齊了全部上述的水果，而他們每人卻選了一種他們喜愛的來吃，並且五人 所吃的均不同。 35.丙所吃的應該是那一種水果呢？（a）香蕉（b）梨子（c）榴槤（d）水蜜桃（e）蘋果 36.戊吃的是那一種水果呢？（a）蘋果（b）橙（c）水蜜桃（d）香蕉（e）梨子 趙、錢、孫、李、吳、王6 位老師圍坐在圓桌開校務會議，若已知孫老師坐在李老師的對面，李老師坐在錢老師的右 手邊，王老師坐在吳老師的左手邊，而趙老師則坐在孫老師的隔壁。 37.若依順時鐘的方向來看，6 人座位的順序應該是： （A）孫王李吳錢趙（B）趙錢李吳王孫 （C）李錢趙孫吳王 （D）趙錢孫李吳王 38. 若6 位老師分別為國文、英文、數學、音樂、體育、美勞等六科的老師，且已知數學老師在美勞老師的對面，體育老 師坐在國文老師的右手邊，音樂老師坐在體育老師的對面。若孫老師是國文老師，則請問李老師是教什麼？（A）英 文（B）數學 （C）體育 （D）音樂 吳宗憲，胡瓜，康康，NONO 及大炳五人來比賽誰私房錢存的多，結果發現吳宗憲的私房錢比大炳多，康康的私房錢比 NONO 少，胡瓜的私房錢比NONO 少，且大炳的私房錢比NONO 多。 39.請問下列敘述何者正確？（A）吳宗憲的私房錢最多（B）大炳的私房錢最少（C）NONO 的私房錢最少（D）以上皆非 40.台灣計程車每輛限乘客4 名，黃家10 人一同外出，應租計程車幾輛？（A）2 輛（B）3 輛（C）4 輛（D）5 輛（E）6 輛 41.下列三組數字有固定之關係，其中漏去一個數字，問漏去的是甚麼數？ 9 7 ? 19 10 3 5 15 13 （A）14（B）15（C）16（D）17（E）18 42.下列三組數字有固定之關係，其中漏去一個數字，問漏去的是甚麼數？ 12 15 28 4 ? 7 3 5 4 （A）15（B）10（C）5（D）3（E）4 43.下列方格內的三組數字互有關係，細心觀察，問漏去的數字應是甚麼？ 32 28 4 38 34 4 41 ? 4 （A）35（B）36（C）37（D）38（E）39 下面還有試題 ▲請在44-46 題的右側選項之中，選擇一個與左側標準圖相同的圖形。 44. (A) (B) (C) (D) 45. (A) (B) (C) (D) 46. (A) (B) (C) (D) . ▲47~50 要拼成正方形，還須要加(A)、(B)、(C)、(D)中的哪一塊。 47 (A) (B) (C) (D) 48 (A) (B) (C) (D) 49. (A) (B) (C) (D) 50. (A) (B) (C) (D) 科目：邏輯推理解答 題號 解答 題號 解答 題號 解答 題號 解答 題號 解答 1 C 11 C 21 C 31 A 41 D 2 A 12 A 22 A 32 E 42 D 3 D 13 B 23 D 33 D 43 C 4 D 14 D 24 C 34 A 44 D 5 D 15 A 25 B 35 C 45 A 6 C 16 D 26 C 36 C 46 D 7 B 17 C 27 C 37 C 47 A 8 D 18 B 28 A 38 A 48 B 9 E 19 D 29 D 39 A 49 C 10 D 20 A 30 C 40 B 50 A 邏輯推理考古題4 注意： ◆ 本試題共計100 分，每題2 分。 ◆ 請將正確答案填寫在「答案卷」或「答案卡」上，並請依題號作答。 ◆ 請考生自行填上准考證號碼，考完後， 「試題」 、 「答案卷」或「答案卡」一併繳回。 1. 500 除以某數，商是29，餘數是7，問該數是多少？（A）15（B）16（C）17（D）18（E）19 2. 80 最少要加上甚麼數，才能成為一個平方數？（A）2（B）4（C）9（D）20（E）25 3. 有五個數，今每個數加2，問平均數增加多少？（A）10（B）5（C）4（D）3（E）2 4. 假如X=2Y，而Y=(1/2)Z，下列五個數式中，哪一個是正確的？（A）X=4Z（B）4X=Z（C）Y>X（D）X=Z （E）Z5/7>3/5 (B)4/5<3/4<2/3 (C)10/11>7/8>4/5 (D)15/13<13/11<11/9 39. 某工作甲需X 天完成，乙需X-4 天，若甲、乙合作兩天後，甲再工作2 天才能完成，問甲獨做需幾天才能完成？(A)6 天 (B)12 天 (C)10 天 (D)8 天 數字推理 40. 1 2 3 4 5 8 7 ？ (A) 9 (B) 12 (C) 6 (D) 16 41. BA, DC, ？, HG, JI, LK. (A) BC (B) ED (C)FE (D)EF (E)CB. 42. M, T, W, ？ F, S, S. (A) A (B) E (C)S (D) T (E) X. 43. 1，1，2，4，7，11，？ (A)14 (B)15 (C) 16 (D)17 (E)18。 下頁還有試題 PDF created with pdfFactory Pro trial version www.pdffactory.com 圖形推理 44. (A) (B) (C) (D) 45. (A) (B) (C) (D) 參考答案 1.C 2.D 3.D 4.D 5.D 6.B 7.B 8.B 9.A 10.A 11.D 12.C 13.B 14.D 15.A 16.C 17.A 18.D 19.D 20.D 21.C 22.D 23.B 24.C 25.B 26.C 27.E 28.A 29.C 30.C 31.B 32.A 33.C 34.B 35.C 36.C 37.B 38.B 39.D 40.C 41.C 42.D 43.C 44.B 45.B PDF created with pdfFactory Pro trial version www.pdffactory.com |
5971 | https://www.forbes.com/sites/investopedia/2013/06/03/depreciation-straight-line-vs-double-declining-methods/ | Depreciation: Straight-Line Vs. Double-Declining Methods
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Depreciation: Straight-Line Vs. Double-Declining Methods
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Depreciation: Straight-Line Vs. Double-Declining Methods
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Salvage Values and Depreciation
One of the consequences of generally accepted accounting principles is that while cash is used to pay for a long-lived asset, such as a semi-trailer to deliver goods, the expenditure is not listed as an expense against revenue at the time. Instead, the cost is placed as an asset onto the balance sheet and that value is steadily reduced over the useful lifetime of the asset. This reduction is an expense called depreciation. This happens because of the matching principle from GAAP, which says expenses are recorded in the same accounting period as the revenue that is earned as a result of those expenses.
For example, suppose the cost of a semi-trailer is $100,000 and the trailer is expected to last for 10 years. If the trailer is expected to be worth $10,000 at the end of that period (salvage value), $9,000 would be recorded as a depreciation expense for each of those 10 years (cost - salvage value/number of years).
Note: This example uses the straight-line method of depreciation and not an accelerated depreciation method that records a larger depreciation expense during the earlier years and a smaller expense in later years. There are also two assumptions built into the depreciation amount: the expected lifetime and the salvage value.
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Long-Term Assets
If you look at the long-term assets, such as property, plant and equipment, on a balance sheet, there are often two lines showing the cost value of those assets and how much depreciation has been charged against that value. (Sometimes, these are combined into a single line such as "PP&E net of depreciation".)
In the above example, $360,000 worth of PP&E was purchased during the year (which would show up under capital expenditures on the cash flow statement) and $150,000 of depreciation was charged (which would show up on the income statement). The difference between the end-of-year PP&E and the end-of-year accumulated depreciation is $2.4 million, which is the total book value of those assets. If the semi-trailer mentioned above had been on the books for three years by this point, then $9,000 of that $150,000 depreciation would have been due to the trailer and the book value of the trailer at the end of the year would be $73,000. It does not matter if the trailer could be sold for $80,000 or $65,000 at this point (market value) - on the balance sheet it is worth $73,000.
Suppose that trailer technology has changed significantly over the past three years and the company wants to upgrade its trailer to the improved version, while selling its old one. There are three scenarios that can occur for that sale. First, the trailer can be sold for its book value of $73,000. In this case, the PP&E asset is reduced by $100,000 and the accumulated depreciation is increased by $27,000 to remove the trailer from the books. (The cash account balance will increase by the sale amount for all cases.)
The second scenario that could occur is that the company really wants the new trailer, and is willing to sell the old one for only $65,000. In this case, three things happen to the financial statements. The first two are the same as above to remove the trailer from the books. In addition, there is a loss of $8,000 recorded on the income statement because only $65,000 was received for the old trailer when its book value was $73,000.
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The third scenario arises if the company finds an eager buyer willing to pay $80,000 for the old trailer. As you might expect, the same two balance sheet changes occur, but this time a gain of $7,000 is recorded on the income statement to represent the difference between book and market values.
Suppose, however, that the company had been using an accelerated depreciation method, such as double-declining balance depreciation. (See Figure 2 below for the difference in depreciation between straight-line and double-declining depreciations on $100,000.) Under the double-declining balance method, the book value of the trailer after three years would be $51,200 and the gain on a sale at $80,000 would be $28,800, recorded on the income statement - quite a one-time boost! Under this accelerated method, there would have been higher expenses for those three years and, as a result, less net income. There would also be a lower net PP&E asset balance. This is just one example of how a change in depreciation can affect both the bottom line and the balance sheet.
Expected lifetime is another area where a change in depreciation will impact both the bottom line and the balance sheet. Suppose that the company is using the straight-line schedule originally described. After three years, the company changes the expected lifetime to a total of 15 years but keeps the salvage value the same. With a book value of $73,000 at this point (one does not go back and "correct" the depreciation applied so far when changing assumptions), there is $63,000 left to depreciate. This will be done over the next 12 years (15-year lifetime minus three years already). Using this new, longer time frame, depreciation will now be $5,250 per year, instead of the original $9,000. That boosts the income statement by $3,750 per year, all else being the same. It also keeps the asset portion of the balance sheet from declining as rapidly, because the book value remains higher. Both of these can make the company appear "better" with larger earnings and a stronger balance sheet.
Similar things occur if the salvage value assumption is changed, instead. Suppose that the company changes salvage value from $10,000 to $17,000 after three years, but keeps the original 10-year lifetime. With a book value of $73,000, there is now only $56,000 left to depreciate over seven years, or $8,000 per year. That boosts income by $1,000 while making the balance sheet stronger by the same amount each year.
Watch For Assumptions
Depreciation is the means by which an asset's book value is "used up" as it helps to generate revenue. In the case of our semi-trailer, such uses could be delivering goods to customers or transporting goods between warehouses and the manufacturing facility or retail outlets. All of these uses contribute to the revenue those goods generate when they are sold, so it makes sense that the trailer's value be charged a bit at a time against that revenue.
However, one can see that how much expense to charge is a function of the assumptions made about both its lifetime and what it might be worth at the end of that lifetime. Those assumptions affect both the net income and book value of the asset. Further, they have an impact on earnings if the asset is ever sold, either for a gain or a loss when compared to its book value.
While companies do not break down the book values or depreciation for investors to the level discussed here, the assumptions they use are often discussed in the footnotes to the financial statements. This is something investors might wish to be aware of. Furthermore, if a company routinely recognizes gains on sales of assets, especially if those have a material impact on total net income, the financial reports should be investigated more thoroughly. Management, which is routinely keeping book value consistently lower than market value, might also be doing other types of manipulation over time to massage the company's results.
Related From Investopedia:
Cleaning Up Dirty Surplus Items On The Income Statement
Understanding The Federal Reserve Balance Sheet
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5972 | https://www.youtube.com/watch?v=MyJB7qNsBWo | Corollary: Two Vectors are Parallel If and Only If Their Cross Product is Zero
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In this video, I go over a corollary to the previous video in which I determined the cross product length as equaling |a| |b| sin θ. The corollary or theorem that follows from the cross product length is that if the vectors a and b are parallel, then the angle between them are either 0 or 180 degrees or π radians. Thus, the sine term vanishes and becomes 0; and so does the cross product! This means that we can determine if any 2 vectors are parallel if and only if their cross product is equal to 0.
This video was taken from my earlier video listed below:
Vectors and the Geometry of Space: The Cross Product:
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Transcript:
so it's going further so corollary or uh or a theorem that follows another one so two non-zero vectors A and B are parallel if and only if their cross products equals to zero so A cross B equals to zero or the zero Vector like that and again this uh corresponds to this sign right here because so when when sine is zero or sine is pi or 180 degrees uh that becomes zero so this whole thing becomes zero and in other words it has to be parallel and we will illustrate that soon and yeah here we'll look at the proof of this so two non-zero vectors A and B are parallel if and only if uh Theta is equal to zero or Pi radians or 180 degrees in either case sine Theta is equal to zero so A cross B is equal to zero and therefore a yeah or the distance I mean there's a distance uh the distance equals zero and therefore A cross B is equal to zero so in other words this right here uh has to equal zero so let's write this here I'll just uh first of all graph this out if we have Vector a like this so you have an A and then uh let's say you have a and then you have a vector B if it's parallel this angle is well this Theta is I'll put that at Theta like this Theta is equal to zero this is B so it's parallel and but if you otherwise the other way to get it uh parallel is if it goes here which is a perfect 90 degrees so let's say you have instead of that so you have B over here this is uh Pi this Theta is equal to I radians like that so in either case let's put this scroll back here we have zero yeah so this basically zero there so what we get is um the distance A cross B is equal to a length of a times length of B and then sine sine zero I'll put a zero or Pi another way it's going to be equal to zero zero or pi equals two well yeah this whole thing has become zero it just becomes made for completeness length of a times length of B times zero which equals to zero so then if it's if the length is zero the cross product is going to be all zero the only way you can get a length of zero of a cross product of a vector is if the vector is a zero Vector you can't have a lot you can't have it pointing out somewhere and then having a length not zero can't uh do that I mean you can't have it pointing out somewhere and the length equals zero you can't do that zero means not playing anywhere this equals to uh zero the zero vector so yes and that's the Corollary from this theorem here so if they're parallel like this because remember this parallel this B is parallel with this I'll make this a bit uh better like that this is parallel as well yeah these are all parallel together |
5973 | https://fiveable.me/ap-calc/unit-8/finding-area-between-curves-that-intersect-at-more-than-two-points/study-guide/QVBQ9TQDM4ZObJl6o0ad | printables
♾️AP Calculus AB/BC Unit 8 Review
8.6 Finding the Area Between Curves That Intersect at More Than Two Points
♾️AP Calculus AB/BC Unit 8 Review
8.6 Finding the Area Between Curves That Intersect at More Than Two Points
Written by the Fiveable Content Team • Last updated September 2025
Verified for the 2026 exam
Verified for the 2026 exam•Written by the Fiveable Content Team • Last updated September 2025
♾️AP Calculus AB/BC
Unit & Topic Study Guides
Unit 8 Overview: Applications of Integration
8.1 Finding the Average Value of a Function on an Interval
8.2 Connecting Position, Velocity, and Acceleration of Functions Using Integrals
8.3 Using Accumulation Functions and Definite Integrals in Applied Contexts
8.4 Finding the Area Between Curves Expressed as Functions of x
8.5 Finding the Area Between Curves Expressed as Functions of y
8.6 Finding the Area Between Curves That Intersect at More Than Two Points
8.7 Volumes with Cross Sections: Squares and Rectangles
8.8 Volumes with Cross Sections: Triangles and Semicircles
8.9 Volume with Disc Method: Revolving Around the x- or y-Axis
8.10 Volume with Disc Method: Revolving Around Other Axes
8.11 Volume with Washer Method: Revolving Around the x- or y-Axis
8.12 Volume with Washer Method: Revolving Around Other Axes
8.13 The Arc Length of a Smooth, Planar Curve and Distance Traveled
8.6 Finding the Area Between Curves That Intersect at More Than Two Points
Welcome back to AP Calculus with Fiveable! In the last two study guides, we discussed how to find the area between curves expressed as functions of x and as functions of y. Today, we’re going to apply what we know from those key topics to curves that intersect at more than two points! 🧠
📌 Want to review the last key topics? Check them out here!
AP Calculus 8.4 Finding the Area Between Curves Expressed as Functions of x
AP Calculus 8.5 Finding the Area Between Curves Expressed as Functions of y
more resources to help you study
practice questionscheatsheetscore calculator
📏 Understanding Finding Area Between Curves
In calculus, finding the area between curves intersecting at more than two points is crucial. This technique involves calculating the definite integral of the absolute value of the difference between the two curves over a given interval. The formula for the area between two curves y=f(x) and y=g(x) from x=a to x=b is given by:
Area =∫ab∣f(x)−g(x)∣dx
Any negative regions are guaranteed to be considered by the absolute value. We can calculate the area between the curves inside the given interval using this approach.
‼️ Keep in mind that the choice of which curve is above or below at different intervals affects the result, so pay attention to the intersection points and the behavior of the curves.
🪜 Finding Area Between Intersecting Curves Steps
Here are some steps you can take when approaching questions asking you to solve for the area between two curves that intersect at more than two points:
👀 Identify the points of intersection by setting the two equations equal to each other.
📈 Optional, but highly recommend: graph the functions.
🤔 Identify what approach to take (top-bottom with vertical slices, or right-left with horizontal slices).
🔢 Set up the integral with different intervals based on intersection points.
💯 Evaluate the integrals to find the area.
Let’s go step by step and walk through an example!
🤓 Area Between Curves Walkthrough
Find the area between the curves y=x3−2x2 and y=x2−2x between the interval 0≤x≤2.
👀 Step 1: Identify the points of intersection.
To find the points of intersection, we set our two equations equal to each other, like so:
x3−2x2=x2−2x
Now simplify to find the points! 🕵️
x3−3x2+2x=0 x(x−1)(x−2)=0
We now know that these two functions intersect at x=0,1,2.
📈 Step 2: Graph the functions.
This helps you visualize what is going on and identify which approach to take in the next step.
Image of two functions: x2−2x and x3−2x2. Image Created with Desmos
🤔 Step 3: Identify what approach to take
Taking a look at the graph, the curves are on top of one another. Therefore, we can use vertical slices and subtract the bottom function from the top function.
In the interval 0≤x≤1, we have x3−2x2 as our f(x) in the blue and x2−2x as our g(x) in the red.
Vice versa is true in the interval 1≤x≤2, since the red curve in the graph above is on top of the blue curve.
🔢 Step 4: Set up the integral with different intervals
So we set up our integral accordingly!
Area=∫01((x3−2x2)−(x2−2x))dx+ ∫12((x2−2x)−(x3−2x2))dx
💯 Step 5: Evaluate the integrals to find the area.
Area=∫01((x3−3x2+2x)dx+ ∫12(−x3+3x2−2x)dx Area=(41x4−x3+x2)01+(−41x4+x3−x2)12 Area=21 unit2
Great work! 🙏
👩🏫 Area Between Curves Practice
Your turn to try a question!
Find the area between 3x2−x2−10x and −x2+2x between the interval −2≤x≤2.
✏️ Solution to Practice Problem
First, set the two equations equal to find the points of intersection. ⬇️
3x3−x2−10x=−x2+2x 3x3−12x=0 3x(x2−4)=0 3x(x−2)(x+2)=0
Therefore, our points of intersection are -2, 0, and 2.
Anddd…here’s the graph of these two functions, so you can determine which approach to take.
Image of two functions: 3x3−x2−10x and −x2+2x. Image created with Desmos
Since the functions are on top of each other, we can set up the integrals with vertical slices in mind.
Area=∫−20((3x3−x2−10x)−(−x2+2x))dx+ ∫02((−x2+2x)−(3x3−x2−10x))dx Area=∫−20(3x3−12x)dx+ ∫02(−3x3+12x)dx =43x4−212x2−20+4−3x4+212x202=24 Area=24 unit2
Great work!
🕺Closing
This walkthrough and practice problems aimed to demystify the process, providing step-by-step guidance on how to approach such problems! We hope you feel more comfortable with the process. 😊
Remember to carefully consider the intervals and points of intersection to calculate the area enclosed by the curves accurately. Happy learning!
Frequently Asked Questions
How do I find the area between two curves that cross each other multiple times?
Find the x-values where the curves intersect, then split the domain at those points so on each subinterval one function is “upper” and the other is “lower.” For vertical slices (most common on the exam) compute area as a sum of definite integrals: area = sum over subintervals ∫[a_k]^{b_k} (top(x) − bottom(x)) dx. Equivalently, area = ∫_A^B |f(x) − g(x)| dx, but on the AP exam you should show the partitioning and which function is upper on each piece. If the curves are easier as functions of y, use horizontal slices and integrate (right − left) dy. Watch for “change of dominance” at every intersection—those x-values are your partition points. Use symmetry when possible to reduce work. This matches CED CHA-5.A: use definite integrals, absolute value of difference, partitioning when dominance changes. For a refresher and worked examples see the Topic 8.6 study guide ( More unit review and practice problems are at ( and (
What's the formula for area between curves when they intersect at more than 2 points?
If two curves cross more than twice, split the x-interval at every intersection so the “top” and “bottom” functions don’t switch inside a subinterval. Then add the areas of each slice: - Let the intersection x-values be a = x0 < x1 < x2 < ... < xn = b. - On each subinterval [x_{k-1}, x_k], identify the upper function U_k(x) and lower function L_k(x). - Area = sum_{k=1}^n ∫_{x_{k-1}}^{x_k} [U_k(x) − L_k(x)] dx. Equivalently, if you prefer one integral form, Area = ∫_a^b |f(x) − g(x)| dx (this produces the same total area because the absolute value handles changes of dominance). Use vertical slices when functions are y = f(x), or do the analogous partitioning with horizontal slices if you integrate with respect to y. Remember AP wording: you may answer either as a sum of definite integrals (piecewise integrand) or as the integral of the absolute value (signed vs total area)—both are CED-allowed approaches. See the Topic 8.6 study guide for examples ( For extra practice, check the Unit 8 overview ( or the 1000+ practice problems (
When do I need to split up the integral into multiple parts for area problems?
Split the integral whenever the “top” and “bottom” functions switch—i.e., where their difference changes sign or where the region’s bounding rule changes. Practically: find all intersection x-values, plot or test which function is upper on each subinterval, then partition the integral at those x-values. Compute area as a sum of integrals of (upper − lower) on each piece (or use ∫ |f − g| if you prefer the absolute-value form). This handles piecewise integrands, signed vs total area, and multiple intersections (CED keywords: intersection points, change of dominance, partitioning the interval, absolute value of difference). On the AP exam, justify your partition (show intersections and which is upper on each interval) and evaluate each definite integral—that’s full credit for CHA-5.A style problems. For a worked guide and examples, see the Topic 8.6 study guide ( For more practice, try the unit problems ( or the large practice set (
I'm confused about how to set up the integral when curves cross each other - do I always subtract the same way?
Short answer: you don’t always subtract the same way—you must subtract (upper function − lower function) on each interval where one function stays on top. How to do it (quick checklist aligned with the CED keywords): - Find all intersection x-values to partition the domain (a, b, c, ...). - On each subinterval, determine which curve is the upper and which is the lower (compare y-values or sketch/plug a test x). - Compute area as sum of definite integrals: ∑ ∫[xi to xi+1] (upper(x) − lower(x)) dx. - Equivalent alternate: integrate the absolute value: ∫ |f(x) − g(x)| dx (but on exams you’ll usually split at intersections so you can evaluate standard integrals). Remember signed area ≠ total area; AP expects either a piecewise sum or the absolute-value approach (CHA-5.A.3). For practice and examples, see the Topic 8.6 study guide ( and try problems at (
How do I know which function is on top when the curves keep switching positions?
Find the x-values where the curves meet (solve f(x)=g(x)). Those intersection points partition the x-interval into subintervals. On each subinterval pick any test x (or use a quick table/graph) and evaluate f(x) and g(x)—the larger value is the upper function there and the smaller is the lower function. If they switch back and forth, you’ll get different “upper–lower” pairs on different subintervals; set up a separate definite integral for each piece and add them: Area = sum over subintervals ∫[a_i to b_i] (upper(x) − lower(x)) dx. Alternately, integrate the absolute value: ∫ |f(x) − g(x)| dx, but on the AP you’ll usually show the partition and write a sum of definite integrals (CHA-5.A.3: change of dominance, piecewise integrand). Remember signed area vs total area: use the difference with upper minus lower to get positive area. For step-by-step examples and practice problems check the Topic 8.6 study guide ( and extra practice (
What's the difference between using one integral vs multiple integrals for area between curves?
Use one integral when one function is the top (upper) and the other is the bottom (lower) everywhere on the interval between the leftmost and rightmost intersection—area = ∫[a to b] (top − bottom) dx. Use multiple integrals when the two curves switch which is on top at one or more interior points: partition the x-interval at each intersection where dominance changes and sum integrals on those subintervals (or equivalently integrate the absolute value of their difference). Why it matters (CED language): when curves intersect more than twice you must account for change of dominance—either make a piecewise integrand (sum of definite integrals) or evaluate ∫ |f(x) − g(x)| dx. Think in terms of vertical slices (or horizontal if easier) and signed vs total area: without absolute value signed areas cancel, so partitioning prevents sign errors. For AP problems, show intersection points, justify your partition, and write either the sum of definite integrals or the absolute-value integral. See the Topic 8.6 study guide for examples ( For more review and practice, check Unit 8 ( and hundreds of practice problems (
Can someone explain step by step how to find intersection points when there are more than 2?
Step-by-step: 1. Solve f(x) = g(x) algebraically (or graphically) to get all x-values where they meet. Don’t stop after two—find every root on the domain (check multiplicities and endpoints). 2. Sort those x-values: a = x0 < x1 < x2 < … < xn = b. These partition the x-interval into subintervals where one function is consistently above the other (change of dominance happens only at intersections). 3. On each subinterval (xi, xi+1) pick a test x (or compare values) to determine the upper function U(x) and lower L(x). 4. Compute area as a sum of definite integrals: Area = Σ ∫_{xi}^{xi+1} (U(x) − L(x)) dx. Equivalently use ∫_{a}^{b} |f(x) − g(x)| dx if you prefer the absolute-value form (you’ll need to split where sign changes). 5. Add numeric values. If curves touch tangentially, check sign changes carefully—touching may not swap dominance. This is exactly what the CED calls for: partition the interval at intersection points and sum definite integrals (or use the absolute value integrand). For a guided walk-through with examples, see the Topic 8.6 study guide ( For extra practice, use the AP Calculus practice bank (
Why do I need to use absolute value sometimes when finding area between curves?
You need absolute value because the definite integral gives signed area: when the “upper” and “lower” functions switch roles (more than two intersections), f(x) − g(x) can be positive on some subintervals and negative on others. To get total area (what the AP wants—CHA-5.A), either: - Partition the x-interval at each intersection and on each piece integrate (top − bottom) dx, or - Integrate |f(x) − g(x)| over the whole interval (conceptually the same). So whenever dominance changes, you must take absolute value (or split the interval) so negative signed areas don’t cancel positive ones. This is just applying vertical slices, checking intersection points, and using a sum of definite integrals or an integral of the absolute value (CED keywords: change of dominance, partitioning the interval, signed area vs total area). For more examples and step-by-step practice, see the Topic 8.6 study guide ( and Unit 8 resources ( More practice problems are at (
How do I determine the limits of integration when curves intersect at 3 or 4 points?
Find all intersection points first (solve f(x)=g(x)). Those x-values partition the x-axis into intervals where one curve is consistently above the other. Use those intersection x-values as your limits: on each subinterval integrate (top − bottom) and sum the results. Equivalently, you can integrate |f(x) − g(x)| on the whole interval, but on the AP exam you’ll usually set up a sum of definite integrals because the “top” can change. Steps: (1) solve f(x)=g(x) to get all x1 For more practice, check Unit 8 ( and the 1000+ practice problems (
I don't understand when to use the absolute value method vs splitting into separate integrals - which is easier?
Use whichever makes the algebra/limits simpler—both are AP-approved ways to get the same total area (CED: CHA-5.A.3). Quick rule: - If one function is the top for the entire interval, just ∫(top − bottom). - If the functions swap who’s on top (change of dominance) inside the interval, either: - Split at each intersection and sum integrals of (upper − lower) on those subintervals; or - Integrate the absolute value: ∫ |f(x) − g(x)| dx over the whole interval. Which is easier? Usually splitting is easier on exams: find intersection x-values, determine which function is upper on each piece, and compute ordinary definite integrals—no extra absolute-value algebra and it matches the AP expectation of partitioning the interval. Use absolute value when symmetry or a neat antiderivative of |f−g| simplifies work, but that’s less common. For more worked examples and AP practice, see the Topic 8.6 study guide ( the Unit 8 overview ( and lots of practice problems (
What happens if I forget to split the integral and just use the whole interval from first to last intersection point?
If you don’t split the integral and just integrate from the first to the last intersection, you’ll get the signed area—not the total geometric area—so parts where the “lower” function is actually above will subtract instead of add. That usually gives the wrong answer when the curves change dominance more than once. What you should do: find all intersection x-values a = x0 < x1 < x2 < … < xn = b, then either - compute area = Σ ∫_{xi-1}^{xi} |f(x) − g(x)| dx, or - on each subinterval determine the upper and lower functions and sum ∫_{xi-1}^{xi} (upper − lower) dx. This is exactly CHA-5.A in the CED: partition the interval where dominance changes or use the absolute-value integrand. On the AP exam you’ll be expected to partition or justify using |f−g| (see Topic 8.6 study guide: For extra practice, try problems from the Unit 8 page ( or the practice bank (
How do I solve for intersection points when one equation has x^3 and gets really complicated?
When one curve gives you a cubic (or any messy polynomial) and you need intersection points, don’t panic—use a mix of algebraic and numeric strategies that AP expects: - Try algebra first: bring everything to one side and attempt to factor (common factors, grouping, Rational Root Theorem). Example: x^3 − x^2 = x^2(x−1). - If factoring fails, use a graphing tool or graphing calculator to locate approximate roots (Part B of the AP exam allows a calculator). Plot f(x)−g(x) to see zero crossings and visually identify intervals. - Use numerical methods for better accuracy: Newton’s method or a calculator’s root finder gives decimals you can use as bounds. - Once you have all intersection x-values, partition the x-intervals at those points and set up integrals of |top−bottom| on each piece (or sum signed integrals where dominance changes) per CHA-5.A. - If you want more practice or walkthroughs for multi-intersection area problems, check the Topic 8.6 study guide ( and try practice problems ( If you tell me the specific equations, I’ll walk through finding the roots and setting up the integrals.
Is there a way to use my calculator to find all the intersection points automatically?
Short answer: yes—but most calculators won’t list every intersection in one click. They find intersections pairwise or find roots of f(x)−g(x) on an interval, so you’ll usually need to scan your window or give it different starting intervals. How to do it efficiently: - Graph y = f(x) and y = g(x) and use the built-in “intersect” or “root” tool (TI-84/83: CALC → intersect; TI-89/Casio: solve/root). Each use finds one intersection near the cursor or between two guesses. - Better: graph h(x) = f(x) − g(x). Then use the root finder on h(x). Roots = intersection x-values. If there are many intersections, scan by: - changing the window (ZoomFit) and re-running roots, or - using a table or sampling x values to find sign changes for h(x) and then root-finding on each interval. - On Desmos you can click intersections directly; it lists them if they’re visible in the window. AP note: calculators are allowed on calculator portions (Part B and some FRQ parts), but you should still show correct setup (partition intervals where dominance changes) when computing areas (CED CHA-5.A keywords: intersection points, change of dominance, partitioning the interval). For more tips and worked examples, see the Topic 8.6 study guide ( and extra practice (
Why does my area come out negative sometimes and how do I fix it?
You’re getting a negative area because definite integrals give signed area—they count area above the x-axis as positive and below as negative, and likewise (f(x) − g(x)) is negative where g(x) > f(x). If your region has more than two intersections, the “top” function can change. Fix: find all intersection x-values, partition the interval at those points, then on each subinterval decide which function is upper. Compute area as the sum of integrals of (upper − lower) on each subinterval: area = Σ ∫_{x_i}^{x_{i+1}} [upper(x) − lower(x)] dx. Equivalently, area = ∫ |f(x) − g(x)| dx, but on the AP you should show the partition and which function is upper on each piece (CHA-5.A / CHA-5.A.3). For step-by-step help and examples, see the Topic 8.6 study guide ( and extra practice at (
What's the step by step process for area problems when curves cross multiple times?
Step-by-step: 1. Graph the curves (or sketch) and find all intersection points—these partition the x- (or y-) interval. Solve f(x)=g(x) to get the exact x-values. 2. Decide slice orientation: use vertical slices (dx) if functions are y = f(x),y = g(x); use horizontal slices (dy) if nicer (functions of y). 3. On each subinterval between consecutive intersections, determine the upper function and lower function (check values or the sign of f−g). The “upper” can change—that’s the key. 4. Compute area as a sum of definite integrals over those subintervals: sum ∫[a_k to a_{k+1}] (upper − lower) dx. Equivalently, one integral of |f−g| works but you must split where f−g changes sign (piecewise integrand). 5. Add the integrals to get total (non-signed) area. Watch units and justify partitioning if asked on the AP. For AP-style practice and examples see the Topic 8.6 study guide ( and extra practice problems ( |
5974 | https://www.wyzant.com/resources/answers/701538/what-is-1-tan-2x | What is 1-tan^2x? | Wyzant Ask An Expert
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Trigonometry
Rahma E.
asked • 07/23/19
What is 1-tan^2x?
Why can't we say that as 1+sec^2x=tan^2x that 1-tan^2x=-sec^2x?
Edit: Here I made a mistake
It is 1+tan^2x = sec^2x
So there isn't a really simple way to get 1-tan^2x.
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Mark M.answered • 07/23/19
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1 + tan 2 x = sec 2 x
So, tan 2 x = sec 2 x - 1
Thus, 1 - tan 2 x = 1 - (sec 2 x - 1) = 2 - sec 2 x
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Rahma E.
Thank you, your answer really helped me.
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07/23/19
Stephen C.answered • 07/23/19
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Well, if we add tan^2x + sec^2x to both sides of your 2nd equation, you get the first equation. So you can say it, and you will be correct. :-)
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Rahma E.
Thank you! Although I wrote it wrong, you taught me a new way to check my answer.
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07/23/19
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5975 | https://cheever.domains.swarthmore.edu/Ref/BinaryMath/NumSys.html | Representation of Numbers
This is an substantially updated version of an older file that can be found here.
The sections below labeled "interactive" contain modules that can let you explore scenarios of your own
choosing.
Contents
When working with any kind of digital system (electronics or computers), it is important to understand the
different ways in which these numbers are represented. Almost without exception, numbers are represented by two
voltage levels which
can represent a one or a zero (an interesting exception to this rule are newer memory
devices that use one four (or more) possible voltage levels, thereby
increasing the amount of information that can be stored by a single memory cell). The number
system based on ones and zeroes is called the binary system (because there are only
two possible digits). Before discussing the binary system, a review of the decimal
(ten possible digits) system is in order because many of the concepts of the binary
system will be easier to understand when introduced alongside their decimal counterpart. The "base" of the system is also called the "radix".
Finding the Decimal Equivalent of the Number with a Different Radix (e.g., binary→decimal)
Positive Decimal Integers
You are familiar with the decimal system. For instance, to
represent the positive integer one hundred and twenty-five as a decimal number, we can
write (with the postivie sign implied). The subscript 10 denotes the number as a base 10
(decimal) number. After the first line, all numbers are implicitly base 10.
Some things to note (that we will be able to apply to the representation of unsigned binary
numbers):
Some observations:
The rightmost digit (5 in this case) is multiplied by 100,
the next digit (3 in this case) to the left is multiplied by 101, and so
on. Each successive digit to the left has a multiplier that is 10 times the previous digit.
With n digits, 10n unique numbers (from 0 to 10n−1) can be
represented. If n=3, 1000 (=103) numbers can be represented 0-999.
To see how many digits a number needs, you can simply take the logarithm (base 10) of
the absolute value of the number, and add 1 to it. The integer part of the result is the
number of digits. For instance, log10(33) + 1 = 2.5; the integer part of that is 2, so 2 digits are
needed.
For this example log10(725) + 1 = 3.86; The integer part of that is 3, so 3 digits are needed.
To multiply a number by 10 you can simply shift it to the left by one digit, and fill in
the rightmost digit with a 0 (moving the decimal place one to the right). To divide a
number by 10, simply shift the number to the right by one digit (moving the decimal place
one to the left).
Negative numbers are handled easily by simply putting a minus sign (−) in front of the
number. This does lead, however, to the somewhat awkward situation where 0=−0. We will
avoid this situation with binary representations, but with a little bit of effort.
Finding the decimal equivalent of an unsigned (positive) binary integer (interactive)
To represent a number in binary, every digit has to be either 0 or 1 (as opposed to decimal numbers where a digit
can take any value from 0 to 9).The subscript 2 denotes a binary (i.e., base 2) number. Each digit in a binary
number
is called a bit.. Likewise we can make a similar set of observations:
To see how the decimal equivalent of an 8 bit unsigned binary number can be calculated, enter an 8 bit unsigned binary number:
.
(Input restrictions).
Note: any spaces in the input are ignored; any other characters that are not 0
or 1 are interpreted as 1, the number is truncated to 8 bits if necessary, and is increased to 8 bits if necessary
(by adding zeros on the left).
Any number can be broken
down this way, by finding all of the powers of 2 that add up to the number in question; you can see this is
exactly analagous to the decimal deconstruction done earlier
The rightmost digit is multiplied by 20,
the next digit to the left is multiplied by 21, and so
on. Each successive digit to the left has a multiplier that is 2 times the previous digit.
With n digits, 2n unique numbers (from 0 to 2n−1) can be
represented. If n=8, 256 (=208) numbers can be represented 0−255.
To see how many digits a number needs, you can simply take the logarithm (base 2) of
the absolute value of the number, and add 1 to it. The integer part of the result is the
number of digits. For instance, log10(53) + 1 = 6.7; the integer part of that is 6, so 6 digits are
needed. Recall that log2(x)=log(x)/log(2) (where log() can be either the natural
log, or log base 10).
To multiply a number by 2 you can simply shift it to the left by one digit, and fill in
the rightmost digit with a 0 (moving the decimal place one to the right). To divide a
number by 2, simply shift the number to the right by one digit (moving the decimal place
one to the left).
At this point we can't represent negative numbers, but will do so soon.
Try this:
Convert 0001 0011 (spaces are ignored but can help
make the number easier to read, much like commas in large decimal numbers) from binary to decimal (you can check yourself by entering it above). Make sure you understand the result.
Enter 0000 0011. Then 0000 0110, then 0000 1100. Note that as the binary number is shifted to the left by one, the value of the
number doubles.
Enter 1111 1111. As expected the result is 255=28−1.
Think of a number, and try to find the binary representation. Later in this document we will find a systematic
way to do this.
Hexadecimal, Octal, Bits, Bytes and Words (interactive).
It is often convenient to handle groups of bits, rather than dealing with theindividually. The most
common grouping is 8 bits, which forms a byte. A single byte can represent 256 (28) numbers. Memory
capacity is usually referred to in bytes. Two
bytes is usually called a word, or short word (though word-length depends on the
application). A two-byte word is also the size that is usually used to represent integers
in programming languages. A long word is usually twice as long as a word. A less common
unit is the nibble which is 4 bits, or half of a byte.
It is cumbersome for humans to deal with writing, reading and remembering the large number
bits, and it takes many of them to represent even fairly small numbers. A number of
different ways have been developed to make the handling of binary data easier for us. The
most common is to use hexadecimal notation. In hexadecimal notation, 4 bits (a nibble) are represented by
a single digit. There is obviously a problem with this since 4 bits gives 16 possible
combinations, and there are only 10 unique decimal digits, 0 to 9. This is solved by using
the first 6 letters (A..F or a..f) of the alphabet as numbers. The table shows the relationship
between decimal, hexadecimal and binary.
| | | |
---
| Decimal | Hexadecimal | Binary |
| 0 | 0 | 0000 |
| 1 | 1 | 0001 |
| 2 | 2 | 0010 |
| 3 | 3 | 0011 |
| 4 | 4 | 0100 |
| 5 | 5 | 0101 |
| 6 | 6 | 0110 |
| 7 | 7 | 0111 |
| 8 | 8 | 1000 |
| 9 | 9 | 1001 |
| 10 | A | 1010 |
| 11 | B | 1011 |
| 12 | C | 1100 |
| 13 | D | 1101 |
| 14 | E | 1110 |
| 15 | F | 1111 |
There are some significant advantages to using hexadecimal when dealing
with electronic representations of numbers (if people had 16 fingers, we wouldn't be
saddled with the awkward decimal system). Using hexadecimal makes it very easy to convert
back and forth from binary because each hexadecimal digit corresponds to exactly 4 bits
(log 2(16) = 4) and each byte is two hexadecimal digit.
In contrast, a decimal digit corresponds to log2(10) =
3.322 bits and a byte is 2.408 decimal digits. Clearly hexadecimal is better suited to the
task of representing binary numbers than is decimal.
As an example, the 16 bit number 0CA316 = 0000 1100 1010 00112 (00002 = 016,
11002 = C16 , 10102 = A16,
00112 = 3 16). It is
convenient to write the binary number with spaces after every fourth bit to make it easier
to read.
Try it yourself: (enter either a value into either the binary or hexadecimal (i.e., hex) text box)
Binary value
Hexadecimal value:
.
(Input
Restrictions)
For the binary number spaces are ignored, 16 bits are assumed, and any character that
is not zero is iunbterpreted as a one. The resulting 16 bit number is displayed with spaces between 4 bit groups, to
make the hex interpretation easier to decipher.
For the hex number spaces are ignored, 4 digits are assumed, and any invalid characters are deleted.
Converting a hexadecimal number to its decimal equivalent is slightly more difficult, but can be done in the same way as before but multiplying each digit by the appropriate power of 16 (instead of powers
of 2 (for binary) or powers of 10 (for decimal)).
Octal notation is yet another compact method for writing binary numbers.
There are 8 octal characters, 0...7. Obviously this can be represented by exactly 3 bits.
Two octal digits can represent numbers up to 64, and three octal digits up to 512
Summary of binary types:
bit: a single binary digit, either zero or one.
byte: 8 bits, can represent positive numbers from 0 to 255.
hexadecimal:
A representation of 4 bits by a single digit 0..9,A..F (or 0..9,a..f). In this way a byte can be
represented by two hexadecimal digits.
long word:
A long word is usually twice as long as a word
nibble (or nybble):
4 bits, half of a byte.
octal:
A representation of 3 bits by a single digit 0..7. This is used much less commonly than
it once was, but it is still seen in special circumstances
word:
Usually 16 bits, or two bytes. But a word can be almost any size, depending on the
application being considered -- 32 and 64 bits are other common sizes.
Exercises (binary types):
| | |
--- |
| Convert 3C from hexadecimal to decimal | |
| Convert 1010 0111 1011 from binary to hexadecimal | |
| Convert 7D0 from hexadecimal to binary | |
| If you shift a hexadecimal number to the left by one digit, how many times larger is the resulting number? | |
Converting a Decimal Number to a Different Radix (e.g., decimal→binary);
Positive Decimal Integers
To begin our discussion on converting decimal numbers to a different radix, let's begin with a discussion of how we could find the individual digits of a decimal number, say 73510. Obviously we can just look at the number in this case (the 1's place is 5, the 10's place is thirty and the 100's place is 7), but let's find an algorithmic way of doing this, so a computer can do it. It will also lead us to a technique for converting to binary (or any other radix).
Find the decimal digits that comprise the number 73510.
We begin by performing an integer division by 10 (recall that decimal is radix 10).
735 ÷ 10 = 73 r 5 (this is read as 735 divided by 10 is 73 with a remainer of 5; the quotient is 73 and the remainder is 5). This tells us that the rightmost digit (i.e., the remainder) is 5.
Repeat process with quotient: 73 ÷ 10 = 7 r 3. So the next digit is 3.
Repeat process with quotient: 7 ÷ 10 = 0 r 7. So the last digit is 7.
Since the new quotient is 0, we are done.
The three digits found are (left→right) 7, 3, and 5, which is also the list of remainders bottom→top.
Finding the unsigned binary equivalent of a positive decimal integer (interactive)
We can now perform conversions from decimal to binary using the same procedure but dividing the number by 2 each time, until the quotient of the division is zero.
Enter a decimal number between 0 and 255:
. (Input
Restrictions)
The procedure is described verbally on the left, and is shown in a more compact tabular form on the right.
The number must be between 0 and 255. Any non-decimal digits will be replaced by 0.
Find the binary digits that comprise the number 18110.
We begin by performing an integer division by 2 (recall that binary is radix 2).
181 ÷ 2 = 90 r 1. This tells us the righmost bit (i.e., the remainder) is 1.
Repeat the process with quotient: 90 ÷ 2 = 45 r 0. So the next digit is 0
Repeat the process with quotient: 45 ÷ 2 = 22 r 1. So the next digit is 1
Repeat the process with quotient: 22 ÷ 2 = 11 r 0. So the next digit is 0
Repeat the process with quotient: 11 ÷ 2 = 5 r 1. So the next digit is 1
Repeat the process with quotient: 5 ÷ 2 = 2 r 1. So the next digit is 1
Repeat the process with quotient: 2 ÷ 2 = 1 r 0. So the next digit is 0
Repeat the process with quotient: 1 ÷ 2 = 0 r 1. So the last (i.e., leftmost) digit is 1
Since new the quotient is 0, we are done.
We have completed the conversion from decimal to binary: 18110 = 101101012.
You can get the bits, left→right, by taking the remainders bottom→top.
| | | |
---
| number | ÷ 2 | |
| quotient | remainder |
| 181 | 90 | 1 |
| 90 | 45 | 0 |
| 45 | 22 | 1 |
| 22 | 11 | 0 |
| 11 | 5 | 1 |
| 5 | 2 | 1 |
| 2 | 1 | 0 |
| 1 | 0 | 1 |
Signed Integers (i.e., the 2's complement representation)
Represented both positive and negative numbers
To represent signed integers we will use what is called the 2's complement representation. Let's start by representing 3 bit unsigned integers on a circle, as shown on the left half of the diagram shown below. This is the method we've been using up until now; only positive numbers (and 0) can be represented.
The image on the left shows the decimal number in red, and the equivalent binary number in blue. For example, the number 2dec (on the right side of the circle) is the same as 0102. Note:
Since there are n=3 bits, there are 2n=23=8 unique numbers that can be represented.
The decimal numbers go from 0 to 2n-1=23−1=7 (i.e., 0→7).
To add, we move clockwise around the circle (i.e., to go from 2dec=0102 to 3dec=0112, we move 1 place clockwise; to go from 2dec=0102 to 5dec=1102, we move 3 places). To subtract, we move counterclockwise.
We get an error in addition or subtraction if we cross the part of the circle with the dashed magenta line (between 7dec and 0dec). For example if we are a 1dec=0012, and subtract 3, then we move 3 spaces counterclockwise and end up at 6dec=1102, which is clearly incorrect because we crossed the dashed magenta line.
To represent negative numbers in 2's complement notation, we refer to the right half of the diagram. We can make similar observations:
Since there are n=3 bits, there are 2n=23=8 unique numbers that can be represented.
The decimal numbers go from −(2n−1)=−(22)=−4 to 2n−1−1=22−1=3 (i.e., −4→3).
To add, we move clockwise (i.e., to go from 2dec=0102 to 3dec=0112, we move 1 place clockwise; to go from −2dec=1102 to 1dec=0012, we move 3 places). To subtract, we move counterclockwise.
We get an error in addition or subtraction if we cross the part of the circle with the dashed magenta line (between 3dec and −4dec). For example if we are a 2dec=0102, and add 3, then we move 3 spaces clockwise and end up at −3dec=1012, which is clearly incorrect.
Note that the binary numbers are identical, it is solely the way that we interpret them that changes. The substantive difference between the two representations is that 2's complement numbers can be either positive or negative as indicated by the leftmost bit. Note that for all of the positive numbers (and 0) the leftmost bit is a zero, and for all of the negative numbers the leftmost bit is a one.
Aside: the diagram below shows the circular representation for 4 bit numbers. To test you knowledge, make sure that you can make equivalent assertions to those listed above for 3 bits (which can represent the numbers −4→3) for the case when there are 4 bits (−8→7).
Finding the decimal equivalent of a 2's complement binary integer (interactive)
The process for finding the decimal equivalent of a 2's complement binary integer is very similar to the way we do it for unsigned numbers (see above), but we must interpret the leftmost bit differently. For an n bit number, if the leftmost bit is 1 we interpret it as a value of −(2n−1), if it is 0 we ignore it. All the other bits are interpreted as before.
4 bit 2's complement:
For a four bit number −(2n−1)=−(24−1)=−(23−1)=−8.
To check your knowledge of signed numbers, explore 4 bit 2's complement numbers. Enter a 4 bit 2's complement number:
.
(Input restrictions).
You can also try entering a 4 bit number here, and then covert it to 8 bits via sign extension, and enter it above.
If the 4 bit binary number 10112=−5dec, how do you represent −5dec with 8 bits?
Note: any spaces in the input are ignored; any other characters that are not 0
or 1 are interpreted as 1, the number is truncated to 4 bits if necessary, and is increased to 4 bits if necessary
(by adding zeros on the left).
Try these things: You may want to check the 4 bit number wheel shown previously.
Enter the number "0011", and the result is 3. Change the leftmost bit to "1" (so the number is "1011"), and the result is −5 (or −8+3; the −8 is from the (leftmost) sign bit, and the number 3 is from the rightmost 3 bits, or "011").
Convert the 2's complement binary number "0111" to decimal. Predict, then check the result if the number is changed to "1111".
Where would these two numbers be on the 4 bit 2's complement number wheel?
Note if any 2's complement binary number is all 1's (i.e., "1111" for four bits, or "1111 1111" for 8 bits) it is always equivalent to the decimal number −1. If you are not sure why, check the number wheels shown previously.
Convert (by hand) the number "1101", then check your result Hint (hover the mouse here for an explanation).
Where would this number be on the number wheel? If you are not sure, check the 4 bit number wheel shown previously.
8 bit 2's complement
For an 8 bit number this means that if the leftmost bit is set, we interpret it as −128 (=−(28−1)=−(27)).
To see how the decimal equivalent of an 8 bit 2's complement binary number can be calculated, enter an 8 bit 2's complement number:
.
(Input restrictions).
Note: any spaces in the input are ignored; any other characters that are not 0
or 1 are interpreted as 1, the number is truncated to 8 bits if necessary, and is increased to 8 bits if necessary
(by adding zeros on the left).
This is almost exactly the way we broke down an unsigned number previously. The only difference is how we treat the leftmost bit. If it is a
"1" we add in −(2n−1)·1=−128·1=128; if it is a 0 we add in −(2n−1)·0=−128·0=0. In other words, if the leftmost bit is "0" we proceed exactly as we did with an unsigned number.
Try these things: (Note: I put a space in the middle of the binary numbers to make them easier to read - you can choose to add these (or not) to the input text box)
Enter the number "0011 0101", and the result is 53. Change the leftmost bit to "1" (so the number is "1011 0101"), and the result is −75 (or −128+53; the −128 is from the sign bit, and the number 53 is from the remainder of the bits).
Convert the 2's complement binary number "0111 1111" to decimal. Predict, then check the result if the number is changed to "1111 1111".
Where would these two numbers be on a number wheel?
Note if any 2's complement binary number is all 1's (i.e., "1111" for four bits, or "1111 1111" for 8 bits) it is always equivalent to the decimal number −1. If you are not sure why, check the number wheels shown previously.
Convert (by hand) the number "1000 0101", then check your result Hint (hover the mouse here for an explanation).
Where would this number be on a number wheel?
Sign Extension (increasing the number of bits in a 2's complement number)
For an unsigned number, increasing the number of bits used to represent a number can be accomplished by simply adding 0's to the left side of a number. For example, the number 3dec can be represented with 3 bits as 0112, or as 4 bits as 00112.
We can continue this process for 8 bits, or even more, and the number represented doesn't change.
The process above works for positive 2's complement numbers, but not for negative numbers. Recall that for negative numbers that the leftmost bit is a "1." To increase the number of bits, we place an additional "1" to the left of the original number. For example, consider the 3 bit representation of the number −3dec=1012 (the leftmost bit represents −4, and the rightmost bit represents 1, and when added together the result is 3). To increase to 4 bits we add a "1" to the left, which yields 11012=−3dec (see the discussion above for 4 bit 2's complement numbers if this is unclear). This is shown below, with the 3 bit version on the top line, and the 4 bit version below.
This can be understood using the image below.
The top line (101) shows the original 3 bit number.
The leftmost bit represents −(2n−1) where n=3, or −(23−1)=−4.
The middle bit is zero and doesn't contribute
The rightmost bit represents 1.
The sum of the 3 bits is −3.
The second line (1101) show the 4 bit representation of the same number.
The leftmost bit represents −(2n−1) where n=4, or −(24−1)=−8.
The next bit represents 22 or 4.
The third bit from the left is zero and doesn't contribute.
The rightmost bit represents 1.
The sum of the 4 bits is −3.
Note that the sum of the 2 leftmost bits in the 4 bit number is equal to −4, which is the same as the leftmost bit, by itself, in the three bit number — so adding the bit to the left has no effect on the final sum. This means we can keep repeating this process to find the 8 (or more) bit representation.
This process of increasing the number of bits used to represent is called "sign extension" because we simply extend the leftmost (sign) bit to fill in the added bits in the larger number. If the sign bit is "1", all of the new bits will be 1; if the sign bit is zero, all of the new bits will be 0. In the image below the sign bit of the 4 bit numbers is underlined, and you can see that it is simply repeated in the new bits of the 8 bit representation.
As a final exercise, you can look at the 3 bit and 4 bit number wheels shown above, and verify that you can convert 3 bit numbers to their 4 bit equivalent by using sign extension.
Convert from 2's complement binary to decimal (interactive).
Converting a 2's complement number from binary to decimal is very similar to converting an unsigned number, but we must account for the sign bit. If the number is positive (i.e., the sign bit is "0") the process is unchanged. If the number is negative, we know the sign bit is 1, and we just need to find the rest of the bits. An example may help to clarify things.
Consider the 4 bit 2's complement number, 11012=−3dec (this is the same number used in the discussion of sign extension). The value of the bits that are set are shown below.
Since the number is negative we know that the sign bit (the leftmost bit) is set to "1". We just need to find the rest of the bits. We see that we can effectively remove the sign bit by adding +8 to the number (+8=2n−1=23, where n is the number of bits in the number — in this case n=4 bits); this eliminates the effect of the sign bit. This process yields a result of 5dec=1012. So the resulting 2's complement number is 11012, where the leftmost bit is the sign bit (i.e., −8), and the other bits represent the number 5 (so the total is −8+5=−3, as desired).
You can try this for yourself by typing in a number that can be represented as an 8 bit 2's complement number (i.e., since the number of bits is n=8, the number must be in the range from −128→+127 (the lower limit is −(2n−1)=−(27)=−128, and the upper limit is 2(n−1)−1=128−1=127).
Enter a number between −127 and 128, and you can see how it is converted to and 8 bit 2's complement number.
. (Input
Restrictions)
The number must be between -128 and 127. Any non-decimal digits will be replaced by 0.
The number is negative. To get the bits of the unsigned part we must first add
128 (i.e., 2n−1, where n=number of bits = 8).
This gives us -75+128=53. We now perform the decimal to binary conversion
on this number (53)
using the same technique that was introduced earlier.
| | | |
---
| number | ÷ 2 | |
| quotient | remainder |
| 53 | 26 | 1 |
| 26 | 13 | 0 |
| 13 | 6 | 1 |
| 6 | 3 | 0 |
| 3 | 1 | 1 |
| 1 | 0 | 1 |
The unsigned part (53) is converted to binary as 0110101
(Note: the rightmost (least significant) bit is the top remainder in the table, and the
leftmost (most significant) bit is the bottom remainder in the table. If the result of the conversion is less then 7
bits (i.e., fewer than 7 rows in the table), it is padded to 7 bits by adding 0's to the left). Then the sign bit (in this case 1 because the number is
negative). The final result is
-75dec=101101012.
this way, by finding all of the powers of 2 that add up to the number in question; you can see this is
exactly analagous to the decimal deconstruction done earlier.
Let's look at how this changes the value of some binary numbers
| | | |
---
| Binary | Unsigned | Signed |
| 0010 0011 | 35 | 35 |
| 1010 0011 | 163 | -93 |
| 1111 1111 | 255 | -1 |
| 1000 0000 | 128 | -128 |
If Bit 7 is not set (as in the first example) the representation of
signed and unsigned numbers is the same. However, when Bit 7 is set, the number is always
negative. For this reason Bit 7 is sometimes called the sign bit. Signed numbers are added
in the same way as unsigned numbers, the only difference is in the way they are
interpreted. This is important for designers of arithmetic circuitry because it means that
numbers can be added by the same circuitry regardless of whether or not they are signed.
To form a two's complement number that is negative you simply take the corresponding
positive number, invert all the bits, and add 1. The example below illustrated this by
forming the number negative 35 as a two's complement integer:
3510 = 0010 00112
invert -> 1101 11002
add 1 -> 1101 11012
So 1101 1101 is our two's complement representation of -35. We can check this by adding
up the contributions from the individual bits
1101 11012 = -128 + 64 +
0 + 16 + 8 + 4 + 0 + 1 = -35.
The same procedure (invert and add 1) is used to convert the negative
number to its positive equivalent. If we want to know what what number is represented by
1111 1101, we apply the procedure again
? = 1111 11012
invert -> 0000 00102
add 1 -> 0000 00112
Since 0000 0011 represents the number 3, we know that 1111 1101 represents the number
-3.
Exercises (binary integers):
Note that a number can be extended from 4 bits to 8 bits by simply repeating the
leftmost bit 4 times. Consider the following examples
| | | |
---
| Decimal | 4 bit | 8 bit |
| 3 | 0011 | 0000 0011 |
| -3 | 1101 | 1111 1101 |
| 7 | 0111 | 0000 0111 |
| -5 | 1011 | 1111 1011 |
Let's carefully consider the last case which uses the number -5. As a 4
bit number this is represented as
1011 = -8 + 2 + 1 = -5
The 8 bit number is
1111 1011 = -128 + 64 + 32 + 16 + 8 + 2 + 1 = -5.
It is clear that in the second case the sum of the contributions from
the leftmost 5 bits (-128 + 64 + 32 + 16 + 8 = -8) is the same as the contribution from
the leftmost bit in the 4 bit representation (-8)
This process is refered to as sign-extension, and can be applied whenever a number is
to be represented by a larger number of bits. Likewise you can remove all but one of the leftmost
bits, as long as they are all the same, so the 8 bit number 000001112=710 can be replaced by
01112. Also 111110112=-510 can be replaced by 10112).
Most processors even have two separate instructions for shifting numbers to the right
(which, you will recall, is equivalent to dividing the number in half). The first
instruction is something like LSR (Logical Shift Right) which simply shifts the bits to
the right and usually fills a zero in as the lefmost bit. The second instruction is
something like ASR (Arithmetic Shift Right), which shifts all of the bits to the
right,
while keeping the leftmost bit unchanged. With ASR 1010 (-6) becomes 1101 (-3). Of course,
there is only one instruction for a left shift (since LSL is equivalent to ASL).
Positive non-integer numbers
Note: If you are only interested in integer numbers you can skip the rest of this page. If you are interested in how non-integer numbers can be respresented in binary, keep reading.
Positive non-integer decimal numbers
Representing positive numbers that are not integers is a simple extension of the representation of integers. To review the concepts involved, let's start with an example using decimal numbers then we will continue with binary numbers. We proceed as we did for positive integers, but we include negative
powers of ten to the right of the decimal point. To wit,
The only pertinent observations here are:
If there are m digits to the right of the decimal point, the smallest number that can be
represented is 10-m. For instance if m=4, the smallest number that can be
represented is 0.0001=10-4.
Positive binary fractional numbers - Binary→Decimal (interactive)
Representing positive numbers that are not integers is a simple extension of binary integers, but we include
negative powers of two to the right of the decimal point.
Enter a number (8 bits with a decimal point),
(Input Restrictions)
Any spaces in the input are ignored; any other characters that are not 0 or
1 are interpreted as 1, the number is truncated to 8 bits if necessary, and is increased to 8 bits if necessary (by
adding zeros on the left). Only one decimal point is allowe.
Q Notation: We define a binary number with a fractional part by the number of bits and with Q equal to the number of bits to the
right of the decimal point.
The number 0011.0101 is an 8 bit, Q4 number.
The number 0.1110101 is an 8 bit Q7 number.
If there are n digits to the left of the decimal point and Q digits to the right of the decimal point (for a total of n+Q bits), the
smallest number that can be
represented is 2-Q, and the largest is 2n-2-Q.
For instance if we have an 8 bit Q5 number then n=3,
the smallest number that can be
represented is 000.000012 = 2-5 = 1/32 = 0.03125 and
the largest is 111.11112 = 7.96875 = 23-2-5
If this isn't clear, try entering binary numbers (with decimal point) in the text box immediately above.
Try this:
Flip the bit immediately to the right of the decimal point. How does the number change?
Move the decimal point one to the right or one to the left. What happens?
Set the decimal point for an 8 bit Q3 number.
What is the smallest number you can represent?
What is the largest number?
Exercises (positive binary fractional numbers):
| | |
--- |
| Convert 37 to binary, shift it right by one and convert back to decimal. What is the result | |
| Convert the 8 bit Q7 binary number 0.100 1001 from to decimal | |
| Convert the 8 bit Q7 binary number 0.111 1111 from to decimal | |
| Convert 0.75 from decimal to an 8 bit Q7 binary fraction | |
| Convert 0.65625 from decimal to an 8 bit Q7 binary fraction | |
| Approximate 0.9 as an 8 bit Q7 binary fraction | |
Signed binary fractions
Signed binary fractions are formed much like signed integers. We will work with a
single digit to the left of the decimal point, and this will represent the number -1 (=
-(20)). The rest of the representation of the fraction
remains unchanged. Therefore this leftmost bit represents a sign bit just as with two's
complement integers. If this bit is set, the number is negative, otherwise the number is
positive. The largest positive number that can be represented is still 1-2-m but the largest negative
number is -1. The resolution is still
1-2-m.
There is a terminology for naming the resolution of signed
fractions. If there are m bits to the right of the decimal point, the number is said to be
in Qm format. For a 16 bit number (15 bits to the right of the decimal point) this
results in Q15 notation.
Exercises (signed binary fractions):
| | |
--- |
| Convert 1.100 1001 from binary to decimal | |
| Convert 1.111 1111 from binary to decimal | |
| Convert -0.75 from decimal to a binary fraction | |
| Convert -0.65625 from decimal to a binary fraction | |
| Approximate -0.9 as a binary fraction (use 8 bits) | |
Signed binary fractions are easily extended to include all numbers by representing the
number to the left of the decimal point as a 2's complement integer, and the number to the
right of the decimal point as a positive fraction. Thus
-6.62510 = (-7+0.375)10 = 1001.0112
Note, that as with two's complement integers, the leftmost digit can be
repeated any number of times without affecting the value of the number.
A Quicker Method for Converting Binary Fractions.
Another way to convert Qm numbers to decimal is to represent the
binary number as a signed integer, and to divide by 2m; this is equivalent to shifting the decimal point
m places to the right. To
convert a decimal number to Qm, multiply the number by 2m and take the rightmost m digits.
Note, this simply truncates the
number; it is more elegant, and accurate, but slightly more complicated, to
round the number.
Examples (all Q7 numbers):
| | |
--- |
| Convert 0.100 1001 to decimal. | Take the binary number 0100 1001 (=7310), and divide by 27=128. The answer is 73/128=0.5703125, which agrees with the result of the previous exercise (Positive Binary Fractions). |
| Convert 1.100 1001 to decimal. | Take the two's complements binary number 1100 1001 (=-5510), and divide by 128. The answer is -0.4296875, which agrees with the result of the previous exercise (Signed Binary Fractions). |
| Convert 0.9 to Q7 format | Multiply 0.9 by 128 to get 115.2. This is represented in binary as 111 0011, so the Q7 representation is 0.111 0011. This agrees with the result of the previous exercise (Positive Binary Fractions). |
| Convert -0.9 to Q7 format | Multiply -0.9 by 128 to get -115.2. The Q7 representation is 1.000 1101. This agrees with the result of the previous exercise (Signed Binary Fractions). |
| | | |
---
| ← | Comments or Questions? Send me email | Erik Cheever Professor Emeritus Engineering Department Swarthmore College | |
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The effect of pressure and temperature on equilibrium
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Use the experiment to introduce Le Chatelier’s principle, or encourage students to apply the principle to make predictions
For this activity, the demonstrator fills a gas syringe with an equilibrium mixture of brown nitrogen dioxide and colourless dinitrogen tetroxide. The effect of pressure on the equilibrium is shown to be in accordance with Le Chatelier’s principle by compressing the mixture and observing the change in colour intensity.
Similarly, the effect of temperature is demonstrated by heating or cooling the mixture and observing the change in colour, or the change in volume of the mixture compared with that of a similar volume of air.
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The demonstration can be used to introduce Le Chatelier’s principle or to ask students to apply it in predicting the changes expected when pressure and temperature are changed.
Preparation must be carried out in a fume cupboard. The filled syringes may be brought out onto a lecture bench in a well ventilated room for better visibility provided they are sealed and do not leak when put under pressure. The nitrogen oxides involved are very toxic and students with breathing problems should avoid inhaling them as they could trigger an asthma attack.
A white background will greatly enhance the visibility of the colour changes.
The time for carrying out the demonstration, including preparation of the dinitrogen tetroxide, should be about 30–40 minutes. More time will be needed for the quantitative extension option.
Equipment
Apparatus
Eye protection (for the demonstrator)
Access to a fume cupboard
Clear glass gas syringes, 100 cm3, x2 or x3 (note 3)
Rubber septum caps to fit syringe tips, x2 or x3 (note 4)
Plastic tubing, short lengths (optional; note 5)
Screw clips (Hoffmann) for rubber tubing, x2 or x3 (optional; note 5)
Boiling tube with a sidearm
Hard glass (borosilicate) boiling tube
One-holed rubber stoppers to fit the boiling tubes, x2
Glass tubing, short lengths, x2 (to fit stoppers and bent as shown in the diagram below)
Spring clip (Mohr) for rubber tubing (optional)
Three-way stopcock or glass (or plastic) T-piece
Beaker, 400 cm3
Beakers, 2 dm3, x2
Thermometer (-10–110 °C)
Bunsen burner
Tripod
Gauze
Bosses, clamps and stands as required
Chemicals
Lead(II) nitrate(V) (TOXIC, DANGEROUS FOR THE ENVIRONMENT), 10–20 g (note 6)
Crushed ice, about 400 cm3
Common salt (sodium chloride) or crushed rock salt, about 100 g
Light lubricating oil (such as WD-40), a few drops
Health, safety and technical notes
Read our standard health and safety guidance.
Wear eye protection and work in a fume cupboard.
Two or three gas syringes is ideal, but the experiment can be done with one.
The syringes should all be airtight under pressure. If not, lubricate the plunger with a few drops of light mineral oil, such as sewing machine oil or WD-40.
Instead of using septum caps to seal the syringes, a short length of plastic tubing carrying a screw clip can be used to connect the syringe to the sidearm tube. The tubing must fit the syringe tightly enough not to be dislodged when the gas is put under pressure. Soften it by warming it in hot water before attaching it to the syringe. However, it must not fit the sidearm tube so tightly that it cannot easily be disconnected from it when the syringe has been filled.
Lead(II) nitrate(V), Pb(NO3)2 (s), and lead(II) oxide, PbO(s), (both TOXIC, DANGEROUS FOR THE ENVIRONMENT) – see CLEAPSS Hazcards HC057a and HC056.
The lead(II) nitrate should be thoroughly dried by heating overnight in an oven set to around 100 °C. Store in a desiccator after heating unless used immediately.
The equation for the decomposition of lead nitrate is Pb(NO3)2 (s) → PbO(s) + 2NO2 (g) + 1/2 O2 (g).
1 g of lead(II) nitrate(V) should therefore produce about 150 cm3 of NO2 gas at room temperature and pressure. Adjust the mass of lead nitrate accordingly (but use at least 5 g) depending on the number of syringes of gas required, bearing in mind the number of times they are flushed (see below) before finally filling.
The lead(II) oxide residue will soften the glass of the test tube and fuse with it. This combined residue should be collected and labelled as hazardous waste, for eventual collection by a licensed contractor – see CLEAPSS Hazcard HC056.
Nitrogen dioxide, NO2 (g), and dinitrogen tetroxide, N2 O4, (both VERY TOXIC) – see CLEAPSS Hazcard HC068B.
Sodium chloride, NaCl(s) – see CLEAPSS Hazcard HC047b.
Procedure
Before the demonstration
In a fume cupboard, set up the apparatus for collection of dinitrogen tetroxide shown in the diagram below. Stand the smaller beaker on a gauze on a tripod and clamp both of the test tubes near the top. Keep the length of the rubber connecting tube as short as possible as nitrogen dioxide attacks rubber.
Source: Royal Society of Chemistry
Equipment required for the preparation of dinitrogen tetroxide.
It is vital that a syringe does not leak under pressure. Test if a syringe is airtight by sealing it using a rubber septum cap, with about 60 cm3 of air in the syringe. Hold the syringe vertically with septum cap resting on a firm support, such as the bench surface. Push in the plunger to compress the gas as far as comfortably possible, hold for a few seconds, then release the pressure. The plunger should return to its original position if the syringe is airtight. If not, check the septum cap, or replace the syringe.
Mix the salt with the crushed ice to produce a freezing mixture and place it in the beaker.
The demonstration
Working in a fume cupboard, heat the lead nitrate gently to decompose it to nitrogen dioxide (and oxygen). Heating too vigorously may decompose the nitrogen dioxide into nitrogen monoxide. As the gas meets the cold wall of the sidearm test tube, it condenses as liquid dinitrogen tetroxide, which may appear greenish in colour due to dissolved water if the lead nitrate was not absolutely dry. The oxygen is lost to the atmosphere.
When about 2 cm3 of liquid has been collected, stop heating, tighten the screw clip and disconnect the tube containing the lead nitrate residue, leaving the screw clip sealing the tube connected to the receiver. The liquefied dinitrogen tetroxide can be kept for some time in the freezing mixture, so that this part of the experiment could be done before the demonstration if desired.
Connect the sidearm of the tube containing the dinitrogen tetroxide to a syringe nozzle via the three-way stopcock or T-piece and short lengths of rubber tubing (see diagram below). If a T-piece is used, the third outlet can be opened or closed using a gloved finger (for preference, use nitrile gloves) or a short length of rubber tubing fitted with a spring clip.
Set the stopcock or T-piece to open the connection between the syringe and the sidearm tube and partly fill the syringe with the brown gas mixture by removing the ice bath and gently warming the sidearm tube in your hand or a beaker of warm water. As the almost colourless dinitrogen tetroxide evaporates, it decomposes to form some brown nitrogen dioxide.
Flush the first filling of gas out of the syringe into the fume cupboard using the three-way tap or the T-piece, and repeat the filling and flushing cycle two or three more times to ensure that there is no air in the system. Finally fill the syringe to the 50–60 cm3 mark, release it from the connecting rubber tubing and quickly seal it with a septum cap. Leave the tap or T-piece open to allow any excess gas to flush out into the fume cupboard (see diagram below).
Source: Royal Society of Chemistry
Equipment required for the preparation of an equilibrium mixture of nitrogen dioxide and dinitrogen tetroxide.
The effect of pressure on the equilibrium
In good view of the class and against a white background, hold the syringe with the septum cap resting on a hard surface. Compress the gas mixture rapidly by pressing in the plunger of the syringe as far as it will comfortably go and hold it there for a few seconds. The colour of the gas mixture will initially become darker as the concentration of the gases increases with the decrease in volume. Within a few seconds, however, it will become paler as the equilibrium responds to the increased pressure and brown nitrogen dioxide is converted into colourless dinitrogen tetroxide.
Pull the plunger out to its starting position. Note the colour changes as the pressure is decreased. They should be the reverse of the above.
The effect of temperature on the equilibrium
Fill one large beaker about two-thirds full with ice-cold water and another with hot water at about 60–70 °C.
Fill one, two or three syringes with 50–60 cm3 of the brown gas mixture as above and seal them with septum caps. If only one syringe is used, let the class observe the colour of its contents against a white background, then place it in the beaker of hot water. The gas will expand and become darker as the equilibrium adjusts to the higher temperature by converting dinitrogen tetroxide to nitrogen dioxide.
Transfer the syringe (or use another syringe full of gas) into the cold water. The gas will contract but become lighter in colour as the equilibrium readjusts to the decrease in temperature. If three syringes of gas are used, one can be kept at room temperature for comparison.
Teaching notes
The gaseous equilibrium studied here is: N2O4(g) ⇌ 2NO2(g) ΔHo = +58 kJ mol–1
Le Chatelier’s principle predicts that the equilibrium will move to the right with an increase in temperature as the forward reaction is endothermic. The brown colour intensifies. The equilibrium constant, Kp, has a value of 48 atm at 400 K (127 °C), and the equilibrium will lie almost completely over to the right at 140 °C.
Increasing the pressure moves the equilibrium to the left as this decreases the number of moles of gas present in the mixture. The brown colour becomes paler.
Remind students that the value of Kp will increase with increase in temperature, whereas it will not be affected by changes in pressure despite a shift in the position of equilibrium.
A quantitative extension: comparison with ideal gas behaviour (optional)
By recording the volume of the gas mixture at intervals as it is warmed up, a plot of volume vs temperature can be obtained – see graph below. This can be compared with volume readings obtained from a syringe containing a similar volume of air.
Clamp two syringes, one containing the mixture of nitrogen oxides and one containing a similar volume of air (about 50 cm3), vertically in a 2 dm3 beaker of water on a tripod and gauze, so that they are immersed up to the 100 cm3 mark.
Note the volume readings of both syringes, which should be as close as possible, and the temperature of the water. Heat the water gently with a Bunsen burner and record the temperature and the volume of gases every 10 °C or so. Before taking each reading, remove the Bunsen burner and stir the water for a couple of minutes to ensure that the temperature of the gases is the same as that of the water. Twist the plungers of the syringes before taking each reading, to ensure that they are not sticking.
Continue taking readings until the temperature is between 70 to 80 °C. Plot graphs of volume against temperature for both gases on the same axes. The nitrogen dioxide/nitrogen tetroxide mixture will expand more than air (see diagram below) as the equilibrium responds to the increase in temperature by producing more nitrogen dioxide. If there is time take further readings as the water cools, to check for leaks.
Source: Royal Society of Chemistry
Comparison of Charles’ Law readings with actual results.
This experiment can be done without a second syringe of air if necessary. The predicted volume of an ideal gas can be worked out using Charles’ Law for each temperature reading and compared with the observed one (see graph above). For example, if the volume of the gas mixture is 50 cm3 at 25 °C (298 K), the predicted volume of an ideal gas at 50 °C (323 K) would be 50 x 323/298 = 54.2 cm3. Students could watch the demonstration and plot the points as the readings are taken. Alternatively they could do the Charles’ Law calculations as the equilibrium mixture is warmed up.
Additional information
This is a resource from the Practical Chemistry project, developed by the Nuffield Foundation and the Royal Society of Chemistry.
Practical Chemistry activities accompany Practical Physics and Practical Biology.
© Nuffield Foundation and the Royal Society of Chemistry
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Equilibrium and Le Chatelier’s principle
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### The effect of pressure and temperature on equilibrium
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AQA Chemistry
Physical Chemistry
Chemical equilibria, Le Chatelier's principle and Kc
Chemical equilibria and Le Chatelier's principle
Students should be able to: use Le Chatelier’s principle to predict qualitatively the effect of changes in temperature, pressure and concentration on the position of equilibrium.
OCR Chemistry A
Module 3: Periodic table and energy
3.2 Physical chemistry
3.2.3 Chemical equilibrium
b) le Chatelier’s principle and its application for homogeneous equilibria to deduce qualitatively the effect of a change in temperature, pressure or concentration on the position of equilibrium
Edexcel Chemistry
Topic 10: Equilibrium I
2. be able to predict and justify the qualitative effect of a change in temperature, concentration or pressure on a homogeneous system in equilibrium
GCSE
AQA Chemistry
4.6 The rate and extent of chemical change
4.6.2 Reversible reactions and dynamic equilibruim
4.6.2.4 The effect of changing conditions on equilibrium (HT only)
The effects of changing conditions on a system at equilibrium can be predicted using Le Chatelier’s Principle.
AQA Combined science: Synergy
4.7 Movement and interactions
4.7.4 The rate and extent of chemical change
4.7.4.10 Factors affecting the position of equilibrium (HT only)
Predict the effect of changing concentration on equilibrium position and suggest appropriate conditions to produce a particular product.
AQA Combined science: Trilogy
5.6 The rate and extent of chemical change
5.6.2 Reversible reactions and dynamic equilibruim
5.6.2.4 The effect of changing conditions on equilibrium (HT only)
The effects of changing conditions on a system at equilibrium can be predicted using Le Chatelier’s Principle.
Edexcel Chemistry
Topic 4 - Extracting metals and equilibria
Reversible reactions and equilibria
4.17 Predict how the position of a dynamic equilibrium is affected by changes in: temperature, pressure, concentration
Edexcel Combined science
Topic 4 - Extracting metals and equilibria
Reversible reactions and equilibria
4.17 Predict how the position of a dynamic equilibrium is affected by changes in: temperature, pressure, concentration
OCR Chemistry B: 21st century
C6 Making useful chemicals
C6.3 What factors affect the yield of chemical reactions?
6.3.3 predict the effect of changing reaction conditions (concentration, temperature and pressure) on equilibrium position and suggest appropriate conditions to produce a particular product, including: catalysts increase rate but do not affect yield; the…
OCR Combined science B: 21st Century
C6 Making useful chemicals
C6.3 What factors affect the yield of chemical reactions?
6.3.3 predict the effect of changing reaction conditions (concentration, temperature and pressure) on equilibrium position and suggest appropriate conditions to produce a particular product, including: catalysts increase rate but do not affect yield; the…
OCR Combined science A: Gateway
C5 Monitoring and controlling chemical reactions
C5.2 Eqilibria
C5.2c predict the effect of changing reaction conditions on equilibrium position and suggest appropriate conditions to produce as much of a particular product as possible
OCR Chemistry A: Gateway
C5 Monitoring and controlling chemical reactions
C5.3 Eqilibria
C5.3c predict the effect of changing reaction conditions on equilibrium position and suggest appropriate conditions to produce as much of a particular product as possible
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B2.32 How the following gas laws describe the behaviour of gases in particular conditions:
Charles’s Law (V₁T₂ = V₂T₁)
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For a given reversible reaction, the effect of altering temperature or pressure or of adding/removing reactants/products can be predicted.
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1.7 Simple equilibria and acid-base reactions
(b) Le Chatelier’s principle in deducing the effect of changes in temperature, concentration and pressure
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CCEA Chemistry
Unit AS 2: Further Physical and inorganic Chemistry and an Introdution to Organic Chemistry
2.10 Equilibrium
2.10.2 deduce the qualitative effects of changes of temperature, pressure, concentration and catalysts on the position of equilibrium for a closed homogeneous system;
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5978 | https://courses.lumenlearning.com/suny-physics/chapter/8-1-linear-momentum-and-force/ | Linear Momentum and Collisions
Linear Momentum and Force
Learning Objectives
By the end of this section, you will be able to:
Define linear momentum.
Explain the relationship between momentum and force.
State Newton’s second law of motion in terms of momentum.
Calculate momentum given mass and velocity.
Linear Momentum
The scientific definition of linear momentum is consistent with most people’s intuitive understanding of momentum: a large, fast-moving object has greater momentum than a smaller, slower object. Linear momentum is defined as the product of a system’s mass multiplied by its velocity. In symbols, linear momentum is expressed as p = mv.
Momentum is directly proportional to the object’s mass and also its velocity. Thus the greater an object’s mass or the greater its velocity, the greater its momentum. Momentum p is a vector having the same direction as the velocity v. The SI unit for momentum is kg · m/s.
Linear Momentum
Linear momentum is defined as the product of a system’s mass multiplied by its velocity:
p = mv
Example 1. Calculating Momentum: A Football Player and a Football
Calculate the momentum of a 110-kg football player running at 8.00 m/s.
Compare the player’s momentum with the momentum of a hard-thrown 0.410-kg football that has a speed of 25.0 m/s.
Strategy
No information is given regarding direction, and so we can calculate only the magnitude of the momentum, p. (As usual, a symbol that is in italics is a magnitude, whereas one that is italicized, boldfaced, and has an arrow is a vector.) In both parts of this example, the magnitude of momentum can be calculated directly from the definition of momentum given in the equation, which becomes p = mv when only magnitudes are considered.
Solution for Part 1
To determine the momentum of the player, substitute the known values for the player’s mass and speed into the equation.
pplayer = (110 kg)( 8.00 m/s) = 880 kg · m/s
Solution for Part 2
To determine the momentum of the ball, substitute the known values for the ball’s mass and speed into the equation.
p ball = (0.410 kg)(25.0 m/s) = 10.3 kg · m/s
The ratio of the player’s momentum to that of the ball is
[latex]\displaystyle\frac{p_{\text{player}}}{p_{\text{ball}}}=\frac{880}{10.3}=85.9\[/latex]
Discussion
Although the ball has greater velocity, the player has a much greater mass. Thus the momentum of the player is much greater than the momentum of the football, as you might guess. As a result, the player’s motion is only slightly affected if he catches the ball. We shall quantify what happens in such collisions in terms of momentum in later sections.
Momentum and Newton’s Second Law
The importance of momentum, unlike the importance of energy, was recognized early in the development of classical physics. Momentum was deemed so important that it was called the “quantity of motion.” Newton actually stated his second law of motion in terms of momentum: The net external force equals the change in momentum of a system divided by the time over which it changes. Using symbols, this law is
[latex]\displaystyle{\mathbf{F}}_{\text{net}}=\frac{\Delta\mathbf{p}}{\Delta t}[/latex],
where Fnet is the net external force, Δp is the change in momentum, and Δt is the change in time.
Newton’s Second Law of Motion in Terms of Momentum
The net external force equals the change in momentum of a system divided by the time over which it changes.
[latex]\displaystyle{\mathbf{F}}_{\text{net}}=\frac{\Delta\mathbf{p}}{\Delta t}[/latex]
Making Connections: Force and Momentum
Force and momentum are intimately related. Force acting over time can change momentum, and Newton’s second law of motion, can be stated in its most broadly applicable form in terms of momentum. Momentum continues to be a key concept in the study of atomic and subatomic particles in quantum mechanics.
This statement of Newton’s second law of motion includes the more familiar Fnet = ma as a special case. We can derive this form as follows. First, note that the change in momentum Δp is given by Δp = Δ(mv).
If the mass of the system is constant, then Δ(mv) = mΔv.
So that for constant mass, Newton’s second law of motion becomes
[latex]\displaystyle{\mathbf{F}}_{\text{net}}=\frac{\Delta\mathbf{p}}{\Delta t}=\frac{m\Delta\mathbf{v}}{\Delta{t}}[/latex]
Because [latex]\frac{\Delta\mathbf{v}}{\Delta{t}}=\mathbf{a}\[/latex], we get the familiar equation Fnet = ma when the mass of the system is constant.
Newton’s second law of motion stated in terms of momentum is more generally applicable because it can be applied to systems where the mass is changing, such as rockets, as well as to systems of constant mass. We will consider systems with varying mass in some detail; however, the relationship between momentum and force remains useful when mass is constant, such as in the following example.
Example 2. Calculating Force: Venus Williams’ Racquet
During the 2007 French Open, Venus Williams hit the fastest recorded serve in a premier women’s match, reaching a speed of 58 m/s (209 km/h). What is the average force exerted on the 0.057-kg tennis ball by Venus Williams’ racquet, assuming that the ball’s speed just after impact is 58 m/s, that the initial horizontal component of the velocity before impact is negligible, and that the ball remained in contact with the racquet for 5.0 ms (milliseconds)?
Strategy
This problem involves only one dimension because the ball starts from having no horizontal velocity component before impact. Newton’s second law stated in terms of momentum is then written as
[latex]\displaystyle{\mathbf{F}}_{\text{net}}=\frac{\Delta\mathbf{p}}{\Delta t}[/latex]
As noted above, when mass is constant, the change in momentum is given by Δ p = mΔv = m ( v f − v i ) .
In this example, the velocity just after impact and the change in time are given; thus, once Δp is calculated, [latex]{\mathbf{F}}_{\text{net}}=\frac{\Delta{p}}{\Delta t}[/latex] can be used to find the force.
Solution
To determine the change in momentum, substitute the values for the initial and final velocities into the equation above.
[latex]\begin{array}{lll}\Delta{p}&=&m(v_{\text{f}}-v{\text{i}})\ &=&(0.057\text{ kg})(58\text{ m/s}-0\text{ m/s})\ &=&3.306\text{ kg}\cdot\text{m/s}\approx3.3\text{ kg}\cdot\text{m/s}\end{array}\[/latex]
Now the magnitude of the net external force can determined by using [latex]{\mathbf{F}}_{\text{net}}=\frac{\Delta{p}}{\Delta t}[/latex]:
[latex]\begin{array}{lll}\mathbf{F}_{\text{net}}&=&\frac{\Delta{p}}{\Delta{t}}=\frac{3.306\text{ kg}\cdot\text{m/s}}{5.0\times10^{-3}\text{ s}}\ &=&661\text{ N}\approx660\text{ N}\end{array}\[/latex]
where we have retained only two significant figures in the final step.
Discussion
This quantity was the average force exerted by Venus Williams’ racquet on the tennis ball during its brief impact (note that the ball also experienced the 0.56-N force of gravity, but that force was not due to the racquet). This problem could also be solved by first finding the acceleration and then using Fnet = ma, but one additional step would be required compared with the strategy used in this example.
Section Summary
Linear momentum (momentum for brevity) is defined as the product of a system’s mass multiplied by its velocity.
In symbols, linear momentum p is defined to be p = mv, where m is the mass of the system and v is its velocity.
The SI unit for momentum is kg · m/s.
Newton’s second law of motion in terms of momentum states that the net external force equals the change in momentum of a system divided by the time over which it changes.
In symbols, Newton’s second law of motion is defined to be [latex]{\mathbf{F}}_{\text{net}}=\frac{\Delta \mathbf{p}}{\Delta t}\[/latex], Fnet is the net external force, Δp is the change in momentum, and Δt is the change time.
Conceptual Questions
An object that has a small mass and an object that has a large mass have the same momentum. Which object has the largest kinetic energy?
An object that has a small mass and an object that has a large mass have the same kinetic energy. Which mass has the largest momentum?
Professional Application.Football coaches advise players to block, hit, and tackle with their feet on the ground rather than by leaping through the air. Using the concepts of momentum, work, and energy, explain how a football player can be more effective with his feet on the ground.
How can a small force impart the same momentum to an object as a large force?
Problems & Exercises
(a) Calculate the momentum of a 2000-kg elephant charging a hunter at a speed of 7.50 m/s. (b) Compare the elephant’s momentum with the momentum of a 0.0400-kg tranquilizer dart fired at a speed of 600 m/s. (c) What is the momentum of the 90.0-kg hunter running at 7.40 m/s after missing the elephant?
(a) What is the mass of a large ship that has a momentum of 1.60 × 109 kg · m/s, when the ship is moving at a speed of 48.0 km/h? (b) Compare the ship’s momentum to the momentum of a 1100-kg artillery shell fired at a speed of 1200 m/s.
(a) At what speed would a 2.00 × 104-kg airplane have to fly to have a momentum of 1.60 × 109 kg · m/s (the same as the ship’s momentum in the problem above)? (b) What is the plane’s momentum when it is taking off at a speed of 60.0 m/s? (c) If the ship is an aircraft carrier that launches these airplanes with a catapult, discuss the implications of your answer to (b) as it relates to recoil effects of the catapult on the ship.
(a) What is the momentum of a garbage truck that is 1.20 × 104 kg and is moving at 10.0 m/s? (b) At what speed would an 8.00-kg trash can have the same momentum as the truck?
A runaway train car that has a mass of 15,000 kg travels at a speed of 5.4 m/s down a track. Compute the time required for a force of 1500 N to bring the car to rest.
The mass of Earth is 5.972 × 1024 kg and its orbital radius is an average of 1.496 × 1011 m. Calculate its linear momentum.
Glossary
linear momentum: the product of mass and velocity
second law of motion: physical law that states that the net external force equals the change in momentum of a system divided by the time over which it changes
Selected Solutions to Problems & Exercises
1. (a) 1.50 × 104 kg ⋅ m/s; (b) 625 to 1; (c) 6.66 × 102 kg ⋅ m/s
3. (a) 8.00 × 104 m/s; (b) 1.20 × 106 kg · m/s; (c) Because the momentum of the airplane is 3 orders of magnitude smaller than of the ship, the ship will not recoil very much. The recoil would be −0.0100 m/s, which is probably not noticeable.
54 s
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5979 | https://math.stackexchange.com/questions/1400530/is-the-function-lnax-b-increasing-decreasing-concave-convex | derivatives - Is the function $\ln(ax + b)$ increasing/decreasing, concave/convex? - Mathematics Stack Exchange
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Is the function ln(a x+b)ln(a x+b) increasing/decreasing, concave/convex?
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h(x)=ln(a x+b)h(x)=ln(a x+b)
NB. Examine your results acccording to values of (a,b)(a,b)
I've differentiated twice in order to get the following:
h′′(x)=−a 2(a x+b)2 h″(x)=−a 2(a x+b)2
I think this proves that h(x)h(x) is concave for all value of a a and b b since h′′(x)≤0 h″(x)≤0. Is this correct?
I don't know how to prove whether it's increasing/decreasing or what the NB really means so any help with that would be great.
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edited Aug 17, 2015 at 18:00
Fabrosi
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asked Aug 17, 2015 at 17:56
James BakerJames Baker
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1 You are correct on the concavity. You can use the same process to figure out increasing/decreasing by looking at where the first derivative is positive/negative, respectively.Michael Dyrud –Michael Dyrud 2015-08-17 17:58:43 +00:00 Commented Aug 17, 2015 at 17:58
@MichaelDyrud h'(x) = a / (ax+b) but how do I know if this is positive or negative without further information? If a, b and x are all positive then it is positive. Otherwise it's hard to tell?James Baker –James Baker 2015-08-17 18:02:58 +00:00 Commented Aug 17, 2015 at 18:02
That's what they're asking you to determine, give restrictions in terms of a a and b b. For instance, we know that a x+b a x+b must be positive, but can we have a a or b b negative? What does that do to the derivative?Michael Dyrud –Michael Dyrud 2015-08-17 18:08:44 +00:00 Commented Aug 17, 2015 at 18:08
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we now that the domain of the function is:
a x+b>0⇒x>−b a so the domain is:(−b a,+∞)a x+b>0⇒x>−b a so the domain is:(−b a,+∞)
f′(x)=a a x+b f′(x)=a a x+b
in the domain of the function since we have x>−b a⇒a x+b>0 x>−b a⇒a x+b>0 the sign of f′(x)=a a x+b f′(x)=a a x+b will be dependent to the sign of a a so:
if a>0⇒f′(x)>0 a>0⇒f′(x)>0 and f(x)f(x) will be increasing in its domain
if a<0⇒f′(x)<0 a<0⇒f′(x)<0 and f(x)f(x) will be decreasing in its domain
Note the phrases in its domain in the above expression, we always study the behaviors of functions in their domain
f′′(x)=−a 2(a x+b)2 f″(x)=−a 2(a x+b)2
as you said without we always have f′′(x)<0 f″(x)<0 so without considering the sign of a a, f(x)f(x) will be a convex function
The value of b b doesnot influence the first and second derivation and so will not affect concavity, convexity, increase or decrease of the function
here is the diagram for f(x)=ln(2 x+1)f(x)=ln(2 x+1) as you see a=2>0 a=2>0 and f(x)f(x) is increasing and convex
here is the diagram for f(x)=ln(−2 x+1)f(x)=ln(−2 x+1) as you see a=−2<0 a=−2<0 and f(x)f(x) is decreasing and convex
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answered Aug 17, 2015 at 18:30
Sepideh AbadpourSepideh Abadpour
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h′(x)=a a x+b>0 h′(x)=a a x+b>0
1 1.if a>0 a>0 a x+b>0 a x+b>0⇒⇒a x>−b a x>−b⇒⇒x>−b a x>−b a function is increasing
2 2.if a<0 a<0 x<−b a x<−b a function is decreasing
And about concavity you are right,find second derivative and check intervals
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answered Aug 17, 2015 at 18:14
haqnaturalhaqnatural
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5980 | https://www.toolazytostudy.com/economics-notes-free-revision/circular-flow-of-income-in-a-closed-economy-and-an-open-economy%3A-the-flow-of-income-between-households%2C-firms-and-government-and-the-international-economy-illustrating-the-circular-flow-of-income-in-closed-and-open-economies. | Circular flow of income in a closed economy and an open economy: the flow of income between households, firms and government and the international economy - Illustrating the circular flow of income in closed and open economies.
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Circular Flow of Income
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Circular flow of income in a closed economy and an open economy: the flow of income between households, firms and government and the international economy - Illustrating the circular flow of income in closed and open economies.
The Circular Flow of Income: A Money-Go-Round for the Economy
Imagine the economy as a giant, never-ending game of economic "pass the parcel." Money flows constantly, moving between different players in this game: households, businesses, and the government. This is the circular flow of income. It’s a simple way to visualize how money moves through the economy and how different sectors interact.
The Closed Economy: A Simple Game
Let's start with a closed economy, one that doesn't interact with other countries. Think of it as a smaller, self-contained game. Here's how it works:
⭐Households: You and your family are households. You work for firms (businesses), earning wages and salaries. You use this money to buy goods and services from firms.
⭐Firms: These are businesses that produce goods and services. They hire households to work and pay them wages. They also buy resources (like land, labor, and capital) from households. In return, they sell goods and services to households.
⭐Government: The government plays a role too. It collects taxes from households and firms. It uses these taxes to provide public services like education, healthcare, and infrastructure.
The Flow:
⭐Income Flow: Money flows from firms to households as wages, salaries, and profits.
⭐Expenditure Flow: Money flows from households to firms as they spend on goods and services.
⭐Government Flow: The government collects taxes from households and firms and uses this money to fund public services.
The Open Economy: A More Complex Game
Now, let's open up the game to include international trade and investment. This is an open economy. We need to add two more key players:
⭐Exports: Goods and services produced domestically and sold to foreigners. Think of this as our economy selling "stuff" to other countries.
⭐Imports: Goods and services purchased from other countries. This is like our economy "buying" stuff from other countries.
The Flow:
⭐Foreign Sector: This sector represents the international economy. It injects money into our economy through exports and withdraws money from our economy through imports.
Impact on the Flow:
⭐Exports: Increase the income flow to firms and households.
⭐Imports: Decrease the income flow to firms and households.
Real World Examples:
⭐A local bakery: This firm hires bakers (households). The bakers earn wages and spend some of it buying bread from the bakery itself. Some of the bakery's earnings are also used to buy ingredients from other firms (farmers who grow wheat).
⭐A car manufacturer: This firm hires workers (households), who earn wages and spend money on cars and other goods. The firm pays taxes to the government, which uses it to fund roads, schools, and hospitals. The firm also imports car parts from other countries, decreasing the income flow in the domestic economy.
Circular Flow of Income: An Economy in Balance
The circular flow of income is a simplified representation of a complex economic system. However, it helps us understand how the different players in the economy interact and how money flows between them. Ideally, the circular flow of income should be in balance. This means that the total income earned should equal the total expenditure on goods and services.
Key Takeaways:
The circular flow of income shows how money moves through the economy.
It involves households, firms, the government, and the international economy.
Exports increase the flow of income, while imports decrease it.
Understanding the circular flow helps us analyze the economic health of a nation.
Compare and contrast the circular flow of income in a closed and an open economy.
The Circular Flow of Income: Closed vs. Open Economies
The circular flow of income model provides a simplified representation of how economic activity flows within an economy. It highlights the continuous interaction between households and firms through the exchange of goods, services, and factors of production. This essay will compare and contrast the circular flow of income in a closed economy, which has no interaction with the outside world, and an open economy, which engages in international trade and financial transactions.
Closed Economy:
In a closed economy, the circular flow model consists of two main sectors: households and firms.
⭐Households: Households supply factors of production (labor, capital, land, and entrepreneurship) to firms in exchange for wages, rent, interest, and profits. They then use this income to purchase goods and services from firms.
⭐Firms: Firms utilize the factors of production provided by households to produce goods and services. These goods and services are sold to households, generating revenue for the firms.
This flow of income and spending creates a continuous cycle. Households' spending fuels firms' production, which in turn generates income for households, leading to further spending.
Open Economy:
The circular flow of income in an open economy is more complex due to the inclusion of international trade and financial transactions.
⭐International Trade: Open economies interact with the rest of the world through exports and imports. Exports represent goods and services produced domestically and sold to foreign consumers, generating income for domestic firms. Imports represent goods and services purchased from foreign countries, leading to leakage from the domestic circular flow.
⭐Financial Transactions: Open economies engage in financial flows, including foreign investment and borrowing. Foreign investment brings capital into the economy, increasing investment expenditure. Borrowing from foreign entities can lead to leakage of income if interest payments are made to foreign lenders.
Comparison and Contrast:
The key difference between the circular flow in closed and open economies lies in the presence of leakage and injection.
⭐Leakage: In an open economy, leakage occurs when income flows out of the circular flow. This includes imports, taxes, and savings that are not re-invested domestically.
⭐Injection: Open economies also experience injections of income, which are flows of money into the circular flow. These include exports, government spending, and investment by foreign entities.
Conclusion:
The circular flow of income model provides a valuable framework for understanding the interconnectedness of economic activity. While the closed economy model offers a simplified view, the open economy model provides a more realistic representation of the complexities of international trade and financial transactions. Understanding the role of leakage and injection in open economies is crucial for analyzing economic growth, balance of payments, and policy implications.
Explain how government spending and taxation affect the circular flow of income.
The Impact of Government Spending and Taxation on the Circular Flow of Income
The circular flow of income model illustrates the continuous flow of money and resources between households and businesses in an economy. Government intervention, through spending and taxation, significantly influences this flow.
Government Spending as an Injection: Government spending acts as an injection into the circular flow of income. It stimulates economic activity by increasing demand for goods and services. This spending can be on various projects like infrastructure development, education, healthcare, or social welfare programs. When the government spends, it directly injects money into the economy, increasing the income of individuals and businesses involved in these projects. This, in turn, leads to higher consumption, stimulating further economic activity.
Taxation as a Leakage: Taxation, on the other hand, acts as a leakage from the circular flow of income. When individuals and businesses pay taxes, money is withdrawn from the circular flow, reducing the amount available for spending and investment. This reduces the overall demand for goods and services, potentially slowing down economic growth.
Government Spending and the Multiplier Effect: Government spending can have a multiplier effect on the economy. When the government spends money, the initial recipients of this spending (e.g., construction workers, teachers) use this income to buy goods and services, further stimulating the economy. This process can continue, with each subsequent round of spending generating additional income and demand. The multiplier effect is influenced by factors like the marginal propensity to consume and the leakages in the economy.
Taxation and the Multiplier Effect: Taxation can also impact the circular flow of income through the multiplier effect. When taxes are raised, disposable income falls, leading to reduced consumption and investment. This reduction in spending can have a multiplier effect, leading to a greater reduction in aggregate demand than the initial tax increase. Conversely, tax cuts can stimulate economic activity through an increased multiplier effect.
Government Spending and Fiscal Policy: Government spending and taxation are key tools of fiscal policy. By adjusting these levers, governments can influence the overall level of economic activity. For example, during a recession, governments often increase spending and cut taxes to stimulate demand and boost economic growth. Conversely, during periods of high inflation, governments may reduce spending and raise taxes to curb demand and control inflation.
The Role of Government in the Circular Flow: Governments play a crucial role in shaping the circular flow of income. Through appropriate spending and taxation policies, governments can promote economic stability, growth, and social welfare. However, it's essential to carefully consider the impact of these policies on the circular flow and ensure they are implemented effectively to achieve desired outcomes.
In conclusion, government spending and taxation are key determinants of the circular flow of income. Government spending acts as an injection, stimulating economic activity, while taxation acts as a leakage, reducing the flow of income. The multiplier effect amplifies the impact of both government spending and taxation on the economy. Governments utilize these tools as part of fiscal policy to influence economic activity and achieve their macroeconomic objectives.
Discuss the role of international trade in the circular flow of income.
The Role of International Trade in the Circular Flow of Income
The Circular Flow of Income: The circular flow of income model is a simplified representation of how economic activity occurs within an economy. It illustrates the continuous flow of money and resources between households and firms. Households provide firms with factors of production (labor, land, capital, and entrepreneurship) in exchange for income. Firms then use these factors to produce goods and services, which they sell to households, generating revenue. This revenue is then used to pay for the factors of production, completing the cycle.
International Trade's Impact: International trade introduces an additional flow of money and goods to the circular flow. When a country exports goods and services, it receives income from foreign buyers. This income, in turn, adds to the domestic circular flow, increasing aggregate demand and contributing to economic growth. Similarly, when a country imports goods and services, it pays for these imports, leading to a leakage from the circular flow. This leakage can reduce domestic demand if not offset by other factors.
Specific Examples:
⭐Exports: A country exporting manufactured goods receives foreign currency, boosting its income and contributing to employment in sectors like manufacturing and transportation.
⭐Imports: A country importing raw materials for its manufacturing industry pays foreign suppliers, leading to a leakage from the circular flow. However, this leakage is offset by the increased output and income generated within the domestic industry.
⭐Foreign Investment: Foreign investment in a country brings in capital, leading to an increase in domestic investment and economic growth.
Benefits of International Trade:
⭐Increased Efficiency and Productivity: International trade allows countries to specialize in producing goods and services where they have a comparative advantage, leading to increased efficiency and productivity.
⭐Access to a Wider Variety of Goods and Services: International trade provides consumers with access to a wider variety of goods and services at potentially lower prices.
⭐Economic Growth: International trade can boost economic growth by increasing exports, attracting foreign investment, and fostering competition.
Challenges of International Trade:
⭐Job Displacement: Imports can lead to job losses in domestic industries that are unable to compete with foreign producers.
⭐Trade Deficits: A significant trade deficit can lead to a depletion of foreign reserves and a weakening of the currency.
⭐Environmental Concerns: International trade can exacerbate environmental problems, particularly if it involves the production and consumption of goods that rely on unsustainable practices.
Conclusion: International trade plays a crucial role in the circular flow of income. It introduces additional flows of money and goods, potentially boosting economic growth and consumer welfare. However, it also presents challenges like job losses and trade deficits. Governments need to manage these challenges through policies that promote fair trade, sustainable practices, and economic diversification.
How does the circular flow of income provide a simplified representation of the complex economic interactions within an economy?
The Circular Flow of Income: A Simplified Model of Economic Interactions
The circular flow of income model is a fundamental concept in macroeconomics, providing a simplified illustration of the intricate relationships between households, businesses, and the government within an economy. This essay will explore how the model effectively represents the complex economic interactions through its portrayal of:
The Flow of Goods and Services: The circular flow model demonstrates the movement of goods and services from businesses to households. Businesses produce goods and services that cater to the needs and wants of households, who in turn consume these products. This exchange represents the core of economic activity, where resources are allocated based on consumer demand.
The Flow of Income and Expenditure: The model highlights the flow of income from businesses to households. As households purchase goods and services, businesses generate revenue, which is then distributed as wages, salaries, profits, and rent to households. These income payments allow households to purchase more goods and services, perpetuating the cycle.
The Role of Government: The circular flow model incorporates the government's role in the economy. The government collects taxes from both households and businesses, using these funds to provide public goods and services, such as healthcare, education, and infrastructure. These expenditures inject money back into the circular flow, stimulating economic activity.
The Savings and Investment Leakages: The model acknowledges that not all income is spent immediately. Households may save a portion of their income, leading to a leakage from the circular flow. This savings can be used for investment by businesses, thus re-entering the flow, or can be held by banks as deposits. The model also includes investment by businesses, which can be financed by borrowed funds or retained earnings, adding to the flow of income.
The Foreign Sector: The model can be expanded to include international trade, representing the flow of goods and services between a country and the rest of the world. Exports add to the circular flow of income, while imports represent a leakage due to outflows of income.
Overall, the circular flow of income model, despite its simplified nature, provides a valuable framework for understanding the interconnectedness of economic agents. It emphasizes the cyclical nature of economic activity, with income generated through production being spent on consumption and investment, thus driving further production. While the model does not capture the full complexity of real-world economic interactions, it serves as a useful tool for explaining basic economic concepts and analyzing the effects of government policies and external factors on the overall economy.
To what extent does the circular flow of income model accurately capture the dynamics of a modern economy? Evaluate its strengths and limitations.
To What Extent Does the Circular Flow of Income Model Accurately Capture the Dynamics of a Modern Economy?
The circular flow of income model is a fundamental tool in economics, providing a simplified visual representation of how money flows through an economy. It depicts the interconnectedness between households, businesses, and the government, highlighting the exchange of goods, services, and money. However, the model’s accuracy in capturing the complexities of a modern economy is debatable, with both strengths and limitations to consider.
Strengths of the Circular Flow of Income Model
⭐Simple and Intuitive: The model's simplicity makes it accessible and easy to understand, even for those with limited economic knowledge. It provides a basic framework for grasping the key players and interactions within an economy.
⭐Illustrates Key Relationships: The model clearly shows the relationship between production, consumption, and income generation. It highlights the role of factors of production, such as labor and capital, in generating income for households, which then fuels spending on goods and services produced by businesses.
⭐Foundation for Macroeconomic Concepts: The model serves as a foundation for understanding core macroeconomic concepts like Gross Domestic Product (GDP), national income, and the multiplier effect.
Limitations of the Circular Flow of Income Model
⭐Oversimplification: The model simplifies a complex reality, neglecting vital aspects of modern economies. It ignores factors like financial markets, international trade, and the informal sector, which play significant roles in economic activity.
⭐Static Nature: The model presents a static view of the economy, not accounting for dynamic changes over time. It ignores the impact of technological advancements, inflation, and changes in consumer preferences that can significantly alter economic flows.
⭐Lack of Realistic Leakages and Injections: The model only includes basic leakages (saving, taxes) and injections (investment, government spending), failing to account for other crucial factors like imports, exports, and foreign investment.
⭐Limited Representation of Government: The model portrays the government as a passive recipient and spender of revenue, neglecting its role in regulating the economy through policies like monetary and fiscal measures.
Conclusion:
While the circular flow of income model provides a valuable starting point for understanding the basic dynamics of an economy, it lacks the complexity to truly capture the nuances of modern economic systems. Its strengths lie in its simplicity and ability to illustrate core relationships, but its limitations in addressing dynamic changes, ignoring important factors, and presenting a limited view of government activity restrict its accuracy. To gain a comprehensive understanding of a modern economy, a more sophisticated and dynamic model that incorporates the complexities of contemporary economic interactions is required.
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5981 | https://math.stackexchange.com/questions/1483130/minimum-number-of-fractions-to-be-summed-up-to-frac45 | algebra precalculus - Minimum number of fractions to be summed up to $\frac45$ - Mathematics Stack Exchange
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Minimum number of fractions to be summed up to 4 5 4 5
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What is the minimum number of fractions having numerator 1 and a natural number as denominator to be summed up to 4 5 4 5?
I have tested with 2 fractions: 1 a+1 b=4 5 1 a+1 b=4 5 and get into the diophantine equation: 5(a+b)=4 a b 5(a+b)=4 a b and it seems this should have some solutions but can't find one!!
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asked Oct 16, 2015 at 13:54
Hamid Reza EbrahimiHamid Reza Ebrahimi
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Considering prime factorization gives that 5∣a b 5∣a b and, by relabeling if necessary, we can assume that 5∣a 5∣a, and in particular 1 a≤1 5 1 a≤1 5. Substituting in the two-fraction equation gives b≤5 3 b≤5 3, so we must have b=1 b=1, but this gives 1 a+1=4 5 1 a+1=4 5, and the solution a a to this equation is negative, so there is no decomposition of 4 5 4 5 as a sum of two such fractions.
On the other hand, if we permit ourselves three such fractions, we can see immediately that one of the fractions must be 1 2 1 2 or 1 3 1 3. (If not, each of the three fractions would be ≤1 4≤1 4 and so would have sum ≤3⋅1 4<4 5≤3⋅1 4<4 5.) So, this reduces the problem to finding decompositions of either 3 10 3 10 or 7 15 7 15 into two such fractions. There turn out to be two solutions. (Incidentally, fractions of the form 1 n 1 n, where n n is a positive integer, are often called Egyptian fractions.)
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answered Oct 16, 2015 at 14:16
Travis WillseTravis Willse
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I have found three unit fractions summing up to 4 5 4 5: 1 2,1 5,1 10 1 2,1 5,1 10 Hamid Reza Ebrahimi –Hamid Reza Ebrahimi 2015-10-16 14:43:15 +00:00 Commented Oct 16, 2015 at 14:43
Very good, can you find the other triple?Travis Willse –Travis Willse 2015-10-16 18:00:28 +00:00 Commented Oct 16, 2015 at 18:00
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For positive a,b a,b, the equation 4 5=1 a+1 b 4 5=1 a+1 b has no solution, for if you allow a,b a,b are integer, 4 5=1 a+1 b 4 5=1 a+1 b has a unique solution that is 4 5=1 1+1−5 4 5=1 1+1−5.
Proof for a,b a,b positive case, that 4/5=1/a+1/b 4/5=1/a+1/b has no solution:
Let 01/a⟹4/5>1/a⟹a>5/4 4/5=1/a+1/b>1/a⟹4/5>1/a⟹a>5/4, because a a is positive integer, so that a≥2⟹1/a≤1/2 a≥2⟹1/a≤1/2, so that 1/b=4/5−1/a≥4/5−1/2=3/10⟹b≤10/3 1/b=4/5−1/a≥4/5−1/2=3/10⟹b≤10/3, recalling that 0<a≤b 0<a≤b and a,b a,b are postive integer. So only possible cases are (a=2,b=2)(a=2,b=2) or (a=2,b=3)(a=2,b=3) or (a=3,b=3)(a=3,b=3), and we can verify all these cases are not the solution. So 4/5 4/5 can't be expanded into two postive unit fractions sum.
Then we can further test if 4/5=1/a+1/b+1/c 4/5=1/a+1/b+1/c has a solution, similar as above method in my proof. We only can find two positive solutions: 4/5=1/2+1/4+1/20=1/2+1/5+1/10 4/5=1/2+1/4+1/20=1/2+1/5+1/10, so that the minimum number of unit fraction is 3, for presentation 4/5 as unit fractions sum.
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edited Mar 13, 2023 at 20:37
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answered Mar 13, 2023 at 13:43
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The equation 4/5=1/a+1/b 4/5=1/a+1/b doesn't have any solutions where a a and b b are positive integers. To prove this, note that either 1/a 1/a or 1/b 1/b must be between 2/5 2/5 and 4/5 4/5, i. e. one of a a or b b must be between 5/4 5/4 and 5/2 5/2. But the only integer in this range is 2, and if a=2 a=2 then the corresponding b b isn't an integer.
So to express 4/5 4/5 as a sum of unit fractions you need at least three fractions. Can you find a way to do it with exactly three?
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answered Oct 16, 2015 at 14:10
Michael LugoMichael Lugo
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Adding to the existing answers, here is another reason why 2 fractions is not enough. With some algebra, you can rearrange your Diophantine equation to:
a=5 b 4 b−5 a=5 b 4 b−5
Now, varying b b, starting at b=1 b=1 we get the following values for a a, rounded to two decimals: −5,3.33,2.14,1.82,…−5,3.33,2.14,1.82,…
After the last value, which is less than 2 2, clearly we can never get an integer, because the function is monotonically decreasing as b b increases, but it is always greater than 5 4 5 4, which is itself greater than 1 1.
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answered Oct 16, 2015 at 14:27
Colm BhandalColm Bhandal
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Let a+b=4 k a+b=4 k, a b=5 k a b=5 k, where k k is a positive integer.
Then, b=4 k−a b=4 k−a, and substitute into a b=5 k a b=5 k, we get:
a(4 k−a)=5 k a(4 k−a)=5 k
a 2−4 k a+5 k=0 a 2−4 k a+5 k=0
Then let Δ=16 k 2−20 k=(4 k)2−20 k=m 2 Δ=16 k 2−20 k=(4 k)2−20 k=m 2, where m m is another positive integer
And we have (8 k−5)2−25 4=m 2(8 k−5)2−25 4=m 2
Clearly (8 k−5)2−25(8 k−5)2−25 should be a perfect square number.
When k=1 k=1, 9−25<0 9−25<0, impossible
When k=2 k=2, 121−25=96 121−25=96 is not perfect square
When k≥3 k≥3, (8 k−5)2−25>(8 k−6)2(8 k−5)2−25>(8 k−6)2.
So there's indeed no positive integer solution to 5(a+b)=4 a b 5(a+b)=4 a b.
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answered Oct 16, 2015 at 14:36
wwwrqnojcmwwwrqnojcm
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There are already some answers, indicating that it can't be done with two, so let's try with three.
The first fractions which comes in mind is 1 2 1 2, which leaves us with:
4 5−1 2 4 5−1 2
=8 10−5 10=8 10−5 10
=3 10=3 10
It's easy to see that 3 10=2 10+1 10=1 5+1 10 3 10=2 10+1 10=1 5+1 10, hence:
4 5=1 2+1 5+1 10 4 5=1 2+1 5+1 10
..., resulting in this answer for the question:
3
:-)
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answered Mar 13, 2023 at 14:22
DominiqueDominique
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5982 | https://www.youtube.com/watch?v=9zjjEBJjmVM | Greatest Common Divisor Proofs
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A proof that the greatest common divisor (gcd) of a set of integers is the smallest positive linear combination of the integers (using integer coefficients) and that any common divisor divides the gcd.
Transcript:
suppose d is a common divisor of the integers a1 a2 up to a n we will call the greatest common divisor of those integers g we define g to be a positive integer then d divides g so any common divisor of a set of integers will divide the greatest common divisor of those integers before proving this theorem we look at an example so we take the integers 24 120 and 720. those integers have common divisors 2 3 and 4. so the theorem says that each of those common divisors will divide the greatest common divisor g of the integers so 2 divides g 3 must divide e and 4 must divide g so that means that g must have factors 2 3 and 4. we can write g as 2 times 3 times 4 times some unknown integer x now g divides the three integers of course so in particular it must divide minus 24 so 24x must divide minus 24. so that means that g must equal 24. g cannot be zero so and x is a integer it's a positive integer so x can only have values 1 or 2 or 3 etc so x must equal 1 in this case if we want g to divide minus 24. the main step in proving this theorem is to prove this statement the gcd of a set of numbers is their smallest positive linear combination so that is if we take the gcd of the set of numbers a1 to a n and call it g then g must be the minimum positive linear combination of all the a's and uh the coefficients in that linear combination the coefficients k i must must be integers so let's look at as an example let's look at the gcd of the integers minus 10 20 and 25 and that's equal to 5 and we can write 5 as a linear combination of minus 10 20 and 25 and you can see the coefficients k1 k2 and k3 are integers so the statement says that if we take the minimum if if we take all positive linear combinations of the integers minus 10 20 and 25 in other words we run we take a b and c as integers but we make sure that this combination is positive we look at all those positive linear combinations we will find the minimum of them will be five proof let l be the minimum positive linear combination of all the ai's with ki and element of z now since the greatest common divisor of all the a's divides obviously divides each of the ais that means that g will divide any linear combination of all those ais if g divides l obviously g has to be less than or equal to l next we apply euclid's division lemma to the integers a i and l so that tells us that there exists integers qi and ri such that we can write a i as qi times l plus ri where ri is non-negative and ri is less than the magnitude of l well l is a positive integer by definition so we don't need to show the magnitude of l if we make r i the subject and substitute in for l we see that ri is actually a linear combination of all the a's as we saw from the division lemma we know that ri is non-negative it's greater than or equal to zero actually we will show that ri is equal to zero to do that we suppose that ri is strictly greater than zero now we just saw that ri is a linear combination of the a's but if ri is greater than zero it means it's a positive linear combination of the a's from the division lemma we also saw that ri is less than l and l is supposed to be the least positive linear combination of a's so here's our contradiction if l is the least positive linear combination of the a's then ri cannot be less than l we conclude that ri must equal zero so if we set r equal to zero we see that q i times l must equal a i which means that l must divide ai if l is a common divisor of all the a's it means l must be less than or equal to the greatest common divisor we showed earlier that l is greater than or equal to g now we've shown that l is less than or equal to g that means that l must equal g now we can get back to the theorem so we want to show that any common divisor d of a set of integers will divide the greatest common divisor g so we show that g is a linear combination of all the a's if d is any common divisor of all the a's clearly d will divide a linear combination of all the a's so d will divide g |
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Nernst equation
Hello, I am getting confused about this question.
A particular cell has a Nernst potential for sodium of 60 mV and a Nernst potential for chloride of -60 mV. If the membrane potential for the cell is 30 mV, what is the expected direction of diffusion of each ion?
A) sodium will diffuse out of the cell; chloride will diffuse into the cell B) sodium will diffuse out of the cell; chloride will diffuse out of the cell C) sodium will diffuse into the cell; chloride will diffuse into the cell D) sodium will diffuse into the cell; chloride will diffuse out of the cell
The correct answer is C
I know that the Nernst equation is E=61.5/z ([Ion outside]/[Ion inside])
If sodium wants to increase the potential to its nernst potential then wouldn't it want to increase the ratio of Ion outside/ion inside, therefore, wouldn't sodium ions flow out of the cell? Conversely, chloride wants to decrease the potential (i.e. make it more negative), so wouldn't it want to increase the ratio of Ion outside/ion inside?
Kaplan explanation: since the membrane potential is less than nernst potential for sodium, sodium will omve in a direction that increases the membrane potential. Since sodium ions are positive, there will be an influx of sodium into the cell. With the same reasoning, chloride will want to move into the cell to decrease membrane potential to the nernst for chloride. Thus C is correct.
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5984 | https://en.wikipedia.org/wiki/Logarithmic_scale | Jump to content
Logarithmic scale
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From Wikipedia, the free encyclopedia
Measurement scale based on orders of magnitude
A logarithmic scale (or log scale) is a method used to display numerical data that spans a broad range of values, especially when there are significant differences among the magnitudes of the numbers involved.
Unlike a linear scale where each unit of distance corresponds to the same increment, on a logarithmic scale each unit of length is a multiple of some base value raised to a power, and corresponds to the multiplication of the previous value in the scale by the base value. In common use, logarithmic scales are in base 10 (unless otherwise specified).
A logarithmic scale is nonlinear, and as such numbers with equal distance between them such as 1, 2, 3, 4, 5 are not equally spaced. Equally spaced values on a logarithmic scale have exponents that increment uniformly. Examples of equally spaced values are 10, 100, 1000, 10000, and 100000 (i.e., 101, 102, 103, 104, 105) and 2, 4, 8, 16, and 32 (i.e., 21, 22, 23, 24, 25).
Exponential growth curves are often depicted on a logarithmic scale graph.
A logarithmic scale from 0.1 to 100
The two logarithmic scales of a slide rule
Common uses
[edit]
The markings on slide rules are arranged in a log scale for multiplying or dividing numbers by adding or subtracting lengths on the scales.
The following are examples of commonly used logarithmic scales, where a larger quantity results in a higher value:
Richter magnitude scale and moment magnitude scale (MMS) for strength of earthquakes and movement in the Earth
Sound level, with the unit decibel
Neper for amplitude, field and power quantities
Frequency level, with units cent, minor second, major second, and octave for the relative pitch of notes in music
Logit for odds in statistics
Palermo technical impact hazard scale
Logarithmic timeline
Counting f-stops for ratios of photographic exposure
The rule of nines used for rating low probabilities
Entropy in thermodynamics
Information in information theory
Particle size distribution curves of soil
The following are examples of commonly used logarithmic scales, where a larger quantity results in a lower (or negative) value:
pH for acidity
Stellar magnitude scale for brightness of stars
Krumbein scale for particle size in geology
Absorbance of light by transparent samples
Some of our senses operate in a logarithmic fashion (Weber–Fechner law), which makes logarithmic scales for these input quantities especially appropriate. In particular, our sense of hearing perceives equal ratios of frequencies as equal differences in pitch. In addition, studies of young children in an isolated tribe have shown logarithmic scales to be the most natural display of numbers in some cultures.
Graphic representation
[edit]
The top left graph is linear in the X- and Y-axes, and the Y-axis ranges from 0 to 10. A base-10 log scale is used for the Y-axis of the bottom left graph, and the Y-axis ranges from 0.1 to 1000.
The top right graph uses a log-10 scale for just the X-axis, and the bottom right graph uses a log-10 scale for both the X axis and the Y-axis.
Presentation of data on a logarithmic scale can be helpful when the data:
covers a large range of values, since the use of the logarithms of the values rather than the actual values reduces a wide range to a more manageable size;
may contain exponential laws or power laws, since these will show up as straight lines.
A slide rule has logarithmic scales, and nomograms often employ logarithmic scales. The geometric mean of two numbers is midway between the numbers. Before the advent of computer graphics, logarithmic graph paper was a commonly used scientific tool.
Log–log plots
[edit]
Main article: Log–log plot
If both the vertical and horizontal axes of a plot are scaled logarithmically, the plot is referred to as a log–log plot.
Semi-logarithmic plots
[edit]
Main article: Semi-log plot
If only the ordinate or abscissa is scaled logarithmically, the plot is referred to as a semi-logarithmic plot.
Extensions
[edit]
A modified log transform can be defined for negative input (y < 0) to avoid the singularity for zero input (y = 0), and so produce symmetric log plots:
for a constant C=1/ln(10).
Logarithmic units
[edit]
A logarithmic unit is a unit that can be used to express a quantity (physical or mathematical) on a logarithmic scale, that is, as being proportional to the value of a logarithm function applied to the ratio of the quantity and a reference quantity of the same type. The choice of unit generally indicates the type of quantity and the base of the logarithm.
Examples
[edit]
Examples of logarithmic units include units of information and information entropy (nat, shannon, ban) and of signal level (decibel, bel, neper). Frequency levels or logarithmic frequency quantities have various units are used in electronics (decade, octave) and for music pitch intervals (octave, semitone, cent, etc.). Other logarithmic scale units include the Richter magnitude scale point.
In addition, several industrial measures are logarithmic, such as standard values for resistors, the American wire gauge, the Birmingham gauge used for wire and needles, and so on.
Units of information
[edit]
bit, byte
hartley
nat
shannon
Units of level or level difference
[edit]
Further information: Level (logarithmic quantity)
bel, decibel
neper
Units of frequency level
[edit]
decade, decidecade, savart
octave, tone, semitone, cent
Table of examples
[edit]
| Unit | Base of logarithm | Underlying quantity | Interpretation |
--- --- |
| bit | 2 | number of possible messages | quantity of information |
| byte | 28 = 256 | number of possible messages | quantity of information |
| decibel | 10(1/10) ≈ 1.259 | any power quantity (sound power, for example) | sound power level (for example) |
| decibel | 10(1/20) ≈ 1.122 | any root-power quantity (sound pressure, for example) | sound pressure level (for example) |
| semitone | 2(1/12) ≈ 1.059 | frequency of sound | pitch interval |
The two definitions of a decibel are equivalent, because a ratio of power quantities is equal to the square of the corresponding ratio of root-power quantities.[citation needed]
See also
[edit]
Mathematics portal
Alexander Graham Bell
Bode plot
Geometric mean (arithmetic mean in logscale)
John Napier
Level (logarithmic quantity)
Log–log plot
Logarithm
Logarithmic mean
Log semiring
Preferred number
Semi-log plot
Scale
[edit]
Order of magnitude
Applications
[edit]
Entropy
Entropy (information theory)
pH
Richter magnitude scale
References
[edit]
^ "Slide Rule Sense: Amazonian Indigenous Culture Demonstrates Universal Mapping Of Number Onto Space". ScienceDaily. 2008-05-30. Retrieved 2008-05-31.
^ Webber, J Beau W (2012-12-21). "A bi-symmetric log transformation for wide-range data" (PDF). Measurement Science and Technology. 24 (2). IOP Publishing: 027001. doi:10.1088/0957-0233/24/2/027001. ISSN 0957-0233. S2CID 12007380.{{cite journal}}: CS1 maint: article number as page number (link)
^ "Symlog Demo". Matplotlib 3.4.2 documentation. 2021-05-08. Retrieved 2021-06-22.
^ Ainslie, M. A. (2015). A Century of Sonar: Planetary Oceanography, Underwater Noise Monitoring, and the Terminology of Underwater Sound.
Further reading
[edit]
Dehaene, Stanislas; Izard, Véronique; Spelke, Elizabeth; Pica, Pierre (2008). "Log or linear? Distinct intuitions of the number scale in Western and Amazonian indigene cultures". Science. 320 (5880): 1217–20. Bibcode:2008Sci...320.1217D. doi:10.1126/science.1156540. PMC 2610411. PMID 18511690.
Tuffentsammer, Karl; Schumacher, P. (1953). "Normzahlen – die einstellige Logarithmentafel des Ingenieurs" [Preferred numbers - the engineer's single-digit logarithm table]. Werkstattechnik und Maschinenbau (in German). 43 (4): 156.
Tuffentsammer, Karl (1956). "Das Dezilog, eine Brücke zwischen Logarithmen, Dezibel, Neper und Normzahlen" [The decilog, a bridge between logarithms, decibel, neper and preferred numbers]. VDI-Zeitschrift (in German). 98: 267–274.
Ries, Clemens (1962). Normung nach Normzahlen [Standardization by preferred numbers] (in German) (1 ed.). Berlin, Germany: Duncker & Humblot Verlag. ISBN 978-3-42801242-8. {{cite book}}: ISBN / Date incompatibility (help) (135 pages)
Paulin, Eugen (2007-09-01). Logarithmen, Normzahlen, Dezibel, Neper, Phon - natürlich verwandt! [Logarithms, preferred numbers, decibel, neper, phon - naturally related!] (PDF) (in German). Archived (PDF) from the original on 2016-12-18. Retrieved 2016-12-18.
External links
[edit]
"GNU Emacs Calc Manual: Logarithmic Units". Gnu.org. Retrieved 2016-11-23.
Non-Newtonian calculus website
Retrieved from "
Categories:
Logarithmic scales of measurement
Non-Newtonian calculus
Hidden categories:
CS1 maint: article number as page number
Articles with short description
Short description matches Wikidata
All articles with unsourced statements
Articles with unsourced statements from December 2019
CS1 German-language sources (de)
CS1 errors: ISBN date
Commons category link from Wikidata |
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Questions 1 - 10
1
A regular tetrahedron has a surface area of . Each face of the tetrahedron has a height of . What is the length of the base of one of the faces?
Explanation
A regular tetrahedron has 4 triangular faces. The area of one of these faces is given by:
Because the surface area is the area of all 4 faces combined, in order to find the area for one of the faces only, we must divide the surface area by 4. We know that the surface area is , therefore:
Since we now have the area of one face, and we know the height of one face is , we can now plug these values into the original formula:
Therefore, the length of the base of one face is .
2
A regular tetrahedron has a surface area of . Each face of the tetrahedron has a height of . What is the length of the base of one of the faces?
Explanation
A regular tetrahedron has 4 triangular faces. The area of one of these faces is given by:
Because the surface area is the area of all 4 faces combined, in order to find the area for one of the faces only, we must divide the surface area by 4. We know that the surface area is , therefore:
Since we now have the area of one face, and we know the height of one face is , we can now plug these values into the original formula:
Therefore, the length of the base of one face is .
3
Calculate the diagonal of a regular tetrahedron (all of the faces are equilateral triangles) with side length .
Explanation
The diagonal of a shape is simply the length from a vertex to the center of the face or vertex opposite to it. With a regular tetrahedron, we have a face opposite to the vertex, and this basically amounts to calculating the height of our shape.
We know that the height of a tetrahedron is where s is the side length, so we can put into this formula:
which gives us the correct answer.
4
What is the surface area of a regular tetrahedron when its volume is 27?
Explanation
The problem is essentially asking us to go from a three-dimensional measurement to a two-dimensional one. In order to approach the problem, it's helpful to see how volume and surface area are related.
This can be done by comparing the formulas for surface area and volume:
We can see that both calculation revolve around the edge length.
That means, if we can solve for (edge length) using volume, we can solve for the surface area.
Now that we know , we can substitute this value in for the surface area formula:
5
Calculate the diagonal of a regular tetrahedron (all of the faces are equilateral triangles) with side length .
Explanation
The diagonal of a shape is simply the length from a vertex to the center of the face or vertex opposite to it. With a regular tetrahedron, we have a face opposite to the vertex, and this basically amounts to calculating the height of our shape.
We know that the height of a tetrahedron is where s is the side length, so we can put into this formula:
which gives us the correct answer.
6
What is the surface area of a regular tetrahedron when its volume is 27?
Explanation
The problem is essentially asking us to go from a three-dimensional measurement to a two-dimensional one. In order to approach the problem, it's helpful to see how volume and surface area are related.
This can be done by comparing the formulas for surface area and volume:
We can see that both calculation revolve around the edge length.
That means, if we can solve for (edge length) using volume, we can solve for the surface area.
Now that we know , we can substitute this value in for the surface area formula:
7
What is the surface area of a regular tetrahedron with a slant height of ?
Cannot be determined
Explanation
If this is a regular tetrahedron, then all four triangles are equilateral triangles.
If the slant height is , then that equates to the height of any of the triangles being .
In order to solve for the surface area, we can use the formula
where in this case is the measure of the edge.
The problem has not given the edge; however, it has provided information that will allow us to solve for the edge and therefore the surface area.
Picture an equilateral triangle with a height .
Drawing in the height will divide the equilateral triangle into two 30/60/90 right triangles. Because this is an equilateral triangle, we can deduce that finding the measure of the hypotenuse will suffice to solve for the edge length ().
In order to solve for the hypotenuse of one of the right triangles, either trig functions or the rules of the special 30/60/90 triangle can be used.
Using trig functions, one option is using .
Rearranging the equation to solve for ,
Now that has been solved for, it can be substituted into the surface area equation.
8
What is the surface area of a regular tetrahedron with a slant height of ?
Cannot be determined
Explanation
If this is a regular tetrahedron, then all four triangles are equilateral triangles.
If the slant height is , then that equates to the height of any of the triangles being .
In order to solve for the surface area, we can use the formula
where in this case is the measure of the edge.
The problem has not given the edge; however, it has provided information that will allow us to solve for the edge and therefore the surface area.
Picture an equilateral triangle with a height .
Drawing in the height will divide the equilateral triangle into two 30/60/90 right triangles. Because this is an equilateral triangle, we can deduce that finding the measure of the hypotenuse will suffice to solve for the edge length ().
In order to solve for the hypotenuse of one of the right triangles, either trig functions or the rules of the special 30/60/90 triangle can be used.
Using trig functions, one option is using .
Rearranging the equation to solve for ,
Now that has been solved for, it can be substituted into the surface area equation.
9
What is the length of an edge of a regular tetrahedron if its surface area is 156?
Explanation
The only given information is the surface area of the regular tetrahedron.
This is a quick problem that can be easily solved for by using the formula for the surface area of a tetrahedron:
If we substitute in the given infomation, we are left with the edge being the only unknown.
10
What is the length of an edge of a regular tetrahedron if its surface area is 156?
Explanation
The only given information is the surface area of the regular tetrahedron.
This is a quick problem that can be easily solved for by using the formula for the surface area of a tetrahedron:
If we substitute in the given infomation, we are left with the edge being the only unknown.
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Proving that $\log_2 7$ is irrational
Prove that $\log_2 7$ is irrational.
Book solution: Suppose $\log_2 7$ is rational. Then $\log_2 7=a/b$, where $a$ and $b$ are integers. We may assume that $a>0$ and $b>0$. We have $2^{a/b}=7$, which implies $2^a=7^b$. But the number $2^a$ is even and the number $7^b$ is odd, a contradiction. Hence, $\log_2 7$ is irrational.
I understand this proof, but my question is whether or not the assumption that $a,b>0$ is really necessary. Is this a necessary condition for the proof to hold? It doesn't seem like it. For $\log_a b=c$, we know $c$ must be positive. Hence, $a/b$ must be positive, meaning $a,b>0$ or $a,b<0$. If $a$ and $b$ were both negative though, the "even and odd" argument at the end would no longer be valid but only semantically. That is, we would have $1/2^{|a|}$ and $1/7^{|b|}$, where you could still argue via parity that the denominators would be different but the numerators the same, hence not equivalent.
Did I miss something or was this assumption not all that necessary?
2 Answers 2
The assumption isn't necessary; they trade that step for the step saying $2^{|a|}=7^{|b|}$ by cross multiplying.
Also note that $\log_ab\geq0$ isn't always the case (your post makes it seems like $\log_ab=c$ implies $c>0$); however $\ln(1/e)=-1$, for example.
First of all, we know that log2(7) is a positive number. So it can be represented as a/b for positive a,b if it is rational. Assuming this just makes it easier so that you have integers on both sides of the equality.
As to the proof: If 2^a = k^b, k has to be a power of 2, or else by the fundamental theorem of arithmetic k^b has primes in its factorization other than 2 and thus is not equal to 2^a. This could help you if you have other numbers instead of 2 and 7.
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5987 | https://undergroundmathematics.org/calculus-of-powers/binomials-are-the-answer/solution-n | underground mathematics
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### Calculus of Powers
Investigation
Binomials are the answer! Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource
Printable/supporting materials Printable version Fullscreen mode Teacher notes
Introduction
Quadratic and cubic
Generalising to $x^n$
Solution for $x^n$
Solution for $x^n$
How can we generalise to find the gradient for any tangent line to a function (y=x^n), where (n) is a positive integer?
We could adapt what we did for the quadratic and cubic curves in order to generalise: how can we find the gradient of the line through the points with coordinates ((x,x^n)) and ((x+h,(x+h)^n))? Where (h) is a very very small positive number.
We can then calculate the gradient of the line going through the points ((x,x^n)) and ((x+h,(x+h)^n)),
[ \frac{y_2-y_1}{x_2-x_1}=\frac{(x+h)^n-x^n}{x+h-x}=\frac{(x+h)^n-x^n}{h}.]
We would like to expand and simplify this expression, but ((x+h)^n) is not so simple to expand when we don’t have a specific value for (n).
However, (x+h) is a binomial. We have a way to expand ((x+h)^n), The Binomial Theorem, for (n) a positive integer:
[(x+h)^n={\binom n 0}x^nh^0+{\binom n 1}x^{n-1}h+{\binom n 2}x^{n-1}h^2+ \dots + {\binom n {n-1}}xh^{n-1}+{\binom n n}x^{0}h^n.]
The binomial coefficients ({\dbinom n k}) are given by (\dfrac{n!}{k!(n-k)!}).
We can now think about how to expand,
[\frac{(x+h)^n-x^n}{h}.]
The numerator is,
[{\binom n 0}x^nh^0+{\binom n 1}x^{n-1}h+{\binom n 2}x^{n-1}h^2+ \dots + {\binom n {n-1}}xh^{n-1}+{\binom n n}x^{0}h^n-x^n]
which simplifies to,
[{\binom n 1}x^{n-1}h+{\binom n 2}x^{n-1}h^2+ \dots + {\binom n {n-1}}xh^{n-1}+{\binom n n}x^{0}h^n.]
The denominator simplifies to (h).
So, the gradient of the line through the points ((x,x^n)) and ((x+h,(x+h)^n)) is
[{\binom n 1}x^{n-1}+{\binom n 2}x^{n-1}h+ \dots + {\binom n {n-1}}xh^{n-2}+{\binom n n}h^{n-1}.]
As we move the point ((x+h,(x+h)^n)) closer and closer to ((x,x^n)), we are making (h) smaller and smaller, finally making (h) zero. This gives the gradient of the line that touches the curve (y=x^n) at the point ((x,x^n)) as ({\binom n 1}x^{n-1}), for (n) a positive integer.
The binomial coefficient (\dbinom{n}{1}) is given by (\dfrac{n!}{1!(n-1)!}=\dfrac{n\times (n-1) \times \dots \times 1}{(n-1)\times (n-2) \times \dots \times 1}=n).
This gives the gradient of the line that touches the curve (y=x^n) at the point ((x,x^n)) as (nx^{n-1}), for (n) a positive integer.
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5988 | https://www.geeksforgeeks.org/dsa/nth-term-of-ap-from-first-two-terms/ | Nth term of AP from First Two Terms - GeeksforGeeks
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Nth term of AP from First Two Terms
Last Updated : 23 Jul, 2025
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Given two integers a1 and a2, the first and second terms of an Arithmetic Series respectively, the problem is to find the n th term of the series.
Examples :
Input : a1 = 2, a2 = 3, n = 4
Output : 5
Explanation :The series is 2, 3, 4, 5, 6, .... , thus the 4th term is 5.
Input : a1 = 1, a2 = 3, n = 10
Output : 19
Explanation:The series is: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21..... Thus,10th term is 19.
Try it on GfG Practice
Table of Content
[Naive Approach] - Using for Loop
[Expected Approach] - Using the Formula for nth Term
[Naive Approach] - Using for Loop
In an Arithmetic Series, the difference between all pair of consecutive terms is same, for example, 2, 5, 8, 11, 14,,,,, The common difference is 3.
Find the common difference of the series, common difference d = a2 - a1
Run a loop to iterate over each term in the series from a1, keep adding common difference d until the n-th term is reached.
C++
```cpp
include
using namespace std;
int nthTermOfAP(int a1, int a2, int n)
{
int nthTerm = a1, d = a2 - a1;
for (int i = 1; i < n; i++){
nthTerm += d;
}
return nthTerm;
}
int main()
{
int a1 = 2, a2 = 3;
int n = 4;
cout << nthTermOfAP(a1, a2, n);
return 0;
}
```
include
using namespace std;int nthTermOfAP(int a1, int a2, int n){ int nthTerm = a1, d = a2 - a1; for (int i = 1; i < n; i++){ nthTerm += d; } return nthTerm;}int main(){ int a1 = 2, a2 = 3; int n = 4; cout << nthTermOfAP(a1, a2, n); return 0;}
C
```c
include
int nthTermOfAP(int a1, int a2, int n)
{
int nthTerm = a1, d = a2 - a1;
for (int i = 1; i < n; i++)
{
nthTerm += d;
}
return nthTerm;
}
int main()
{
int a1 = 2, a2 = 3;
int n = 4;
printf("%d", nthTermOfAP(a1, a2, n));
return 0;
}
Javajava
public class Main {
public static int nthTermOfAP(int a1, int a2, int n)
{
int nthTerm = a1, d = a2 - a1;
for (int i = 1; i < n; i++) {
nthTerm += d;
}
return nthTerm;
}
public static void main(String[] args)
{
int a1 = 2, a2 = 3;
int n = 4;
System.out.println(nthTermOfAP(a1, a2, n));
}
}
Pythonpython3
def nthTermOfAP(a1, a2, n):
nthTerm = a1
d = a2 - a1
for i in range(1, n):
nthTerm += d
return nthTerm
a1 = 2
a2 = 3
n = 4
print(nthTermOfAP(a1, a2, n))
C#csharp
using System;
class Program {
static int nthTermOfAP(int a1, int a2, int n)
{
int nthTerm = a1;
int d = a2 - a1;
for (int i = 1; i < n; i++) {
nthTerm += d;
}
return nthTerm;
}
static void Main()
{
int a1 = 2, a2 = 3;
int n = 4;
Console.WriteLine(nthTermOfAP(a1, a2, n));
}
}
JavaScriptjavascript
function nthTermOfAP(a1, a2, n)
{
let nthTerm = a1;
let d = a2 - a1;
for (let i = 1; i < n; i++) {
nthTerm += d;
}
return nthTerm;
}
let a1 = 2, a2 = 3;
let n = 4;
console.log(nthTermOfAP(a1, a2, n));
```
Output5
Time Complexity- O(n)
Auxiliary Space -O(1)
[Expected Approach] - Using the Formula for nth Term
To find the n th term in the Arithmetic Progression series we use the simple formula .
We know the Arithmetic Progression series is like = 2, 3, 4, 5, 6. …. …
In this series 2 is the first term and 3 is the second term of the series .
Common difference = a2 - a1 = 3 – 2 = 1 (Difference common in the series).
so we can write the series as :
t 1 = a 1
t 2 = a 1 + (2-1) d
t 3 = a 1 + (3-1) d
.
.
.
t N = a 1 + (n-1) d
t N = a 1 + (n-1) (a2-a1)
C++
```cpp
include
using namespace std;
int nthTermOfAP(int a1, int a2, int n)
{
// using formula to find the
// Nth term t(n) = a(1) + (n-1)d
return (a1 + (n - 1) (a2 - a1));
}
int main()
{
int a1 = 2, a2 = 3;
int n = 4;
cout << nthTermOfAP(a1, a2, n);
return 0;
}
```
include
using namespace std;int nthTermOfAP(int a1, int a2, int n){ // using formula to find the // Nth term t(n) = a(1) + (n-1)d return (a1 + (n - 1) (a2 - a1));}int main(){ int a1 = 2, a2 = 3; int n = 4; cout << nthTermOfAP(a1, a2, n); return 0;}
C
```c
include
int nthTermOfAP(int a1, int a2, int n)
{
// using formula to find the
// Nth term t(n) = a(1) + (n-1)d
return (a1 + (n - 1) (a2 - a1));
}
int main()
{
int a1 = 2, a2 = 3;
int n = 4;
printf("%d", nthTermOfAP(a1, a2, n));
return 0;
}
Javajava
public class Main {
public static int nthTermOfAP(int a1, int a2, int n)
{
// using formula to find the
// Nth term t(n) = a(1) + (n-1)d
return (a1 + (n - 1) (a2 - a1));
}
public static void main(String[] args)
{
int a1 = 2, a2 = 3;
int n = 4;
System.out.println(nthTermOfAP(a1, a2, n));
}
}
Pythonpython3
def nthTermOfAP(a1, a2, n):
# using formula to find the
# Nth term t(n) = a(1) + (n-1)d
return a1 + (n - 1) (a2 - a1)
a1 = 2
a2 = 3
n = 4
print(nthTermOfAP(a1, a2, n))
C#csharp
using System;
class Program {
static int nthTermOfAP(int a1, int a2, int n)
{
// using formula to find the
// Nth term t(n) = a(1) + (n-1)d
return (a1 + (n - 1) (a2 - a1));
}
static void Main()
{
int a1 = 2, a2 = 3;
int n = 4;
Console.WriteLine(nthTermOfAP(a1, a2, n));
}
}
JavaScriptjavascript
function nthTermOfAP(a1, a2, n)
{
// using formula to find the
// Nth term t(n) = a(1) + (n-1)d
return a1 + (n - 1) (a2 - a1);
}
let a1 = 2, a2 = 3;
let n = 4;
console.log(nthTermOfAP(a1, a2, n));
```
Output5
Time Complexity- O(1)
Auxiliary Space -O(1)
Comment
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5989 | https://dokumen.pub/biographical-memoirs-9780309041980-0309041988.html | Biographical memoirs. 9780309041980, 0309041988 - DOKUMEN.PUB
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###### Biographical Memoirs : Volume 53 [1 ed.] 9780309599047, 9780309032872 Biographic Memoirs: Volume 53 contains the biographies of deceased members of the National Academy of Sciences and bibli 301 36 4MB Read more
###### Biographical Memoirs : Volume 57 [1 ed.] 9780309597289, 9780309037297 This distinguished series contains the biographies of deceased members of the National Academy of Sciences and bibliogra 319 33 5MB Read more
###### Biographical Memoirs : Volume 66 [1 ed.] 9780309564373, 9780309052375 Biographic Memoirs: Volume 66 contains the biographies of deceased members of the National Academy of Sciences and bibli 249 85 8MB Read more
###### Biographical Memoirs : Volume 61 [1 ed.] 9780309563451, 9780309047463 Biographic Memoirs: Volume 61 contains the biographies of deceased members of the National Academy of Sciences and bibli 571 107 6MB Read more
###### Biographical Memoirs : Volume 76 [1 ed.] 9780309569132, 9780309064347 Biographic Memoirs: Volume 76 contains the biographies of deceased members of the National Academy of Sciences and bibli 431 101 10MB Read more
###### Biographical Memoirs : Volume 69 [1 ed.] 9780309588577, 9780309053464 Biographic Memoirs: Volume 69 contains the biographies of deceased members of the National Academy of Sciences and bibli 480 65 5MB Read more
###### Biographical Memoirs : Volume 67 [1 ed.] 9780309598590, 9780309052382 Biographic Memoirs: Volume 67 contains the biographies of deceased members of the National Academy of Sciences and bibli 240 129 4MB Read more
###### Biographical Memoirs : Volume 47 [1 ed.] 9780309598927, 9780309022453 Biographic Memoirs: Volume 47 contains the biographies of deceased members of the National Academy of Sciences and bibli 280 44 4MB Read more
###### Biographical Memoirs: Volume 87 [1 ed.] 9780309548410, 9780309095792 Biographical Memoirs is series of essays containing the life histories and selected bibliographies of deceased members o 213 7 27MB Read more
###### Biographical Memoirs: Volume 89 [1 ed.] 9780309113731, 9780309113724 On March 3, 1863, Abraham Lincoln signed the Act of Incorporation that brought the National Academy of Sciences into bei 249 11 71MB Read more
_Table of contents :
Biographical Memoirs
Copyright
Contents
Preface
Biographical Memoirs
Frederic C. Bartter September 10, 1914-May 5, 1983
EARLY LIFE
ACCOMPLISHMENTS IN BIOMEDICAL RESEARCH
PERSONAL QUALITIES
HONORS
CHRONOLOGY
Postgraduate Training And Fellowships
Professional Appointments
Memberships
Selected Bibliography
George Wells Beadle October 22, 1903-June 9, 1989
EDUCATION AND EARLY LIFE
DROSOPHILA STUDIES: CROSSING OVER, VERMILION AND CINNABAR
BEADLE AND TATUM
NEUROSPORA CRASSA AND GENE ACTION
BEADLE AS LABORATORY HEAD
CHAIRMAN OF THE DIVISION OF BIOLOGY, CALIFORNIA INSTITUTE OF TECHNOLOGY
HONORS AND DISTINCTIONS
Honorary Degrees
Doctor of Science
Legum Doctor (LL. D.)
Litterarum Humaniorum Doctor (L.H.D.)
Doctor of Public Service
Awards
Professional And Honorary Societies
Selected Bibliography
Solomon A. Berson April 22, 1918-April 11, 1972
Selected Bibliography
Raymond Thayer Birge March 13, 1887-March 22, 1980
EARLY LIFE
BERKELEY
CHAIRMAN OF THE DEPARTMENT OF PHYSICS
HONORS AND DISTINCTIONS
RAYMOND BIRGE, THE MAN
Selected Bibliography
William Henry Chandler July 31, 1878-October 29, 1970
EARLY LIFE AND EDUCATION
TREES IN TWO CLIMATES
Death by Freezing
The Rest Period
FRUIT TREE NUTRITION—THE ZINC STORY
REFLECTIONS, CONVICTIONS, AND FAITH
Selected Bibliography
Gertrude Mary Cox January 13, 1900-October 17, 1978
EARLY YEARS
SOUTHERN VENTURE
PROFESSIONAL ACTIVITIES AND HONORS
TRAVELS
CLOSING REMARKS
Selected Bibliography
Conrad Arnold Elvehjem May 27, 1901-July 27, 1962
EARLY YEARS
RETURN TO WISCONSIN
NICOTINIC ACID
THE B VITAMINS AND AMINO ACIDS
PUBLIC SERVICE
UNIVERSITY ADMINISTRATION
AWARDS
Selected Bibliography
Gottfried Samuel Fraenkel April 23, 1901-October 26, 1984
EARLY LIFE
BRITAIN
AMERICA
CLOSING REMARKS
HONORS AND DISTINCTIONS
Selected Bibliography
Haldan Keffer Hartline December 22, 1903-March 18, 1983
EDUCATION AND EARLY LIFE
PROFESSIONAL CAREER
MAJOR SCIENTIFIC CONTRIBUTIONS
Single Optic Nerve Fibers
The Receptive Field
The Generator Potential
The Hartline-Ratliff Equations
A SENSE OF HUMOR
HONORS AND AWARDS
HOME AND FAMILY
CONCLUDING REMARKS
Selected Bibliography
Mark Kac August 16, 1914-October 25, 1984
AMERICA
MATHEMATICAL WORK
Independence and the Normal Law
Brownian Motion and Integration in Function Space
Statistical Mechanics
PERSONAL APPRECIATION
HONORS, PRIZES, AND SERVICE
Selected Bibliography
Aldo Starker Leopold October 22, 1913-August 23, 1983
PROFESSIONAL AND PUBLIC SERVICE
HONORS AND DISTINCTIONS
Selected Bibliography
Manfred Martin Mayer June 15, 1916-September 18, 1984
EDUCATION AND EARLY LIFE
SCIENTIFIC CONTRIBUTION
TEACHER AND MENTOR
HONORS AND DISTINCTIONS
Professional And Academic Positions
Learned Societies
Honorary Memberships
Other Professional Activities
Prizes And Awards
Selected Bibliography
Walsh McDermott October 24, 1909-October 17, 1981
EDUCATION AND EARLY LIFE
THE MCDERMOTT LABORATORY
Penicillin
Antimicrobial Therapy for Infections in Animals
More Penicillin and the Role of Drugs in Combination
EDITORIAL WORK
CHANGE IN FOCUS: PUBLIC HEALTH
WORK IN THE JOHNSON FOUNDATION
PROFESSIONAL MEMBERSHIPS AND OTHER ACTIVITIES
FRIEND AND COLLEAGUE
CREATION OF THE INSTITUTE OF MEDICINE
AWARDS
ETHICS, THE MEDICAL PROFESSION, AND MODERN SCIENCE
Selected Bibliography
Theophilus Shickel Painter August 22, 1889-October 5, 1969
EARLY LIFE AND EDUCATION
CHROMOSOME CYTOLOGY AND SEX CHROMOSOMES
DROSOPHILA CYTOGENETICS
UNIVERSITY ADMINISTRATION
RETURN TO SCIENCE
Selected Bibliography
George Polya December 13, 1887-September7, 1985
ORIGINS AND CAREER
PROBABILITY
COMPLEX ANALYSIS
REAL ANALYSIS, APPROXIMATION THEORY, NUMERICAL ANALYSIS
COMBINATORICS
MATHEMATICAL PHYSICS
TEACHING AND LEARNING MATHEMATICS
Selected Bibliography
Edward Lawrie Tatum December 14, 1909-November 7, 1975
EDUCATION AND EARLY LIFE
THE STANFORD YEARS (1937-1945)
Neurospora and the One Gene-One Enzyme Theory
Anticipating the One Gene-One Enzyme Theory
Tryptophane and E. coli K-12
Tatum and Lederberg—Genetic Recombination in Bacteria
RETURN TO STANFORD (1948-1956)
THE ROCKEFELLER INSTITUTE (1957-1975)
THE NOBEL PRIZE (1958)
IN CONCLUSION
Selected Bibliography
Cornelis Bernardus Van Niel November 4, 1897-March 10, 1985
EDUCATION AND EARLY LIFE
DELFT: WORKING WITH KLUYVER
PACIFIC GROVE: HOPKINS MARINE STATION
PHOTOSYNTHESIS STUDIES
METHANE PRODUCTION AND CARBON DIOXIDE UTILIZATION
Bacterial Taxonomy
DENITRIFICATION
VAN NIEL THE GENERALIST
TEACHER AND COLLEAGUE
RETIREMENT
HONORS AND DISTINCTIONS
Degrees And Honorary Degrees
Fellowships And Professional Appointments
Awards And Honors
Learned Societies
Selected Bibliography
Robert H. Whittaker December 27, 1920-October 20, 1980
EDUCATION AND EARLY LIFE
SCIENTIFIC WORK
The Continuum of Plant Species Distribution
Plant and Insect Population Patterns, and Element Cycling
Dimension Analysis and the Classification of the Kingdoms
Desert and Forest: Structure and Function
Species Diversity, Ordination Methods
TEACHER, DIPLOMAT, HONORED RESEARCHER
HEALTH PROBLEMS
IN CONCLUSION
Selected Bibliography
Maxwell Myer Wintrobe October 27, 1901-December 9, 1986
EDUCATION AND EARLY LIFE
THE TULANE YEARS (1927-1930): ''ANEMIA OF THE SOUTH," NORMAL BLOOD VALUES, THE WINTROBE HEMATOCRIT,...
JOHNS HOPKINS (1930-1943)
THE UTAH YEARS (1943-1986)
The Utah Group and the Wintrobe Legacy
Clinical Hematology; Principles Of Internal Medicine; Blood, Pure And Eloquent
RETIREMENT FROM THE CHAIR OF MEDICINE
Selected Bibliography
Cumulative Index_
Citation preview
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i
Biographical Memoirs
NATIONAL ACADEMY OF SCIENCES
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ii
Nationd Academy Press
The National Academy Press was created by the National Academy of Sciences to publish the reports issued by the Academy and by the National Academy of Engineering, the Institute of Medicine, and the National Research Council, all operating under the charter granted to the National Academy of Sciences by the Congress of the United States.
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iii
National Academy of Sciences of the United States of America
Biographical Memoirs Volume 59
NATIONAL ACADEMY PRESS WASHINGTON, D.C. 1990
About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as the authoritative version for attribution.
iv Disclaimer: This book contains characters with diacritics. When the characters can be represented using the ISO 8859-1 character set ( netLibrary will represent them as they appear in the original text, and most computers will be able to show the full characters correctly. In order to keep the text searchable and readable on most computers, characters with diacritics that are not part of the ISO 8859-1 list will be represented without their diacritical marks. The National Academy of Sciences was established in 1863 by Act of Congress as a private, nonprofit, self-governing membership corporation for the furtherance of science and technology, required to advise the federal government upon request within its fields of competence. Under its corporate charter the Academy established the National Research Council in 1916, the National Academy of Engineering in 1964, and the Institute of Medicine in 1970. INTERNATIONAL STANDARD BOOK NUMBER 0-309-04198-8 LIBRARY OF CONGRESS CATALOG CARD NUMBER 5-26629 Available from NATIONAL ACADEMY PRESS 2101 CONSTITUTION AVENUE, N.W. WASHINGTON, D.C. 20418 PRINTED IN THE UNITED STATES OF AMERICA
About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as the authoritative version for attribution. CONTENTS v
Contents
Preface
Frederic C. Bartter By Jean D. Wilson And Catherine S. Delea vii 3
George Wells Beadle By Norman H. Horowitz 27
Solomon A. Berson By J. E. Rall 55
Raymond Thayer Birge By A. Carl Hemlholz 73
William Henry Chandler By Jacob B. Biale 87
Gertrude Mary Cox By Richard L. Anderson 117
Conrad Arnold Elvehjem By R. H. Burris, C. A. Baumann, And Van R. Potter 135
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CONTENTS
vi
Gottfried Samuel Fraenkel By C. Ladd Prosser, Stanley Friedman, And Judith H. Willis
169
Haldan Keffer Hartline By Floyd Ratliff
197
Mark Kac By H. P. Mckean
215
Aldo Starker Leopold By Robert A. Mccabe
237
Manfred Martin Mayer By K. Frank Austen
257
Walsh McDermott By Paul B. Beeson
283
Theophilus Shickel Painter By Bentley Glass
309
George Pólya By R. P. Boas
339
Edward Lawrie Tatum By Joshua Lederberg
357
Cornelis Bernardus Van Niel By H. A. Barker And Robert E. Hungate
389
Robert H. Whittaker By Walter E. Westman, Robert K. Peet, And Gene E. Likens
425
Maxwell Mayer Wintrobe By William N. Valentine
447
Cumulative Index
473
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PREFACE
vii
Preface
On March 3, 1863, Abraham Lincoln signed the Act of Incorporation that brought the National Academy of Sciences into being. In accordance with that original charter, the Academy is a private, honorary organization of scientists, elected for outstanding contributions to knowledge, who can be called upon to advise the federal government. As an institution the Academy's goal is to work toward increasing scientific knowledge and to further the use of that knowledge for the general good. The Biographical Memoirs, begun in 1877, are a series of volumes containing the life histories and selected bibliographies of deceased members of the Academy. Colleagues familiar with the discipline and the subjects' work prepare the essays. These volumes, then, contain a record of the life and work of our most distinguished leaders in the sciences, as witnessed and interpreted by their colleagues and peers. They form a biographical history of science in America—an important part of our nation's contribution to the intellectual heritage of the world. PETER H. RAVEN HOME SECRETARY ELIZABETH J. SHERMAN EDITOR
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1
Biographical Memoirs VOLUME 59
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FREDERIC C. BARTTER
3
Frederic C. Bartter September 10, 1914-May 5, 1983 By Jean D. Wilson and Catherine S. Delea Frederic Crosby Bartter was born in the Philippine Islands on September 10, 1914, and died in Washington, D.C., on May 5, 1983, of complications resulting from a cerebral hemorrhage that occurred while he was attending the annual meeting of the National Academy of Sciences. With his death, clinical science lost one of its most imaginative investigators and charismatic personalities. His achievements were both broad and deep. He devoted a major portion of his career to investigating the interrelation between the kidney and various endocrine systems and contributed to aspects of clinical science as diverse as chronobiology, the physiology of taste and smell, and mushroom poisoning. At the National Institutes of Health he collaborated with more than a hundred investigators (friends), enriching the lives and scientific stature of each through his ability to stimulate, guide, and enhance the talents of others. EARLY LIFE George Bartter, an Anglican minister from England, and his wife, Frances Buffington, an American teacher, had two children—George and Frederic— both born in Manila and raised in the remote mountain village of Baguio, Philippine Islands, which became the family home. Bartter's early edu
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FREDERIC C. BARTTER
4
cation was supervised by his father, his mother (a Smith college graduate and classical scholar), and priests at a nearby Catholic monastery. Early in life he acquired a love of poetry and good writing and, in later years, was able to recite from memory long passages from Shakespeare, St. Teresa, and Rupert Brooke. At age thirteen he and his brother were sent to the United States and enrolled at the Lennox School in Lennox, Massachusetts, from which he graduated in 1930. He returned to the Philippine Islands for a year and worked in the English school before entering Harvard College. After receiving a Bachelor of Arts degree in 1935, Bartter spent a year in the Department of Physiology, Harvard School of Public Health. His interest in an investigative career and his first paper on lymph sugar stemmed from this experience. He obtained his M.D. degree from Harvard Medical School in 1940 and spent his internship at Roosevelt Hospital in New York from 1941 to 1942. ACCOMPLISHMENTS IN BIOMEDICAL RESEARCH Bartter's first paper after graduation from medical school resulted from his service as an officer in the U.S. Public Health Service during World War II. The paper concerned plasma volume and the speed with which plasma is reconstituted after donation of blood, the control of blood volume being an important topic throughout his subsequent career. The Public Health Service then assigned him to the Pan American Sanitary Bureau to investigate the physiology of parasitic diseases, one result of which was a pioneering study of the treatment of onchocerciasis. There can be no doubt that both the style and the focus of his investigative career were profoundly influenced by his subsequent association with Fuller Albright, first as a research fellow from 1946 to 1950, then as a junior member of the faculty at the Massachusetts General Hospital and the
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FREDERIC C. BARTTER
5
Harvard Medical School. Many have described the unique environment Fuller Albright created on Ward Four at the Massachusetts General Hospital, including Bartter, who wrote several moving accounts.1 At least three distinguishing features of Bartter's work stem directly from his relationship with Albright. First, he performed virtually all of his work directly on humans. Indeed, his bibliography of over 400 papers lists only a few studies using experimental animals and fewer still experiments in vitro. Though clinical physiologists usually draw clinical deductions from animal studies, Albright, whose model Bartter followed, deduced physiological principles from physiology deranged by the disease process. Secondly, Bartter used few patients in each study, but every patient was studied intensively over a long period of time with the most advanced methodologies and techniques. Finally, Bartter benefitted from his mentor's remarkable breadth of interests that encompassed electrolyte and renal physiology, endocrinology, intermediary metabolism, the control of blood pressure, biological rhythms, and neurophysiology. The last of a school of clinical investigation built on the metabolic balance technique, Bartter was yet uniquely adept at applying new technologies to in vivo studies, from isotope dilution to radioimmunoassay procedures. During his years with Albright, Bartter developed a number of interests that would continue throughout his career: the metabolic effects of ACTH in man, parathyroid pathophysiology and bone metabolism, the control of blood volume in disease, and the metabolic effects of androgens, estrogens, and adrenocortical steroids in various disorders. An outstanding example of Bartter and Albright's joint
1 See ''Fuller Albright," in The Massachusetts General Hospital, 1955-1980 (Boston: Little Brown & Company, 1981), p. 86; and "Fuller Albright," Endocrinology 87 (1970):1109.
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FREDERIC C. BARTTER
6
creativity was their deduction that the common virilizing form of adrenal hyperplasia is fundamentally a type of adrenocortical insufficiency arising from a metabolic error in the biosynthetic pathway for cortisol. To compensate for the deficiency in cortisol secretion, they reasoned, the pituitary secretes excessive quantities of ACTH leading to excessive secretion of other classes of adrenal steroids by the adrenals themselves. Bartter and Albright proved their thesis by treating affected patients with cortisone to correct the hypersecretion of virilizing steroids—undoubtedly the single greatest stroke of genius in understanding and controlling adrenal hyperplasia. In 1951, Bartter's move from Boston to the National Institutes of Health, initially in Baltimore and then Bethesda, broadened the focus of his studies of the pathophysiology of disease. When "electrocortin" (aldosterone) was discovered in 1953, it was immediately apparent to Bartter that this new hormone must be of critical importance in cardiovascularrenal physiology. He turned his attention to determining its role in health and disease and the factors controlling its secretion. Without neglecting the importance of other aldosterone regulatory factors, Bartter, together with Grant Liddle, reasoned that extracellular fluid volume is a major determinant of aldosterone secretion. This deduction ultimately led several groups to the discovery that the aldosterone regulatory influence of extracellular volume is mediated by the reninangiotensin system. In 1960, Bartter described the syndrome of hyperplasia of the juxtaglomerular complex—in which hyperaldosteronism and hypokalemic alkalosis coexist with normal blood pressure: now commonly termed Bartter's syndrome. His findings added to the growing body of evidence that adrenal cortical secretion is influenced by the renin-angiotensin system. He further proposed a hypothesis for the paradox of
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FREDERIC C. BARTTER
7
normal blood pressure in the presence of high concentrations of aldosterone and angiotensin, a paradox still being investigated today. Adrenal hyperplasia, with all its complexities, held a continuing fascination for Bartter. He realized that a third of all the patients he studied with primary aldosteronism also had adrenal hyperplasia. Originally it was hoped that plasma renin determinations might differentiate between aldosteronism produced by tumor from that produced by hyperplasia. The low plasma renin values measured in several patients with proven adrenal hyperplasia suggested that, in these patients, all adrenal tissue responds to a tropic stimulus other than ACTH or the renin-angiotensin system. This, too, continues to be an active field of investigation. While many of the seventy papers on calcium and phosphorus metabolism coauthored by Bartter relate to the diagnosis and treatment of hyperparathyroidism, pseudohypoparathyroidism, and metabolic bone diseases, several significant studies deal with the renal handling of phosphorus and calcium under the influences of parathyroid hormone, vitamin D, large doses of phosphate, and calcium infusions. Bartter's laboratory also explored the physiology of thyrocalcitonin and its relation to disease states, the solubility and composition of bone mineral, and the gastrointestinal absorption of calcium and its role in metabolic diseases. In the late 1960s he and Charles Y. C. Pak began a pioneering series of studies on the classification, pathogenesis, and treatment of kidney stones. During these years at the NIH, Bartter's studies covered a broad range of metabolic topics: renal concentrating mechanisms, steroid-hormone binding and transport, urinary acidification mechanisms, regulation of aldosterone biosynthesis, the effect of adrenal hormones on taste and auditory thresholds, vitamin D metabolism and action, phosphorus
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depletion, mechanisms of edema formation, cystine metabolism, magnesium metabolism, radiographic measurements of bone minerals, prostaglandin and catecholamine metabolism, and collagen formation in bone. The common theme in all these studies was Bartter's passion for analyzing the disease process. A highlight of his investigative career came in 1957, when—with William B. Schwartz of Tufts University—he described the syndrome of inappropriate secretion of antidiuretic hormone (ADH, or vasopressin). Hyponatremia and renal sodium loss unrelated to renal or adrenal disease were seen in two patients with bronchogenic carcinoma. The data from a series of studies of these patients suggested overexpansion of the body fluids, probably as a result of sustained, inappropriate secretion of ADH. Bartter and Schwartz characterized this clinical entity, now known to occur in a variety of pathophysiological settings, in a trenchant series of clinical experimental and didactic studies developed over more than two decades. The syndrome is found with various tumors; in disorders affecting the central nervous system or the lungs; and in adrenal, thyroid, or pituitary insufficiency. It is now known that the tumors produce an antidiuretic substance directly and that some of the other disorders are associated with an abnormal release of ADH from the pituitary gland. From its immediate impact upon medicine, Bartter's description of inappropriate ADH secretion was perhaps his most important discovery. During the last decade of his scientific career, Bartter focused on the control of blood pressure and the derangements that underlie the hypertensive disorders of man—a line of investigation that continued after his 1978 move to the University of Texas Health Science Center in San Antonio and was cut short by his untimely death. It is an irony that he discovered his own hypertension during these studies.
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PERSONAL QUALITIES Fred Bartter's curiosity and quest for intellectual expansion extended well beyond his professional interests. He had a great love and knowledge of music and sang with several musical groups, an interest he shared with his family. A devotee of mathematician and philosopher Alfred North Whitehead (under whom he studied), he read widely in philosophy and poetry. He was a perpetual student who insisted, both in his public speaking and writing, that clarity of expression reflects clarity of thought. He was a strict adherent of correct grammar, and everyone who worked with him became aware of his meticulous attention to detail. Yet his subtle sense of humor, his joy in and excitement about life on the day-to-day level, made him particularly endearing. His warmth and sensitivity gained him the respect and loyalty of his patients, whom he treated as an integral part of the investigative team. Delighting in the diagnostic pursuit of a disease, he yet never lost sight of the person. One of Bartter's many interests deserves special comment. During a summer vacation he picked up a book belonging to his mother-in-law, who had been a botany major at Smith College, about mushrooms. Its beautiful illustrations and the complex classification system of species and subvariants fascinated him, and he began looking for mushrooms in the woods and lawns back home. Pursuing this subject with the same intellectual vigor he applied to his work, Bartter became an authority on the subject. He could identify more than 200 varieties, and for many years he combined his avocation with his professional career, giving lectures on mycology and on the symptoms and treatment of mushroom poisoning. Following Czech reports of lipoic acid as an antidote for Amanita mushroom poisoning, Bartter—and Charles Becker of the University of California, San Francisco—obtained an
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investigational permit from the Food and Drug Administration to use lipoic acid as a treatment for patients who had eaten supposedly lethal mushrooms.2 The toxins of the "Death Caps" (or "Destroying Angels," as deadly Amanitas are called) attack the liver, causing hepatitis and acute yellow atrophy that may progress to liver failure. Bartter and Becker treated many patients who had ingested the mushrooms, and were therefore at risk, with the agent. Although the precise therapeutic role of lipoic acid—as opposed to other supportive features of the experimental regimen—was never clarified, the treatment was successful. Bartter's experience with treating mushroom poisoning enhanced his zest as a mushroom collector, and he delighted in instructing others and in serving as a resident expert on mushroom identification. HONORS Fred Bartter was a member of numerous professional and scientific societies, including the Endocrine Society, the American Society for Clinical Investigation, the Association of American Physicians, the Royal Society of Medicine, the Royal College of Physicians of London, the Peripatetic Club, and the National Academy of Sciences, to which he was elected in 1979. He received the Sandoz Contemporary Man in Medicine Award, the Modern Medicine Distinguished Achievement Award, the Fred C. Koch Award of the Endocrine Society, and the Meritorious Service Medal from the National Institutes of Health. These honors were followed by election as the 1981 honorary faculty member of the Epsilon Chapter of Alpha Omega Alpha—the medical honorary society at the
2 See B. J. Culliton, "The Destroying Angel: A Story of a Search for an Antidote," Science 185(1974):600; and "Dr. Bartter Tries Thioctic Acid as Antidote to Fascinating Fatal Wild Mushrooms," NIH Record (November 4, 1975):6.
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University of Texas Health Science Center in San Antonio—and, in 1982, election as an Honorary Fellow of the American College of Cardiology. In 1982, the American College of Physicians conferred on him the John Phillips Memorial Award "in recognition of his outstanding career as an investigator and teacher and for his memorable contribution to the understanding of hormonal regulation of renal function and salt and water homeostasis." Bartter was also asked to give many honorary lectures, including the 1980 Arthur B. Corcoran Award of the High Blood Pressure Council and the 1982 Fuller Albright Lecture of the Peripatetic Club. The San Antonio Veterans Administration Medical Center named its Bartter Clinical Research Center in his memory—a posthumous tribute that surely would have pleased him. Fred Bartter is survived by his wife, the former Jane Lillard; three children, Frederic C. Bartter, Jr., of Baltimore, Dr. Thaddeus C. Bartter of Boston, and Mrs. George (Pamela) Reiser of Lincoln, Massachusetts; and three grandchildren. Fred Bartter will be remembered by his associates for his persistence, imagination, endless curiosity, and bottomless fund of knowledge. The ability to perceive a disease in a set of slightly aberrant numbers, the unshakable faith that, in metabolic balance studies, what goes in must eventually come out, and the optimism that all is eventually discoverable—this is "Bartter's Syndrome," and we are all the better for having been exposed to it.
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CHRONOLOGY Postgraduate Training And Fellowships 1941-1942 Medical intern, Roosevelt Hospital 1942-1945 Medical officer, U.S. Public Health Service 1945-1946 Staff member, Laboratory of Tropical Diseases, National Institutes of Health 1946-1948 Research Fellow in Medicine, Massachusetts General Hospital 1968-1969 Overseas Fellow, Churchill College, University of Cambridge Professional Appointments 1948-1950 Assistant in Medicine, Massachusetts General Hospital 1951 Associate in Medicine, Massachusetts General Hospital 1951-1973 Chief, Endocrinology Branch, National Heart and Lung Institute, National Institutes of Health 1970-1976 Clinical Director, National Heart and Lung Institute, National Institutes of Health 1973-1978 Chief, Hypertension, Endocrine Branch, National Heart and Lung Institute, National Institutes of Health 1958-1978 Associate Professor and Professor of Pediatrics, Howard University 1960-1978 Associate Professor and Clinical Professor of Medicine, Georgetown University 1978-1983 Professor of Medicine, University of Texas Health Science Center, San Antonio, and Associate Chief of Staff for Research, Audie L. Murphy Memorial Veterans Administration Hospital, San Antonio Memberships Endocrine Society Laurentian Hormone Conference American Society for Clinical Investigation Association of American Physicians Salt and Water Club
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Peripatetic Society American Physiological Society Royal Society of Medicine-Endocrinology Section Royal College of Physicians of London National Academy of Sciences Alpha Omega Alpha
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Selected Bibliography
1935 With J. W Heim and R. S. Thomson. Lymph sugar. Am. J. Physiol., 113:548. 1944 With F. Co Tui, A. M. Wright, and R. B. Holt. Red cell reinfusion and the frequency of plasma donations: A preliminary report of multiple donations in eight weeks by each of six donors. J. Am. Med. Assoc., 124:331. 1949 With F. Albright and A. Forbes. The fate of human serum albumin administered intravenously to a patient with idiopathic hypoalbuminemia and hypoglobulinemia. Trans. Assoc. Am. Physicians, 62:204. 1950 With P. Fourman, F. Albright, A. P. Forbes, W. Jeffries, G. Griswold, et al. The effect of adrenocorticotropic hormone in panhypopituitarism. J. Clin. Invest., 29:950. With P. Fourman, F. Albright, E. Dempsey, E. Carroll, and J. Alexander. Effects of 17hydroxycorticosterone (compound F) in man. J. Clin. Invest., 29:1462. With T. Elrick, F. Albright, A. P. Forbes, and J. D. Reeves. Further studies on pseudohypoparathyroidism: Report of four new cases. Acta Endrocrinol., 5:199. 1951 With F. Albright, A. P. Forbes, A. Leaf, E. Dempsey, and E. Carroll. The effects of adrenocorticotropic hormone and cortisone in the adrenogenital syndrome associated with congenital adrenal hyperplasia: An attempt to explain and correct its disordered hormonal pattern. J. Clin. Invest., 30:237. 1952 With P. Fourman, E. C. Reifenstein, Jr., E. J. Kepler, E. Dempsey, and F. Albright. Effect of desoxycorticosterone acetate on electrolyte metabolism in normal man. Metabolism, 1:242.
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1953 With A. Leaf, R. F. Santos, and O. Wrong. Evidence in man that urinary electrolyte loss induced by pitressin is a function of water retention. J. Clin. Invest., 32:868. 1956 With G. W. Liddle and L. E. Duncan, Jr. Dual mechanism regulating adrenocortical function in man. Am. J. Med., 21:380. With L. E. Duncan, Jr., G. W Liddle, and K. Buck. The effect of changes in body sodium on extracellular fluid volume and aldosterone and sodium excretion by normal and edematous man. J. Clin. Invest., 35:1299. The role of aldosterone in normal homeostasis and in certain disease states. Metabolism, 5:369. With G. W. Liddle, L. E. Duncan, Jr., J. K. Barber, and C. Delea. The regulation of aldosterone secretion in man. The role of fluid volume. J. Clin. Invest., 35:1306. 1957 The role of aldosterone in the regulation of body fluid volume and composition. Scand. J. Clin. Lab. Invest., 10:50. With W. B. Schwartz, W Bennett, and S. Curelop. A syndrome of renal sodium loss and hyponatremia probably resulting from inappropriate secretion of antidiuretic hormone. Am. J. Med., 33:529. 1958 With R. S. Goldsmith, P. J. Rosch, W H. Meroney, and E. G. Herndon. ''Primary aldosteronism" associated with significant edema. J. Clin. Endocrinol., 18:323. With W. E. Schatten, A. G. Ship, and W. J. Pieper. Syndrome resembling hyperparathyroidism associated with squamous cell carcinoma. Ann. Surg., 148:890. 1959 With R. S. Gordon, Jr., and T. Waldmann. Idiopathic hypoalbuminemias: Clinical staff conference at the National Institutes of Health. Ann. Intern. Med., 51:553. With J. Orloff, M. Walser, and T. J. Kennedy, Jr. Hyponatremia. Circulation, 19:284.
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With M. M. Pechet and B. Bowers. Metabolic studies with a new series of 1,4-diene steroids. I. Effects in Addisonian subjects of prednisone, prednisolone, and the 1,2-dehydro analogues of corticosterone, desoxycorticosterone, 17-hydroxy-11-desoxycorticosterone, and 9 αfluorocortisol. J. Clin. Invest., 38:681. 1960 With H. P. Schedl. An explanation for and experimental correction of the abnormal water diuresis in cirrhosis. J. Clin. Invest., 39:248. With I. H. Mills, H. P. Schedl, and P. S. Chen, Jr. The effect of estrogen administration on the metabolism and protein binding of hydrocortisone. J. Endocrinol., 20:515. With W. B. Schwartz and D. Tassell. Further observations on hyponatremia and renal sodium loss probably resulting from inappropriate secretion of antidiuretic hormone. N. Engl. J. Med., 262:743. With R. S. Goldsmith and W. H. Meroney. Prominent peripheral edema associated with primary aldosteronism due to an adrenocortical adenoma. J. Clin. Endocrinol., 20:1168. 1961 With J. R. Gill, Jr. On the impairment of renal concentrating in prolonged hypercalcemia and hypercalciuria in man. J. Clin. Invest., 40:716. With J. P. Thomas. Relation between diuretic agents and aldosterone in cardiac and cirrhotic patients with sodium retention. Br. Med. J., 1:1134. With P. S. Chen, Jr., and I. H. Mills. Ultrafiltration studies of steroid-protein binding. J. Endocrinol., 23:129. With A. G. T. Casper, C. S. Delea, and J. D. H. Slater. On the role of the kidney in control of adrenal steroid production. Metabolism, 10:1006. With J. Steinfeld, T. Waldmann, and C. S. Delea. Metabolism of infused serum albumin in the hypoproteinemia of gastrointestinal protein loss and in analbuminemia. Trans. Assoc. Am. Physicians, 74:180. With J. P. Thomas. Blood volume measurements in normal subjects and in patients with cirrhosis or cardiac disease. Clin. Sci., 21:301.
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1962 With P. Fourman. The different effects of aldosterone-like steroids and hydrocortisone-like steroids on urinary excretion of potassium and acid. Metabolism, 11:6. With N. M. Kaplan. The effect of ACTH, renin, angiotensin II and various precursors on biosynthesis of aldosterone by adrenal slices. J. Clin. Invest., 41:715. With J. R. Gill, Jr., and D. S. Gann. Restoration of water diuresis in Addisonian patients by expansion of the volume of extracellular fluid. J. Clin. Invest., 41:1078. With P. Pronove, J. R. Gill, Jr., R. C. MacCardle, and E. Diller. Hyperplasia of the juxtaglomerular complex with hyperaldosteronism and hypokalemic alkalosis. Am. J. Med., 33:811. With D. S. Gann, J. F. Cruz, and A. G. T. Casper. Mechanism by which potassium increases aldosterone secretion in the dog. Am. J. Physiol., 202:991. 1963 With R. I. Henkin and J. R. Gill, Jr. Studies on taste thresholds in normal man and in patients with adrenal corticol insufficiency: The role of adrenal cortical steroids and of serum sodium concentration. J. Clin. Invest., 42:727. With N. H. Bell and E. S. Gerard. Pseudohypoparathyroidism with osteitis fibrosa cystica and impaired absorption of calcium. J. Clin. Endocrinol., 23:759. With J. D. H. Slater, B. H. Barbour, H. Henderson, and A. G. T. Casper. Influence of the pituitary and the renin-angiotensin system on the secretion of aldosterone, cortisol and corticosterone. J. Clin. Invest., 42:1504. 1964 With N. H. Bell and H. Schedl. An explanation for abnormal water retention and hypoosmolality in congestive heart failure. Am J. Med., 36:351. With N. H. Bell and J. R. Gill, Jr. On the abnormal calcium absorption in sarcoidosis. Am. J. Med., 36:500. With J. R. Gill, Jr., B. H. Barbour, and J. D. H. Slater. Effect of angiotensin II on urinary dilution in normal man. Am J. Physiol., 206:750.
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With J. R. Gill, Jr., J. M. George, and A. Solomon. Hyperaldosteronism and renal sodium loss reversed by drug treatment for malignant hypertension. N. Engl. J. Med., 270:1088. With G. T. Bryan, B. Kliman, and J. R. Gill, Jr. Effect of human renin on aldosterone secretion rate in normal man and in patients with the syndrome of hyperaldosteronism, juxtaglomerular hyperplasia and normal blood pressure. J. Clin. Endocrinol., 24:729. With D. Hellman and R. Baird. Relationship of maximal tubular reabsorption to filtration rate in the dog. Am. J. Physiol., 207:89. With D. S. Gann, C. S. Delea, J. R. Gill, Jr., and J. P. Thomas. Control of aldosterone secretion by change of body potassium in normal man. Am. J. Physiol., 207:104. 1965 With J. D. H. Slater, B. H. Barbour, H. H. Henderson, and A. G. T. Casper. Physiological influence of the kidney on the secretion of aldosterone, corticosterone and cortisol by the adrenal cortex. Clin. Sci., 28:219. With G. T. Bryan and B. Kliman. Impaired aldosterone production in "salt-losing" congenital adrenal hyperplasia. J. Clin. Invest., 44:957. With Y. H. Pilch and W. S. Kiser. A case of villous adenoma of the rectum with hyperaldosteronism and unusual renal manifestations. Am. J. Med., 39:483. With D. E. Hellman and W. Y. W. Au. Evidence for a direct effect of parathyroid hormone on urinary acidification. Am. J. Physiol., 209:643. With R. L. Ney, W. Y. W. Au., G. Kelly, and I. Radde. Actions of parathyroid hormone in the vitamin D-deficient dog. J. Clin. Invest., 44:2003. With G. T. Bryan and R. C. MacCardle. Hyperaldosteronism, hyperplasia of the juxtaglomerular complex, normal blood pressure, and dwarfism: Report of a case. Pediatrics, 37:43. 1966 With J. R. Gill, Jr. Adrenergic nervous system in sodium metabolism. II. Effects of guanethidine on the renal response to sodium deprivation in normal man. N. Engl. J. Med., 275:1466.
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1967 With J. R. Gill, Jr., A. A. Carr, L. E. Fleischmann, and A. G. T. Casper. Effects of pentolinium on sodium excretion in dogs with constriction of the vena cava. Am. J. Physiol., 212:191. With J. R. Gill, Jr., and N. H. Bell. Effect of parathyroid extract on magnesium excretion in man. J. Appl. Physiol., 22:136. With R. A. Melick, J. R. Gill, Jr., S. A. Berson, R. S. Yalow, J. T. Potts, and G. D. Aurbach. Antibodies and clinical resistance to parathyroid hormone. N. Engl. J. Med., 276:144. With R. I. Henkin, R. E. McGlond, and R. Daly. Studies on auditory thresholds in normal man and in patients with adrenal cortical steroids . J. Clin. Invest., 46:429. With W. W. Davis, H. H. Newsome, L. D. Wright, W. G. Hammond, and J. Easton. Bilateral adrenal hyperplasia as a cause of primary aldosteronism with hypertension, hypokalemia and suppressed renin activity. Am. J. Med., 42:642. With C. Y. C. Pak. Ionic interaction with bone mineral. I. Evidence for an isoionic calcium exchange with hydroxyapatite. Biochim. Biophys. Acta, 141:401. 1968 With M. Lotz and E. Zisman. Evidence of a phosphorus-depletion syndrome in man. N. Engl. J. Med., 278:409. With R. L. Ney and G. Kelly. Actions of vitamin D independent of parathyroid glands. Endocrinology, 82:760. With W. W. Davis, L. R. Burwell, and A. G. T. Casper. Sites of action of sodium depletion on aldosterone biosynthesis in the dog. J. Clin. Invest., 47:1425. With R. I. Henkin and G. T. Bryan. Aldosterone hypersecretion in non-salt-losing congenital adrenal hyperplasia. J. Clin. Invest., 47:1742. With J. M. George and L. Gillespie. Aldosterone secretion in hypertension. Ann. Intern. Med., 69:693. With C. Y. C. Pak, M. R. Wills, and G. W Smith. Treatment with thyrocalcitonin of the hypercalcemia of parathyroid carcinoma. J. Clin. Endocrinol., 28:1657. With G. S. Stokes, J. T. Potts, Jr., and M. Lotz. Mechanisms of action of d-Penicillamine and nAcetyl-d-penicillamine in the therapy of cystinuria. Clin. Sci., 35:467.
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With R. J. Wurtzman, A. G. T. Casper, and L. A. Pohorecky. Impaired secretion of epinephrine in response to insulin among hypophysectomized dogs. Proc. Natl. Acad. Sci. USA, 61:522. With R. D. Gordon, J. Spinks, A. Dulmanis, B. Hudson, and F. Halberg. Amplitude and phase relations of several circadian rhythms in human plasma and urine: Demonstration of rhythm for tetrahydrocortisol and tetrahydrocorticosterone. Clin. Sci., 35:307. 1969 With W. W. Davis and L. R. Burwell. Inhibition of the effects of angiotensin II on adrenal steroid production by dietary sodium . Proc. Natl. Acad. Sci. USA, 63:718. With M. R. Wills, C. Y. C. Pak, and W. G. Hammond. Normocalcemic primary hyperparathyroidism. Am. J. Med., 47:384. With M. R. Wills and J. R. Gill, Jr. The interrelationships of sodium and calcium excretion. Clin. Sci., 37:621 1970 With J. M. George, L. Wright, N. H. Bell, and R. Brown. The syndrome of primary aldosteronism. Am. J. Med., 48:343. With M. R. Wills, J. Wortsman, and C. Y. C. Pak. The role of parathyroid hormone in the gastrointestinal absorption of calcium. Clin. Sci., 39:39. 1971 With A. P. Simpoulos, J. R. Marshall, and C. S. Delea. Studies on the deficiency of 21hydroxylation in patients with congenital adrenal hyperplasia. J. Clin. Endocrinol. Metab., 32:438. With H. H. Newsome, Jr., and M. S. Kafka. Intrarenal blood flow in dogs with constriction of the inferior thoracic vena cava. Am. J. Physiol., 221:48. With J. R. Gill, Jr., and C. S. Delea. A role for sodium-retaining steroid in the regulation of proximal tubular sodium reabsorption in man. Clin. Sci., 42:423. With I. B. Transbol, J. R. Gill, Jr., M. Lifschitz, and C. S. Delea. Intestinal absorption and renal excretion of calcium in metabolic acidosis and alkalosis. Acta Endocrinol. (suppl.) (Copenhagen), 155:217.
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1972 With J. R. Gill, Jr., and T. A. Waldmann. Idiopathic edema. I. The occurrence of hypoalbuminemia and abnormal albumin metabolism in women with unexplained edema. Am. J. Med., 52:445. With J. R. Gill, Jr., J. W. Cox, and C. S. Delea. Idiopathic edema. II. Pathogenesis of edema in patients with hypoalbuminemia. Am. J. Med., 52:452. With C. Y. C. Pak, D. A. East, L. H. Sanzenbacher, and C. S. Delea. Gastrointestinal calcium absorption in nephrolithiasis. J. Clin. Endocrinol. Metab., 35:261. 1973 With S. Middler, C. Y. C. Pak, and F. Murad. Thiazide diuretics and calcium metabolism. Metabolism, 22:139. With J. B. Gross. Effects of prostaglandins, E1, A1, and F2α on renal handling of salt and water. Am. J. Physiol., 225:218. With L. A. Pohoreck, B. S. Baliga, and R. J. Wurtzman. Adrenocortical control of catecholamine metabolism in the dog adrenal medulla: Relationship to protein synthesis. Endocrinology, 93:566. 1974 With C. Y. C. Pak and C. S. Delea. Successful treatment of recurrent nephrolithiasis (calcium stones) with cellulose phosphate. N. Engl. J. Med., 290:175. With W. L. Miller and W. J. Meyer III. Intermittent hyperphosphatemia, polyuria, and seizures— new familial disorder. J. Pediatr., 86:233. 1975 With J. Walton and M. Dominguez. Effects of calcium infusions in patients with postmenopausal osteoporosis. Metabolism, 24:849. With H. Zimbler, G. L. Robertson, C. S. Delea, and T. Pomeroy. Ewing's sarcoma as a cause of the syndrome of inappropriate secretion of antidiuretic hormone. J. Clin. Endocrinol. Metab., 41:390. With B. Stripp, A. A. Taylor, J. R. Gillette, D. L. Loriaux, R. Easley,
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and R. H. Menard. Effect of spironolactone on sex hormones in man. J. Clin. Endocrinol. Metab., 41:777. 1976 With R. H. Menard and J. R. Gillette. Spironolactone and cytochrome P-450: Impairment of steroid 21-hydroxylation in the adrenal cortex. Arch. Biochim. Biophys., 173:395. With J. R. Gill, J. C. Frolich, R. E. Bowden, A. A. Taylor, H. R. Keiser, et al. Bartter's syndrome: A disorder characterized by high urinary prostaglandins and a dependence of hyperreninemia on prostaglandin synthesis. Am. J. Med., 61:43. With C. E. Becker, T. G. Tong, U. Boerner, R. L. Roe, R. A. T. Scott, and M. B. MacQuarrie. Diagnosis and treatment of amanita phalloides-type mushroom poisoning. West. J. Med., 125:100. With J. D. Baxter, M. Schambelan, D. T. Matulich, B. J. Spindler, and A. A. Taylor. Aldosterone receptors and the evaluation of plasma mineralocorticoid activity in normal and hypertensive states. J. Clin. Invest., 58:579. With W. J. Meyer III, E. C. Diller, and F. Halberg. The circadian periodicity of urinary 17ketosteroids, corticosteroids, and electrolytes in congenital adrenal hyperplasia. J. Clin. Endocrinol. Metab., 43:1122. 1977 With J. Yun, G. Kelly, and H. Smith, Jr. Role of prostaglandins in the control of renin secretion in the dog. Circ. Res., 40:459. With A. E. Broadus, J. E. Mahaffey, and R. M. Neer. Nephrogenous cyclic adenosine monophosphate as a parathyroid function test. J. Clin. Invest., 60:771. With N. Radfar, R. Easley, J. Kolins, N. Javadpour, and R. J. Sherins. Evidence for endogenous LH suppression in a man with bilateral testicular tumors and congenital adrenal hyperplasia. J. Clin. Endocrinol. Metab., 45:1194. 1978 With A. E. Broadus and L. J. Deftos. Effects of the intravenous administration of calcium on nephrogenous cyclic AMP: Use as a parathyroid suppression test. J. Clin. Endocrinol. Metab., 46:477.
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With A. E. Broadus and M. Dominguez. Pathophysiological studies in idiopathic hypercalciuria: Use of an oral calcium tolerance test to characterize distinctive hypercalciuric subgroups. J. Clin. Endocrinol. Metab., 47:751. With J. M. Vinci, J. R. Gill, R. E. Bowden, J. J. Pisano, J. L. Izzo, et al. The Kallikrein-kinin system in Bartter's syndrome and its response to prostaglandin synthetase inhibition. J. Clin. Invest., 61:1671. 1979 With M. S. Kafka, C. R. Lake, H. G. Gullner, J. F. Tallman, and T. Fujita. Adrenergic receptor function is different in male and female patients with essential hypertension. Clin. Exp. Hypertens., 1:613. With A. A. Licata, E. Bou, and J. Cox. Effects of dietary protein on urinary calcium in normal subjects and in patients with nephrolithiasis. Metabolism, 28:895. With H. G. Gullner, C. R. Lake, and M. S. Kafka. Effect on inhibition of prostaglandin synthesis on sympathetic nervous system function in man . J. Clin. Endocrinol. Metab., 49:552. With H. G. Gullner, C. Cerletti, J. B. Smith, and J. R. Gill. Prostacyclin overproduction in Bartter's syndrome. Lancet, 2:767. 1980 With H. G. Gullner, J. R. Gill, Jr., R. Lake, and D. J. Lakatua. Correction of increased sympathoadrenal activity in Bartter's syndrome by inhibition of prostaglandin synthesis. J. Clin. Endocrinol. Metab., 50:857. With T. Fujita, W. L. Henry, C. R. Lake, C. S. Delea. Factors influencing blood pressure in saltsensitive patients with hypertension. Am. J. Med., 69:334. With H. G. Gullner, J. R. Gill, Jr., and R. Dusing. The role of the prostaglandin system in the regulation of renal function in normal women. Am. J. Med., 69:718. 1981 With C. M. Chan. Weight reduction: Renal mineral and hormonal excretion during semistarvation in obese patients. J. Am. Med. Assoc., 245:371. With S. Broder, T. R. Callihan, E. S. Jaffe, V. T. DeVita, W. Strober,
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and T. A. Waldmann. Resolution of longstanding protein-losing enteropathy in a patient with intestinal lymphangiectasia after treatment for malignant lymphoma. Gastroenterology, 80:166. C. R. Lake, H. G. Gullner, R. J. Polinsky, M. H. Evert, and M. G. Ziegler. Essential hypertension: Central and peripheral norepinephrine. Science, 211:955. H. G. Gullner and J. R. Gill. Correction of hypokalemia by magnesium repletion in familial hypokalemic alkalosis with tubulopathy. Am. J. Med., 71:578. J. R. Gill. Overproduction of sodium-retaining steroids by the zona glomerulosa is adrenocorticotropin-dependent and mediates hypertension in dexamethasone-suppressible aldosteronism. J. Clin. Endocrinol. Metab., 53:331. With H. G. Gullner, W E. Nicholson, M. G. Wilson, and D. N. Orth. The response of plasma immunoreactive adrenocorticotropin, beta-endorphin/beta-lipotropin, gamma lipotropin and cortisol to experimentally induced pain in normal subjects. Clin. Sci., 63:397. N. C. Lan, B. Graham, and J. D. Baxter. Binding of steroids to mineralocorticoid receptors: Implications for in vivo occupancy by glucocorticoids . J. Clin. Endocrinol. Metab., 54:332.
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George Wells Beadle October 22, 1903-June 9, 1989
By Norman H. Horowitz George Beadle was a giant in the field of modern genetics. He initiated the great series of advances made between 1941 and 1953 that brought the era of classical genetics to a close and launched the molecular age. For this achievement he received many honors, including the Nobel Prize. He was elected to the National Academy of Sciences in 1944 and served on its Council from 1969 to 1972. Beadle also had a distinguished career as an academic administrator. When he retired in 1968, he was President of The University of Chicago. Long years in administration, however, did not dampen his love of experimental genetics, and after his retirement he resumed experimental work on a favorite subject— the origin of maize. In 1981, he gave up research altogether because of increasing disability from the Alzheimer's disease that eventually ended his life. EDUCATION AND EARLY LIFE Beadle—his oldest friends usually called him by his boyhood nickname, ''Beets"—was born in Wahoo, Nebraska, to Hattie Albro and Chauncey Elmer Beadle, and he died in Pomona, California, at age eighty-five. He grew up on his father's forty-acre farm near Wahoo. The farm was a model for farms its size and was so designated by the U. S. Depart
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ment of Agriculture in 1908. Beets' mother died when he was four years old, and he and his brother and sister were raised by a succession of housekeepers. As a boy he worked on the farm, and he retained the skills he learned as a gardener and beekeeper there, and his handiness with tools, for the rest of his life. Gardening remained one of his greatest pleasures, and the victory garden he grew around his home at Stanford during the War produced enough for two families. This garden included beehives, but Beets wouldn't eat the honey, saying he had been stung too many times as a boy. He loved corn, on the other hand, and raised several kinds, including a small Mexican variety that gave his garden the distinction of having the earliest sweet corn at Stanford. After his retirement to Pomona in 1982, he derived much pleasure from growing flowers, a hobby he pursued as long as his health permitted. Beets did well in school and was inspired to go on to college by his high school science teacher, Bess MacDonald (the debt to whom he acknowledged more than once in later years). Despite his father's opinion that a farmer did not need all that education, he entered the University of Nebraska College of Agriculture in 1922. He graduated in 1926 with a B. S. degree and stayed on for another year to work for a master's degree with Franklin D. Keim. His first scientific publication, with Keim, dealt with the ecology of grasses. At some point along the way under Keim's beneficent influence, Beets became interested in fundamental genetics and was persuaded to apply to the graduate school at Cornell University instead of going back to the farm. He entered Cornell in 1927 with a graduate assistantship and shortly afterward joined R. A. Emerson's research group on the cytogenetics of maize. Corn genetics was new and exciting for Beets, and Emerson and his team— which included Barbara McClintock and
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Marcus Rhoades—were inspiring. The result was that in the following five years, Beets published no fewer than fourteen papers dealing with investigations on maize, all begun while he was a graduate student at Cornell. In 1928 he married Marion Hill, a graduate student in botany at Cornell, who assisted him with some of his early corn research. Their son, David, was born in 1931. Beets received his Ph.D. in 1931 and was awarded a National Research Council Fellowship to do postdoctoral work in T. H. Morgan's Division of Biology at the California Institute of Technology. At Caltech, while finishing the work on maize cytogenetics he had started at Cornell—on genes for pollen sterility, sticky chromosomes, failure of cytokinesis, and chromosome behavior in maize-teosinte hybrids (a subject he would return to in his retirement)— Beadle also began doing research on Drosophila. Out of it would come one of the most interesting investigations of his career. DROSOPHILA STUDIES: CROSSING OVER, VERMILION AND CINNABAR Beadle's Drosophila studies at Caltech were concerned with the results of crossing over within various chromosomal rearrangements. The important study of crossing over in attached-X chromosomes he conducted with Sterling Emerson showed that exchanges occur at random between any two non-sister chromatids. Another, reported jointly with A. H. Sturtevant (in a paper called "monumental" by E. B. Lewis), was the first systematic investigation of crossing over and disjunction in chromosomes bearing inversions. In 1934, Boris Ephrussi arrived at Caltech from Paris to study Drosophila genetics with Morgan and Sturtevant. He was just two years older than Beadle and they became close friends. Ephrussi soon communicated to Beadle his own interest in the problem of gene action, and the two planned a
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collaborative study on Drosophila that would use Ephrussi's skill in the techniques of tissue culture and transplantation. In mid-1935, the two men went to Paris to carry out experiments in Ephrussi's laboratory at the Institut de Biologie. Though their attempts to grow imaginal discs in tissue culture failed, they succeeded in devising a method for transplanting discs from one larva to another that allowed the discs to continue to develop. Before year's end, they had gone as far as they could with this methodology and had worked out a hypothesis to account for the interaction they observed between the vermilion and cinnabar genes in transplanted flies. The results, they showed, could be explained by the following assumptions: (1) the normal alleles of the two genes control the production of two specific substances, called the v+- and cn+-substances, both necessary for brown eye pigment formation; (2) the v+-substance is a precursor of the cn+-substance; and (3) gene mutation blocks formation of the corresponding substance. It was not clear until much later that the two substances are actually precursors of the pigment, and Ephrussi and Beadle frequently referred to them as "hormones." At the time, this small step was a great advance in the science of genetics, for it suggested that development could be broken down into series of genecontrolled chemical reactions—an idea that cried out for further investigation. It implanted in Beadle the germ of the one gene-one enzyme idea that he later brought to full flower. But first, the two eye color substances had to be identified, a process that took five years. By that time, Beets was hunting bigger game. Following his return from Paris, Beadle moved to Harvard University as an assistant professor. There, on a few brief occasions, he met a young woman who would later become my wife and who remembered him fondly afterwards as the only member of the Harvard faculty who spoke to Radcliffe undergraduates at Biology Departmental teas.
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BEADLE AND TATUM Beadle left Harvard the following year (1937) for Stanford University, where he had accepted an appointment as professor of biology. He was joined by biochemist Edward L. Tatum (1909-1975) as a research associate.1 Over the next three years, Tatum contributed his skills to the work of isolating and identifying the two eye color substances. With others, they established that the two substances were derivatives of tryptophan. By 1940, Tatum had obtained a crystalline preparation of the v+-substance, but he and Beadle were beaten to the identification by Butenandt, Weidel, and Becker, who had adopted the simple procedure of testing known metabolites of tryptophan for their biological activity. These researchers found that kynurenine is active as the v+-substance and that OH-kynurenine is active as the cn+-substance. Much later it was shown that condensation of two molecules of OH-kynurenine forms brown pigment. Despite this setback in the laboratory, the years from 1937 to 1939 were not wasted for Beadle. During this period, he joined A. H. Sturtevant in writing a superlative textbook, An Introduction to Genetics (1939,5), praised by J. A. Moore as "the complete statement of classical genetics." NEUROSPORA CRASSA AND GENE ACTION As a result of his Drosophila experience it became clear to Beadle that an entirely different method was needed to make headway with the problem of gene action. No other nonautonomous traits were known in Drosophila, and the autonomous ones—of which there were many—were of such towering complexity from the biochemical standpoint that it was hopeless to attempt to reduce them to their individual chemical steps. Beets enjoyed telling how the solution to this problem came to him while he was listening to Tatum lecture in a
1
See p. 356 for Joshua Lederberg's memoir of Tatum.
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course on comparative biochemistry. Microbial species, Beets learned, differ in their nutritional requirements even though they share the same basic biochemistry. If these differences were genetic in origin, he thought to himself, it should be possible to induce gene mutations that would produce new nutritional requirements in the test organism. Such an approach, if successful, would allow the researchers to identify genes governing known biochemical compounds immediately, as opposed to the years needed to identify the unknown substances controlled by the usual kinds of genes, including most of those then known. What was needed for such an undertaking was an organism that was genetically workable that could be grown on a chemically defined medium. Beadle knew just the organism. While still a graduate student at Cornell, he had heard about Neurospora crassa, the red bread mold. B. O. Dodge had come to the campus from the New York Botanical Garden to give a lecture on Neurospora. Beets remembered clearly that the lecture dealt with the genetics of the organism, including results on first-and second-division segregations of the mating-type and other loci. Even years later Beets was pleased to recall that he and a few other graduate students had been able to explain to the skeptical Dodge that his data could be explained by crossing over—or the lack of it— between the gene and its centromere. Dodge had played an important role in the history of Neurospora. It was he who discovered that the ascospores could be germinated by heat, thus closing its life cycle and making the organism accessible for genetic study. He also did basic studies on its genetics and was enthusiastic about its possibilities for genetic research. He convinced T. H. Morgan, a close friend, to take some cultures with him to Pasadena when, in 1928, Morgan went out to found the Division of Biology at Caltech. Dodge, according to Beadle, told
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Morgan that Neurospora would be "more important than Drosophila some day," and, in Pasadena, Morgan assigned the cultures to graduate student Carl Lindegren, for his thesis in genetics. Lindegren studied the relation between first-and second-division segregations and crossing over. He completed his thesis in 1931, the year Beadle arrived at Caltech. In 1940 the question of the nutritional requirements of Neurospora was still an open one. Previous workers had used nutrient agar as the growth medium, but this would not do for the experiment Beadle had in mind. Related fungi, however, were known to have simple requirements, and Tatum soon showed that Neurospora would grow on a synthetic medium containing sugar, salts, and a single growth factor—biotin—thenceforth referred to as "minimal medium." Fortunately, purified concentrates of biotin had recently become available, and nothing now stood in the way of an experimental test of Beadle's idea. The final step was to clear the Drosophila cultures out of the Stanford lab and convert it into a laboratory for Neurospora genetics. The plan was to x-ray one parent of a cross and collect offspring (haploid ascospores isolated by hand) onto a medium designed to satisfy the maximum number of possible nutritional requirements (so-called "complete medium"). The resulting cultures would next be transferred to minimal medium. Growth on complete medium, combined with failure to grow on minimal medium, was to be taken as presumptive evidence of an induced nutritional requirement. The requirement would be identified, if possible, and the culture would be crossed to wild type to determine its heritability. This scheme, in its time, was breathtakingly daring. Some nongeneticists still suspected that genes governed only trivial biological traits, such as eye color and bristle pattern, while important characters were determined in the cytoplasm by
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an unknown mechanism. Many geneticists believed that gene action was far too complex to be resolved by any simple experiment. Indeed, the outcome of Beadle and Tatum's trial run was so uncertain that they agreed at the outset to test 5,000 ascospores before giving up the project and—to avoid early disappointment—isolated and stored over a thousand spores before testing any of them. Success came with spore no. 299, which gave rise to a culture that grew on complete but not on minimal medium unless this was supplemented with pyridoxine. This mutant was followed by others showing requirements for thiamine and p-aminobenzoic acid, respectively. All three requirements were inherited as single-gene defects in crosses to wild type. These mutants were the subject of the first Neurospora paper by Beadle and Tatum (1941,2). Before long, mutants requiring amino acids, purines, and pyrimidines were also found, and the science of biochemical genetics had been born. Beadle recognized that he and Tatum had discovered a new world of genetics and that more hands would be needed to explore it. Early in the fall of 1941 he came to Caltech to give a seminar on the new discoveries and to recruit a couple of research associates to join the enterprise. Since the first BeadleTatum paper on Neurospora had yet to be published, no one in the audience had an inkling of what was to come. The seminar was memorable. I recorded my recollection of it in an article written in honor of Beadle's seventieth birthday: "The talk lasted only half an hour, and when it was suddenly over, the room was silent. The silence was a form of tribute. The audience was thinking: Nobody with such a discovery could stop talking about it after just thirty minutes—there must be more. Superimposed on this thought was the realization that something historic had happened. Each one of us, I suspect, was mentally surveying, as best he could, the consequences of the revolution that had just taken place. Finally, when it became clear that
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Beadle had actually finished speaking, Prof. Frits Went—whose father had carried out the first nutritional studies on Neurospora in Java at the turn of the century—got to his feet and, with characteristic enthusiasm, addressed the graduate students in the room. This lecture proved, said Went, that biology is not a finished subject—there are still great discoveries to be made!" Neurospora Newsletter 20(1973):4-6
BEADLE AS LABORATORY HEAD David Bonner and I accepted appointments with Beadle and joined his group at Stanford the following year. Later, H. K. Mitchell and Mary Houlahan (Mitchell) came. There were also graduate students (including A. H. Doermann and Adrian Srb) and a steady turnover of visitors in the lab. The next four years were the most exciting of my life, and I imagine the same was true for everyone else in the lab. Before the Neurospora revolution, the idea of uniting genetics and biochemistry had been only a dream with a few scattered observations. Now, biochemical genetics was a real science, and it was all new. Incredibly, we privileged few had it all to ourselves. Every day brought unexpected new results, new mutants, new phenomena. It was a time when one went to work in the morning wondering what new excitement the day would bring. Beadle presided over this scientific paradise with the enthusiasm, intelligence, and good humor that characterized everything he did. He was a popular and much admired boss. He worked in the lab with everyone else. He especially enjoyed working with his hands, and he had plenty of opportunity to indulge himself in this regard. The laboratories were located in the basement (the "catacombs") of Jordan Hall, a location that gave them a certain remoteness from the campus. There were a bench and lathe in the lab, and Beets used these to make small equipment and do minor repairs around the place; he called the campus shops only for major work and did as much as possible him
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self. I came to work early one morning and found him painting one of the rooms. All this was in addition to his research and his teaching duties as a professor of biology. He always did more than anyone else. I recall going to a lab picnic at the beach one summer day, over the coast range of hills from the Stanford campus. We were bicycling to save gas, huffing and wheezing (we had no gears then); Beets differed from the rest of us only in that he was carrying a watermelon on his handlebars. Beets knew his responsibilities and took them seriously. It was wartime, and he concerned himself with all that implied for the pursuit of fundamental research. He had to find financial support for the program while trying to keep his group together. He succeeded on both scores, obtaining support from both the Rockefeller and Nutrition foundations—support that continued throughout the war and even afterwards. The Committee on Medical Research of the Office of Scientific Research and Development classified the Neurospora program as essential to the war effort. As I recall, no senior researcher or graduate student was drafted, although some of us were called up for physical examinations. Practical applications of Neurospora research were of potential utility to the war effort—in developing bioassays for vitamins and amino acids in preserved foods, and in searching for new vitamins and amino acids. Although the major thrust of the lab remained basic science, we worked on both these applications during the war years. Toward the end of World War II, Beadle was asked by the War Production Board to devote part of the effort of the lab to seeking mutants of Penicillium with increased yields of penicillin. He complied, of course, but we were not successful in this endeavor. The biochemical and genetic studies carried out between 1941 and 1945 on Neurospora mutants in the Stanford laboratory showed that the biosynthesis of any given substance
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of the organism is under the control of a set of nonallelic genes. Mutation of any one of these genes results in loss of the synthesis, due to blocking of a single step in the biosynthetic pathway. Beadle summarized the whole field of biochemical genetics in an historic article in Chemical Reviews (1945,4). He proposed that the biochemical actions of genes could be explained by assuming that genes are responsible for enzyme specificity, the relation being that ''a given enzyme will usually have its final specificity set by one and only one gene. The same is true of other unique proteins, for example, those functioning as antigens." This statement became known as the "one gene-one enzyme" hypothesis of gene action and is Beadle's major legacy to fundamental genetics. Controversial at first (the controversy itself is an interesting reflection on the state of genetics at the time), it was eventually proved to be correct. Yet, important though this summary statement of the Neurospora findings is in the history of science, there is little doubt that Beets' most inspired contribution to genetics was the method he devised with Tatum to produce the mutants from which the theory was derived. He showed how an important class of lethal mutations could be recovered by the use of a microorganism with known nutritional requirements. The same method, as Tatum later found, could be applied to bacteria. Tatum's student Joshua Lederberg, using the resulting mutants, demonstrated genetic recombination in E. coli and thereby founded modern bacterial genetics. Beadle, Tatum, and Lederberg shared the Nobel Prize in 1958. CHAIRMAN OF THE DIVISION OF BIOLOGY, CALIFORNIA INSTITUTE OF TECHNOLOGY With the war drawing to a close in 1945, Tatum departed Stanford for Yale University and the team of Beadle and
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Tatum dissolved. The following year Beadle returned to Caltech—this time to succeed T. H. Morgan as chairman of the Division of Biology. Morgan had died, and there was a need to find as his successor a first-rate biologist who would continue the Morgan tradition: a strong emphasis on experimental, quantitative, and chemical biology. (Biochemistry had been included in Caltech's Biology Division since its inception.) It is interesting that the key figure in the negotiations on the Caltech side was the chemist Linus Pauling. Pauling had a lively interest in the new genetics, understood its importance, and later made important contributions to it. Beadle was the ideal choice, but it is doubtful if he would have made the move if it were not for Pauling's intercession. For a time after returning to Caltech, Beets continued with laboratory research, but administrative matters began to absorb his attention and finally swallowed him up. He stopped working in the lab. His last research paper on Neurospora was published with David Bonner in 1946 (1946,1). After that, and for the next thirty years, his scientific writings consisted of reviews, lectures, historical essays, and a prizewinning book for young people, The Language of Life: An Introduction to the Science of Genetics (1966,1). The book was coauthored with his second wife, Muriel Barnett, a writer, whom he married in 1953 following his divorce. (Muriel's first husband having died, Beadle legally adopted her son, Redmond.) In an autobiographical sketch published in 1974, Beets made the following revealing statement about his decision to give up laboratory research: "In my own situation, I tried a quarter of a century ago what I thought of as an experiment in combining research in biochemical genetics with a substantial commitment to academic administration. I soon found that, unlike a number of my more versatile colleagues, I could not do justice to
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both. Finding it increasingly difficult to reverse the decision I had made, I saw the commitment to administration through as best I could, often wondering if I could have come near keeping up with the ever increasing demands of research had I taken the other route. My doubts increased with time."
He finally did in fact return to research, but only after his retirement. As successor to the legendary Morgan, Beadle fulfilled all expectations. Faculty appointments made during his tenure as chairman included Max Delbrück, Renato Dulbecco, Ray Owen, Robert Sinsheimer, and Roger Sperry. These men were not only eminent, their appointment set the direction of the Biology Division's post-war growth toward molecular, cellular, and behavioral biology—a direction the Division has followed ever since. In addition, the material wealth of the Division increased considerably during Beadle's tenure. Not the least of the additions were two new laboratory buildings. Informal, unaffected, and open, Beets was later described as a chairman who steered the Division without actually seeming to run it. He was at the same time hardheaded and witty, and his insights often took the form of memorable quips. My favorite of these—because it is both true and pure (unmistakable) Beadle—was: "It's hard to make a good theory—a theory has to be reasonable, but a fact doesn't." I quoted this saying to great effect at a meeting on the origin of life held in Moscow in 1957, and I told Beets about it when I got back to Pasadena. He, as usual, could not remember saying it. In 1961 Beadle left Caltech to become president of The University of Chicago. Why he took this job and what he did after he arrived in Chicago was for years a mystery to me and, I suspect, to most (if not all) of his old scientific friends. Everybody who visited The University at that time knew that it was in trouble because of urban decay in the surrounding
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neighborhood. A little later, we heard about the spectacular student sit-ins, but none of this seemed to connect with the George Beadle we knew. The mystery was cleared up in 1972—four years after Beadle's retirement— with the publication of a book by Muriel Beadle entitled Where Has All The Ivy Gone?, an honest and highly entertaining account of the Beadles' years at The University of Chicago. It explains why The University wanted Beets as its president (to restore its academic standing after difficult years that saw the loss of many first-class faculty members); why he took the job (it was put to him as a challenge by a persuasive Dean of the Law School, Edward Levi, who later succeeded Beadle as president); and what he did there (a great deal). Friends of mine at The University have informed me that Beadle was much admired as president and that he did stanch the loss of faculty, particularly in the sciences and in medicine. He is also remembered by many for the garden he established on the campus near the president's house, where he could be observed working in the early morning. (Some were surprised to discover that this man was the president of the University, having thought him a hired gardener!) In 1968 Beadle attained mandatory retirement age. He and Muriel decided to remain in Chicago and bought a home in Hyde Park, one of the neighborhoods saved by the urban renewal program they had both worked hard on. Beets then returned to research, after twenty-three years in the wilderness. The problem he chose to investigate—the origin of maize—was one he was familiar with from his Cornell days. Maize is a cultivated plant that cannot survive in the wild. How did it arise? R. A. Emerson and Beadle showed that it is closely related genetically and cytologically to teosinte, a plant that grows wild in Mexico and Guatemala. They considered teosinte the most plausible ancestor of maize. In 1939, Beadle found that
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teosinte seeds—which are enclosed in a hard coat that makes them inedible— can be popped like ordinary popcorn, which would give prehistoric Americans an incentive to grow teosinte as a food plant. Once it was under cultivation, mutations could have been selected that, in time, would have transformed it into maize. This theory was criticized by Paul Mangelsdorf, primarily because there was little archaeological evidence to support it. In its place he proposed that maize evolved from a hypothetical wild corn, now extinct. In his retirement Beadle decided to gather more evidence on the question. He displayed the same vigor and inventiveness in this undertaking that had distinguished the researches of his younger days. In a lecture he delivered at Caltech in 1978 on the occasion of the fiftieth anniversary of the founding of the Biology Division, he summarized his findings. It was a brilliant tour de force, touching on every aspect of the subject: genetics, linguistics, palynology, archaeology, folklore, animal behavior. (What does a squirrel do when given seeds of maize and teosinte?) He described an experiment he had made on himself to decide whether teosinte meal was edible. He was informative, witty, and persuasive, his conclusion unambiguous: "Just when and where the American Indians transformed teosinte into corn we do not know, but it was surely the most remarkable single plant-breeding achievement of all time." This must have been one of Beets's last public lectures. As a finale to a scientific life it could hardly have been better. George Beadle has passed into history now. His papers are rarely read anymore; his lively presence is no longer felt. But the changes he brought about in biology are permanent. No scientist could ask for a grander memorial than that. For their comments on the manuscript, I am indebted to Muriel Beadle, Elizabeth Bertani, James Bonner, Edward and
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Pamela Lewis, and Ray Owen. For answering my questions on a variety of matters, I thank Marion Beadle, Walton Galinat, Barbara McClintock, Oliver Nelson, Jane Overton, and Bernard Strauss. And I thank the personnel of the Caltech Archives for their unfailing courtesy.
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HONORS AND DISTINCTIONS Honorary Degrees Doctor of Science 1947 Yale University 1949 University of Nebraska 1952 Northwestern University 1954 Rutgers University 1955 Kenyon College 1956 Wesleyan University 1959 Birmingham University 1959 Oxford University 1961 Pomona College 1962 Lake Forest College 1963 University of Rochester 1963 University of Illinois 1964 Brown University 1964 Kansas State University 1964 University of Pennsylvania 1966 Wabash College 1967 Syracuse University 1970 Loyola University, Chicago 1971 Hanover College 1972 Eureka College 1973 Butler University 1975 Gustavus Adolphus College 1976 Indiana State University Legum Doctor (LL. D.) 1962 University of California, Los Angeles 1963 University of Miami 1963 Brandeis University 1966 Johns Hopkins University 1966 Beloit College 1969 University of Michigan
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Litterarum Humaniorum Doctor (L.H.D.) 1966 Jewish Theological Seminary of America 1969 DePaul University 1969 University of Chicago 1969 Canisius College 1969 Knox College 1971 Roosevelt University 1971 Carroll College Doctor of Public Service 1970 Ohio Northern University Awards 1950 Lasker Award 1951 Dyer Award 1953 Emil Christian Hansen Prize (Denmark) 1958 Albert Einstein Commemorative Award in Science 1958 Nobel Prize in Physiology or Medicine (with E. L. Tatum and J. Lederberg) 1959 National Award, American Cancer Society 1960 Kimber Genetics Award 1967 Priestley Memorial Award 1967 Edison Prize, Best Science Book for Youth (with Muriel Beadle) 1972 Donald Forsha Jones Medal 1984 Thomas Hunt Morgan Medal
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GEORGE WELLS BEADLE
Professional And Honorary Societies Genetics Society of America (president, 1945) American Association for the Advancement of Science (president, 1955) National Academy of Sciences (Council, 1969-1972) American Philosophical Society American Academy of Arts and Sciences Royal Society Danish Royal Academy of Sciences Japan Academy Instituto Lombardo di Scienze e Lettre (Milan) Genetical Society of Great Britain Indian Society of Genetics and Plant Breeding Indian Natural Science Academy Chicago Horticultural Society (president, 1968-1971) Phi Beta Kappa Sigma Xi
45
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Selected Bibliography
1927 With F. D. Keim. Relation of time of seeding to root development and winter survival of fall seeded grasses and legumes. Ecology, 8:251-64. 1928 With B. McClintock. A genic disturbance of meiosis in Zea mays. Science, 68:433. 1929 Yellow stripe-A factor for chlorophyll deficiency in maize located in the Pr pr chromosome. Am. Nat., 68:189-192. A gene for supernumerary mitoses during spore development in Zea mays. Science, 70:406-7. 1930 Heritable characters in maize. J. Hered., 21:45-48. Genetical and cytological studies of Mendelian asynapsis in Zea mays . Cornell Univ. Memoir, 129:3-23. A fertile tetraploid hybrid between Euchlaena perennis and Zea mays . Am. Nat., 69:190-92. 1931 A gene in Zea mays for failure of cytokinesis during meiosis. Cytol., 3:142-55. A gene in maize for supernumerary cell divisions following meiosis. Cornell Univ. Memoir, 135:3-12. 1932 A possible influence of the spindle fibre on crossing-over in Drosophila. Proc. Natl. Acad. Sci. USA, 18:160-65. A gene in Zea mays for failure of cytokinesis during meiosis. Cytol., 3:142-55. Genes in maize for pollen sterility. Genet., 17:413-31. The relation of crossing over to chromosome association in Zea-Euchlaena hybrids. Genet., 17:481-501. Studies of Euchlaena and its hybrids with Zea. I. Chromosome be
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havior in Euchlaena mexicana and its hybrids with Zea mays. ZIAVA, 62:291-304. With R. Emerson. Studies of Euchlaena and its hybrids with Zea. II. Crossing over between the chromosomes of Euchlaena and those of Zea . ZIAVA, 62:305-15. A gene for sticky chromosomes in Zea mays. ZIAVA, 63:195-217. 1933 Studies of crossing-over in heterozygous translocations in Drosophila melanogaster. ZIAVA, 65:111-28. With S. Emerson. Crossing-over near the spindle fiber in attached-X chromosomes of Drosophila melanogaster. ZIAVA, 65:129-40. Further studies of asynaptic maize. Cytol., 4:269-287. Polymitotic maize and the precocity hypothesis of chromosome conjugation. Cytol., 5:118-21. 1934 Crossing-over in attached-X triploids of Drosophila melanogaster. J. Genet., 29:277-309. 1935 Crossing over near the spindle attachment of the X chromosomes in attached-X triploids of Drosophila melanogaster. Genet., 20:179-91. With S. Emerson. Further studies of crossing-over in attached-X chromosomes of Drosophila melanogaster. Genet., 20:192-206. With R. A. Emerson and A. C. Fraser. A summary of linkage studies in maize. Cornell Univ. Memoir, 180:3-83. With A. H. Sturtevant. X chromosome inversions and meiosis in Drosophila melanogaster. Proc. Natl. Acad. Sci. USA, 21:384-90. With B. Ephrussi. La transplantation des disques imaginaux chez la Drosophile. C. R. Acad. Sci., 201:98. With B. Ephrussi. Différenciation de la couleur de l'oeil cinnabar chez la Drosophile. C. R. Acad. Sci., 201:620. With B. Ephrussi. La transplantation des ovaires chez la Drosophile. Bull. Biol. Belg., 69:492-502. With B. Ephrussi. Sur les conditions de l'auto-différenciation des caractères mendéliens. C. R. Acad. Sci., 201:1148. With B. Ephrussi. Transplantation in Drosophila. Proc. Natl. Acad. Sci. USA, 21:642-46.
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GEORGE WELLS BEADLE
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1936 With B. Ephrussi. A technique of transplantation for Drosophila. Am. Nat., 70:218-25. With B. Ephrussi. The differentiation of eye pigments in Drosophila as studied by transplantation. Genet., 21:225-47. With B. Ephrussi. Development of eye colors in Drosophila: Transplantation experiments with suppressor of vermilion. Proc. Natl. Acad. Sci. USA, 22:536-40. With B. Ephrussi and C. W. Clancy. Influence de la lymphe sur la couleur des yeux vermilion chez la Drosophile. C. R. Acad. Sci., 203:545. With A. H. Sturtevant. The relation of inversions in the X chromosome of Drosophila melanogaster to crossing-over and disjunction. Genet., 21:554-604. With Th. Dobzhansky. Studies on hybrid sterility IV. Transplanted testes in Drosophila pseudoobscura. Genet., 21:832-40. With B. Ephrussi. Development of eye colors in Drosophila: Studies of the mutant claret. J. Genet., 33:407-10. 1937 With B. Ephrussi. Development of eye colors in Drosophila: Transplantation experiments on the interaction of vermilion with other eye colors. Genet., 22:65-75. With B. Ephrussi. Development of eye colors in Drosophila: Diffusible substances and their interrelations. Genet., 22:76-85. With B. Ephrussi. Développement des couleurs des yeux chez la Drosophile: Influence des implants sur la couleur des yeux de l'hôte. Bull. Biol. Belg., 71:75-90. With B. Ephrussi. Développement des couleurs des yeux chez la Drosophile: Revue des experiences de transplantation. Bull. Biol. Belg., 71:54-74. With C. W. Clancy. Ovary transplants in Drosophila melanogaster: Studies of the characters singed, fused, and female-sterile. Biol. Bull, 72:47-56. With B. Ephrussi. Development of eye colors in Drosophila: The mutants bright and mahogany. Am. Nat., 71:91-95. The development of eye colors in Drosophila as studied by transplantation. Am. Nat., 71:120-26. With K. V. Thimann. Development of eye colors in Drosophila:
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GEORGE WELLS BEADLE
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Extraction of the diffusible substances concerned. Proc. Natl. Acad. Sci. USA, 23:143-46. Development of eye colors in Drosophila: Fat bodies and Malpighian tubes as sources of diffusible substances. Proc. Natl. Acad. Sci. USA, 23:146-52. With C. W. Clancy and B. Ephrussi. Development of eye colours in Drosophila: Pupal transplants and the influence of body fluid on vermilion. Proc. R. Soc. London, 122:98-105. The inheritance of the color of Malpighian tubes in Drosophila melanogaster . Am. Nat., 71:277-79. With B. Ephrussi. Ovary transplants in Drosophila melanogaster. Meiosis and crossing-over in superfemales. Proc. Natl. Acad. Sci. USA, 23:356-60. With B. Ephrussi. Development of eye colors in Drosophila: Production and release of cn+ substance by the eyes of different eye color mutants. Genet., 22:479-83. Chromosome aberration and gene mutation in sticky chromosome plants of Zea mays. Cytol. Fujii Jubilee, pp. 43-56. Development of eye colors in Drosophila: Fat bodies and Malpighian tubes in relation to diffusible substances. Genet., 22:587-611. 1938 With L. W. Law. Influence on eye color of feeding diffusible substances to Drosophila melanogaster. Proc. Soc. Exp. Biol. Med., 37:621-23. With R. Anderson and J. Maxwell. A comparison of the diffusible substances concerned with eye color development in Drosophila, Ephestia and Habrobracon. Proc. Natl. Acad. Sci. USA, 24:8085. With E. L. Tatum. Development of eye colors in Drosophila: Some properties of the hormones concerned. J. Gen. Physiol., 22:239-53. With E. L. Tatum and C. W. Clancy. Food level in relation to rate of development and eye pigmentation in Drosophila melanogaster. Biol. Bull., 75:447-62. 1939 Physiological aspects of genetics. Annu. Rev. Physiol., 1:41-62. Teosinte and the origin of maize. J. Hered., 30:245-47.
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With E. L. Tatum and C. W. Clancy. Development of eye colors in Drosophila: Production of v+ hormone by fat bodies. Biol. Bull., 77:407-14. With E. L. Tatum. Effect of diet on eye color development in Drosophila melanogaster. Biol. Bull., 77:415-22. With A. H. Sturtevant. An Introduction to Genetics. Philadelphia: W. B. Saunders. 1940 With E. L. Tatum. Crystalline Drosophila eye color hormone. Science, 91:458. 1941 With E. L. Tatum. Experimental control of development and differentiation. Am. Nat., 75:107-16. With E. L. Tatum. Genetic control of biochemical reactions in Neurospora. Proc. Natl. Acad. Sci. USA, 27:499-506. 1942 With E. L. Tatum. Genetic control of biochemical reactions in Neurospora: An ''aminobenzoicless" mutant. Proc. Natl. Acad. Sci. USA, 28:234-43. 1943 With N. H. Horowitz. A microbiological method for the determination of choline by use of a mutant of Neurospora. J. Biol. Chem., 150:325-33. With D. Bonner and E. L. Tatum. The genetic control of biochemical reactions in Neurospora: A mutant strain requiring isoleucine and valine. Arch. Biochem., 3:71-91. With F. J. Ryan and E. L. Tatum. The tube method of measuring the growth rate of Neurospora. Am. J. Bot., 30:784-99. 1944 With E. L. Tatum and D. Bonner. Anthranilic acid and the biosynthesis of indole and tryptophan by Neurospora. Arch. Biochem., 3:477-78. With V. L. Coonradt. Heterocaryosis in Neurospora crassa. Genet., 29:291-308.
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An inositolless mutant strain of Neurospora and its use in bioassays. J. Biol. Chem., 156:683-89. 1945 With H. K. Mitchell and D. Bonner. Improvements in the cylinder-plate method for penicillin assay. J. Bacteriol., 49:101-4. With E. L. Tatum. Biochemical genetics of Neurospora. Ann. Mo. Bot. Gard., 32:125-29. With N. H. Horowitz, D. Bonner, H. K. Mitchell, and E. L. Tatum. Genic control of biochemical reactions in Neurospora. Am. Nat., 79:304-17. Biochemical genetics. Chem. Rev., 37:15-96. Genetics and metabolism in Neurospora. Physiol. Rev. 25:643-63. With E. L. Tatum. Neurospora. II. Methods of producing and detecting mutations concerned with nutritional requirements. Am. J. Bot., 32:678-86. The genetic control of biochemical reactions. Harvey Lect., 40:179-94. Genes and the chemistry of the organism. Am. Sci., 34:31-53, and 76. 1946 With D. Bonner. Mutant strains of Neurospora requiring nicotinamide or related compounds for growth. Arch. Biochem., 11:319-28. High-frequency radiation and the gene. Science Life in the World, New York: McGraw-Hill, Vol. 2, pp. 163-93. 1959 Genes and chemical reactions in Neurospora. In: Les Prix Nobel, pp. 147-59. Also in: Science, 129:1715-19. 1960 Evolution in microorganisms, with special reference to the fungi. ANL, 47:301-19. 1963 Genetics and modern biology. Jayne Lectures for 1962. Am. Philos. Soc., 57:1-73.
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1966 With Muriel B. Beadle. The Language of Life: An Introduction to the Science of Genetics. New York: Doubleday (Doubleday Anchor-book, 1967). Biochemical genetics: Some recollections. In: Phage and the Origins of Molecular Biology, eds. J. Cairns, G. S. Stent, and J. D. Watson, Cold Spring Harbor Laboratory of Quantitative Biology, pp. 23-32. 1972 The mystery of maize. Field Mus. Nat. Hist. Bull., 43:2-11. 1973 Thomas Hunt Morgan. In: Dictionary of American Biography, Supplement 3 (1941-1945), New York: Chas. Scribners Sons, pp. 538-41. 1974 Recollections. Annu. Rev. Biochem. 43:1-13. 1980 The origin of maize. In: Genes, Cells, and Behavior, eds. N. H. Horowitz and E. Hutchings, Jr., San Francisco: W. H. Freeman and Co., pp. 81-87. Ancestry of corn. Sci. Am., 242(1):112-19. 1981 Origin of corn: Pollen evidence. Science, 213:890-92.
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Solomon A. Berson April 22, 1918-April 11, 1972
By J. E. Rall Solomon A. Berson was born April 22, 1918, in New York City. His father, a Russian émigré who studied chemical engineering at Columbia University, went into business and became a reasonably prosperous fur dyer and the owner of his own company. He was a competent mathematician, enjoyed chess, and played duplicate bridge sufficiently well to become a life master. Solomon Berson—Sol to his many friends—was the eldest of three children: Manny, the second, became a dentist; Gloria, the youngest, married Aaron Kelman, a physician and a friend of Sol's. In 1942 Sol married Miriam (Mimi) Gittleson. They had two daughters whom Sol adored, and a happy, warm family life. Sol discovered a taste and aptitude for music early in life. He played in chamber music groups in high school and developed into an accomplished violinist. My impression has always been that he liked the presto movements best —he clearly led his entire life at a presto pace. He also played chess in high school and became sufficiently expert to play multiple games blindfolded. In 1934 he entered the City College of New York and, in 1938, received his degree. At that time Sol decided he wanted to study medicine. He applied to twenty-one different medical schools but was
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turned down by every one. Instead, he went to New York University, where he earned a Master of Science degree (in 1939) and a fellowship to teach anatomy at the NYU Dental School, where his brother, Manny, was a student. He was finally admitted to New York University Medical School in 1941 and, a member of Alpha Omega Alpha honorary medical fraternity, received his M. D. degree in 1945. Sol interned at Boston City Hospital from 1945 to 1946, then joined the Army. Serving from 1946 to 1948, he went from first lieutenant to captain. He spent 1948 to 1950 at the Bronx Veterans Administration Hospital for further training in internal medicine, then decided to go into research. In the spring of 1950, Rosalyn Yalow, assistant chief of the Radioisotope Service in the Radiotherapy Department at the Bronx V. A. Hospital, was looking for a physician qualified in internal medicine and asked Bernard Strauss, Chief of Medicine, to recommend someone. He suggested Solomon Berson, though Sol had already arranged to go to the V. A. Hospital in Bedford, Massachusetts. Strauss nevertheless encouraged Yalow to interview Berson, and, during the interview, Sol presented her with a series of mathematical puzzles. Since Ros Yalow is not a bad mathematician and has a sense of humor, she offered Sol the position, and he accepted. So began a collaboration that lasted until Sol's death in 1972. For about a year, while working full-time in the Radioisotope Service, Sol "moonlighted" in the private practice of medicine. He found clinical practice gratifying and his patients adored him, but his work at the V. A. became too engrossing and he gave up his practice. In 1954 when the Radioisotope Service became independent of Radiotherapy, Sol became its chief. The Radioisotope Service was the forerunner of the modern Nuclear Medicine Service, and the thyroid clinic he established there in 1950 continues even now to function as he planned it.
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One of Berson and Yalow's early papers in thyroid physiology exemplifies the method that would characterize their research for the next few decades. The research was clinical, involving both normal and diseased subjects; it was mathematically and logically precise; it went beyond specification of the technical requirements of the study to make assumptions inherent in the measurements explicit. In this early study, Berson and Yalow answered precisely the question of what the human thyroid's so-called "uptake" of radioactive iodine represents. To do this they focused on the quantity of iodide the thyroid clears from the blood per unit of time, having first determined that this was the only physiological constant they could measure that also described one of the functions of the thyroid. In 1952 they published their classic paper on the subject, which is still quoted. It is particularly remarkable that Berson, who lacked extensive formal training in mathematics and physical chemistry, used both—to such good effect— in his research. About this time Berson and Yalow decided that an excellent way to investigate the metabolism of a variety of biologically interesting compounds was to label them with a radioactive isotope. They were among the first to label serum albumin with radioactive iodine to study its metabolism. This work, reported in 1953, was one of the earliest studies to show how long albumin lasted in the circulation and the kinetic processes governing its synthesis and degradation. Shortly thereafter the two researchers used insulin labeled with radioactive iodine to test the hypothesis that diabetes of the maturity-onset type was due to an excessively rapid degradation of normally-secreted insulin. They found that when labeled insulin was given to subjects who had been treated with insulin either for diabetes or as shock therapy for schizophrenia, it disappeared more slowly than did insulin administered to normal subjects. They surmised that
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this was due to the formation of soluble antigen antibody complexes that were metabolized more slowly than free insulin. Analyzing serum by paper electrophoresis, they showed that in subjects previously treated with insulin, labeled insulin added later was found in the B-γ region of serum proteins rather than as free insulin. This observation, published in 1956, led Berson and Yalow to consider the reversible equilibrium between a binding protein and a ligand, and they soon realized that a method using binding equilibria could be developed to measure very small amounts of material. They then developed the general method of radioimmunoassay on the theory that—if a substance (in their early work, an antibody) can be produced that binds a ligand—the following situation obtains:
One must be able to separate bound ligand (PL + PL') from free P (protein or antibody) and free ligand, L (in this case, insulin). The actual assay is performed with the experimental solution containing a small but unknown amount of ligand, to which an extremely small amount of radioactively labeled ligand (L') is added. After attaining equilibrium and after electrophoretic separation, the bound and free amounts of radioactivity are measured. A series of standard reactions containing labeled ligand and progressively increasing amounts of unlabeled ligand is prepared simultaneously exactly as above. With increasing amounts of unlabeled ligand, progressively increasing amounts of labelled ligand will be displaced from the antibody. Interpolation of the experimental results on the standard curve then permits accurate estimation of the amount of ligand in the experimental solutions.
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The researchers had to solve several additional problems, however, before their method could be accepted as both sensitive and accurate. Most scientists at that time believed that insulin did not produce antibodies, though Berson and Yalow, building on the work of others, had demonstrated that animal insulins used for the treatment of diabetes did, in fact, produce antibodies in man. (In guinea pigs they had observed specific, high-affinity antibodies to animal insulins that reacted well with human insulin.) It was also important to label the insulin so that there were no degradation products and it could be separated out as a clean component after labeling. When, in 1959, these procedures were finally perfected, Berson and Yalow were able to report the success of their method for measuring insulin concentration in human plasma. This accomplishment led to a series of studies on insulin secretion and the effect of human diabetes on insulin concentration in plasma. It had been known for many years that there were differences between individuals who developed diabetes as youngsters—who were more likely to go into ketoacidosis—and older diabetics with a tendency to obesity, who rarely went into ketoacidosis. The younger group of patients generally exhibited greatly reduced quantities of insulin in the pancreas and bloodstream and were, therefore, insulin deficient. By radioimmunoassay of insulin, Berson and Yalow showed that many older diabetics had normal or even elevated levels of insulin in the bloodstream. The defect, therefore, was not in the secretion of insulin but in some subsequent step. A complicating factor in all these measurements are the antibodies most patients treated with insulin develop to it (as Berson and Yalow had demonstrated), and precisely where the defect occurs in what is now called "Type II" diabetes is still not completely understood. The high degree of specificity of the immune system
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makes radioimmunoassay capable of distinguishing closely related compounds such as thyroxine and triiodothyronine, or cortisol and corticosterone that differ only in a single hydroxyl group. The general principle, furthermore, can be extended to any system in which a specific binding material is available, such as thyroxine-binding globulin for the measurement of thyroxine, or intrinsic factor for measurement of vitamin B12. Over the next few years, the Radioisotope Service at the Bronx VA Hospital saw an enormous burst of activity as Berson and Yalow adapted radioimmunoassay to the analysis of parathyroid hormone, growth hormone, ACTH, and gastrin, which were until then impossible to measure in blood with any degree of accuracy. In 1963, for example, Berson and Yalow showed for the first time that the secretion of growth hormone was acutely regulated by stimuli such as hypoglycemia and exercise. They also found parathyroid hormone in the blood in several forms that could be differentiated by antibodies with different specificities. They measured gastrin, then a newly-discovered hormone that stimulates secretion of stomach acid, and showed that it existed in several forms of varying size in human plasma. Radioimmunoassay has since been adapted to the measurement of literally hundreds of different substances, ranging from steroids, to thyroid hormones, to the hepatitis B surface antigen, and the tubercle bacilli. The possibility of radioimmunoassay analysis of substances present in concentrations of 10-9 to 10-13 molar has enormously accelerated progress in many fields of biomedical research. Dr. Berson received numerous awards for this work, including the 1971 Gairdner Award, the 1971 Dickson Prize, the 1965 Banting Memorial Lecture and Banting Medal of the American Diabetes Association, the 1960 William S. Middleton Medical Research Award, and the American Diabetes
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Association's first Eli Lilly Award in 1957. In 1977, five years after his death, Dr. Rosalyn Yalow received the Nobel Prize for the ''development of radioimmunoassays of peptide hormones." Drs. Berson and Yalow patented neither the general concept of radioimmunoassay nor any of the procedures they had developed to make it so precise and sensitive an assay. But while numerous commercial laboratories made large sums of money for performing radioimmunoassays, Berson remained unconcerned. His salary at the Veterans Administration was anything but munificent. Yet, wrote Dr. Jesse Roth, one of Berson's early postdoctoral fellows, ". . . Seymour Glick and I didn't have any travel grants included in our fellowships, nor did the laboratory provide any travel funds, so our meeting expenses were paid for out of Dr. Berson's pocket." Dr. Berson continued his research at the Radioisotope Service until his death, but in 1968 accepted the professorship and chairmanship of the Department of Medicine, at the Mount Sinai School of Medicine of the City University of New York. In this position, he influenced many medical students and house staff. When he had an argument with the administration at Mount Sinai and threatened to resign, the entire house staff on the medical service agreed to resign en masse if he were to leave. Needless to say, the dispute was adjudicated and both Dr. Berson and the house staff stayed on. In spite of the heavy demands of being professor of medicine and chairman of the department in a large medical school, Dr. Berson retained close ties with Dr. Yalow and their laboratory, and the productivity of their scientific collaboration continued unabated. As is the case with many great and busy scientists, Dr. Berson was on the editorial boards of numerous journals to which he gave a surprising amount of time, carefully and
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thoughtfully reviewing articles. He was a member of many boards and advisory councils, several with the National Institutes of Health. In April 1972, the month he was elected to the National Academy of Sciences, Sol Berson died while attending a meeting of the Federation of the American Societies for Experimental Biology in Atlantic City.
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Selected Bibliography
1951 With R. S. Yalow. The use of K42-tagged erythrocytes in blood volume determinations. Science, 114:14-15. 1952 With R. S. Yalow. The effect of cortisone on the iodine accumulating function of the thyroid gland in euthyroid subjects. J. Clin. Endocrinol. Metab., 12:407-22. With R. S. Yalow, J. Sorrentino, and B. Roswit. The determination of thyroidal and renal plasma I131 clearance rates as a routine diagnostic test of thyroid dysfunction. J. Clin. Invest., 31:141-58. With R. S. Yalow. The use of K42 or P32 labeled erythrocytes and I131 tagged human serum albumin in simultaneous blood volume determinations. J. Clin. Invest., 31:572-80. With R. S. Yalow, A. Azulay, S. Schreiber, and B. Roswit. The biological decay curve of P32-tagged erythrocytes. Application to the study of acute changes in blood volume. J. Clin. Invest., 31:581-91. 1953 With R. S. Yalow, J. Post, L. H. Wisham, K. N. Newerly, M. J. Villazon, and O. N. Vazquez. Distribution and fate of intravenously administered modified human globin and its effect on blood volume. Studies utilizing I131-tagged globin. J. Clin. Invest., 32:22-32. With R. S. Yalow, S. S. Schreiber, and J. Post. Tracer experiments with I131-labeled human serum albumin: Distribution and degradation studies. J. Clin. Invest., 32:746-68. 1954 With R. S. Yalow. The distribution of I131-labeled human serum albumin introduced into ascitic fluid: Analysis of the kinetics of a three compartment catenary transfer system in man and speculations on possible sites of degradation. J. Clin. Invest., 33:377-87. With S. S. Schreiber, A. Bauman, and R. S. Yalow. Blood volume
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alterations in congestive heart failure. J. Clin. Invest., 33:578-86. With R. S. Yalow. Quantitative aspects of iodine metabolism. The exchangeable organic iodine pool, and the rates of thyroidal secretion, peripheral degradation and fecal excretion of endogenously synthesized organically bound iodine. J. Clin. Invest., 33:1533-52. 1955 With R. S. Yalow. Critique of extracellular space measurements with small ions: Na24 and Br82 spaces. Science, 121:34-36. With R. S. Yalow. The iodide trapping and binding functions of the thyroid. J. Clin. Invest., 34:186-204. With M. A. Rothschild, A. Bauman, and R. S. Yalow. Tissue distribution of I131-labeled human serum albumin following intravenous administration. J. Clin. Invest., 34:1354-58. With A. Bauman, M. A. Rothschild, and R. S. Yalow. Distribution and metabolism of I131-labeled human serum albumin in congestive heart failure with and without proteinuria. J. Clin. Invest., 34:1359-68. 1956 With R. S. Yalow, A. Bauman, M. A. Rothschild, and K. Newerly. Insulin-I 131 metabolism in human subjects: demonstration of insulin binding globulin in the circulation of insulintreated subjects. J. Clin. Invest., 35:170-90. 1957 With R. S. Yalow. Chemical and biological alterations induced by irradiation of I131-labeled human serum albumin. J. Clin. Invest., 36:44-50. With M. A. Rothschild, A. Bauman, and R. S. Yalow. The effect of large doses of desiccated thyroid on the distribution and metabolism of albumin-I131 in euthyroid subjects. J. Clin. Invest., 36:422-28. With R. S. Yalow. Serum protein turnover in multiple myeloma. J. Lab. Clin. Med., 49:386-94. With R. S. Yalow. Ethanol fractionation of plasma and electrophoretic identification of insulinbinding antibody. J. Clin. Invest., 36:642-47.
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With R. S. Yalow. Apparent inhibition of liver insulinase activity by serum and serum fractions containing insulin-binding antibody. J. Clin. Invest., 36:648-55. With R. S. Yalow, S. Weisenfeld, M. G. Goldner, and B. W Volk. The effect of sulfonylureas on the rates of metabolic degradation of insulin-I131 and glucagon-I131 in vivo and in vitro. Diabetes, 6:54-60. With A. Bauman, M. A. Rothschild, and R. S. Yalow. Pulmonary circulation and transcapillary exchange of electrolytes. J. Appl. Physiol., 11:353-61. With R. S. Yalow. Studies with insulin-binding antibody. Diabetes, 6:402-7. 1958 With R. S. Yalow. Insulin antagonists, insulin antibodies and insulin resistance. Am. J. Med., 25:155-59. With A. B. Gutman, T. F. Yü, H. Black, and R. S. Yalow. Incorporation of glycine-1-C14, glycine-1C14 and glycine-N15 into uric acid in normal and gouty subjects. Am. J. Med., 25:917-32. 1959 With S. Weisenfeld and M. Pascullo. Utilization of glucose in normal and diabetic rabbits. Effects of insulin, glucagon and glucose. Diabetes, 8:116-27. With R. S. Yalow. Quantitative aspects of reaction between insulin and insulin-binding antibody. J. Clin. Invest., 38:1996-2016. With R. S. Yalow. Species-specificity of human anti-beef, pork insulin serum. J. Clin. Invest., 38:2017-25. With R. S. Yalow. Assay of plasma insulin in human subjects by immunological methods. Nature, 184:1648-49. With R. S. Yalow. Recent studies on insulin-binding antibodies. Ann. N.Y. Acad. Sci., 82:338-44. 1960 With R. S. Yalow. Immunoassay of endogenous plasma insulin in man. J. Clin. Invest., 39:1157-75. With R. S. Yalow. Plasma insulin in man (Editorial). Am. J. Med., 29:1-8. With R. S. Yalow. Plasma insulin concentrations in nondiabetic and early diabetic subjects. Diabetes, 9:254-60.
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With R. S. Yalow, H. Black, and M. Villazon. Comparison of plasma insulin levels following administration of tolbutamide and glucose. Diabetes, 9:356-62. 1961 With R. S. Yalow. The effects of x-radiation of I131-labeled iodotyrosines in solution: the significance of reducing and oxidizing radicals. Radiat. Res., 14:590-604. With R. S. Yalow. Immunologic specificity of human insulin: application to immunoassay of insulin. J. Clin. Invest., 40:2190-98. With R. S. Yalow. Immunoassay of plasma insulin in man. Diabetes, 10:339-44. With R. S. Yalow. Preparation and purification of human insulin-I131 binding to human insulinbinding antibodies. J. Clin. Invest., 40:1803-8. With R. S. Yalow. Immunochemical distinction between insulins with identical amino acid sequences from different mammalian species (pork and sperm whale insulins). Nature, 191:1392-93. With R. S. Yalow. Plasma insulin in health and disease. Am. J. Med., 31:874-81. 1962 With R. S. Yalow. Diverse applications of isotopically labeled insulin. Trans. N.Y. Acad. Sci., 24:487-95. With R. S. Yalow. Insulin antibodies and insulin resistance. Diabetes Dig., 1:4. 1963 With R. S. Yalow. Iodine metabolism and the thyroid gland. N.Y. State J. Med., 62:35-42. With R. S. Yalow. Antigens in insulin: Determinants of specificity of porcine insulin in man. Science, 139:844-85. With R. S. Yalow, G. D. Aubarch, and J. T. Potts, Jr. Immunoassay of bovine and human parathyroid hormone. Proc. Natl. Acad. Sci. USA, 49:613-17. With J. Roth, S. M. Glick, and R. S. Yalow. Hypoglycemia: A potent stimulus to secretion of growth hormone. Science, 140:987-88. With J. Roth, S. M. Glick, and R. S. Yalow. Secretion of human growth hormone: Physiologic and experimental modification. Metabolism, 12:577-79.
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With S. M. Glick, J. Roth, and R. S. Yalow. Immunoassay of human growth hormone in plasma. Nature, 199:784-87. 1964 With J. Roth, S. M. Glick, and R. S. Yalow. Antibodies to human growth hormone (HGH) in human subjects treated with HGH. J. Clin. Invest., 43:1056-65. With R. S. Yalow. The present status of insulin antagonists in plasma. Diabetes, 13:247-59. With R. S. Yalow, S. M. Glick, and J. Roth. Immunoassay of protein and peptide hormones. Metabolism, 13:1135-53. With R. S. Yalow. Reaction of fish insulins with human insulin antiserums: Potential value in the treatment of insulin resistance. N. Engl. J. Med., 270:1171-78. With J. Roth, S. M. Glick, and R. S. Yalow. The influence of blood glucose and other factors on the plasma concentration of growth hormone. Diabetes, 13:355-61. With R. S. Yalow, S. M. Glick, and J. Roth. Radioimmunoassay of human plasma ACTH. J. Clin. Endocrinol. Metab., 24: 1219-25. 1965 With S. M. Glick, J. Roth, and R. S. Yalow. The regulation of growth hormone secretion. In: Recent Progress in Hormone Research, ed. G. Pincus, New York: Academic Press, vol. 21, pp. 241-83. With R. S. Yalow. Dynamics of insulin secretion in hypoglycemia. Diabetes, 14:341-49. With R. S. Yalow, S. M. Glick, and J. Roth. Plasma insulin and growth hormone levels in obesity and diabetes. (Conference on Adipose Tissue Metabolism and Obesity.) Ann. N.Y. Acad. Sci., 131:357-73. With R. S. Yalow. Some current controversies in diabetes research. Diabetes, 14:549-72. 1966 With R. S. Yalow. Insulin in blood and insulin antibodies. Am. J. Med., 40:676-90. With R. S. Yalow. Iodoinsulin used to determine specific activity of Iodine-131. Science, 152:205-7.
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With R. S. Yalow. Labeling of proteins—problems and practices. Trans. N.Y. Acad. Sci., 28:1033-44. With R. S. Yalow. Deamidation of insulin during storage in frozen state. Diabetes, 15:875-79. With G. Roselin, R. Assan, and R. S. Yalow. Separation of antibody-bound and unbound peptide hormones labeled with iodine131 by talcum powder and precipitated silica. Nature, 212:355-57. With R. S. Yalow. Purification of I131-parathyroid hormone with microfine granules of precipitated silica. Nature, 212:357-58. With R. S. Yalow. Parathyroid hormone in plasma in adenomatous hyperparathyroidism, uremia, and bronchogenic carcinoma. Science, 154:907-9. With R. S. Yalow. State of human growth hormone in plasma and changes in stored solutions of pituitary growth hormone. J. Biol. Chem., 241:5745-49. 1967 With R. S. Melick, J. R. Gill, Jr., R. S. Yalow, F. C. Bartter, J. T. Potts, Jr., and G. D. Aurbach. Antibodies and clinical resistance to parathyroid hormone. N. Engl. J. Med., 276: 144-47. With R. S. Yalow. Radioimmunoassays of peptide hormones in plasma. N. Engl. J. Med., 277:640-47. 1968 With R. S. Yalow. Peptide hormones in plasma. In: The Harvey Lectures, New York: Academic Press, ser. 62, 1966-1967, pp. 107-63. With R. S. Yalow. Immunochemical heterogeneity of parathyroid hormone. J. Clin. Endocrinol., 28:1037-47. With R. S. Yalow. Radioimmunoassay of ACTH in plasma. J. Clin. Invest., 47:2725-51. 1969 With R. S. Yalow, N. Varsano-Aharon, and E. Echemendia. HGH and ACTH secretory responses to stress. Horm. Metab. Res., 1:3-8. With R. S. Yalow and S. J. Goldsmith. Influence of physiologic fluctuations in plasma growth hormone on glucose tolerance. Diabetes, 18:402-8.
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With R. S. Yalow. Significance of human plasma insulin sephadex fractions. Diabetes, 18:834-39. 1970 With R. S. Yalow. Radioimmunoassay of gastrin. Gastroenterology, 58:1-14. With N. Varsano-Aharon, E. Echemendia, and R. S. Yalow. Early insulin responses to glucose and to tolbutamide in maturity-onset diabetes. Metabolism, 19:409-17. With R. S. Yalow. Size and charge distinctions between endogenous human plasma gastrin in peripheral blood and heptadecapeptide gastrins. Gastroenterology, 58:609-15. With S. J. Goldsmith and R. S. Yalow. Effects of 2-deoxy-d-glucose on insulin-secretory responses to intravenous glucose, glucagon, tolbutamide and arginine in man. Diabetes, 19:453-57. With J. H. Walsh and R. S. Yalow. Detection of Australia antigen and antibody by means of radioimmunoassay techniques. J. Infect. Dis., 121:550-54. 1971 With J. H. Walsh and R. S. Yalow. The effect of atropine on plasma gastrin response to feeding. Gastroenterology, 60:16-21. With R. S. Yalow. Further studies on the nature of immunoreactive gastrin in human plasma. Gastroenterology, 60:203-14. With R. S. Yalow. Nature of immunoreactive gastrin extracted from tissues of gastrointestinal tract. Gastroenterology, 60:215-22. With G. M. A. Palmieri and R. S. Yalow. Adsorbent techniques for the separation of antibodybound from free hormone in radioimmunoassay. Horm. Metab. Res., 3:301-5. With R. S. Yalow, T. Saito, and I. J. Selikoff. Antibodies to "Alcalase" after industrial exposure. N. Engl. J. Med., 284:688-90. With R. S. Yalow. Gastrin in duodenal ulcer. N. Engl. J. Med., 284:445. With R. S. Yalow. Size heterogeneity of immunoreactive human ACTH in plasma and in extracts of pituitary glands and ACTH-producing thymoma. Biochem. Biophys. Res. Commun., 44:439-45. 1972 With R. S. Yalow. Radioimmunoassay in gastroenterology. Gastroenterology, 62:1061-84.
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With G. Nilsson, J. Simon, and R. S. Yalow. Plasma gastrin and gastric acid responses to sham feeding and feeding in dogs. Gastroenterology, 63:51-59. With R. S. Yalow. And now, "big, big" gastrin. Biochem. Biophys. Res. Commun. 48:391-95. 1973 With R. S. Yalow. "Big, big insulin." Metabolism, 22:703-13. With R. S. Yalow. Characteristics of "Big ACTH" in human plasma and pituitary extracts. J. Clin. Endocrinol. Metab., 36:415-23.
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Raymond Thayer Birge March 13, 1887-March 22, 1980
By A. Carl Helmholz Raymond Thayer Birge, a member of the National Academy of Sciences from 1932, died in Berkeley, California, on March 22, 1980, at the age of ninety-three. Prominent in the field of physics from 1920 to 1955, he retired as chairman of the Department of Physics at the University of California, Berkeley, after a tenure of twenty-three years. His determination of the precise values of physical constants and his work in spectroscopy established his excellence as a physicist, while his key role in building a world class Department of Physics at Berkeley established him as an equally outstanding administrator. EARLY LIFE Birge was born on March 13, 1887, in Brooklyn, New York, into an old New England family. His father worked in river transport until 1898, when he became an executive in the laundry machine business and moved the family to Troy, New York. Raymond finished grammar school in Troy and graduated from Troy High School in 1905, valedictorian of his class. His lifelong interest in physics had already begun, but the honors he won were in Latin, there being no honors in science at that time. Although Ray had planned to attend college, his father's
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business failed just before his graduation from high school, and he entered the local business college instead to study bookkeeping. His accuracy and aptitude for numbers were evident, and he was soon asked to teach night classes to the beginning sections. Fortunately, at this time, Ray's uncle, Charles T. Raymond, offered to pay his expenses for a college education. Delighted, Ray chose the University of Wisconsin at Madison, where another uncle, Edward Asahel Birge, was dean of the Faculty. Edward Birge, a pioneer in the field of limnology, served as president of the University from 1918 to 1925. Ray, entering in 1906, began studying physics immediately and soon decided to major in it. He received his A.B. degree in 1909 after three-and-ahalf years and a summer session, writing his senior thesis under L. R. Ingersoll on the reflecting power of metals. ''As an experimental physicist," he later recorded, "my talents were perfectly circumscribed. I could take a piece of optical equipment, put it in perfect adjustment, and get with it as precise or more precise readings than had ever been gotten before. But I was quite unable to construct such equipment." Since his academic work was excellent and he liked both Madison and the Physics Department, Birge decided to continue on to the Ph.D. He received his M.A. degree in 1910, and the Astrophysical Journal published his thesis, "Formulae for the Spectral Series for the Alkali Metals and Helium." His Ph.D. thesis (1913,1), in which he photographed the band spectrum of nitrogen at high dispersion, was supervised by C. E. Mendenhall, the bestknown member of the Department. Vigilant regarding possible sources of error and intent on achieving high resolution, he kept the temperature constant to better than 0.1°C and compensated for changes in atmospheric pressure by purposeful changes in temperature. His exposures ran up to five days. "For forty days I lived
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in the laboratory," he wrote, "leaving it only for meals, and reading a number of thermometers every few hours, day and night." He finished this work at the end of the summer of 1913, though his degree was not officially awarded until 1914. During his years in Madison Birge was one of the founders of a walking club, through which he met Irene Adelaide Walsh, who had come to Madison from Redfield, South Dakota. They were married on August 12, 1913. It was a very happy marriage. They remained devoted to each other, and Irene died just three weeks before Raymond. After their marriage the young couple moved to Syracuse, New York, where Ray had accepted a position as instructor at Syracuse University, hoping to work with the well-known spectroscopist F. A. Saunders (Russell-Saunders coupling). Saunders, unfortunately, was away on sabbatical leave during the 1913-14 academic year, and in 1914 left Syracuse to join the physics faculty at Vassar College. Birge never got the chance to work with him. Syracuse was a sterile place for a young and ambitious person, eager to teach and do research. Nevertheless, Ray stayed there for five years, winning promotion after two to assistant professor. While at Syracuse he published several papers: on temperature effects in the use of concave gratings; on "Mathematical Structure of Band Series," an extension of his thesis work; and, in the Journal of the New York State Teachers Association, on "Some Popular Misconceptions in Physics." With S. F. Acree of the New York State College of Forestry, he also published on the theory of chemical indicators and on the precise value of the Rydberg constant. This last short paper, which appeared in Science, showed his grasp of the importance of determining the value of this constant with extreme accuracy—in this case, to about five parts per million.
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BERKELEY In 1918, unhappy with his situation at Syracuse, Birge wrote to E. P. Lewis. Then the new head of the Physics Department at Berkeley, Lewis, a fellow spectroscopist, had two openings for instructors for the next year. He raised the salary of one and offered it to Birge, who accepted with alacrity. Ray and Irene moved to Berkeley in the summer of 1918 and thus began his distinguished career at the University of California of thirty-seven years. Possibly stimulated by Gilbert Lewis (no relation), who had been appointed Dean of the College of Chemistry in 1912, Lewis was interested in promoting research in physics. Leonard Loeb, William H. Williams, Victor Lenzen, and Raymond Birge were the founding members of this ultimately outstanding Department. The first task to which Birge set himself was the introduction of the Bohr theory of the atom. Gilbert Lewis had, with others such as Langmuir, formulated the cubical model of the atom that held sway on the campus. Over the next few years, with patience and persistence, Birge won over the Physics and then the Chemistry Departments—a feat he later described as one of his most important achievements. When asked what was the difference between chemistry and physics, he replied with a smile. "When Giauque and Johnston discovered the isotopes 17 and 18 of oxygen, that was chemistry because it was done in Gilman Hall. When King and I discovered the isotope 13 of carbon, that was physics because it was done in LeConte Hall." Since Birge started it in 1918, the cooperation of physics and chemistry in research and teaching has expanded and borne fruit. During his early years in Berkeley, Birge published a steady stream of papers, many on band spectra, some on the accurate values of the physical constants. From 1920 to 1925, for example, he produced eleven publications and thirteen
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abstracts of talks at American Physical Society meetings. In 1921 he published "The Balmer Series of Hydrogen and the Quantum Theory of Line Spectra" (1921,1), cited by Sommerfeld in his third edition of Atombau und Spectrallinien. This helped to build Berkeley's reputation, and in 1927, a committee appointed by the National Research Council with E. C. Kemble as chairman wrote the 400-page Molecular Spectra in Gases—about half of which was contributed by Birge (1926,1). In 1925 Dr. Hertha Sponer (later Mrs. James Franck) came to Berkeley on an international fellowship and worked with Birge in the field of band spectra. Together they produced "Heat of Dissociation of Nonpolar Molecules" (1926,2). Theirs was the first quantitative method of determining this important constant, accomplished by extrapolating vibrational spectra to the limit of zero frequency. Hertha's knowledge of chemistry and Ray's of spectra made this a most profitable collaboration. Birge realized, for example, that the Rydberg constant was determined by an exact relation of e, h, and m, and consequently that the best values of these constants determined by other means (in the case of m by e/m) must fit the best value of the Rydberg from experiment. A number of his papers previous to 1928 had been concerned with best values of the physical constants. In 1928 he submitted "Molecular Constants Derived from Band Spectra of Diatomic Molecules" to the International Critical Tables (1929,1), and it was natural that he should also submit a review paper, "Probable Values of the General Physical Constants" (1929,3), to the new Physical Review Supplement . Shortly after it was published, the name of this journal was changed to Reviews of Modern Physics, so that this important paper—probably the most important that Birge ever wrote—is the first article to appear in this journal. "Probable Values of the General Physical Constants" is a
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remarkable work, covering constants from the velocity of light through the mechanical equivalent of heat, and Avogadro's number to Planck's constant. The article was openly critical of the work of others, and Birge said in interviews that he might not have any friends left once it was published. Careful and painstaking, it established a whole new field in which he was clearly the leader, until DuMond and Cohen used his methods with computers in the late 1940s. Regarding the propagation of errors, Birge wrote with J. D. Shea on least squares solutions of polynomials in 1924, and, eight years later, "Calculation of Errors by the Method of Least Squares" (1932,1). He then produced "On the Statistical Theory of Errors" (1934,1) and later, "Least Squares' Fitting of Data by Means of Polynomials,'' with a mathematical appendix by J. W. Weinberg, which appeared in Reviews of Modern Physics (1947,1). Birge himself considered this final paper a satisfactory conclusion to his long-standing work. An interesting story is associated with Birge and King's discovery of the isotope of carbon of mass 13. A. S. King, of Mt. Wilson Observatory, was in Berkeley in July, 1929, at a meeting of the American Physical Society. At about 4 o'clock, after the close of the meeting, King showed Birge a plate he had taken of the Swan bands of carbon from a carbon furnace and questioned him about the possible origins of some faint lines in the bands. Birge, with recent experience on the isotopes of oxygen, immediately realized that these might be due to an isotope of carbon of mass 13. He measured them on his comparator, performed the calculations, wrote the paper with King, and by one o'clock the next morning mailed the article to Nature and to Physical Review (1929,2)—the fastest paper, he used to say, he had ever written. Birge also investigated isotopes with D. H. Menzel, their work resulting in "The Relative Abundance of the Oxygen Isotopes and the Atomic Weight System" (1931,1). When
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Menzel, at the Lick Observatory, consulted with Birge about measurements of spectra from the sun and the abundance of the isotopes, Birge noticed that the atomic weight of hydrogen measured in mass spectrographs did not agree with the chemically measured atomic weight. He noted that this discrepancy could be explained by the presence in hydrogen of mass 2, thereby prompting a flurry of experiments in the race to find the isotope; Harold C. Urey, then at Columbia, eventually won this race and was awarded the 1934 Nobel Prize in Chemistry for the discovery, but Birge's prediction on an abundance of 1 in 6000 for deuterium is quite close to the present value. CHAIRMAN OF THE DEPARTMENT OF PHYSICS When E. P. Lewis died in 1926, Birge and E. E. Hall were appointed to the search committee for a new physics chairman. Hall was eventually named to the post and remained chairman until his death in November 1932. Because Hall was not widely acquainted with physicists, the responsibility for making new faculty appointments fell mainly to Birge and Loeb. This spectacular period saw the additions of E. O. Lawrence, J. R. Oppenheimer, R. B. Brode, F. A. Jenkins, and H. E. White to the Berkeley staff. When Hall died, Birge was named acting chairman, then chairman, a post he occupied until his retirement in 1955. During his tenure both faculty and graduate students quadrupled in number so that, shortly after he retired, 300 graduate students were enrolled. Birge himself took great care with the quality of instruction and when chairman continued to teach graduate courses in physical optics and in reduction of observations. His very high standards for his department, both in teaching and research, account in very large part for the growth in distinction of physics at Berkeley. Birge was presented with a new set of problems during
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the years of World War II. His faculty dispersed to various locations around the country to do war work. Fortunately, he was able to find a number of substitutes among the physicists Lawrence brought to Berkeley for his Manhattan Project work. But he was faced with the problem of staffing a great many undergraduate courses for Army, Navy, and Air Force recruits in addition to Berkeley's regular quota of students, all with different schedules. With the help of his faithful secretary, Rebecca Young, he somehow managed in this exacting job. At war's end, Berkeley's faculty returned, and the school became for many returning GI's the first choice for graduate work in physics. Classes and research work had to be started up again and expanded. Lawrence was anxious to build the 184-inch cyclotron, McMillan to build the synchrotron, and Alvarez to build the proton linear accelerator. Many graduate students were employed, and the faculty introduced several new fields of research. Nierenberg was brought to start work in atomic beams, Kittel and Kip in solid state theoretical and experimental physics, Reynolds in mass spectroscopy, Knight and Jeffries in magnetic resonance—all in addition to new faculty members in nuclear, high energy, and theoretical physics. Oppenheimer never returned to Berkeley fulltime and left in October 1947 to become the Director of the Institute for Advanced Study at Princeton, Serber and Wick (with younger members like Chew and Lewis) managed the theoretical program. A new building was necessary, and with help from Harvey White, new LeConte Hall—joined to old LeConte—was opened in late 1950. HONORS AND DISTINCTIONS The Department continued to grow after Birge's retirement in 1955, and the American Physical Society, meeting in Berkeley on December 21, 1964, set aside the afternoon to
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dedicate another new building, Birge Hall. Birge was deeply moved by this tribute, one that so rarely comes to a person during his life. He was also proud of his election to the National Academy of Sciences in 1932. The first physicist ever elected from Berkeley, it is a tribute to his administrative skill that his Department now contains more Academy members than any other physics department in the country. Throughout his years on the faculty, Birge was active in the Academic Senate, and in 1946 the faculty voted him their highest honor—faculty research lecturer. He was also an active member of the Committee on the Calendar and served as chairman for several years. Campus wags used to accuse Birge of arranging the calendar so that physicists could attend the meetings of the American Physical Society and of the National Academy during the spring vacation. He himself liked to tell of one ideal calendar suggestion he had received, whose only fault was a schedule of fifty-three weeks in the year. Deeply dismayed by the "oath dispute" in 1949 and 1950, he did his best in the Academic Senate to avoid the imposition of a loyalty oath. He failed, and though he himself signed, several of his faculty refused to do so and left Berkeley because of it. From 1942 to 1947 he served as Pacific Coast Secretary of the American Physical Society. This was before the days of air travel, and very few physicists came to Pacific Coast meetings. When Ray retired from this position, K. K. Darrow, secretary of the Society and a close friend, cited his outstanding work and commended him for saving postage and paper by writing small and singlespacing everything! In 1954 he was elected vice president of the Physical Society and succeeded to the presidency in 1955. He faithfully attended all the meetings and ran the Council sessions with expert fairness. In his retiring address, "Physics and Physicists of the
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Past Fifty Years" (1956,1), he reviewed the evolving state of his science throughout his long career. When Birge retired in 1955 the University of California awarded him the LL. D. degree, a fitting tribute to his long and distinguished service to the institution. He kept an office in LeConte Hall for a number of years and finished a history of the Physics Department from 1868, the year the University opened, to 1955. This admittedly not very readable work is yet packed with information, and is, for many matters, the only reference source available. Birge had known every Physics Department chairman except the first, John LeConte. He also contributed an oral history to the Bancroft Library in Berkeley. RAYMOND BIRGE, THE MAN Ray and Irene Birge had two children, Carolyn Elizabeth (Mrs. E. D. Yocky) and Robert Walsh (married to Ann Chamberlain). Each had three children, and the Yocky's have three grandchildren. Raymond Birge was a man of outstanding honesty and integrity. Reserved to most, he was yet loving to his family. His last scientific paper (1957,1) was read at the hundredth anniversary of the death of Avogadro in Turin, Italy, in September 1956. His opening remarks at this conference (1957,2) express, better than any biographer could, his lifelong reverence for and joy in science: "Now, to me, the study of science is, in a sense, a religion. For there can scarcely be anything more marvelous than the structure of nature, nor anything more satisfying than to aid, even in the smallest way, in the gradual unfolding of the intricacies of our universe. From the beginning of the human race, man has speculated on the wonders of his environment, but there is and can be nothing in even his wildest speculation in any way comparable to the actual facts of nature. For just this reason, the true objective study of science offers a never-ending and wholly satisfying human endeavor: at least I have found it so."
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Selected Bibliography
1913 The first Deslandres' group of the positive band spectrum of nitrogen under high dispersion. Astrophys. J., 39:50-88. 1921 The Balmer series of hydrogen and the quantum theory of line spectra. Phys. Rev., 17:589-607. 1926 Report on molecular spectra in gases. Bull. Nat. Res. Counc. (U.S.), 57:69-259. With H. Sponer. The heat of dissociation of nonpolar molecules. Phys. Rev., 28:259-83. 1929 Molecular constants derived from band spectra of diatomic molecules. Int. Critical Tables V, 409-18. With A. S. King. An isotope of carbon, mass 13. Phys. Rev., 34:376 (July 15, 1929); Nature, 124:127. Probable values of the general physical constants. Phys. Rev. (Suppl. 1), 1-73. 1931 With D. H. Menzel. The relative abundance of the oxygen isotopes, and the basis of the atomic weight system. Phys. Rev., 37:1669-71. 1932 The calculation of errors by the method of least squares. Phys. Rev., 40:207-27. 1934 With W. E. Deming. On the statistical theory of errors. Rev. Mod. Phys., 6:119-61.
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1947 Least squares' fitting of data by means of polynomials. Rev. Mod. Phys., 19:298-347. (With appendix by J. W. Weinberg, 348-60.) 1956 Physics and physicists of the past fifty years. Address of Retiring President of American Physical Society, delivered in New York City, February 2, 1956. Phys. Today, 9:20-28. 1957 A survey of the systematic evaluation of the universal physical constants. Address at the Avogadro Congress, Turin, Italy, September 1956. No. 1, Suppl. 6, Ser. X Nuovo Cimento, pp. 39-67. Words spoken in memory of Amadeo Avogadro and for the opening of the Congresses. No. 1, Suppl. 6, Ser. X Nuovo Cimento, pp. 35-38.
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William Henry Chandler July 31, 1878-October 29, 1970
By Jacob B. Biale "Because of this reserve of dormant buds," said W. H. Chandler, lecturing during the dark days of World War II, "a tree is more dependable in a destructive world. It can be broken to pieces pretty badly and will grow new parts to replace the lost ones" (1944,1). Trees with buds at rest, keeping the secrets of dormancy; trees with buds bursting to bloom and to fruit; trees of different climates and of varied behavior fascinated Chandler and served as his dependable companions throughout a long, productive, and humane life. Delving into their complex functioning, he unraveled the story of their response to internal and external environment. Esteemed worldwide for transforming horticulture from an art into a science, he— with his reservations about the validity of classifying horticulture or agriculture as distinct sciences—would surely have rejected any such claim. But his original research papers and books, filled with knowledge and deep insight, continue to bring him international recognition. In addition to advancing the field of horticulture generally, Chandler helped elucidate the mechanism by which frost kills plant tissue. He was the codiscoverer of the fact that zinc deficiency causes a number of physiological disorders, including little leaf and mottle leaf. He introduced a system of
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pruning that resulted in maximum yield. He developed hybrids of temperate zone trees that grow, flower, and produce fruits satisfactorily in climates with mild winters. The university community, experiment station workers, and extension staff all valued Chandler's ideas on research, teaching, and communicating results. He inspired promising investigators, then helped place them where they could contribute the most to horticulture and plant physiology. Anyone who had the good fortune to know him—whether professionally or socially—was left with the impression of a man of sturdy character, mild manner, and no pretensions. His convictions were strong, yet he was open to others' views. He was cultured, appreciating history, poetry and novels. Not blind to human shortcomings, he yet had an idealistic trust in the future of mankind. EARLY LIFE AND EDUCATION Bill Chandler, the oldest of eight children, was born in Butler, Missouri, in a little log house where the dog went in and out freely through the open door. Many years later he recalled that, during his childhood, all eight children slept in a single room in trundle beds that were stored away under larger beds during the day. His father, who came from the hill country bordering Virginia and Tennessee, disliked farming and often allowed weeds to displace planted crops. When Bill was ten years old, the family moved to a somewhat larger house and smaller farm, incurring a large debt. They lost the property three years later, and from then on the family was forced to live on rented farms. From the age of fifteen, the responsibility for maintaining his family through farming rested on Bill with the help of a younger brother. Seriously restricted in the time he could devote to schooling, Bill attended the country school only during the six
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months of autumn and winter. The remainder of the year he worked full-time on the farm. At eighteen he went to stay with his uncle at the county seat, where he studied for two semiannual periods at the Academy. From 1898 to 1901 he taught in a single-room country school, while aspiring at the same time to study farming at the University of Missouri College of Agriculture. When he divulged his ambitions to his uncles, who were successful farmers, they ridiculed the young dreamer. ''It isn't what I don't know that loses me money," one told him. "It's what I know and don't do." Chandler disregarded the advice of his relatives and enrolled in agriculture at the University in the fall of 1901. The five-year course led to the B.S. degree in 1905, and a year later he received the M.S. degree. Partly due to the influence of Dr. J. C. Whitten, then head of the Department, he specialized in tree horticulture, though—in later years—he regretted that the program of study had not included required courses in physics and chemistry. As a student, Chandler was inspired by the teaching of plant physiologist B. M. Duggar. For his doctoral dissertation topic he elected to study the killing of plant tissue by low temperature, a major problem in agriculture in Missouri as well as in many other regions, which continued to interest him throughout later appointments as assistant (1906-1908), instructor (1908-1909), and assistant professor (1910-1913) in horticulture. Due to a technical regulation, he was not officially awarded the Ph.D. until 1914, when he was no longer affiliated with the University of Missouri. In 1913, Chandler was invited to join the faculty of the College of Agriculture at Cornell University as professor of pomology. Better pay and research support, the presence of Liberty Hyde Bailey as dean of the College, and the greater distinction of the University made the offer extremely attractive, and he accepted. Once there, he found that the climatic
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conditions and the widespread growing of apples in New York State stimulated his interest in winter injury to fruit trees, and he extended his observations to the relation of winter frost damage to growth responses during the preceding summer. Local farmers cooperated willingly with the research and extension staff of Cornell's agricultural experiment station. Yet not everyone was equally enthusiastic about the program. "It's the farmer's conservatism that saves him," a skeptical Dean Bailey was reported to say. "If he'd done everything that you [the research men] recommend, he'd be ruined." Chandler shared Bailey's respect for the innate intelligence and good sense of the farmer. Working on field plots with New York growers, he found their attitude to farm life more wholesome than that of farmers in Missouri, so that the area remained relatively free of land speculation and real estate promotion. "You could not buy a farm at any price," he remarked at the time, "from a man who had a son to take his place." Like Bailey, too, he was skeptical about the quality of knowledge imparted by teachers of agriculture and the worthiness of certain agriculture research projects. L. H. MacDaniels, a Cornell graduate student at that time, reported that when Chandler arrived he was assigned to teach a course in the culture of nut trees that a number of football players took to lighten their load. After delivering a half-dozen lectures, he dismissed the class for the rest of the semester, saying that he had covered all that was known about the subject that was backed by evidence. Chandler insisted that the pomology program be related to plant physiology and the basic sciences, arguing that preparation for trees research should lead to a Ph.D. in plant physiology or in another related field that could serve as a background for horticulture. He often directed his graduate
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students to study under other professors, and—though many investigators credit him for inspiring and directing their horticultural or physiological research— chaired, in fact, only one doctoral committee, that of A. J. Heinicke. During his decade at Cornell, Chandler chaired the Department of Pomology from 1915 to 1920 and, as vice-director for research, administered research funds from 1920 to 1923. This last task, at times frustrating because of the limited funds supporting a number of meritorious projects, allowed him to broaden his contacts with his colleagues. He enjoyed his dealings with members of the general faculty on campus and life in the small, charming community of Ithaca. During this period, he also established his professional standing as the pomologist best able to analyze and understand the complex responses of fruit trees. This ability found its fullest expression in Fruit Growing, a textbook written and revised with great care and precision during his Cornell years, though published after he left there permanently for the West. In 1922, Chandler was invited to tour various regions of California in connection with the dedication of a building of the University of California at Davis. Once there, he observed a wealth of horticultural problems that did not exist in New York, where fruit trees had grown for hundreds of years and many of the intricacies of their culture were known. California, on the other hand, with its great range of climatic zones and wide spectrum of horticultural materials, was unique. In addition to the innate interest to an agriculturalist, C. B. Hutchison, an administrator in the College of Agriculture at Davis who had been Chandler's associate at Missouri and Cornell, also played a major role in his decision to transfer. Chandler came to California in 1923 as professor of pomology and chairman of the Department at both Berkeley and
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Davis, with headquarters in Berkeley. He later considered his fifteen at Berkeley highly, both professionally and personally, crediting his accomplishments in part to D. R. Hoagland, professor and chairman of the Division of Plant Nutrition. Hoagland's Division was noted for its research on the nutritional requirements of plants, and especially their need for trace nutrients: copper, zinc, molybdenum, manganese, and boron. Using special laboratory apparatus free of contaminating elements, the Division staff developed a procedure for purifying chemicals to a high degree. In this atmosphere, Chandler investigated physiological disorders known as "little leaf" in peaches, "rosette" in apples and pears, and "mottle leaf" in citrus. His training in both horticulture and plant chemistry enabled him to identify a zinc deficiency as the cause of all of these disorders, thereby solving a problem that had baffled fruit growers since the beginning of the century. Chandler viewed these zinc-deficiency studies as the most significant economic and scientific contribution of his career. He attributed his gratifying results to the combined efforts of his team members, whose diverse talents allowed them to focus on the problem from different angles, and to methodical experimentation using advanced procedures of purification and analysis. As a result of this cooperative venture, Chandler and Hoagland established a long-lasting friendship. They shared similar outlooks on research and university affairs and advocated harmonious interaction between applied and basic research. Both men had unusual personal qualities that inspired those students and colleagues who had the good fortune to be associated with them. In 1938, with the zinc work partially completed, Chandler was persuaded to accept the assistant deanship of the University of California's College of Agriculture and to establish his headquarters on the Los Angeles campus of the Univer
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sity. Administrative duties held no great attraction for him, but he yielded to the urgent pleas of C. B. Hutchison, then statewide dean of the College. As assistant dean, Chandler's function was to harmonize relations between the Los Angeles and Riverside Departments of the College and to strengthen UCLA's program in plant science. He also identified profitable directions for research in plant biology within the constraints imposed by field work on a campus in an urban setting. He focused on studies not requiring much land that could be conducted in greenhouses and in laboratories, and on plants with rapid growth rates, as the most suitable for graduate thesis work. He was, consequently, instrumental in establishing a Department of Floriculture and Ornamental Horticulture at UCLA. Knowing, from past experience, the benefits of administrative association between botany and agriculture, he further made a special effort to transfer the Botany Department from the College of Letters and Sciences to the College of Agriculture. This action was later credited with enriching UCLA's offerings in plant science, particularly at the graduate level. Arranging this transfer was Chandler's last major administrative act before he relinquished the deanship in 1943. He continued on at UCLA as a professor of horticulture until he officially retired in 1948. During retirement, Chandler thoroughly revised his two textbooks, Deciduous Orchards and Evergreen Orchards. To collect source materials for his books he traveled to the West Indies, Trinidad, and Central America. UCLA's unofficial advisor for campus landscaping, he maintained his interest in plant physiology and regularly attended seminars. In 1966, the Chandler's moved from Beverly Hills to Berkeley so that they could live closer to their three daughters. In November 1969, he suffered a mild stroke and a year later died at the age of 92.
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Chandler and his wife of sixty-three years, Nancy Caroline, were married in 1905 when he was starting his graduate studies at Missouri. The Chandler home exuded a spirit of tranquility, hospitality, and good comradeship. In his affection for his wife, Chandler named a wisteria after her and dedicated to her several of his books. Mrs. Chandler died in Berkeley in 1968. Their son, William Lewis (wife Eleanor), a microbiologist, established his home in Altadena. Their daughters, Carolyn Geraldine Cruess and Ruth Steele Lewis, live in Berkeley, and Mary Martha Honeychurch has her home in Orinda, California. Chandler is survived by four children, eleven grandchildren, and ten great-grand-children. William Henry Chandler was awarded many honors during his lifetime. He was elected president of the American Society for Horticultural Science in 1921, member of the National Academy of Sciences in 1943, and Faculty Research Lecturer at UCLA in 1944. He won the Wilder Medal of the American Pomological Society in 1948 and, in the same year, was named one of three outstanding American horticulturists by the American Fruit Grower magazine. The American Society of Plant Physiologists bestowed on him the Charles Barnes Life Membership in 1951. In 1949 he received the honorary LL. D. degree from UCLA. He held membership in the American Association for the Advancement of Science, American Society for Horticultural Science, American Society of Plant Physiologists, Botanical Society of America, and Sigma Xi. TREES IN TWO CLIMATES On March 21, 1944, four years before his retirement at the age of seventy, Professor Chandler delivered a talk on this subject as the annual UCLA Faculty Research Lecture (1945,2). By that time, he had spent two decades in the mild
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climate of California, with the last six years in the subtropical environment of the southern region of the state. Growing conditions and responses of fruit trees in the West differed dramatically from those he had observed during the first two decades of his career in Missouri and New York. The time was ripe for him to summarize his rich experiences with fruit trees of various climatic zones and to analyze the effects of temperature on cellular events as the major factor determining their growth. Death by Freezing Chandler was searching for the mechanism of cellular death by freezing. The killing of plant tissue by low temperature had been the subject of his dissertation at the University of Missouri, while in California he had been attracted to the problem of why certain fruit trees required these same chilling temperatures to grow. Shortly after transferring from Missouri to Cornell, Chandler began observing the response of deciduous fruit trees to extremely low temperatures. In the early morning hours of January 14, 1914, the temperature of -34°F (-36.7° C) was recorded in an orchard in upstate New York in which Northern Spy apples were grown. Several days earlier ice had begun to form at the outer surfaces of some cells. From his own research and the work of others, Chandler knew that the gradual lowering of temperature facilitated the movement of water from the interior of cells to the intercellular spaces where ice crystals were formed. He further discovered that, although water expands as it freezes, air in these spaces gave way to ice so that the frozen tree actually shrank. He estimated that seventy to eighty percent of the water in the tree was converted to ice and that a third or more of the weight of the above-ground portion of the tree was ice.
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Microscopic observation showed, furthermore, that shrunken cells as protoplasm became a thin layer between flattened walls. The pressure of ice particles present in the intercellular spaces appeared to cause distortions of a magnitude that suggested severe injury. As judged from the luxuriant growth during the spring and summer following the severe winter, however, this was not the case, and the tree's survival suggested that these had been, in fact, the proper conditions for hardening. Through field observations and laboratory tests Chandler discovered a decreasing order of resistance to freezing temperatures in the various tissues of hardened trees. Most resistent was cambium, which, when not well-hardened, turned out to be as sensitive as other tissues; then came bark, sapwood, and pith. He further observed that above-ground portions of a tree were more resistant than roots; that flower buds, generally more sensitive than vegetative buds, were less sensitive when trees were not fully mature; that resistance diminished in some species whose flower buds reached an advanced stage of differentiation by the beginning of winter. With great precision, he described how frost resistance developed, singling out two ways—"maturing" and "hardening"—deciduous trees and shrubs became resistant to cold. Maturing of wood and buds begins after growth ceases in the summer. It is characterized by the accumulation of carbohydrates, decline of water content, increase in osmotic pressure, thickening of cell walls, and a marked drop in the succulence of newly formed tissues. At the end of the maturing process—the time of natural leaf abscission—some deciduous trees can withstand temperatures of -17° to -25°C. Hardening of mature wood occurs with exposure to freezing or nearfreezing temperatures, with immature wood requiring a longer time to harden. Once hardened, some varieties can withstand temperatures ten degrees lower. Even a relatively short warm period can undo this increased resis
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tance, but it can be regained—unless growth started during the warm spell— with repeated exposure to low temperatures. Chandler found that internal tissue changes during hardening included increased osmotic pressure (from starch hydrolysis) and greater holding capacity for unfrozen water at temperatures above the eutectic point. In some hardy species, vacuolar sap also contained colloids that held water against freezing and osmotic activity, while, in other species, expansion of cytoplasm and the consequent reduction of the vacuoles might accompany hardening. The most resistant, living cells in well-hardened deciduous wood turned out to be nonvacuolated, meristematic cells in leaf buds and cambium. These cells in hardened plants could survive more shrinkage and the loss of a larger proportion of their water to ice masses than could cells of unhardened plants. The protoplasm of hardened cells, furthermore, seemed less easily ruptured than that of unhardened cells. These extensive observations on the responses of plants and plant parts to temperature stress led Chandler to probe the fundamental question of how freezing kills plant tissue. Well aware of the variety of centuries-old opinions on the subject, he compiled a list of established facts regarding plant death by freezing. Foremost was the phenomenon of ice formation—in tender tissue primarily within cells and, in cold-resistant material, in intercellular spaces— subjected to relatively slow temperature fall (the case during a normal cold wave). Rapid temperature drop, on the other hand, caused ice to form within the cells and raised the temperature at which death occurs. Ice formation and death occurred rapidly at killing temperatures, unlike other chemical changes, which were markedly suppressed under such conditions. Death by freezing can best be seen in thawed tissue, which
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darkens and takes on a water-soaked appearance coupled with a rapid rate of evaporation. In most tissues the rate of thawing does not influence the level of the killing temperature, but some tissues show symptoms of death before thawing begins. Ice formed in the plant is pure water, while the solution left in the cells is highly concentrated. Sap solutes tend to hold some water unfrozen at temperatures below the eutectic point. Such concentrated sap may be toxic to protoplasm at room temperature but is a source of protection at freezing temperatures; it also often contains water-binding colloids. Chandler also noted that bacterial spores, seeds, and pollen grains in the proper state of dehydration could withstand temperatures of liquid hydrogen and remain viable. Any explanation of the mechanism of freezing to death would have to account for these observations, as well as for supercooling as a means of protecting tissue from injury at freezing temperatures so long as ice formation did not occur. After examining the various hypotheses concerning the mechanism of death by freezing (including disorganization of protoplasm through water loss and toxicity through concentration of the sap), he arrived at the conclusion that plants were most probably killed by the pressure of the ice masses on plasma membranes. The Rest Period When he moved to California, Chandler's concern with low temperature as a limiting factor in the growth of fruit trees took a different turn—rather causing losses from freezing, low temperatures in fall and winter were necessary to some California plants if they were to develop normal shoot and flower buds the subsequent spring. In a subtropical as opposed to a harsh climate, the limiting factor for growing apples, pears, apricots, peaches, and plums was the absence of sufficient days at moderately low temperatures to "break
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the rest,'' a condition known in horticulture as the "chilling requirement." In three of his four textbooks and in several special papers Chandler described the phenomenon of the "rest period" precisely. In the spring a hormonal substance produced in the tip prevents newly formed buds along the shoot from growing. Early in the season, if this apical inhibitor is removed, these buds will grow. Later in the summer, however, the buds enter the rest period and absence of the apical inhibitor does not cause growth. Rest period is, therefore, the period when the plant, or a portion of the plant, will not grow even when temperature, moisture, and nutrient conditions are favorable for growth. It is different from "dormancy," a state of inactivity brought on by any cause. An apple tree, for example, might be said to be dormant in February because the temperature is too low for growth, or it might fail to grow in December, not because of the temperature, but because it is in the rest period. In some fruit trees this rest period is attained as early as five to seven weeks after the start of spring growth. In the warm winters typical of the coastal regions of California, on the other hand, buds on some varieties of deciduous trees do not grow until the middle of the following summer, and even then only a small percentage will grow. To demonstrate his point, Chandler used the striking example of a Northern Spy apple tree in Berkeley that had experienced a rest period in which no buds grew for two seasons. Yet, Chandler maintained, if the same tree had been put at 5°C in the fall of the first year, its rest period could have been reduced from two years to six months. Placing a number of branches of a cherry tree at 0°C for two months, he showed that their buds opened a month earlier than buds on the unchilled tree. In another experiment, he subjected peach trees to tem
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peratures ranging from -1° to 0°C for two and one-half months during the fall, then transferred them to a warm greenhouse (15°C), keeping control trees continuously at the higher temperature. The chilled buds grew as much in fourteen days as the unchilled buds in 133 days. Chandler's experiments showed that emergence from rest was a function of both temperature and time; that spring growth was more rapid when buds were previously subjected to temperatures of 5° to 10°C for fifty to sixty days; that the more vigorous and later the growth during the preceding summer, the greater the chilling requirement. He found that insufficient chilling caused some buds to open before others, and many to fail to open altogether. Inadequate chilling, furthermore, affected flower buds as much as leaf buds, causing many to fall off before they had fully opened. In some trees, flower initials died in the buds before opening, while apple and pear trees, whose buds are mixed (consisting of both flower and shoot initials), insufficient chilling led to the production of leafy shoot only, or of leafy shoots with a reduced number of flowers. As for the biological role of the rest period, he pointed out that delay in spring budding lessened the danger from spring frosts and opening to occur in weather more favorable for pollination and fruit setting. Seeking the cause of the rest period in trees, Chandler suggested that a hormonal substance might be involved and cited changes in ether-extractable auxins in buds upon emergence from the rest. Treatments with rest-breaking substances such as ethylene chlorhydrin tended to reduce the auxin levels in plant tissue. Fully acknowledging the lack of verifiable data, Chandler advanced the idea that a bound form of auxin might be responsible for keeping buds from growing during the deep part of the rest, and that the rate of retardation of bud opening was determined by the balance between the bound and free forms.
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FRUIT TREE NUTRITION—THE ZINC STORY Regarding the nutritional requirements for optimal growth and yield in fruit trees, Professor Chandler's investigations ranged from experiments with specific nutrients to analysis of a complex biological system dependent on minerals derived from a highly variable medium. Before Chandler, experiment station researchers tended to concern themselves with annuals. In their orchard-fertilizer experiments, they applied different quantities and combinations of required elements over a number of years, then analyzed the results statistically. Chandler questioned the reliability of field trials where experimenters seeking to minimize error increased the size of their samples, necessarily using larger and more variable soil plots. He also called attention to errors caused by such frequently overlooked variables as bud variation, differences in the vigor of seedling stock growth, the cumulative effect of injuries sustained with age, and —in measuring growth and yield—the number of branches with which a tree started. He pointed out that the outbreak of disease (as happened when mottle leaf blighted certain experimental citrus trees) could vitiate years of carefully planned fertilizer experimentation that depended on uniformity of plots to test differential treatments. Cognizant of these difficulties, Chandler designed a new approach to field testing with fruit trees. Shortly after he arrived in California, orchards in a variety of climatic zones both inland and along the entire Pacific Coast suffered great losses from a tree disease known since the beginning of the century. This disease, affecting both deciduous and evergreen trees (and walnuts and grapes as well), is called "little leaf" in stone fruits—almond, apricot, cherry, peach, plum; "mottle leaf" in citrus; and "rosette" in apples and pears.
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The disease is most dependably characterized by stiff, narrow leaves— about five percent of normal size—that appear in the spring. Each of the small tufts, or rosettes, of these abnormally small leaves originates from a bud that would normally produce a shoot. Leaves are also mottled, with yellow streaks and splashes between veins, while the veins themselves, and some adjoining tissue, are green. These symptoms are most conspicuous in spring. Later in the season, healthy shoots may grow from buds lower on the branch. In severe cases, distorted yellow leaves form even late in the summer. Fruit size in all species is reduced and, in some, the fruit is also strikingly distorted. Moderately affected pome and stone fruits may live for many years producing fruit of inferior quality and yield. In some soils trees grow well for the first few years but then develop symptoms rapidly and die. Chandler undertook to study this problem together with two members of Berkeley's Plant Nutrition Division—plant physiologist and soil chemist D. R. Hoagland and chemist P. L. Hibbard. Before starting trials of treatments he carefully observed conditions in various districts of California. He sought out the experiences of farm advisors, extension specialists, and orchardists. He noted that while trees in deep, well drained, sandy soils with low clay content were the most readily affected by the disease, in some regions little leaf also affected trees in loam soils. He paid special attention to orchards on land formerly used as corrals for livestock. On these soils, with high nitrogen content, the disease was rampant. Quickly ruling out deficiencies of nitrogen, phosphorus, potassium, calcium, and magnesium, Chandler proceeded to test for iron. A preliminary mid-winter trial with large quantities of a commercial grade of ferrous sulphate resulted in normal leaves in summer. Chandler first thought the iron sulphate worked by reducing the alkalinity of the soil, but other pH-lowering sub
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stances such as sulphur had no effect. When he then applied chemically pure ferrous sulphate, the results were equally negative, and it was immediately apparent that an impurity in the commercial grade of ferrous sulphate might account for its effectiveness. Chemical analysis showed that the sulphate contained one percent of zinc and several other elements in small amounts. Further tests with zinc sulphate gave positive results, though the amounts required varied widely and a broader range of trials seemed called for. Chandler decided against concentrating his efforts in a single area, opting instead for a wide range of soils—twenty-six locations in ten counties. Leaving several severely diseased trees in each locality as controls, he treated some 2,000 others. It soon became evident to him that the degree of correction was a function of the solubility and dosage of the zinc compounds used. Yet extreme variability in the effectiveness of the treatment also suggested significant differences in the zinc sulphate. To find out whether zinc was essential to fruit tree nutrition or had a secondary, soil-related function (such as correcting for undesirable flora), it was necessary to circumvent the soils and apply zinc directly to the trees. This Chandler accomplished in a variety of ways. He put dry zinc sulphate in gelatine capsules in holes in tree trunks, getting earlier, longer-lasting benefits than from soil treatments. He found that trees would absorb zinc from metallic zinc nails driven into the trunk or branches, and—though this treatment caused some injury to the wood—injured areas usually filled with callous tissue if the nails were not too close together. Trees cured of zinc deficiency symptoms by these direct methods, moreover, remained healthy for six years or more after a single application, though with certain citrus and stone-fruit trees, spraying trees with a zinc sulphate solution got the earliest beneficial results. Chandler favored the idea that zinc, a nutrient required
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in minute amounts, acted as a catalyst for some biochemical process. In view of his observation that the demand was greatest when respiration was likely to be most rapid, he suggested that this catalysis might be an essential step in the respiratory pathway. REFLECTIONS, CONVICTIONS, AND FAITH I cherish the privilege of having had Professor Chandler as my teacher and mentor during my student days at Berkeley and as my colleague and friend after his move to UCLA. During the decade preceding his retirement, and for a considerable time thereafter, he expressed many thoughts (often unorthodox) on matters within and outside his immediate professional interests. He was particularly concerned with the position of the university in society, the role of the investigator and teacher in agricultural schools, and the responsibilities of scientists—both as citizens and as members of the human race. Many of these opinions were delivered in speeches to meetings of faculty, students, extension workers, and fruit growers. Copies of Prof. Chandler's speeches, which I was privileged to receive, serve as the main background for the comments in this section. To Chandler, work for an institution of higher learning where scholars joined together in the attempt to find truth was a great cause deserving of the highest loyalty. For loyalty to survive the confusing vicissitudes of life, he added, its object had to be too important to be blamed for failures. "I may serve my cause ill," he quoted the philosopher Josiah Royce. "I may conceive it erroneously. I may lose it in the thicket of world transient experience. My every human endeavor may involve a blunder. My mortal life may seem one long series of failures. But I know that my cause liveth." Chandler singled out universities as the greatest cooperative enterprise the world has ever known, for the investi
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gator—engaged in solving a problem of his choosing—collaborated not only with his contemporaries but with generations of seekers of knowledge from the past as well. Chandler's own work, for instance, depended on that of those brave, "determined souls" of the Dark Ages who recorded unorthodox findings at their own peril. From this historical view of communication's significance to science, Chandler particularly emphasized precise and careful reporting as essential to the great cooperative enterprise of learning. He remarked that, as methodology becomes more refined and thinking more rigorous, the presentation of data becomes more concise. "Where opinions are published in the most words and where there is most argument," he observed, there is the greatest accumulation of ignorance most likely to be found. Chandler admired the brief, precise reports—targeted to a specific audience and unencumbered by lengthy discussions—common to the physical sciences. By contrast, agricultural experimentalists often failed to address their most interested readers, being more concerned about a paper's reception in peripheral scientific fields than its usefulness to other horticulturists. They published too often, he maintained, in too much detail, included exhaustive reviews of the literature, and got lost in wordy theoretical explanations. He particularly objected to experimental stations publishing special editions of technical papers, which tended to be lengthy, cumbersome, costly, of limited reader access, and poorly edited. He favored, rather, publication in society journals, which had a wide circulation and were reviewed by peers capable of independent judgment. The issue of priority of authorship in scientific publishing also failed to impress Chandler. Since, he said, investigators were rarely responsible for the same data in a paper, priority played little role in their professional standing among their
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peers. For all that he admonished his colleagues to be especially vigilant in fully crediting their research associates—including assistants and graduate students— for their contributions. Finally, Chandler cautioned agricultural experiment stations against possessiveness with regard to research projects. While major responsibility and funding should go to the best qualified investigators, he contended, others should be encouraged to test promising leads. Researchers should also welcome the cooperation of county farm advisors and extension specialists. These people, who knew local conditions best, could help by testing laboratory results on the farm or arranging for the use of outside growers' field plots. Chandler further advised laboratory people to present their findings to farmers through agricultural agents rather than direct contact. He saw no discredit in a researcher attending so diligently to his research that he had no time to learn applied aspects of the work necessary for giving the best practical advice. He himself had intimate personal knowledge of working with trees that yielded publishable data but rarely and practical advice for growers even less. In real life, according to Chandler, farmers "harassed by a whole range of nature's reactions" posed challenging questions to horticultural researchers. Yet attempts to solve a problem with fruit trees required the convergence of several disciplines, and those who "discovered" a practical remedy might be no more deserving of credit than the many earlier researchers whose earlier experiences had suggested the solution. It was often, he contended, a matter of good fortune to come to a problem when just a few added experiences were needed to supply the solution. In a dinner talk delivered in 1941 to the western section of the American Society for Horticultural Science (1942,2), Chandler reflected on the merits of studying plants.
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"The material we work with has character," he stated, considering himself fortunate in both the trees and the people with whom he had worked. Citing literary references to the sturdy character and earthy beauty of the apple tree, he went on to say that to him fruit and vegetables were not merely a mass of materials but a collection of individuals. Trees and plants, furthermore, were not merely objects worthy of admiration, they also exerted an influence on the behavior of the people who tended them. "As the apple tree is among the trees of the wood," he quoted from the Song of Songs, "so is my beloved among the sons. I sat down under his shadow and his fruit was sweet to me." Chandler suggested that Thomas Jefferson's ability to endure the rigors and criticism of political life might be attributed to the comfort and encouragement he derived from the extensive time he spent on his farm working with his trees. Chandler discovered that, in the Scandinavian countries perhaps more than anywhere else, the beauty of flowers and trees, both ornamental and fruitbearing, was associated with efforts for the general good that he himself called "effective human love." When he visited Denmark he was told that preference in police recruitment was given to horticultural school graduates who were known for their even tempers. In Sweden, trained agriculturalists were put in charge of urban housing projects in recognition of the importance of plants for social contentment. Chandler expressed his faith in the Tree of Knowledge and in humankind in the following words: "The God of Nature reveals his laws, I believe, very rarely to the propagandist or to the pompous, or even to the merely zealous, but rather to him who trains diligently in the technique and the records of a system of knowledge, who records his own observations clearly and briefly for the benefit of all workers, who reviews and reorganizes his knowledge frequently in the light of new discoveries, who consults as frequently as possible with workers in his field and related fields, hoping for a vision that
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points to a safe advance in human welfare, and who is meek enough to see a vision unobscured by projects of himself. "Truth discovered by research enters into the lives of the people and its beauty is recorded for all time in literature and art; the drudgery of the laboratory today becomes beauty in the soul of humanity tomorrow. Because our discoveries enter the basic part, the masonry of the soul of humanity, we should report them with modest reverence. We want a foundation not of spongy lava thrown up by workers—each anxious to strut about the biggest pile, even if it is the trashiest—but rather of dressed stone, each piece placed carefully where it belongs in the structure. "We can have faith in the triumph of good in humanity in spite of the evil we know exists; in fact, life is richer because of the imperfections in it. I liked the part in one of George Bernard Shaw's plays where the Bishop advised people always to give the devil a chance to state his case, for I have come to believe that the devil has a rather strong case. He stands for selfishness, and a degree of selfishness is socially necessary for the most diligent care of each individual. Furthermore, we need something to struggle against. If in man the instinct of self-preservation, selfishness, and the group instinct, human love, were so nicely balanced that there would be no conflict, so that we could just enjoy our goodness comfortably like pigs enjoy their fatness, would life be very interesting? "Perhaps the richest part of life is knowledge of the great people that have been in it. If selfishness were no problem, we should never have heard of the thundering righteousness of the Hebrew prophets or of Jesus; they would have been just other nicely balanced men. And what use would we have had for Thomas Jefferson or Lincoln or Horace Greeley, or for the thousands of supporters who made their work possible, dormant-bud Jefferson's and Lincoln's and Greeley's out among the people? The only changes I want to see in man are those he makes himself—struggling upward in response to the soul of humanity and his group instinct. "The emblem of my faith is the tree and its system of dormant buds that can grow only if buds that happen to be in more favorable positions for growth are removed. If ends of branches are removed, shoots will grow out of the older wood from buds that have grown each year only enough to keep their tips in the bark. Then when their opportunity comes, they grow vigorously. Because of this reserve of dormant buds a tree is more dependable in a destructive world. It can be broken to pieces pretty badly and will grow new parts to replace the lost ones. "This condition in the tree symbolizes my faith in humanity, my con
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viction that society, at least in those countries that have been able to maintain order without despotism most of the time, cannot long change in any direction except toward a richer life for the average person: For I know there are many dormant buds in human society also.''
William Chandler shared his sturdy faith in humanity with the renowned fellow-botanist Liberty Hyde Bailey. Both lived to a ripe and productive old age, and I include, in conclusion, a stanza from "My Great Oak Tree," a poem by Bailey that Chandler greatly cherished: "And thrice since then far over the sea Have I journeyed alone to my old oak tree And silently sat in its brotherly shade And I felt no longer alone and afraid; I was filled with strength of its brawny-ribbed bole And the leaves slow-whispered their peace in my soul."
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Selected Bibliography
1904 Result of girdling peach trees. West. Fruit Grower, 15: 191. 1907 The winter killing of peach buds as influenced by previous treatment. Mo. Agric. Exp. Stn. Bull., 74:1-47. 1908 Hardiness of peach buds, blossoms and young fruits as influenced by the care of the orchard. Mo. Agric. Exp. Stn. Circ., 31:1-31. Instructions for spraying. Mo. Agric. Exp. Stn. Circ., 34:1-16. 1911 Cooperation among fruit growers. Mo. Agric. Exp. Stn. Bull., 97:3-58. 1912 Combating orchard and garden enemies. Mo. Agric. Exp. Stn. Bull., 102:237-90. 1913 The killing of plant tissue by low temperature. Mo. Agric. Exp. Stn. Res. Bull., 8:141-309. Commercial fertilizers for strawberries. Mo. Agric. Exp. Stn. Bull., 113:297-305. 1914 Sap studies with horticultural plants. Mo. Agric. Exp. Stn. Res. Bull., 14:491-553. Some problems connected with killing by low temperature. Proc. Am. Soc. Hortic. Sci., 11:56-63. Osmotic relationships and incipient drying with apples. Proc. Am. Soc. Hortic. Sci., 11:112-16. 1915 Some peculiar forms of winter injury in New York State during the winter of 1914-15. Proc. Am. Soc. Hortic. Sci., 12:118-21.
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1916 Influence of low temperature on fruit growing in New York State. Cornell Countryman, 13:373-77. 1918 Influence of low temperature on fruit growing in New York State. N.Y. State Fruit Grow. Assoc. Prod., 16:186-94. Winter injury in New York State during 1917-18. Proc. Am. Soc. Hortic. Sci., 15:18-24. 1919 Pollination. Ind. Hortic. Sci. Trans. for 1918:11-120, 173-75. The effect of the cold winter of 1917-18 on the fruit industry. Ind. Hortic. Sci. Trans. for 1918:91-103. Pruning-its effect on production. Ind. Hortic. Sci. Trans. for 1918:137-45, 156-61. Some results as to the response of fruit trees to pruning. Proc. Am. Soc. Hortic. Sci., 16:88-101. 1920 Winter injury to fruit trees. Mass. Dep. Agric. Circ., 24:11. Some preliminary results from pruning experiments. N.Y. State Hortic. Soc. Proc., 2:77-84. Some responses of bush fruits to fertilizers. Proc. Am. Soc. Hortic. Sci., 17:201-4. 1921 The trend of research in pomology. Proc. Am. Soc. Hortic. Sci., 18:233-40. 1922 The outlook of agricultural research. (Address delivered at the dedication of the Dairy Industry and Horticulture buildings, University Farm, Davis: 24-37.) 1923 Results of some experiments in pruning fruit trees. N.Y. Agric. Exp. Stn. Cornell Bull., 415:5-74.
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1924 The advantages and disadvantages of organization and standardization in horticultural research. Proc. Am. Soc. Hortic. Sci., 21:259-63. 1925 Fruit Growing. Boston: Houghton Mifflin Co. Polarity in the formation of scion roots. Proc. Am. Soc. Hortic. Sci., 22:218-22. With A. J. Heinicke. Some effects of fruiting on growth of grape vines. Proc. Am. Soc. Hortic. Sci., 22:74-80. 1926 With A. J. Heinicke. The effect of fruiting on the growth of Oldenburg apple trees. Proc. Am. Soc. Hortic. Sci., 23:36-46. 1928 North American Orchards, Their Crops and Some of Their Problems. Philadelphia: Lea & Febiger. 1931 Freezing of pollen: evidence as to how freezing kills plant cells. Am. J. Bot., 18:892. With D. R. Hoagland and P. L. Hibbard. Little leaf or rosette of fruit trees. Proc. Am. Soc. Hortic. Sci., 28:556-60. 1932 With D. R. Hoagland and P. L. Hibbard. Little leaf or rosette of fruit trees. II. Effect of zinc and other treatments. Proc. Am. Soc. Hortic. Sci., 29:255-63. With D. R. Hoagland. Some effects of deficiencies of phosphate and potassium on the growth and composition of fruit trees under controlled conditions. Proc. Am. Soc. Hortic. Sci., 29:267-71. 1933 With D. R. Hoagland and P. L. Hibbard. Little leaf or rosette of fruit trees. III. Proc. Am. Soc. Hortic. Sci., 30:70-86. With W P. Tufts. Influence of the rest period on opening of bud
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of fruit trees in spring and on development of flower buds of peach trees. Proc. Am. Soc. Hortic. Sci., 30:180-86. 1934 The dry matter residue of trees and their products in proportion to leaf area. Proc. Am. Soc. Hortic. Sci., 31:39-56. With D. R. Hoagland and P. L. Hibbard. Little leaf or rosette of fruit trees. IV. Proc. Am. Soc. Hortic. Sci., 32:11-19. 1935 With A. S. Hildreth. Evidence as to how freezing kills plant tissue. Proc. Am. Soc. Hortic. Sci., 33:27-35. With D. R. Hoagland and P. L. Hibbard. Little leaf or rosette of fruit trees. V. Effects of zinc on the growth of plants of various types in controlled soil and water culture experiments. Proc. Am. Soc. Hortic. Sci., 33:131-41. 1936 With D. R. Hoagland and P. R. Stout. Little leaf or rosette of fruit trees. VI. Further experiments bearing on the cause of the disease. Proc. Am. Soc. Hortic. Sci., 34:210-12. With M. H. Kimball, G. L. Philp, W. P. Tufts, and G. P. Weldon. Chilling requirements of opening of buds on deciduous orchard trees and some other plants in California. Calif. Agric. Exp. Stn. Bull., 611:3-63. 1937 Zinc as a nutrient for plants. Bot. Gaz., 98:625-46. 1938 The winter chilling requirements of deciduous fruit trees. Blue Anchor, 5:2-5. Our work. (Address to the Synapsis Club of the Citrus Experiment Station at Riverside, California, October 3, 1938, pp. 1-13.) Rolling of leaves on Oriental plum trees, apparently caused by cool summers. Proc. Am. Soc. Hortic. Sci., 26:259-60. 1940 Some problems of pruning, with special application to shade trees. In: Proc. Seventh Western Shade Tree Conference, pp. 50-64.
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Teaching in a college of agriculture. (Address before the annual conference of the Agricultural Extension Service, January 2, 1940, pp. 1-8.) 1942 Deciduous Orchards. Philadelphia: Lea & Febiger. Forty years of helping the farmer with knowledge. Science, 95:563-67. Sermons. (Address before the Western Section of the American Society for Horticultural Science, June 20, 1941.) Proc. Am. Soc. Hortic. Sci., 41:387-97. 1943 Some responses of trees in a few subtropical evergreen species to severe pruning. Proc. Am. Soc. Hortic. Sci., 42:646-51. 1944 Sturdy faith and dormant buds. (Address before joint meeting of the Synapsis Club, Citrus Experiment Station, and the American Society for Horticultural Science. January 3, 1944, pp. 1-6.) 1945 Trees in two climates. (Faculty Research Lecture, University of California, Los Angeles, March 21, 1944. Univ. of Calif. Press, Berkeley and Los Angeles, pp. 1-22.) 1946 With D. R. Hoagland and J. C. Martin. Little leaf or rosette of fruit trees. VIII . Zinc and copper deficiency in corral soils. Proc. Am. Soc. Hortic. Sci., 47:15-19. With D. Appleman. Little leaf or rosette of fruit trees. IX. Attempt to produce corral injury with constituents of urine. Proc. Am. Soc. Hortic. Sci., 47:25. 1949 Pruning trials on wisteria vines. Proc. Soc. Hortic. Sci., 54:482-84. Evergreen Orchards. Philadelphia: Lea & Febiger.
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1951 Deciduous Orchards, 2d ed. Philadelphia: Lea & Febiger. With D. S. Brown. Deciduous orchards in California winters. Calif. Agric. Ext. Serv. Circ., 179:3-39. 1952 With R. D. Cornell. Pruning ornamental trees, shrubs, and vines. Calif. Agric. Ext. Serv. Circ., 183:1-44. 1954 Cold resistance in horticultural plants: A review. Proc. Am. Soc. Hortic. Sci., 64:552-72. 1955 Twenty-five years' progress in California fruit production. (Address delivered at University of California, Davis, October 29, 1955, pp. 1-15.) 1957 Deciduous Orchards, 3d ed. Philadelphia: Lea & Febiger. 1958 Evergreen Orchards, 2d ed. Philadelphia: Lea & Febiger. 1959 Plant physiology and horticulture. (Prefatory chapter.) Annu. Rev. Plant Physiol., 10:1-12. 1961 Some studies of rest in apple trees. Proc. Am. Soc. Hortic. Sci., 76:1-10. 1965 Reminiscences. Oral History Program. University of California, Los Angeles, pp. 1-39.
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Gertrude Mary Cox January 13, 1900-October 17, 1978
By Richard L. Anderson This is a final tribute to a fellow statistician, fellow graduate student, employer, and—above all—best friend and well-wisher, the confidante and constant companion of my wife and children. Gertrude Mary Cox had that rare combination of administrative strength and love for her fellow man we so desperately need at the present time. A gracious, patient, tenacious visionary, she brought out the best in people. As a pioneer in the development of statistics she was a servant to science who never lost her touch with people.1 EARLY YEARS Gertrude Cox was born on a farm near Dayton, Iowa, where she spent several years "roaming in the woods by the river," as she put it, "and wandering over the hills." The family then moved to the small town of Perry, Iowa, where Gertrude attended public school. A lover of competitive sports, she played on the high school basketball team. (Iowa was the center of girls' basketball in those days.)
1 Much of what is printed here is excerpted from a 1979 obituary I prepared with Robert Monroe and Larry Nelson of North Carolina State University, "Gertrude Cox—A Modern Pioneer in Statistics," Biometrics 35(1979):3-7. I have also included remarks from a letter Gertrude Cox wrote to me on October 10, 1975.
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The Coxes were a close-knit, midwestern family with four children—two boys and two girls. Gertrude was especially close to her mother, Emma, and later wrote of her: "I learned from my mother the value and joy of doing for other people. She nursed the sick for miles around and raised us to be active church workers." During those early years Gertrude also learned to like making bread— perhaps because she was allowed to sell one pan of biscuits. Her excellent cinnamon rolls were famous. She always served them to us whenever we visited, and, when we left, provided one package for our son, Bill, and another for the rest of us. Gertrude loved children and always joined us on Christmas morning to see our two youngsters open their gifts. Gertrude's early ambition was to help others. She took a two-year course in social science, then spent another two years as a housemother for sixteen small orphan boys in Montana. As preparation for becoming the superintendent of the orphanage, she decided to enroll at Iowa State College. Majoring in mathematics because it was easy for her, she elected courses in psychology, sociology, and crafts—courses useful to her in her chosen career. In 1929 she received her B.S degree. To help pay her college expenses Gertrude did computing, George Snedecor —her calculus professor—having asked her to work with the comptometers in his computing laboratory. Speculating (forty-six years later) as to why he had chosen her for this work, she told the Raleigh News and Observer in May 1975, that he had probably hoped that she, the only woman in the class, would have more patience for detail work than the men. Perhaps because of this computing experience, Gertrude became interested in statistics. But the Mathematics Department at that time would not award an assistantship to a
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woman, and she financed her graduate work with assistantships in psychology and art. In 1931, she received the first Master's degree ever given by Iowa State in statistics but was turned down for a job teaching high school mathematics because she did not have the required courses in education. She decided to continue her graduate career. Because of her love of people and her desire to learn what ''made them tick," Gertrude chose psychology as her research area. With a graduate assistantship at the University of California, Berkeley, she began work on a doctorate in psychological statistics. Unfortunately for the field of psychology, she stayed only two years. In 1933, Iowa State established its Statistical Laboratory under the direction of George Snedecor, Gertrude's former mentor, and he persuaded her to return home to help him. Back in Iowa, she continued her interest in psychology and worked with several members of the Psychology Department—including its chairman (later, dean of the School of Industrial Science), Harold Gaskill—on the evaluation of aptitude tests, test scoring procedures, and the analysis of psychological data. At the same time she was put in charge of establishing a Computing Laboratory and consulted in and taught experimental designs. In 1934 she began to teach "Design of Experiments"—a course that would become renowned —to follow Snedecor's "Statistical Methods." Most graduate students in agriculture were required to take this sequence, a requirement that was later extended to a number of other disciplines at Iowa State and was my own introduction to experimental statistics. Both the Snedecor and Cox courses were originally taught from mimeographed materials. In 1937, Snedecor's material came out in book form, but Gertrude only published her design material in 1950, when it came out as a collaborative effort with W. G. Cochran (1950,7). Gertrude's course was built around a multitude of specific
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examples (many of which I still keep in my files) in a variety of areas of experimentation. Members of her computing staff analyzed all of the data, which were then completely checked by Gertrude and the hundreds of graduate students in her course. Despite the fact that these experiments were conducted four or five decades ago, they could still furnish the basis for a solid course on the design of experiments, especially in biology and agriculture. In her later "Advanced Experimental Design," Gertrude concentrated on her three basic principles for setting up an experiment: (1) Experiment objectives should be set forth clearly at the outset, the experimenter having answered the following questions regarding his or her experiment: Is it a preliminary experiment to determine the course of future research or is it intended to furnish answers to immediate questions? Are the results to be put to immediate practical use or are they intended to help clarify theoretical questions? Does the researcher wish to obtain estimates or to test for significance? Over what range of experimental conditions do the results extend? (2) The experimenter should describe the experiment in detail, clearly defining proposed treatments, size, and materials: Is a control treatment necessary for comparison with past results? Will the funds available support an experiment of sufficient size to yield useful results? Are the materials necessary for the experiment available? (3) The experimenter should draw up an outline analyzing the data before starting the experiment. Both as a teacher and a consultant, Gertrude particularly emphasized randomization, replication, and experimental controls as procedures essential to experimental design: "Randomization is somewhat analogous to insurance in that it is a precaution against disturbances that may or may not occur and that may or may not be serious if they do occur. It is generally advisable to take the trouble to randomize even when it is not expected that there will be any serious bias from failure to randomize. The experimenter is thus protected against unusual events that upset his expectations. Of course in experiments
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where a great number of physical operations are involved, the application of randomization to every operation becomes time-consuming, and the experimenter may use his judgment in omitting randomization where there is real knowledge that the results will not be vitiated. It should be realized, however, that failure to randomize at any stage may introduce bias unless either the variation introduced in that stage is negligible or the experiment effectively randomizes itself." (1950,7, p.8)
As she pointed out, replication not only increases the accuracy of treatment comparison, it also enables the experimenter to obtain a valid estimate of the magnitude of experimental error. She also offered the following ways to increase accuracy by improving the control of experimental techniques: (1) (2) (3) (4)
Select the best experimental design for the proposed experiment; Ascertain the optimal size and shape of the experimental unit; Use uniform methods for applying treatments to experimental units; In order that every treatment operate under conditions as nearly the same as possible, exercise control over external influences; (5) Devise unbiased methods for increasing treatment effects; (6) Take additional measurements (covariates) often to help explain final results; (7) Provide checks to avoid gross errors in recording and analyzing data. Though Gertrude was enrolled in a Ph.D. program in mathematics at Iowa State, her teaching and consulting duties did not leave her enough time to write a dissertation. An "assistant" from 1933, she was appointed research assistant professor in 1939, though her design course was listed under Professor Snedecor's name. In 1940 Snedecor was asked to recommend candidates to head the new Department of Experimental Statistics in the School of Agriculture at North Carolina State College. "Why didn't you put my name on the list?" Gertrude asked when
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he showed her his all-male list of candidates, and her name was added to the accompanying letter in the following postscript: "If you would consider a woman for this position, I would recommend Gertrude Cox of my staff." This terse note was to have far-reaching consequences for statistics, for not only was Gertrude considered, she was selected. Her resignation led to a heart-rending session with Dean Gaskill, she later told me, in which he tried to convince her that she was being disloyal to her native state and to Iowa State College, and that a woman would never be accepted as a department head in a southern state. SOUTHERN VENTURE Gertrude Cox became the head of North Carolina State's Department of Experimental Statistics on November 1, 1940. The Board of Trustees of the Consolidated University of North Carolina authorized the establishment of the Department and confirmed Professor Cox as its head on January 22, 1941. She had strong support from the U.S. Bureau of Agricultural Economics, which had been instrumental in establishing the Department, and, in particular, from the Raleigh-based Division of Agricultural Statistics of its North Carolina Research Office. She encouraged researchers in the School of Agriculture to attend her experimental design course and recruited capable applied statisticians to develop and teach basic statistical methods. She made these statisticians available to consult with researchers on procedures for designing experiments and analyzing data. Most faculty had been trained in one of these disciplines and acquired some statistical training as a minor area. To secure at least one faculty member for every agricultural discipline, she had to start from scratch. "There weren't any statisticians to hire when I first started," she later wrote. "I had to choose from other fields and train them."
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By the time Gertrude left Ames, I had had enough of Iowa's winters and told her I would like to join her group whenever a mathematical statistician position became available. In 1942, I transferred from North Carolina State's Mathematics Department to handle statistical consulting with agricultural economists. Gertrude had decided that it was necessary to bolster the methods courses with courses in statistical theory; a graduate program was in the offing. Another innovative feature of the Cox statistics program was a series of one-week working conferences on specific topics, such as agricultural economics and rural sociology, biological and nutritional problems, agronomic and horticultural problems, plant sciences, animal sciences, quality control, nutrition, industrial statistics, soil science, and plant breeding. Gertrude later obtained outside funds to hold two summer conferences in the mountains of North Carolina, which were attended by statisticians from throughout the United States and abroad. In addition to experimental and mathematical statistics, these conferences covered many research areas involving statistics, including life testing, operations research, clinical trials, surveys, pasture and rotation experiments, and genetics. Many were held during World War II. Gertrude, realizing the importance of quality control methods to the war effort, included engineering statisticians on the faculty. During this period Gertrude realized still another dream. She had become a close friend of Frank Graham, the University's president, who had been instrumental in starting the statistics program in 1940. In 1944, Dr. Graham helped her get a grant from the General Education Board, founded by John D. Rockefeller, to establish and direct an Institute of Statistics to improve statistical competency in the South. This grant enabled her to add six faculty members to her department, including W. G. Cochran, who was to develop a grad
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uate program. In 1945, the General Education Board made an additional grant to establish a Consolidated University of North Carolina Institute of Statistics, with a Department of Mathematical Statistics at Chapel Hill, to concentrate on graduate training and research in statistical theory. With complementary graduate programs, the two departments produced many outstanding applied and theoretical statisticians. Gertrude remained as head of the Experimental Statistics Department in Raleigh until 1949, when she decided to administer the Institute almost fulltime, with the exception of teaching her course in experimental design. In the School of Public Health at Chapel Hill, she helped establish the Biostatistics Department, the Social Science Statistical Laboratory in the Institute for Research in Social Science, and the Psychometric Laboratory in the Department of Psychology. These two laboratories were a culmination of Gertrude's lifelong interest in the use of statistics to study human relationships. During this time, North Carolina State statisticians began visiting a number of experimental stations to assist research programs in the use of statistical methods. Cox's Institute coordinated a number of short courses for researchers in industry and the physical sciences. One of her most important accomplishments was her successful effort, along with Boyd Harshbarger of Virginia Polytechnic Institute, to establish the Southern Regional Education Board's Committee on Statistics to develop cooperative programs for statistics teaching, research, and consulting in the South. This Committee contributed tremendously to sound statistical programs throughout the South and fostered the spirit of cooperation that Gertrude envisaged. From 1954 to 1973 it sponsored a continuing series of six-week summer sessions and is now conducting an annual one-week Summer
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Research Conference modeled on the original Gordon conferences.2 Gertrude's first contact with statistics came in the computing laboratory, and she remained a strong advocate of the integral connection between statistical analysis and an up-to-date computing facility. At Iowa State she had developed an excellent computing laboratory. Early in 1941 she persuaded Robert Monroe, one of her chief associates there, to come to Raleigh to develop a similar facility. I remember those old Hollerith machines at Ames and Raleigh— and the tremendous leap forward when IBM entered the electronic age. Gertrude Cox, naturally, had one of the first IBM 650s on a college campus, and North Carolina State subsequently designed for the 650 the best statistical software. From then on, Gertrude made certain that the Institute was in the forefront when it came to statistical software, and Raleigh statisticians designed the initial SAS programs. No account of Gertrude Cox's meteoric success at the University of North Carolina would be complete without mentioning her unique ability to secure outside financial support. Though her Institute was originally funded by General Education Board grants, Gertrude Cox persuaded the Rockefeller Foundation to support a substantial program in statistical genetics. She obtained funds from the Ford Foundation for a joint program in dynamic economics with the London School of Economics. Finally, in 1952, she obtained a large grant from the General Education Board (matched by 1958) for a revolving research fund enabling the Institute to finance fundamental, nonsponsored statistical research for many years thereafter.
2 The Committee Cox and Harshbarger founded was still operating as of 1990 under the name of the Southern Regional Committee on Statistics. Though no longer affiliated with the Southern Regional Education Board, it continues to sponsor summer research conferences.
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Starting in 1958, Gertrude and seven other members of the North Carolina State statistics faculty worked out procedures for establishing a Statistical Division in the proposed not-for-profit Research Triangle Institute (RTI) between Raleigh and Chapel Hill. RTI was established in 1959, and Gertrude retired from North Carolina State in 1960 to direct its Statistics Research Division, whose major component was the sample survey unit. Retiring from that post in 1965, she continued on as a consultant for many years, even occasionally teaching her design course at North Carolina State. During her fiveyear tenure, RTI—and especially the Statistics Division—became an internationally recognized consulting and research organization. Gertrude Cox was a consultant to the Pineapple Research Institute of Hawaii, the World Health Organization in Guatemala, the U. S. Public Health Service, the government of Thailand, the Pan American Health Organization, and many other organizations overseas. She served on a number of government committees including the U. S. Bureau of the Budget's Advisory Committee on Statistical Policy (1956-1958); the National Institutes of Health's Agricultural Marketing Service, Epidemiology, and Biometry Committees (1959-1964); and the National Science Foundation's Office of Education (1963-1964) and Teacher Education Section (1966). Even after retirement she served on advisory committees to the Secretary of Health, Education and Welfare (1970-1973), the Bureau of the Census, and the Department of Agriculture (1974). PROFESSIONAL ACTIVITIES AND HONORS Gertrude Cox's major contribution to science was her ability to organize and administer programs, but her early accomplishments in psychological statistics and experimental design were widely recognized.
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Gertrude was a founding member of the International Biometric Society in 1947, served as editor of its journal, Biometrics, from 1947 to 1955, and was a member of its Council three times and president from 1968 to 1969. She was proud that she had attended every international meeting of the society, and, in 1964, was awarded an honorary life membership. She was also active in the International Statistical Institute and was a member of its Council in 1949, treasurer from 1955 to 1961, and chairman of the Education Committee from 1962 to 1968. She was president of the American Statistical Association (ASA) in 1956. She was a fellow of the American Public Health Association, the American Association for the Advancement of Science, the Institute of Mathematical Statistics, and the ASA. She was also a member of the Psychometric Society, the Royal Statistical Society, and the Inter-American Statistical Institute. In recognition of her international reputation she was named honorary vicepresident of the South African Statistical Association, honorary member of the Société Adolphe Quetelet of Belgium, and the Thai Statistical Association, and an honorary fellow of the Royal Statistical Society. She was a member of the honor societies Delta Kappa Gamma (education), Gamma Sigma Delta (agriculture), Pi Mu Epsilon (mathematics), Phi Kappa Phi (scholastic), and Sigma Xi (science). In 1958, Gertrude Cox's alma mater, Iowa State University, conferred upon her an honorary Doctorate of Science as a "stimulating leader in experimental statistics . . . outstanding teacher, researcher, leader and administrator . . . Her influence is worldwide, contributing to the development of national and international organizations, publications, and councils of her field." In 1959 she received the highest recognition the Consolidated University of North Carolina can confer upon its faculty—the 0. Max Gardner Award. The
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citation named her a "statistical frontierswoman"—a phrase suggested by the title of her ASA presidential address, "Statistical Frontiers." In 1970, North Carolina State University honored her once again by designating the building in which the Statistics Department is located Cox Hall, and in 1977 a Gertrude M. Cox Fellowship Fund was established for outstanding graduate students in statistics. Her most treasured honor came in 1975, when she was elected to the National Academy of Sciences. TRAVELS Gertrude Cox was a world traveller who particularly enjoyed working in developing countries where she could offer advice and inspiration. All of Gertrude's trips were carefully planned, usually with reservations at excellent hotels. Fascinated by Egypt, she helped establish a statistical program at Cairo University and, during the months she spent there, toured many historical sites. She was especially excited by her visits to the Sinai and to Abu Simbel. Thailand was another of her particular favorites, and I was touched, when I visited Bangkok in 1982, by how much the Thais loved her. She loved wearing dresses she had had made from colorful Thai silk, and—a grower of orchids since her visits to Hawaii in the late 1940s—she struck up a close friendship with Rapee Sagarik, Thailand's principal orchid expert. (She grew these beautiful orchids for pleasure, not profit, and enjoyed giving them to her friends, as my own family can attest.) CLOSING REMARKS Gertrude Cox loved people, especially children. She always brought back gifts from her travels and was especially generous at Christmas time. She considered the faculty mem
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bers and their families to be her family and entertained them frequently. She was an excellent cook and had two hobbies that she indulged during her travels: collecting dolls and silver spoons. She learned chip carving and block printing at an early age and spent many hours training others in these arts. She loved gardening, and, when she had had a particularly hard day with administrators, would work off her exasperation in the garden. She had a fine appreciation for balance, design, and symmetry. In 1976, Gertrude learned that she had leukemia but remained sure that she would conquer it up to the end. She even continued construction of a new house, unfortunately not completed until a week after her death. While under treatment at Duke University Hospital she kept detailed records of her progress, and her doctor often referred to them. With characteristic testy humor she called herself ''the experimental unit," and died as she had lived, fighting to the end. To those of us who were fortunate to be with her through so many years, Raleigh will never be the same.
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Selected Bibliography
1930 A statistical study of industrial science students of the class of 1926. Iowa Acad. Sci. Proc., 37:337-41. 1931 The use of the individual parts of the aptitude test for predicting success of students. Iowa Acad. Sci. Proc., 38:225-27. 1933 With C. W. Brown and P. Bartelme. The scoring of individual performance on tests scaled according to the theory of absolute scaling. J. Educ. Psychol., 24:654-62. 1935 With G. W. Snedecor. Disproportionate subclass numbers in tables of multiple classification. Iowa Agric. Exp. Stn. Res. Bull., 180:233-72. Index number of Iowa farm products prices. Iowa Agric. Exp. Stn. Bull., 336:297-328. 1936 With G. W. Snedecor. Covariance used to analyze the relation between corn yield and acreage. J. Farm Econ., 18:597-607. 1937 With H. Gaskill. Patterns in emotional reactions: I. Respiration. The use of analysis of variance and covariance in psychological data. J. Gen. Psychol., 16:21-38. With G. W Snedecor. Analysis of covariance of yield and time to first silks in maize. J. Agric. Res., 54:449-59. With W. P. Martin. Use of discriminate function for differentiating soils with different asotabacter populations. Iowa State Coll. J. Sci., 11:323-32. 1939 The multiple factor theory in terms of common elements. Psychometrika, 4:59-68.
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With M. G. Weiss. Balanced incomplete block and lattice square designs for testing yield differences among large numbers of soybean varieties. Iowa Agric. Exp. Stn. Res. Bull., 257:290-316. 1940 Enumeration and construction of balanced incomplete block configurations. Ann. Math. Stat., 11:72-85. With R. C. Eckhardt and W. G. Cochran. The analysis of lattice and triple lattice experiments in corn varietal tests. Iowa Agric. Exp. Stn. Res. Bull., 281:1-66. 1941 With H. V. Gaskill. Patterns in emotional reactions. II. Heart rate and blood pressure. J. Gen. Psychol., 23:409-21. 1942 With H. McKay, et al. Length of the observation period as a factor in variability in calcium retentions. J. Home Econ., 34:679-81. With H. McKay, et al. Calcium, phosphorus, and nitrogen metabolism of young college women. J. Nutr., 24:367-84. 1944 Modernized field designs at Rothamsted. Soil Sci. Soc. Am. Proc., 8:20-22. Statistics as a tool for research. J. Home Econ., 36:575-80. 1945 Opportunities for teaching and research. J. Am. Stat. Assoc., 229:71-74. 1946 With W. G. Cochran. Designs of greenhouse experiments for statistical analysis. Soil Sci., 62:87-98. 1950 The function of designs of experiments. Ann. N.Y. Acad. Sci., 52(Art. 6):800-7.
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With W. G. Cochran. Experimental Designs. New York: John Wiley & Sons, Inc. 1953 Elements of an effective inter-American training program in agricultural statistics. Estadist., 11:120-28. 1957 Statistical frontiers. J. Am. Stat. Assoc., 52:1-12. (Institute of Statistics Reprint Series, no. 99.) With W. G. Cochran. Experimental Designs. 2d ed. New York: John Wiley & Sons. 1964 With W. S. Connor. Methodology for estimating reliability. Ann. Inst. Stat. Math. The Twentieth Anniversary, 16:55-67. 1972 The Biometric Society: The first twenty-five years (1947-1972). Biometrics, 28(2):285-311. 1975 With Paul G. Homeyer. Professional and personal glimpses of George W. Snedecor. Biometrics, 31(2):265-301.
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Conrad Arnold Elvehjem May 27, 1901-July 27, 1962
By R. H. Burris, C. A. Baumann, And Van R. Potter The work of Conrad Elvehjem—a major contributor to the golden era of nutritional research—touched most aspects of animal nutrition, advancing, in particular, our understanding of the B vitamins, the phenomenon of amino acid imbalance, and the identification of trace minerals needed in the diet. Elvehjem made the major discovery that nicotinic acid functions as the antipellagra vitamin. Elvehjem was also a superb administrator, an efficient man who channeled his great energy with seemingly little effort. On the local scene he served the University of Wisconsin as chairman of the Department of Biochemistry, dean of the Graduate School, and, finally, as president of the University. On the national level he helped make policy decisions concerning the level of vitamins and other nutrients required for health. The implementation of his cure for pellagra was international in scope. EARLY YEARS Conrad Elvehjem was born in 1901 to Ole Johnson Elvehjem and Christine Lewis Elvehjem on a modest farm near McFarland, Wisconsin. In this primarily Norwegian area (May 17 is still celebrated as Norwegian Independence Day in Stoughton), Elvehjem grew up and attended high
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school. He would spend his adult life within a few miles of his birth place, for Madison's capitol building is visible from the farm. The Elvehjem children were expected to do their share of the farm chores, and—while there was little time for nonsense—education was encouraged. In those days Wisconsin farm boys usually did not go to high school and college, but his family made sure he was able to do so. In 1919, Elvehjem enrolled in the University of Wisconsin's College of Agriculture, already recognized for its research in agricultural chemistry, genetics, plant pathology, and bacteriology. Elvehjem majored in agricultural chemistry, a field in which Babcock, Hart, Steenbock, McCollum, and Peterson had all done, or were doing, meritorious work at Madison. He did his undergraduate research under the direction of Harry Steenbock and wrote his senior thesis jointly with W. P. Elmslie on "buckwheat itch," a light-induced disturbance in animals. As to Elvehjem's early motivation in the choice of his career, in 1957 he answered a thirteen-year-old boy who had questioned him on this subject as follows: "I chose the field of biochemistry because as a youngster I was interested in what made plants grow and develop. I was very intrigued by the rapid growth of the corn plant, and I was interested in knowing what reactions took place within the plant to allow such rapid growth." Of his achievements, he said: "My achievements cover work on many of the B vitamins—including the isolation and identification of nicotinic acid as the antipellagra factor, also work on a number of trace mineral elements showing that they have specific functions in nutrition and metabolism. I also pioneered in work demonstrating the relationship between vitamins and enzymes. Today I am more interested in amino acids in nutrition." It never occurred to him, apparently, to mention his many administrative successes! In 1923 Elvehjem began graduate work as a teaching as
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sistant under Professor E. B. Hart, his major professor until he received his Ph.D. degree in 1927. In 1924 he published his first paper with Hart and Steenbock on dietary factors influencing calcium metabolism (1924,1). But his graduate work centered mainly on iron deficiency in rats, including a demonstration that copper must accompany iron in the diet to cure this type of anemia. Hart's encouragement of Elvehjem during his student days is just one example of his remarkable capacity to pick winners. This was before the period when talented students were being attracted to agricultural chemistry in large numbers, yet Hart had staffed his small department with a remarkable group of investigators. He supported them through administrative difficulties, had a building constructed for their teaching and research, and offered them whatever he could given the limited resources available at that time. As long as Hart lived, he and Elvehjem worked together on many joint research projects. Indeed, approximately half of Elvehjem's long list of publications contains Hart's name as well. On June 30, 1926, Elvehjem married Constance Waltz, a journalism student at the University of Wisconsin and the daughter of a Rockford, Illinois, dentist. This was a happy union, and the two Connies—called Mr. Connie and Mrs. Connie by their friends—complemented each other. He was relatively quiet, while she bubbled with enthusiasm, meeting people easily with charm and grace. She was a source of strength to her husband throughout all stages of his career, and most particularly when he held administrative positions. From 1927 to 1929, after receiving his Ph.D. degree, Elvehjem held an instructorship in agricultural biochemistry. In 1929, he received a National Research Council Fellowship to study in the biochemistry laboratories at Cambridge University, England. This was the only substantial period in his career that
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Elvehjem spent away from Madison, and he and Mrs. Connie took full advantage of it. As Elvehjem himself described it, they arrived in England, took a guided tour through London that allowed him to spot the laboratories he wanted to visit later, searched for housing in Cambridge, and met Dr. and Mrs. C. G. King—kindred souls with whom they would share many experiences. According to Elvehjem, the Biochemistry Department at Cambridge was a lively spot in 1929 and 1930, and he describes Sir Frederick Hopkins' lively welcome back as the recently announced recipient of the Nobel Prize. At Cambridge Elvehjem worked under the tutelage of Dr. David Keilin, who was then busy with the cytochromes and the role of iron in cytochrome c. Copper and iron in cytochrome oxidase were of particular interest to Elvehjem, whose own work had shown that animals deficient in copper were also deficient in cytochrome oxidase. By the 1930s, a role for copper in cytochrome oxidase was widely accepted. At Wisconsin Elvehjem had studied nutritional anemia in rats on a diet very low in iron (viz., milk). The addition of relatively large amounts of inorganic iron salts to such milk failed to prevent this type of anemia. Testing crude materials protective against anemia—and later the ash of those most potent for supplementing iron in a milk diet—Elvehjem, Steenbock, Hart, and Waddell found that traces of inorganic copper were necessary for the incorporation of iron into hemoglobin, even though hemoglobin contains no copper. In this way, the idea of catalysis in life-processes was brought forcibly to Elvehjem's attention. Elvehjem later published two papers on his work at the Biochemical Laboratory in Cambridge with acknowledgments to Hopkins "for his interest and advice" and to Keilin "for many helpful suggestions." The first, ''Factors Affecting the Catalytic Action of Copper in the Oxidation of Cysteine" (1930,1), clearly derived from a project suggested by Keilin.
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The second, "The Role of Iron and Copper in the Growth and Metabolism of Yeast" (1931,1), contained observations that, though yeast contains no hemoglobin, its respiration requires iron-containing pigments; and that copper increases the levels of cytochrome a, presumably cytochrome oxidase. Both studies gave Elvehjem experience with manometric techniques, and he spent a busy year visiting laboratories, doing research on several problems, and aiding in a laboratory course. RETURN TO WISCONSIN At Cambridge, Elvehjem—in the forefront of nutrition research—worried less about finding new vitamins than about understanding how these substances functioned in the metabolism of the living cell. He developed a new research strategy—parallel studies on respiratory enzymes and on deficiency-producing diets, especially designed to be assayed for new growth factors and trace elements. This new methodology would further allow him to isolate the new substances and determine their action. He was, therefore, particularly intrigued by the Barcroft respirometer. This instrument permitted accurate measurements of oxidative enzymatic activity with small samples of tissue, enabling researchers to define differences in the responses of normal versus deficient tissue and the responses of deficient tissue to added compounds. While he was away, Elvehjem maintained a correspondence with Hart, and their exchange of letters concerning salary is interesting. On April 23, 1930, Hart wrote Elvehjem: "I understood today that the Board of Regents had passed the budget which appoints you as an Assistant Professor at $3,000 for the academic year. You ought to be very happy over this because it was very difficult to get an increment of $600 for you in the present state of Wisconsin finances. You are young, and with summer pay and gradual increments, and an
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opportunity for research the position you will hold with us ought to be very attractive."
Elvehjem to Hart, May 16, 1930: "I was glad to have your letter and to learn that there would be a job waiting for me when I return.... I can't say that I am exceedingly happy over the salary but we can talk about that later. What I am wondering about now is, if you will buy a Barcroft for me. If we are going to continue to work on the minor inorganic elements it will come in very handy. In fact there are a thousand things to do in regard to the catalytic action of copper before leaving it in favor of other elements."
Hart agreed to let Elvehjem purchase his device, and he brought a set of respirometers back with him on his return to Wisconsin. In Madison it soon became a treasured possession, and each noninterchangeable flask was carefully guarded. The Potter-Elvehjem homogenizer remains still to remind investigators of the days when Elvehjem was actively studying respiratory enzymes. Elvehjem immediately put his Barcroft respirometer to good use studying the respiration of minced tissues from normal and from vitamindeficient experimental animals. He also continued his joint researches with Professor Hart and a number of students on the mineral requirements—zinc, manganese, and molybdenum—in the rat, chicken, dog, and pig and began a large program on the vitamin B complex, a relatively neglected area at Wisconsin at that time. In the early 1930s techniques available for nutritional studies left much to be desired. Deficient diets were usually lacking to varying degrees in more than one essential, and curative preparations contained a number of different vitamins. A typically crude (but useful) method of producing deficiencies was to damage a mixed diet with dry or moist heat, destroying vitamins differentially. The sources used for growth factors were yeast, milk, liver, or fractions of liver left
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over from the commercial preparation of extracts for the treatment of pernicious anemia. "Success" meant restoration of the growth rate—by means of a supplemented diet—that had decreased on a defective diet. Elvehjem's approach was similar to that of others working on the B vitamins except that his graduate students worked simultaneously on different growth factors or with different species, so that when one achieved a preparation active against his particular deficiency, others could test a similar preparation for those deficiencies that were their own primary concern. This insured quick determination of the effects of a given concentrate on the various deficiencies under study. NICOTINIC ACID Elvehjem was particularly skillful in coordinating experiments and crosschecking results, and he was never timid about postulating the existence of new growth factors. One of these, "Factor W," represented what, in addition to the established B vitamins, remained in a liver concentrate. His recognition of nicotinic acid as the antipellagra principle was typical of his thoroughness and his ability to combine information gleaned from various sources, with data produced by his own students, and of his active collaboration with academic and commercial colleagues. In 1912, exactly twenty-five years before nicotinic acid's true status as a vitamin was established, Casmir Funk—in one of the more curious twists of nutritional history—attempted to cure a vitamin deficiency by feeding it to polyneuritic birds. The results were unexciting. The substance came into its own as an important biochemical, however, in 1936, when Warburg and Christian identified it as one of the components of "coferment" (NADP). Discovery of its presence in cozymase (NAD) followed quickly. About the same time, several
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investigators reported it to be essential for the growth of certain microorganisms. Elvehjem and Douglas Frost, feeding nicotinic acid to rats deficient in "Factor W," reported a slight growth response, though much less than that obtained with crude liver preparations. The isolation of nicotinic acid came about primarily through the fractionation of liver extracts. By means of successive solvent extractions, Carl Koehn had converted 400 grams of liver extract to 2.5 grams of a powder active against canine black tongue. Robert Madden achieved further concentrations by means of adsorption on an appropriate charcoal. Elvehjem had for some time been receiving liver extracts for these studies from the Wilson and Abbott Laboratories. Then Dr. Rhodehamal of the Eli Lilly Company, working according to the Koehn and Elvehjem procedure, furnished a concentrate from seventeen kilograms of liver. The next big step was Frank Strong's sublimation of this concentrate in a molecular still. Almost immediately Wayne Woolley obtained crystals from the distillate and, on Karl Link's microapparatus, H. Campbell determined the percentages of C, H, and N. The response to these crystals in deficient dogs was dramatic, and the correlation between the analytical values and the theory for nicotinamide was close enough to lead Woolley to take a mixed melting point and perform the appropriate characterization reactions—all in a matter of a few days. Synthetic nicotinic acid and amide were then fed to other dogs and found to be highly active. The research community lost little time in applying these results to human pellagra. Elvehjem's first published notice of his laboratory's findings appeared in September 1937, in a "letter to the editor" of the Journal of the American Chemical Society. Van Potter, who shared an office with him at the time, recalls that Elvehjem sent telegrams to a number of clinical
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investigators interested in pellagra—including Tom Spies. Before the end of 1937, Elvehjem's results with dogs had been confirmed by six independent investigators. By the time his more complete paper on the subject appeared in 1938, it was possible to add the following: "Spies has used nicotinic acid in four cases of classical pellagra and reports (personal communication) that the fiery red color associated with pellagrous dermatitis, stomatitis, and vaginitis improved promptly." The Wisconsin paper (1938,1) not only summarized the known biochemical facts on nicotinic acid, it even expressed concern about possible toxicity in its application!1 THE B VITAMINS AND AMINO ACIDS Nicotinic acid, however, was not the only B vitamin to occupy Elvehjem and his research team. As his list of publications shows, his laboratory investigated every B vitamin at one time or another, though occasionally under a different name until its true nature was established. The clarification and disentanglement of the B complex occupied many investigators worldwide for years, during which the Elvehjem group made substantial contributions to our present understanding. But the latter years of his laboratory career were spent on amino acids, an interest that had grown out of the pellagra problem. Pellagra occurred in areas where people consumed inadequate diets high in corn, and Elvehjem's studies on black tongue in dogs also involved a diet high in corn. The diets used for studies of the B complex in rats and chicks, on the
1 Because of Elvehjem's generosity in disseminating his laboratory's findings widely, the medical implications of nicotinic acid in the treatment of pellagra became apparent almost immediately. On January 22, 1938, Tom Spies's November 5, 1937, report of his own experiments to the Central Society for Clinical Research was the subject of an editorial in the Journal of the American Medical Association. For his dramatically successful use of nicotinic acid to treat pellagra in humans, Time magazine named Spies 1938's "Man of the Year."
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other hand, were so-called "semisynthetic"—usually based on casein, starch, sugar, etc. Rats fed this diet never developed nicotinic acid deficiency, nor did administration of nicotinic acid improve their growth. But when corn was used to replace forty percent of such a diet, growth was depressed and could be restored by supplements of nicotinic acid or of tryptophan—an amino acid that is relatively lacking in corn. Further studies in a number of laboratories clarified the mechanism by which tryptophan is converted to niacin in the body. Working with A. E. Harper, Elvehjem carried out experiments on requirements for other amino acids that presaged an extensive investigation of amino acid imbalance an investigation that ceased, however, when he became president of the University. The coenzyme connection to nicotinic acid (NAD and NADP) was important in motivating Elvehjem and Thorfin Hogness, of The University of Chicago, to organize a "Symposium on Respiratory Enzymes" in Madison on September 11-13, 1941, and one on "The Biological Action of the Vitamins," held at The University of Chicago on September 1519. David H. Smith noted at the vitamin symposium that Elvehjem's observations on the relation of nicotinic acid to canine black tongue (published in the September issue of the Journal of the American Chemical Society [1937,4]) were verified promptly and extended to human pellagra by a number of investigators. Subsequent to the spectacular conquest of pellagra, Elvehjem was invited to Cornell University Medical School to be interviewed for the chairmanship of the Department of Biochemistry—the only position outside of the University of Wisconsin he ever considered. A modest and humble man, Elvehjem had simple tastes and more than a touch of austerity. Potter recalls his dismay on entering their shared office one Saturday shortly after the Cornell trip to find Elvehjem
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on his knees scrubbing the frayed maroon linoleum—last renewed during Stephen Moulton Babcock's earlier tenancy. Bob Burris, Connie's successor as department chairman, recalls only two occasions of being "chewed out" by him— once when he acquired a new office desk to replace Connie's old one, and once when he approved shifting the time of departmental seminar from 8 A.M. on Saturdays. The attack on Pearl Harbor in December 1941, followed on the heels of Elvehjem and Hogness's joint symposia in September. As a member of the National Research Council's Food and Nutrition Board, Connie strongly recommended fortifying bread with vitamins on a national scale. Writing to Potter on August 17, 1983, Jean St. Clair, the archivist for the National Research Council, recalled the high regard Elvehjem enjoyed among his colleagues throughout the nation: "[In] March, 1958 . . . Dr. Elvehjem, Chairman of the Board, had sent word that he could not attend the Friday meeting but that he hoped to attend the dinner and the Saturday morning session. The appointment of Dr. Elvehjem to the presidency of the University of Wisconsin, effective July 1, had been announced at the Friday meeting, as had his decision that, under that circumstance, he would be unable to continue as chairman of the Board. "The speaker for the dinner was George McGovern, Congressman from South Dakota, [who] had decided, via his membership on the House Committee on Education and Labor, to develop a guide to inform the American people on what to eat to be healthy. . . . The Board's plan was to listen to what he had to say, and then tactfully offer the Congressman its assistance. "In Dr. Elvehjem's absence, Dr. Grace Goldsmith, vice chairman of the Board, had just introduced McGovern, who had delivered a sentence or two of his speech, when Dr. Elvehjem entered the room. Applause broke out and McGovern said, 'Well, I can't compete with that,' and sat down. A standing ovation followed for 'Connie' Elvehjem, not so much for his new assignment at Wisconsin as for himself as a person. It was a remarkable show of affection and respect. "Dr. Elvehjem succeeded in restoring quiet. He reintroduced
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McGovern who, after his speech, had to leave for another appointment. It was just as well, because it turned out to be the wrong night for an outsider. I don't think the planners of the dinner meeting had anticipated what would happen if Dr. Elvehjem were to appear in the middle of the program, and it was evident that Elvehjem was surprised as well. "At the close of the dinner session, Dr. Elvehjem was presented with a signed scroll which read: 'On the occasion of his movement into a new orbit, members and friends of the Food and Nutrition Board join in expressing to Conrad A. Elvehjem their appreciation of his scientific leadership, his sustained wisdom, his common sense, and his good humor as a member of the Board from its beginning, and as its chairman from 1955 to 1958."'
In the years just prior to the discovery of the antipellagra vitamin and for many years thereafter, Elvehjem and Professor Perry Wilson conducted an enzyme seminar with their most interested students and colleagues. This bore fruit in the form of a manual on respiratory enzymes in which sixteen local Wisconsin students and faculty described the state of the field in the period just prior to the date of publication, 1939—a book that remains interesting for its historical introduction by Elvehjem. In 1945 he attended a national "Conference on Intracellular Enzymes of Normal and Malignant Tissues" at Hershey, Pennsylvania. There he and Van Potter, his former student, discussed the fact that opportunities for postdoctoral study in Europe that had proved so important for Wisconsin biochemists in the past (Hart, Steenbock, Peterson, Link, Elvehjem, Johnson, Baumann, and Strong) would no longer be available to them in the immediate postwar years. The two hit on the idea of a postdoctoral training facility, and—with support from Dean W. S. Middleton of the Medical School, the Wisconsin Alumni Research Foundation, and from the Rockefeller Foundation—the Enzyme Institute became a reality. Elvehjem traveled to St. Louis in an attempt to recruit Carl
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F. Cori for the Institute, but Cori's timely (or untimely) receipt of the Nobel Prize made him impossible to move. Instead, David Green, who had a brilliant record at Cambridge and Columbia universities, was persuaded to become the new Enzyme Institute's first team leader. Following its original plan, the Institute had several established investigators leading their own group of postdoctoral fellows. The second team leader recruited was Professor Henry Lardy, who moved from the Biochemistry Department to the Enzyme Institute, retaining his privilege of training Ph.D. candidates. Green and Lardy were subsequently named codirectors of the Institute, which was guided by an Enzyme Committee with the dean of the Graduate School (at that time, Elvehjem) as chairman ex officio. PUBLIC SERVICE Elvehjem's successful researches early in his career brought him many invitations to lecture and to join committees and learned associations. He was a member of the Food and Nutrition Board from its inception until 1961. The Board, as a measure to improve the national food supply during World War II, developed guidelines for the fortification of foods with vitamins and minerals. It subsequently provided estimates of human nutritional requirements—the so-called "recommended dietary allowances"—that are now regarded as national standards. Elvehjem was also an active member of the American Medical Association's Council on Food and Nutrition (1941-1958) and served on advisory committees of the Nutrition Foundation and the National Science Foundation. In 1960 he was consultant to the President's Science Advisory Committee and president of the American Institute of Nutrition. Elvehjem also received a number of honorary degrees. After his death, the American Institute of Nutrition created
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the Conrad Elvehjem Award to honor those of its members who were remarkable for their distinguished public service. The Wisconsin Alumni Research Foundation provided funds for an Elvehjem Professorship in the Life Sciences, first held by Gobind Khorana who, four years later, received the Nobel Prize. UNIVERSITY ADMINISTRATION A scientist at heart, Elvehjem early demonstrated his talents as an administrator managing a large research staff. He grasped concepts with remarkable speed, marshalled evidence pertinent to the problem, and reached logical conclusions without delay. As one rather slow-spoken faculty member remarked, ''Connie answers your problem before you have completed stating your question." This is not to say that his conclusions were snap judgments— they were consistently sound and were respected considering his remarkable record for being right. He was also scrupulously honest in his dealings. Intrigue was foreign to him; he trusted his colleagues and they trusted him. On the assumption that he was dealing with reasonable people who were seeking solutions, he willingly used his remarkable perceptiveness and breadth of understanding to help formulate and implement those solutions. Initially Elvehjem had done research with Harry Steenbock but then shifted to E. B. Hart's group. When Hart stepped down in 1944, after thirtyeight years as chairman of the Department of Biochemistry, Elvehjem was clearly the staff's choice to succeed him. As chairman, Elvehjem followed the pattern set by Hart and did not allow the job to overwhelm him. He treated trivia as trivia. He examined and solved substantive problems with minimal wasted effort. He delegated tasks, and people were pleased to aid so decisive a
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man whom they admired. Although some were jealous of his uncanny ability to get things done, Connie had few enemies. Elvehjem's scientific standing and administrative talents were widely recognized locally, and when the position of graduate dean came open, he was asked to fill it. When he accepted the challenge of the deanship, it was probably assumed that he would then relinquish the chairmanship of the Biochemistry Department. But biochemistry was home to Connie and the base for his research, and he carried both jobs. He still appeared daily at the Department well before 8:00 A.M. to make the rounds of the rooms and quiz his students on their latest observations. The mornings sufficed for administering the Biochemistry Department, maintaining a productive research program, and writing technical papers. The afternoons were spent at the Graduate School office on the central campus. Elvehjem kept operations under control, and one heard no complaints that he was neglecting either biochemistry or the graduate deanship. This was the more remarkable in that Wisconsin's graduate biochemistry program was then the largest in the country and the leading grantor of Ph.D. degrees. Administration at Wisconsin's Graduate School had long been dominated by people from the sciences, and other sectors of the University felt some trepidation when yet another scientist was selected as its head. Certainly there was nothing in Connie Elvehjem's background to suggest empathy with the arts and humanities. But as graduate dean, he made it his policy to channel flexible supporting funds to areas outside the hard sciences while continuing to support basic science with funds that could not be shifted. This policy reflected his inherent fairness and his clear perception that a university without breadth and balance could not be a great university. For some years, for example, the Graduate Dean and his Research Committee had administered a substantial
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block-grant from the Wisconsin Alumni Research Foundation. Under Elvehjem, more of this grant went to the Department of History than to Biochemistry—the department that had generated the patents from which ninety-four percent of the Foundation's funds (amplified by skillful investment) had come. Elvehjem's concern for maintaining and enhancing his great university came through clearly during his tenure as president from 1958 to 1962. He encouraged the establishment of an Institute for Research in the Humanities and found funds for other efforts in the humanities and social sciences. He supported efforts to create a worthy art gallery, and though it came to fruition only after his death, it was named the Elvehjem Museum of Art. Elvehjem's tenure as president was only four years, but it was a period of substantial change during which the University of Wisconsin-Milwaukee grew rapidly and gained new stature. Without dominating this growth, he helped guide it. Though the physical plant was expanded during his tenure, Elvehjem clearly felt that a great institution is built primarily on people. He encouraged the recruitment of promising scholars in a variety of fields. President Elvehjem was stricken with a heart attack at his desk on the morning of July 27, 1962, at the age of sixty-one and died within the hour. Although he left tasks unfinished, he also left a great legacy of accomplishment and affection. The faculty memorial resolution on the death of Conrad Arnold Elvehjem catches the character of the man: "Such basic traditions of the University as academic freedom, enthusiasm for the pursuit of truth, concern for the individual student, and service of the whole state were not only fostered but exemplified by him. . . . He could make allowances for weaknesses he did not share. . . . He even tried to understand the untidy desk but never quite succeeded . . . . Few men changed more than Elvehjem; yet few remained as constant. In his direct
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CONRAD ARNOLD ELVEHJEM
ness and honesty, in his unswerving devotion to high religious and moral standards, in his regard for the rights of others, in his complete dedication to learning and the University of Wisconsin as a home of learning, the undergraduate who became the president was the same man. Both humility and self-confidence were natural to him. He had an iron will which he used to control himself rather than others, a will which turned his natural impatience into an asset and drove his splendid brain from one accomplishment to another. "For one of the constants of his character was the ability to grow. He could value what he did not himself savor. In the breadth of his sympathies, in the understanding of the foibles of others and of himself, in the appreciation of those of less talent, he grew at each stage of his career. What had been the tolerance of the specialist was at the close of his life ripening into genuine catholicity of interest."
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AWARDS
1939 Mead Johnson Award, American Institute of Nutrition 1942 Grocery Manufacturers of America Award 1943 Willard Gibbs Award, American Chemical Society 1948 Nicholas Appert Medal, Institute of Food Technologists 1950 Osborne-Mendel Award, American Institute of Nutrition 1952 Lasker Award in Medical Research, American Public Health Association 1956 Charles Spencer Award, American Chemical Society 1957 American Institute of Baking Award
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Selected Bibliography
In addition to the original research papers listed below, Professor Elvehjem published eighty-five commentaries and reviews under his sole authorship. 1924 With E. B. Hart and H. Steenbock. Dietary factors influencing calcium assimilation. V. The effect of light upon calcium and phosphorus equilibrium in mature lactating animals. J. Biol. Chem., 62:117. 1925 With E. B. Hart, H. Steenbock, and J. Waddell. Iron in nutrition. I. Nutritional anemia on whole milk diets and the utilization of inorganic iron in hemoglobin building. J. Biol. Chem., 65:67. With H. Steenbock, E. B. Hart, and S. W. F. Kletzien. Dietary factors influencing calcium assimilation. VI. The antirachitic properties of hays as related to climatic conditions with some observations on the effect of irradiation with ultra-violet light. J. Biol. Chem., 66:425. 1926 With E. B. Hart. Iron in nutrition. II. Quantitative methods for the determination of iron in biological materials. J. Biol. Chem., 67:43. 1927 With R. C. Herrin and E. B. Hart. Iron in nutrition. III. The effect of diet on the iron content of milk. J. Biol. Chem., 71:255. With W. H. Peterson. The iron content of animal tissues. J. Biol. Chem., 74:433. With E. B. Hart, J. Waddell, and R. C. Herrin. Iron in nutrition. IV. Nutritional anemia on whole milk diets and its correction with the ash of certain plant and animal tissues or with soluble iron salts. J. Biol. Chem., 72:299. 1928 With W. H. Peterson. The iron content of plant and animal foods. J. Biol. Chem., 78:215.
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With J. Waddell, H. Steenbock, and E. B. Hart. Iron in nutrition. V. The availability of the rat for studies in anemia. J. Biol. Chem., 77:769. With J. Waddell, H. Steenbock, and E. B. Hart. Iron in nutrition. VI. Iron salts and iron-containing ash extracts in the correction of anemia. J. Biol. Chem., 77:777. With E. B. Hart, H. Steenbock, and J. Waddell. Iron in nutrition. VII. Copper as a supplement to iron for hemoglobin building in the rat. J. Biol. Chem., 77:797. 1929 With C. W. Lindow. The determination of copper in biological materials. J. Biol. Chem., 81:435. With E. B. Hart. The copper content of feedingstuffs. J. Biol. Chem., 82:473. With H. Steenbock and E. B. Hart. The effect of diet on the copper content of milk. J. Biol. Chem., 83:27. With H. Steenbock and E. B. Hart. Is copper a constituent of the hemoglobin molecule? The distribution of copper in blood. J. Biol. Chem., 83:21. With E. B. Hart. The relation of iron and copper to hemoglobin synthesis in the chick. J. Biol. Chem., 84:131. With J. Waddell, H. Steenbock, and E. B. Hart. Iron in nutrition. IX. Further proof that the anemia produced on diets of whole milk and iron is due to a deficiency of copper. J. Biol. Chem., 83:251. 1930 Factors affecting the catalytic action of copper in the oxidation of cysteine. Biochem. J., 24:415. With E. B. Hart, A. R. Kemmerer, and J. G. Halpin. Does the practical chick ration need iron and copper additions to insure normal hemoglobin building? Poult. Sci., 9:92. With E. B. Hart, H. Steenbock, G. Bohstedt, and J. M. Fargo. A study of the anemia of young pigs and its prevention. J. Nutr., 2:277. 1931 The role of iron and copper in the growth and metabolism of yeast. J. Biol. Chem., 90:111.
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The so-called autoxidation of cysteine. Science, 74:567. With A. R. Kemmerer and E. B. Hart. Studies on the relation of manganese to the nutrition of the mouse. J. Biol. Chem., 92:623. 1932 The relative value of inorganic and organic iron in hemoglobin formation. J. Am. Med. Assoc., 98:1047-50. With V. F. Neu. Studies in vitamin A avitaminosis in the chick. J. Biol. Chem., 97:71. With F. J. Stare. The phosphorus partition in the blood of rachitic and non-rachitic calves. J. Biol. Chem., 97:511. With W. C. Sherman. The action of copper in iron metabolism. J. Biol. Chem., 98:309. With O. L. Kline, J. A. Keenan, and E. B. Hart. The use of the chick in vitamin B1 and B2 studies. J. Biol. Chem., 99:295. 1933 With F. J. Stare. Cobalt in animal nutrition. J. Biol. Chem., 99:473. With F. J. Stare. Studies on the respiration of animal tissues. Am. J. Physiol., 105:655. With M. O. Schultze. The relation of iron and copper to the reticulocyte response in anemic rats. J. Biol. Chem., 102:357. With J. A. Keenan, O. L. Kline, E. B. Hart, and J. G. Halpin. New nutritional factors required by the chick. J. Biol. Chem., 103:671. 1934 With W. R. Todd and E. B. Hart. Zinc in the nutrition of the rat. Am. J. Physiol., 107:146. With E. B. Hart and W. C. Sherman. The limitations of cereal-milk diets for hemoglobin formation. J. Pediatr., 4:65. With O. L. Kline, J. A. Keenan, and E. B. Hart. Studies on the growth factor in liver. J. Biol. Chem., 107:107. With Eugene Cohen. The relation of iron and copper to the cytochrome and oxidase content of animal tissues. J. Biol. Chem., 107:97. With M. O. Schultze. The mechanism of the blood changes during the treatment of secondary and pernicious anemia. J. Lab. Clin. Med., 20:13.
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1935 With C. J. Koehn, Jr. Non-identity of vitamin B, and flavines. Nature, 134:1007. With F. E. Stirn and E. B. Hart. The indispensability of zinc in the nutrition of the rat. J. Biol. Chem., 109:347. With A. Siemers and D. R. Mendenhall. Effect of iron and copper therapy on hemoglobin content of the blood in infants. Am. J. Dis. Child., 50:28. With L. E. Clifhorn and V. W. Meloche. The absorption of carbon monoxide with reduced hematin and pyridine hemochromogen. J. Biol. Chem., 111:399. 1936 With C. J. Koehn, Jr. Studies on vitamin G(B2) and its relation to canine black tongue. J. Nutr., 2:67. With V. R. Potter. The effect of selenium on cellular metabolism. The rate of oxygen uptake by living yeast in the presence of sodium selenite. Biochem. J., 30:189. With W. C. Sherman. In vitro studies on lactic acid metabolism in tissues from polyneuritic chicks. Biochem. J., 30:785. With V. R. Potter. A modified method for the study of tissue oxidations. J. Biol. Chem., 114:495. With W. C. Sherman. In vitro action of crystalline vitamin B1 on pyruvic acid metabolism in tissues from polyneuritic chicks. Am. J. Physiol., 117:142. With M. O. Schultze and E. B. Hart. Studies on the copper content of the blood in nutritional anemia. J. Biol. Chem., 116:107. With A. Arnold, O. L. Kline, and E. B. Hart. Further studies on the growth factor required by chicks. The essential nature of arginine. J. Biol. Chem., 116:699. 1937 With V. R. Potter. The effect of inhibitors on succinoxidase. J. Biol. Chem., 117:341. With C. J. Koehn, Jr. Further studies on the concentration of the antipellagra factor. J. Biol. Chem., 118:693. With E. B. Hart and G. O. Kohler. Does liver supply factors in addition to iron and copper for hemoglobin regeneration in nutritional anemia? J. Exp. Med., 66:145.
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With R. J. Madden, F. M. Strong, and D. W. Woolley. Relation of nicotinic acid and nicotinic acid amide to canine black tongue. J. Am. Chem. Soc., 59:1767. 1938 With R. J. Madden, F. M. Strong, and D. W. Woolley. The isolation and identification of the anti-black tongue factor. J. Biol. Chem., 123:137. With M. A. Lipschitz and V. R. Potter. The relation of vitamin B1 to cocarboxylase. Biochem. J., 32:474. With P. L. Pavcek and W. H. Peterson. Factors affecting the vitamin B1 content of yeast. Ind. Eng. Chem., 30:802. With D. W. Woolley, F. M. Strong, and R. J. Madden. Anti-black tongue activity of various pyridine derivatives. J. Biol. Chem., 124:715. With F. M. Strong and R. J. Madden. The ineffectiveness of ß-aminopyridine in black tongue. J. Am. Chem. Soc., 60:2564. With D. W. Woolley, H. A. Waisman, and O. Mickelsen. Some observations on the chick antidermatitis factor. J. Biol. Chem., 125:715. With V. R. Potter and E. B. Hart. Anemia studies with dogs. J. Biol. Chem., 126:155. 1939 With J. J. Oleson, H. R. Bird, and E. B. Hart. Additional nutritional factors required by the rat. J. Biol. Chem., 127:23 The vitamin B complex in practical nutrition. J. Am. Diet. Assoc., 15:6. With A. E. Axelrod. Effect of nicotinic acid deficiency on the cozymase content of tissues. Nature, 143:281. With D. W. Woolley and H. A. Waisman. Nature and partial synthesis of the chick antidermatitis factor. J. Am. Chem. Soc., 61:977. With A. Arnold. Influence of the composition of the diet on the thiamin requirement of dogs. Am. J. Physiol., 126:289. With D. W Woolley and H. A. Waisman. Studies on the structure of the chick antidermatitis factor. J. Biol. Chem., 129:673. With A. E. Axelrod and R. J. Madden. The effect of a nicotinic acid deficiency upon the coenzyme I content of animal tissues. J. Biol. Chem., 131:85.
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With J. M. McKibbin, R. J. Madden, and S. Black. The importance of vitamin B6 and factor W in the nutrition of dogs. Am. J. Physiol., 128:102. With H. D. Anderson and J. E. Gonce, Jr. Vitamin E deficiency in dogs. Proc. Soc. Exp. Biol. Med., 42:750. 1940 With D. V. Frost, V. R. Potter, and E. B. Hart. Iron and copper versus liver in treatment of hemorrhagic anemia in dogs on milk diets. J. Nutr., 19:207. With M. A. Lipton. Mechanism of the enzymatic phosphorylation of thiamin. Nature, 145:226. With E. J. Schantz and E. B. Hart. The comparative nutritive value of butter fat and certain vegetable oils. J. Dairy Sci., 23:181. With A. E. Axelrod and E. S. Gordon. The relationship of the dietary intake of nicotinic acid to the coenzyme I content of blood. Am. J. Med. Sci., 199:697. With D. M. Hegsted, J. J. Oleson, and E. B. Hart. The essential nature of a new growth factor and vitamin B6 for chicks. Poult. Sci., 19:167. With H. A. Sober and M. A. Lipton. The relation of thiamine to citric acid metabolism. J. Biol. Chem., 134:605. With E. Hove and E. B. Hart. The relation of zinc to carbonic anhydrase. J. Biol. Chem., 136:425. With M. I. Wegner, A. N. Booth, and E. B. Hart. Rumen synthesis of the vitamin B complex. Proc. Soc. Exp. Biol. Med., 45:769. 1941 With L. W. Wachtal, E. Hove, and E. B. Hart. Blood uric acid and liver uricase of zincdeficient rats on various diets. J. Biol. Chem., 138:361. With D. M. Hegsted, R. C. Mills, and E. B. Hart. Choline in the nutrition of chicks. J. Biol. Chem., 138:459. With T. W. Conger. The biological estimation of pyridoxine (vitamin B6). J. Biol. Chem., 138:555. With H. A. Waisman. Chemical estimation of nicotinic acid and vitamin B6. Ind. Eng. Chem., 13:221. With M. I. Wegner, A. N. Booth, and E. B. Hart. Rumen synthesis
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of the vitamin B complex on natural rations. Proc. Soc. Exp. Biol. Med., 47:90. With S. Black and J. M. McKibbin. Use of sulfaguanidine in nutrition experiments. Proc. Soc. Exp. Biol. Med., 47:308. With E. Nielsen. Cure of spectacle eye condition in rats with biotin concentrates. Proc. Soc. Exp. Biol. Med., 48:349. 1942 With L. M. Henderson, J. M. McIntire, and H. A. Waisman. Pantothenic acid in the nutrition of the rat. J. Nutr., 23:47. With A. E. Axelrod and V. R. Potter. The succinoxidase system in riboflavin-deficient rats. J. Biol. Chem., 142:85. With D. M. Hegsted, R. C. Mills, G. M. Briggs, and E. B. Hart. Biotin in chick nutrition. J. Nutr., 23:175. With L. J. Teply and F. M. Strong. Nicotinic acid, pantothenic acid and pyridoxine in wheat and wheat products. J. Nutr., 24:167. With S. Black, R. S. Overman, and K. P. Link. The effect of sulfaguanidine on rat growth and plasma prothrombin. J. Biol. Chem., 145:137. With D. Orsini and H. A. Waisman. Effect of vitamin deficiencies on basal metabolism and respiratory quotient in rats. Proc. Soc. Exp. Biol. Med., 51:99. With J. D. Teresi and E. B. Hart. Molybdenum in the nutrition of the rat. Am. J. Physiol., 137:504. 1943 With G. M. Briggs, Jr., T. D. Luckey, R. C. Mills, and E. B. Hart. Effect of p-aminobenzoic acid when added to purified chick diets deficient in unknown vitamins. Proc. Soc. Exp. Biol. Med., 52:7. With O. K. Gant, B. Ransone, and E. McCoy. Intestinal flora of rats on purified diets containing sulfonamides. Proc. Soc. Exp. Biol. Med., 52:276. With J. B. Field and C. Juday. A study of the blood constituents of carp and trout. J. Biol. Chem., 148:261. With H. A. Waisman, A. F. Rasmussen, Jr., and P. F. Clark. Studies on the nutritional requirements of the rhesus monkey. J. Nutr., 26:205.
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With L. W. Wachtel and E. B. Hart. Studies on the physiology of manganese in the rat. Am. J. Physiol., 140:72. 1944 With L. J. Teply. Use of germicidal quaternary ammonium salt in nutritional studies. Proc. Soc. Exp. Biol. Med., 55:59. With H. C. Lichstein, H. A. Waisman, and P. F. Clark. Influence of pantothenic acid deficiency on resistance of mice to experimental poliomyelitis. Proc. Soc. Exp. Biol. and Med., 56:3. With B. S. Schweigert, J. M. McIntire, and F. M. Strong. The direct determination of valine and leucine in fresh animal tissues. J. Biol. Chem., 155:183. With J. H. Shaw, B. S. Schweigert, J. M. McIntire, and P. H. Phillips. Dental caries in the cotton rat. II. Methods of study and preliminary nutritional experiments. J. Nutr., 28:333. With S. R. Ames. Inhibition of the succinoxidase system by cysteine and cystine. Arch. of Biochem., 5:191. With W. A. Krehl and F. M. Strong. The biological activity of a precursor of nicotinic acid in cereal products. J. Biol. Chem., 156:13. 1945 With H. A. Waisman and K. B. McCall. Acute and chronic biotin deficiencies in the monkey (Macaca mulatta). J. Nutr., 29:1. With L. J. Teply. The titrimetric determination of ''Lactobacillus Casei factor" and "Folic acid." J. Biol. Chem., 157:303. With W. A. Krehl and L. J. Teply. Corn as an etiological factor in the production of a nicotinic acid deficiency in the rat. Science, 101:283. With James H. Shaw, B. S. Schweigert, and Paul H. Phillips. Dental caries in the cotton rat. II. Production and description of the carious lesions. J. Dent. Res., 23:417. With B. S. Schweigert and I. E. Tatman. The leucine, valine, and isoleucine content of meats. Arch. Biochem., 6:177. With W. A. Krehl, L. J. Teply, and P. S. Sarma. Growth-retarding effect of corn in nicotinic-acidlow rations and its counteraction by tryptophane. Science, 101:489. With B. S. Schweigert, J. M. McIntire, and L. M. Henderson. Intestinal synthesis of B vitamins by the rat. Arch. Biochem., 6:403.
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With B. Schweigert, J. H. Shaw, and P. H. Phillips. Dental caries in the cotton rat. III. Effect of different dietary carbohydrates on the incidence and extent of dental caries. J. Nutr., 29:405. With J. H. Shaw, B. S. Schweigert, and P. H. Phillips. Dental caries in the cotton rat. IV. Inhibitory effect of fluorine additions to the ration. Proc. Soc. Exp. Biol. Med., 59:89. With H. C. Lichstein, H. A. Waisman, K. B. McCall, and P. F. Clark. Influence of pyridoxine, inositol, and biotin on susceptibility of Swiss mice to experimental poliomyelitis. Proc. Soc. Exp. Biol. Med., 60:279. 1946 With W. A. Krehl, P. S. Sarma, and L. J. Teply. Factors affecting the dietary niacin and tryptophane requirement of the growing rat. J. Nutr., 31:85. With W. H. Ruegamer and E. B. Hart. Potassium deficiency in the dog. Proc. Soc. Exp. Biol. Med., 61:234. With A. Evenson, Elizabeth McCoy, and B. R. Geyer. The cecal flora of white rats on a purified diet and its modification by succinylsulfathiazole. J. Bacteriol., 51:513. With A. E. Schaefer and C. K. Whitehair. Purified rations and the importance of folic acid in mink nutrition. Proc. Soc. Exp. Biol. Med., 62:169. With B. A. McLaren and E. F. Herman. Nutrition of rainbow trout; studies with purified rations. Arch. Biochem., 10:433. With S. R. Ames. Enzymatic oxidation of glutathione II. Studies on the addition of several cofactors. Arch. Biochem., 10:443. With P. S. Sarma and E. E. Snell. The vitamin B6 group. VIII. Biological assay of pyridoxal, pyridoxamine, and pyridoxine. J. Biol. Chem., 165:55. With S. P. Ames and A. J. Ziegenhagen. Studies on the inhibition of enzyme systems involving cytochrome c. J. Biol. Chem., 165:81. With S. R. Ames. Determination of aspartic-glutamic transaminase in tissue homogenates. J. Biol. Chem., 166:81. With W. A. Krehl, L. M. Henderson, and J. de la Huerga. Relation of amino acid imbalance to niacin-tryptophane deficiency in growing rats. J. Biol. Chem., 166:531.
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1947 With L. J. Teply and W. A. Krehl. The intestinal synthesis of niacin and folic acid in the rat. Am. J. Physiol., 148:91. With S. R. Ames and P. S. Sarma. Transaminase and pyridoxine deficiency. J. Biol. Chem., 167:135. With T. D. Luckey, P. R. Moore, and E. B. Hart. Growth of chicks on purified and synthetic diets containing amino acids. Proc. Soc. Exp. Biol. Med., 64:423. With B. A. McLaren and E. F. Herman. Nutrition of trout: Studies with practical diets. Proc. Soc. Exp. Biol. Med., 65:97. With A. E. Schaefer and C. K. Whitehair. The importance of riboflavin, pantothenic acid, niacin and pyridoxine in the nutrition of foxes. J. Nutr., 34:131. With H. A. Lardy and R. L. Potter. The role of biotin in bicarbonate utilization by bacteria. J. Biol. Chem., 169:451. With G. W. Newell, T. C. Erickson, W. E. Gilson, and S. N. Gershoff. Role of "agenized" flour in the production of running fits. J. Am. Med. Assoc., 135:760. 1948 With A. E. Schaefer, S. B. Tove, and C. K. Whitehair. The requirement of unidentified factors for mink. J. Nutr., 35:157. With V. H. Barki, H. Nath, and E. B. Hart. Production of essential fatty acid deficiency symptoms in the mature rat. Proc. Soc. Exp. Biol. Med., 66:474. With E. J. Wakeman, J. K. Smith, W. B. Sarles, and P. H. Phillips. A method for quantitative determinations of microorganisms in carious and noncarious teeth of the cotton rat. J. Dent. Res., 27:41. With E. M. Sporn and W. R. Ruegamer. Studies with monkeys fed army combat rations. J. Nutr., 35:559. With O. E. Olson, E. E. C. Fager, and R. H. Burris. The use of a hog kidney conjugase in the assay of plant materials for folic acid. Arch. Biochem., 18:261. With L. V. Hankes, L. M. Henderson, and W. L. Brickson. Effect of amino acids on the growth of rats on niacin-tryptophan-deficient rations. J. Biol. Chem., 174:873.
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1949 With A. Sreenivasan and A. E. Harper. The use of conjugase preparations in the microbiological assay of folic acid. J. Biol. Chem., 177:117. With C. A. Nichol, L. S. Dietrich, and W. W. Cravens. Activity of vitamin B12 in the growth of chicks. Proc. Soc. Exp. Biol. Med., 70:40. With K. H. Maddy. Studies on growth of mice fed rations containing free amino acids. J. Biol. Chem., 177:577. With C. A. Nichol and A. E. Harper. Effect of folic acid, liver extract, and vitamin B12 on hemoglobin regeneration in chicks. Proc. Soc. Exp. Biol. Med., 71:34. With G. B. Ramasarma and L. M. Henderson. Purified amino acids as a source of nitrogen for the growing rat. J. Nutr., 38:177. With V. H. Barki, R. A. Collins, and E. B. Hart. Relation of fat deficiency symptoms to the polyunsaturated fatty acid content of the tissues of the mature rat. Proc. Soc. Exp. Biol. Med., 71:694. With V. H. Barki, P. Feigelson, R. A. Collins, and E. B. Hart. Factors influencing galactose utilization. J. Biol. Chem., 181:565. 1950 With H. T. Thompson, P. E. Schurr, and L. M. Henderson. The influence of fasting and nitrogen deprivation on the concentration of free amino acids in rat tissues. J. Biol. Chem., 182:47. With P. Roine. Significance of the intestinal flora in nutrition of the guinea pig. Proc. Soc. Exp. Biol. Med., 73:308. With J. N. Williams, Jr., and P. Feigelson. A study of xanthine metabolism in the rat. J. Biol. Chem., 185:887. With A. E. Denton and J. N. Williams, Jr. The influence of methionine deficiency on amino acid metabolism in the rat. J. Biol. Chem., 186:377. With L. S. Dietrich and W. J. Monson. Effect of sulfasuxidine on the interrelation of folic acid, vitamin B12 and vitamin C. Proc. Soc. Exp. Biol. Med., 75:130. With L. V. Hankes and R. L. Lyman. Effect of niacin precursors on growth of rats fed tryptophanlow rations. J. Biol. Chem., 187:547.
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1951 With J. N. Williams, Jr., P. Feigelson, and S. S. Shahinian. Interrelationships of vitamin B6, niacin, and tryptophan in pyridine nucleotide formation. Proc. Soc. Exp. Biol. Med., 76:441. With P. Feigelson and J. N. Williams, Jr. Inhibition of diphosphopyridine nucleotide-requiring enzymes by nicotinamide. J. Biol. Chem., 189:361. With R. J. Sirny and L. T. Cheng. An arginine-proline interdependence in Leuconostoc mesenteroides P-60. J. Biol. Chem., 190:547. With J. N. Williams, Jr., and G. Litwack. Studies on rat liver choline oxidase: an assay method. J. Biol. Chem., 192:73. With S. N. Gershoff. Studies of the biological effects of methionine sulfoximine . J. Biol. Chem., 192:569. With R. L. Lyman. Further studies on amino acid imbalance produced by gelatin in rats on niacintryptophan-low ration. J. Nutr., 45:101. 1952 With A. R. Taborda, L. C. Taborda, and J. N. Williams, Jr. A study of the ribonuclease activity of snake venoms. J. Biol. Chem., 194:227. With D. V. Tappan, U. J. Lewis, and U. D. Register. Niacin deficiency in the rhesus monkey. J. Nutr., 46:75. With M. Constant and P. H. Phillips. Dental caries in the cotton rat. XIII. The effect of whole grain and processed cereals on dental caries production. J. Nutr., 46:271. With J. P. Kring, K. Ebisuzaki, and J. N. Williams, Jr. The influence of vitamin B6 on the formation of liver pyridine nucleotides. J. Biol. Chem., 195:591. With S. S. Shahinian, K. Ebisuzaki, J. P. Kring, and J. N. Williams, Jr. The action of threonine in inducing an amino acid imbalance. Proc. Soc. Exp. Biol. Med., 80:146. With L. S. Dietrich and W. J. Monson. Utilization of pteroylglutamic acid conjugates in the in vitro synthesis of L. citrovorum activity. J. Am. Chem. Soc., 74:3705. With W. L. Davies, W. L. Pond, S. C. Smith, A. F. Rasmussen, Jr., and P. F. Clark. The effect of certain amino acid deficiencies on Lansing poliomyelitis in mice. J. Bacteriol., 64:571.
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1953 With H. Nino-Herrera, M. Schreiber, and R. A. Collins. Dermatosis in weanling rats fed lactose diets. II. Histological studies. J. Nutr., 49:99. With A. E. Denton. Enzymatic liberation of amino acids from different proteins. J. Nutr., 49:221. With G. Litwack and J. N. Williams, Jr. The roles of essential and nonessential amino acids in maintaining liver xanthine oxidase. J. Biol. Chem., 201:261. With S. C. Smith, A. F. Rasmussen, Jr., and P. F. Clark. Influence of hyper-and hypothyroidism on susceptibility of mice to infection with Lansing poliomyelitis virus . Proc. Soc. Exp. Biol. Med., 82:269. With J. N. Williams, Jr., A. Sreenivasan, and S. C. Sung. Relationship of the deposition of folic and folinic acids to choline oxidase of isolated mitochondria. J. Biol. Chem., 202:233. With J. N. Williams, Jr., W. J. Monson, A. Sreenivasan, L. S. Dietrich, and A. E. Harper. Effects of a vitamin B12 deficiency on liver enzymes in the rat. J. Biol. Chem., 202:151. With A. E. Harper, W. J. Monson, and D. A. Benton. The influence of protein and certain amino acids, particularly threonine, on the deposition of fat in the liver of the rat. J. Nutr., 50:383. 1954 With A. E. Denton. Amino acid concentration in the portal vein after ingestion of amino acid. J. Biol. Chem., 206:455. With W. L. Loeschke. Prevention of urinary calculi formation in mink by alteration of urinary pH. Proc. Soc. Exp. Biol. Med., 85:42. With W. J. Monson, A. E. Harper, and M. E. Winje. A mechanism of the vitamin-sparing effect of antibiotics. J. Nutr., 52:627. With A. E. Harper, D. A. Benton, and M. E. Winje. Leucineisoleucine antagonism in the rat. Arch. Biochem. Biophys., 51:523. With M. E. Winje, A. E. Harper, D. A. Benton, and R. E. Boldt. Effect of dietary amino acid balance on fat deposition in the livers of rats fed low protein diets. J. Nutr., 54:155.
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1955 With L.-E. Ericson and J. N. Williams, Jr. Studies on partially purified betaine-homocysteine transmethylase of liver. J. Biol. Chem., 212:537. With D. A. Benton and A. E. Harper. Effect of isoleucine supplementation on the growth of rats fed zein or corn diets. Arch. Biochem. Biophys., 57:13. With Selma Hayman, S. S. Shahinian, and J. N. Williams, Jr. Effect of 3-acetylpyridine on pyridine nucleotide formation from tryptophan and niacin. J. Biol. Chem., 217:225. With P. D. Deshpande, A. E. Harper, and Felipe Quiros-Perez. Further observations on the improvement of polished rice with protein and amino acid supplements. J. Nutr., 57:415. With H. R. Heinicke and A. E. Harper. Protein and amino acid requirements of the guinea pig. I. Effect of carbohydrate, protein level and amino acid supplementation. J. Nutr., 57:483. 1956 With D. A. Benton, A. E. Harper, and H. E. Spivey. Leucine, isoleucine, and valine relationships in the rat. Arch. Biochem. Biophys., 60:147. With D. A. Benton and A. E. Harper. The effect of different dietary fats on liver fat deposition. J. Biol. Chem., 218:693. With L. E. Ericson, A. E. Harper, and J. N. Williams, Jr. Effect of diet on the betaine-homocysteine transmethylase activity of rat liver. J. Biol. Chem., 219:59. With A. E. Harper, L. E. Ericson, and R. E. Boldt. Effect of thyroid active substances on the betainehomocysteine transmethylase activity of rat liver. Am. J. Physiol., 184:457. With R. F. Wiseman, W. B. Sarles, D. A. Benton, and A. E. Harper. Effects of dietary antibiotics upon numbers and kinds of intestinal bacteria used in chicks. J. Bacteriol., 72:723. 1957 With P. D. Deshpande, A. E. Harper, and Macie Collins. Biological availability of isoleucine. Arch. Biochem. Biophys., 67:341. With P. D. Deshpande and A. E. Harper. Nutritional improvement of white flour with protein and amino acid supplements. J. Nutr., 62:503.
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With J. D. Gupta. Biological availability of tryptophan. J. Nutr., 62:313. With F. N. Hepburn and E. W. Lewis, Jr. The amino acid content of wheat, flour, and bread. Cereal Chem., 34:312. 1958 With M. M. Chaloupka, J. N. Williams, Jr., and May S. Reynolds. Relative roles of niacin and tryptophan in maintaining blood pyridine nucleotides, nitrogen balance and growth in adult rats. J. Nutr., 63:361. With P. D. Deshpande and A. E. Harper. Amino acid imbalance and nitrogen retention. J. Biol. Chem., 230:335. With J. D. Gupta, A. M. Dakroury, and A. E. Harper. Biological availability of lysine. J. Nutr., 64:259. With Narindar Nath and A. E. Harper. Dietary protein and serum cholesterol. Arch. Biochem. Biophys., 77:234. With A. Yoshida and A. E. Harper. Effect of dietary level of fat and type of carbohydrate on growth and food intake. J. Nutr., 66:217. 1959 With Narindar Nath, Ruta Wiener, and A. E. Harper. Diet and cholesteremia. I. Development of a diet for the study of nutritional factors affecting cholesteremia in the rat. J. Nutr., 67:289. With A. J. Bosch and A. E. Harper. Factors affecting liver pyridine nucleotide concentration in hyperthyroid rats. Soc. Exp. Biol. Med., 100:774. With W. L. Loeschke. The importance of arginine and methionine for the growth and fur development of mink fed purified diets. J. Nutr., 69:147. With W. L. Loeschke. Riboflavin in the nutrition of the chinchilla. J. Nutr., 69:214.
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Gottfried Samuel Fraenkel April 23, 1901-October 26, 1984
By C. Ladd Prosser, Stanley Friedman, and Judith H. Willis Gottfried Fraenkel was elected to the National Academy of Sciences in 1968 for his contributions to insect physiology. Although one might attribute his success as a pioneer in diverse areas—behavior, endocrinology, nutrition, insect-plant interaction—to his living in a period with few scientists and many uncharted fields, a reading of this biographical sketch reveals that many of his discoveries came during periods of political upheaval, economic hardship, and conflict with bosses—not conditions generally considered optimal for the advancement of basic research. His published contributions to musicology indicate that he was also adept at finding treasures in well-mined areas. One can only conclude that he was an exceptional person with an uncanny sense of what problems were interesting, important, and solvable. EARLY LIFE Gottfried Samuel Fraenkel was born in Munich, Germany. His father was a Justizrat and the family typically middle-class Jewish, with interests far from the science that Fraenkel was later to take up so successfully. As a boy he devoted much time to music—both piano playing and singing—but as a young man his major preoccupation was the Zionist cause.
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He continued to pursue these activities throughout his life, but in his early years his belief in the Zionist movement was so strong that he made the decision to spend a part of his life living and working in Palestine. To prepare for this goal, he enrolled in a teaching degree program at the University of Munich. There he attended lectures and engaged in laboratory exercises under R. C. Hertwig, Karl Von Goebel, Richard Martin, Richard Willstätter, Wilhelm Konrad, W. K. Röntgen, and Karl von Frisch. He became attracted to the field of hydrobiology and—having decided to take a doctoral degree—began to study the life histories of certain leeches on fish. When all of his tank specimens died as the result of a laboratory accident, he took a short trip to the Zoological Station in Naples to obtain fresh material. Once there, he was immediately and irrevocably charmed by the enormous variety and beauty of Mediterranean invertebrate fauna. Already knowledgeable about marine invertebrates from his course with Wolfgang von Buddenbrock at Helgoland he began to experiment, in the short time available, with some jellyfish blown into the Naples harbor by a storm. Within two weeks he worked out and successfully tested his idea that the medusa statocysts functioned as gravity receptors—a theory totally contrary to the dogma of the time. He returned to Munich, and, being advised that his discovery was a suitable dissertation thesis, arranged for Professor O. Koehler to ''direct" it. His talent for quickly defining and completing a project, an ability that was to remain with him throughout his life, was already highly developed at this early stage in his career. Having received his doctorate, he returned to Naples on a Rockefeller Foundation grant and, within a year, produced six publications on various aspects of sensory physiology and orientation of marine invertebrates. He also spent a short period with Alfred Kuhn in Göttingen and found time to
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visit the marine stations at Roscoff and Plymouth. The stay with Kuhn resulted in his first paper on insects, a behavioral analysis of the response of bees to color. This predilection for travel and marine stations became a lifelong passion. After these academic adventures, Fraenkel concluded that it was time to fulfill his Zionist commitment. With the small amount of money remaining from his fellowship, he boarded a ship for Palestine. Upon arrival he called on the distinguished entomologist F. S. Bodenheimer, who immediately offered him a job as his assistant at the newly founded Zoology Laboratory of the Hebrew University in Jerusalem. There were not many young, vigorous, and experienced zoologists at that time ready to work under the conditions prevailing in Palestine. While visiting friends in the period before the job began, he met—and shortly thereafter married—the Lithuanian-born Rachel Sobol, daughter of a family of well-known and politically active settlers. As he later put it, the family was not overly impressed with this scientist and "latecomer" to Palestine. It was during this sojourn in Palestine that he became involved with the animals he would study the rest of his working life. In those days Jerusalem was considered a "long distance" from the sea, and Fraenkel was attracted to the only water around—the papyrus pond on the grounds of the university—and to its myriad insect visitors. This pond provided subject matter for a number of fundamental studies on insect tracheal respiration, but it was a major invasion of locusts in 1929 that finally determined Fraenkel's fate. All of the research in the zoology laboratory was turned toward the problem of locust control, and Fraenkel's work in the desert on locust behavior and sensory physiology became the basis of attempts to hold locusts in check. The investigations are classics of their kind and are still widely quoted today. The work also resulted in a falling out with Bodenheimer over author
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ship, finally ending in Fraenkel's departure from his job and from Palestine. BRITAIN He returned to Germany in 1932 when the Nazis were already on the rise. Fraenkel felt himself fortunate to find a position as Private Dozent in Frankfurt's Zoology Department. But as soon as Hitler came to power in 1933, he was dismissed. Fortunately, his reputation was already sufficiently established that he was offered a position in England. It is worthy of note that this was one of the many positions awarded to German refugee scientists through the Academic Assistance Council, funded by contributions from English scientists out of their meager, depression-level wages. Fraenkel never forgot this help and often spoke about it as having saved his family. He came to University College, London, as a research associate in 1933. His life and future scientific activity were immediately influenced by the fly, Calliphora erythrocephala, that he saw come in through an open window and deposit its eggs on a small piece of meat. He watched the larvae emerge from the eggs and grow and—amid all the difficulties of a new language, a new culture, a new family, almost no salary, and using the most primitive tools—conceived the idea that, within a period of two months, resulted in the discovery of the bloodborne factor we now know to be the insect molting hormone, ecdysterone. He submitted his paper on this to Nature and it was printed three weeks later—his first paper in English. (Twenty-one years after Fraenkel's discovery, the structure of the molting prehormone was identified by Peter Karlson using Fraenkel's bioassay method.) Fraenkel's encounters with British scientists during these early years led to three seminal cooperative ventures. He and John Pringle showed that the halteres, which replace the sec
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ond pair of wings on the adult fly, actually function as miniature gyroscopes, or balance organs. Fraenkel and the physical chemist Kenneth M. Rudall analyzed the strange changes occurring in the cuticle of the larval fly at pupariation, using, among other methods, X-ray diffraction. This study provided the basis for work on insect cuticle that continues to this day, forty years later. Finally, at a chance meeting with a behaviorist, Donald L. Gunn, Fraenkel found a willing audience for his data and ideas on insect behavior—information that would eventually appear in their classic text, The Orientation of Animals, published in 1940. By 1936, Fraenkel's reputation was such that he was offered a post in insect physiology—perhaps the first full-time teaching position ever established in this discipline—in the Department of Zoology and Applied Entomology at Imperial College of the University of London. When World War II came, the Department was evacuated to Slough, and, to aid the war effort, the Pest Infestation Laboratory was created. Professor J. W. Munro wanted Fraenkel to work on insecticides, but Fraenkel chose to take the view that understanding stored-grain pests would develop the intelligence with which to deal with them successfully. Published as a series of detailed diet studies, his findings showed that insects have the same nutritional requirements as man, except for the beetle Tenebrio molitor, that needed an additional but as yet undefined component in its standard diet. By war's end Fraenkel could conjecture that he had found a new vitamin. It is doubtful whether Fraenkel's work contributed directly to ending the war, but his experiments and their results shaped the fields of insect nutrition and applied entomology for years to come. His nutritional expertise, furthermore, extended well beyond insects. As a member of a committee organized by the Fabian Society, he investigated problems of British agriculture after the war and wrote the chapter on
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Britain's nutrient requirements in the committee report. He also gained a certain notoriety in Britain as one of the designers of the British National (bread) Loaf— the size, shape, and composition of which were standardized during World War II. AMERICA In 1947 Fraenkel paid his first visit to the United States as a lecturer at the University of Minnesota. In 1948, after meeting with various American entomologists, he accepted an offer of a position in the Department of Entomology at the University of Illinois. After his experiences with restrictions on research in Palestine and Slough, he later confided, the freedom to pursue his own objectives in and of itself justified the move to Illinois. At Illinois, with its strong chemistry department, he began a collaboration with Herbert Carter on his new "vitamin" that led to the isolation, crystallization, and identification of the Tenebrio growth factor. He was disappointed when the "vitamin" turned out to be a molecule—carnitine—that had been isolated and identified fifty years earlier from mammalian muscle. Still, no biological role had been assigned to it in the interim, and the work of Fraenkel and his collaborators succeeded in establishing its universality of occurrence and its importance in Coenzyme A transfer reactions. It is worth noting that Fraenkel himself was never satisfied that the full spectrum of its action had yet been elucidated. Continuing to mine the vein of insect nutrition, he next posed an important question. If, as he had shown, all insects had the same dietary requirements, and if, as was well known, plant leaves generally contained all of the required compounds, why were so many insects restricted in the plants they would eat? By 1958, he had examined enough of the literature to recognize that the socalled "secondary" plant
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compounds, of many different structures, might provide a clue to the evolution of host selection. For years, botanists and chemists had been isolating different classes of these compounds associated with different plant families but were unable to establish functions for most of them. Fraenkel opened a new field— insect-plant coevolution based upon chemical and sensory interactions. He described the raison d'être of secondary plant compounds "as only . . . to repel and attract insects." Some regarded this as a flash of insight, but it is, as documented in his 1959 Science paper, the result of a long, thoughtful process tempered by extensive experience in nutrition and behavior. In a comment made much later, when this paper was chosen as a Citation Classic in 1984, he described the initial resistance to his idea as follows: "Perhaps it seemed implausible that such a simple explanation could be virtually new and at the same time correct." In 1961, when this idea was beginning to cause ferment in ecological circles, Fraenkel received one of the few Research Career Awards ever given in his field by the U. S. Public Health Service. In collaboration with a number of his students, he then examined the chemical basis of host selection, solidifying the theory. Fraenkel's early work with flies still intrigued him. Using modern techniques developed long after those halcyon days of string and wax, he reexamined the tanning of adult flies after emergence from the puparium, and promptly discovered a new hormone, bursicon, which was proven responsible for post-ecdysial activities. Interest in this hormone grew, and by 1968 much of Fraenkel's work had been corroborated and extended to other insects. He was elected to the National Academy of Sciences in that year. By the time he retired in 1972 he had also vindicated his old Calliphora assay. Responding to a challenge by Carroll M. Williams and associates, Fraenkel and a Czech colleague, Jan
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Zdarek, discovered additional factors that accelerate puparium formation. Thanks to the enlightened policy of the University of Illinois in supporting the continued research of emeritus professors, Fraenkel embarked upon a research program after his retirement that included such topics as interactions among nutritional states, developmental hormones, behavioral changes accompanying metamorphosis, and aging. Over the next twelve years he worked with a number of senior colleagues but most often employed bright undergraduate students as his hands. In this way he helped train a number of disciplined investigators. Throughout that time Fraenkel's manual Smith Corona typewriter continued to pound out research articles on diverse topics. He also maintained a steady flow of correspondence with far-flung colleagues until only a few weeks before his death at age eighty-four. CLOSING REMARKS Fraenkel's travels, both to meetings and for research purposes, took him all over the world. He studied inter-tidal snails on Bimini, leather pests in Yemen, rice leaf folders in Sri Lanka, and silkworm nutrition in Japan. He had a collector's eye for art objects and—with his love of music—the decorative title pages of sheet music. In 1968 he published a book deriving from this avocation, Decorative Music Title Pages (Dover Press). He also turned up a rare and instructive edition of Hector Berlioz's Les Troyens and published a paper on its significance. He was a skilled pianist and made a practice of seeking other musicians wherever he went. But Fraenkel's first love was biology—a love he communicated to his two sons. Gideon Fraenkel is now professor of chemistry at Ohio State University and Dan, professor of microbiology and molecular genetics at Harvard Medical
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School. Their father died a few months after the death of his devoted wife of fifty-six years, Rachel Sobel Fraenkel, herself an accomplished sculptor. Gottfried Fraenkel had that rare ability to recognize important questions and solve them with direct and simple techniques. Ever ready to exploit the materials at hand, his work was seminal to diverse areas of insect biology that have since become major fields of study. We wish to thank Robert Metcalf for his help in discussing Professor Fraenkel's scientific contributions.
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HONORS AND DISTINCTIONS 1926-1927 Fellow, International Education Board, Rockefeller Foundation 1955 Honorary Fellow, Royal Entomological Society, London 1962-1972 Research Career Awardee, U. S. Public Health Service 1968 Member, National Academy of Sciences 1972 Fellow, American Association for the Advancement of Science 1980 Honorary Doctor, François Rabelais University, Tours 1982 Honorary Fellow, The Linnean Society, London 1984 Honorary Doctor, The Hebrew University, Jerusalem
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Selected Bibliography
1925 Der statische Sinn der Medusen. Z. Vgl. Physiol., 2:658-90. 1927 Phototropotaxis bei Meerestieren. Die Naturwissenschaften, 14: 117-22. Beiträge zur Biologie eines Arcturiden. Zoo. Anz., 69:219-22. Beiträge zur Geotaxis und Phototaxis von Littorina. Z. Vgl. Physiol., 5:585-97. Die Grabbewegungen der Soleniden. Z. Vgl. Physiol., 6:167-220. Über Photomenotaxis bei Elysia viridis. Mont. Z. Vgl. Physiol., 6:385-401. Biologische Beobachtungen an Janthina. Morphol. Oekol. Tiere, 7:597-608. With Kuhn, A. Über das Unterscheidungsvermögen der Bienen für Wellenlängen im Spektrum. Nachr. Ges. Wiss. Göttingen. Math. Phys. K.:1-6. 1928 Über den Auslösungsreiz des Umdrehreflexes bei Seestsernen und Schlangensternen. Z. Vgl. Physiol., 7:365-78. 1929 Über die Geotaxis von Convoluta roscoffensis. Z. Vgl. Physiol., 10:237-47. Untersuchungen über Lebensgewohnheiten, Sinnesphysiologie und Sozialpsychologie der wandernden Larven der afrikanischen Wanderheuschrecke Schistocerca gregaria (Forsk.). Biol. Zentralbl., 46:657-80. 1930 Der Atmungsmechanismus des Skorpions. Z. Vgl. Physiol., 11:656-61. Die Orientierung von Schistocerca gregaria zu strahlender Wärme. Z. Vgl. Physiol., 13:300-13. Beiträge zur Physiologie der Atmung der Insekten. Atti 11. Congr. Int. Zool. Pavoda, pp. 905-21. With F. S. Bodenheimer, K. Reich, and N. Segal. Studien zur
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Epidemiologie, Ökologie und Physiologie der afrikanischen Wanderheuschrecke. (Schistocerca gregaria Forsk.) Z. Angew. Entomol., 15:435-557. 1931 Die Mechanik der Orientierung der Tiere im Raum. Biol. Rev., 6:36-87. 1932 Untersuchungen über die Koordination von Reflexen und automatisch-nervösen Rhythmen bei Insekten. I. Die Flugreflexe der Insekten und ihre Koordination. Z. Vgl. Physiol., 16:371-93. Untersuchungen über die Koordination von Reflexen und automatisch-nervösen Rhythmen bei Insekten. II. Die nervöse Regulierung der Atmung während des Fluges. Z. Vgl. Physiol., 16:394-417. Untersuchungen über die Koordination von Reflexen und automatisch-nervösen Rhythmen bei Insekten. III. Das Problem des gerichteten Atemstromes in den Tracheen der Insekten. Z. Vgl. Physiol., 16:418-43. Untersuchungen über die Koordination von Reflexen und automatisch-nervösen Rhythmen bei Insekten. IV. Uber die nervösen Zentren der Atmung und die Koordination ihrer Tätigkeit. Z. Vgl. Physiol., 16:444-62. Die Wanderungen der Insekten. Ergeb. Biol., 9:1-238. 1934 Der Atmungsmechanismus der Vögel während des Fluges. Biol. Zentralbl., 54:96-101. Pupation of flies initiated by a hormone. Nature, 133:834. 1935 A hormone causing pupation in the blowfly Calliphora erythrocephala . Proc. R. Soc. London Ser. B., 118:1-12. 1936 Observations and experiments on the blowfly (Calliphora erythrocephala ) during the first day after emergence. Proc. Zool. Soc. London, pp. 893-904.
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Utilization of sugars and polyhydric alcohols by the adult blowfly. Nature, 137:237. 1938 With G. V. B. Herford. The respiration of insects through the skin. J. Exp. Biol., 15:266-80. With J. W. S. Pringle. Halteres of flies as gyroscopic organs of equilibrium. Nature, 141:919. Temperature adaptation and the physiological action of high temperatures. Kongressbericht II. 16th Int. Physiol. Congr. Zurich. (Abstract). With J. L. Harrison. Irregular abdomina in Calliphora erythrocephala (Mg.) . Proc. R. Entomol. Soc. London (A), 13:95-96. The evagination of the head in the pupae of cyclorrhaphous flies (Diptera). Proc. R. Entomol. Soc. London (A), 13:137-39. The number of moults in the cyclorrhaphous flies (Diptera). Proc. R. Entomol. Soc. London (A), 13:158-60. 1939 The function of the halteres of flies (Diptera). Proc. Zool. Soc. London (A), 109:69-78. 1940 Utilization and digestion of carbohydrates by the adult blowfly. J. Exp. Biol., 17:18-29. With K. M. Rudall. A study of the physical and chemical properties of the insect cuticle. Proc. R. Soc. London (B), 129:1-35. With H. S. Hopf. The physiological action of abnormally high temperatures on poikilothermic animals. I. Temperature adaptation and the degree of saturation of the phosphatides. Biochem. J., 34:1085-92. With G. V. B. Herford. The physiological action of abnormally high temperatures on poikilothermic animals. II. The respiration at high sublethal and lethal temperatures. J. Exp. Biol., 17:386-95. With R. A. Davis. The oxygen consumption of flies during flight. J. Exp. Biol., 17:402-7. With D. L. Gun. The Orientation of Animals. Oxford: Clarendon Press. 352 pp.
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1941 With J. A. Reid and M. Blewett. The sterol requirements of the larva of the beetle, Dermestes vulpinus Fabr. Biochem. J., 35:712-20. With M. Blewett. Deficiency of white flour in riboflavin (tested with the flour beetle Tribolium confusum). Nature, 147:716. 1942 With M. Blewett. Biotin as a possible growth factor for insects. Nature, 149:301. With M. Blewett. Biotin, B1 riboflavin, nicotinic acid, B6, and pantothenic acid as growth factors for insects. Nature, 150:177. 1943 With M. Blewett. Vitamins of the B-group required by insects. Nature, 151:703. With M. Blewett. Intracellular symbionts of insects as a source of vitamins. Nature, 152:506. Insect nutrition. J. R. Coll. Sci., 13:59-69. With M. Blewett. The basic food requirements of several insects. J. Exp. Biol, 20:28-34. With M. Blewett. The vitamin B-complex requirements of several insects. Biochem. J., 37:686-92. With M. Blewett. The sterol requirements of several insects. Biochem. J., 37:692-95. With M. Blewett. The natural foods and the food requirements of several species of stored products insects. Trans. R. Entomol. Soc. London, 93:457-90. 1944 With M. Blewett. Intracellular symbiosis and vitamin requirements of two insects, Lasioderma serricorne and Sitodrepa panicea. Proc. R. Soc. London (B), 132:212-21. With M. Blewett. Stages in the recognition of biotin as a growth factor for insects. Proc. R. Entomol. Soc. London (A), 19:30-35. With M. Blewett. The utilization of metabolic water in insects. Bull. Entomol. Res., 35:127-39.
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1945 With M. Blewett. Linoleic acid, α-tocopherol and other fat-soluble substances as nutritional factors for insects. Nature, 155:392. 1946 With M. Blewett. The dietetics of the clothes moth, Tineola bisselliella . Hum. J. Exp. Biol., 22:156-61. With M. Blewett. The dietetics of the caterpillars of three Ephestia species, E. kuehniella, E. elutella and E. cautella, and of a closely related species, Plodia interpunctella. J. Exp. Biol., 22:162-71. With M. Blewett. Linoleic acid, vitamin E and other fat-soluble substances in the nutrition of certain insects, (Ephestia kuehniella, E. elutella , E. cautella and Plodia interpunctella [Lep.]). J. Exp. Biol., 22:172-90. With M. Blewett. Folic acid in the nutrition of certain insects. Nature, 157:697. Britain's nutritional requirements. In: Towards a Socialist Agriculture. Studies by a Group of Fabians, ed. F. W. Bateson, London: Victor Gollancz, pp. 42-76. 1947 With K. M. Rudall. The structure of insect cuticles. Proc. R. Soc. London (B), 134:111-43. With M. Blewett. The importance of folic acid and unidentified members of the vitamin B complex in the nutrition of certain insects. Biochem. J., 41:469-75. With M. Blewett. Linoleic acid and arachidonic acid in the metabolism of two insects, Ephestia kuehniella (Lep.) and Tenebrio molitor (Col.). Biochem. J., 41:475-78. With P. Ellinger and M. M. Abdel Kader. The utilization of nicotinamide derivatives and related compounds by mammals, insects and bacteria. Biochem. J., 41:559-68. 1948 BT, a new vitamin of the B-group and its relation to the folic acid group and other anti-anemia factors. Nature, 161:981-83. The effects of a relative deficiency of lysine and tryptophane in the diet of an insect, Tribolium confusum. Biochem. J., 43:Proceedings XIV.
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Evidence for the need, by certain insects, for three chemically unidentified factors of the vitamin Bcomplex. Br. J. Nutr. 2, Abstr. Commun., 1:ii. 1950 The nutrition of the meal worm, Tenebrio molitor L. (Tenebrionidae, Coleoptera). Physiol. Zool., 23:92-108. With N. C. Pant. The function of the symbiotic yeasts of two insect species, Lasioderma serricorne F. and Stegobium (Sitodrepa) paniceum L. Science, 112:498-500. 1951 With H. R. Stern. The nicotinic acid requirements of two insect species in relation to the protein content of their diets. Arch. Biochem., 30:438-44. Effect and distribution of vitamin BT. Arch. Biochem. Biophys. 34:457-67. Isolation procedures and certain properties of vitamin BT. Arch. Biochem. Biophys., 34:468-77. 1952 With H. E. Carter, P. K. Bhattacharyya, and K. Weidman. The identity of vitamin BT with carnitine. Arch. Biochem. Biophys., 35:241-42. With M. I. Cooper. Nutritive requirements of the small-eyed flour beetle, Palorus ratzeburgi Wissman (Tenebrionidae, Coleoptera), Physiol. Zool., 25:20-28. The role of symbionts as sources of vitamins and growth factors for their insect hosts. Tijdschr. Entomol., 95:183-95. The nutritional requirements of insects for known and unknown vitamins. Trans. 9th Int. Congr. Entomol. Amsterdam, 1951, 1:277-80. 1953 The nutritional value of green plants for insects. Trans. 9th Int. Congr. Entomol. Amsterdam, 1951, 2:90-100. With H. E. Gray. Fructomaltose, a recently discovered trisaccharide isolated from honeydew. Science, 118:304-5. Studies on the distribution of vitamin BT (carnitine). Biol. Bull., 104:359-71.
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With S. Friedman and P. K. Bhattacharyya. Function of carnitine (BT). Fed. Proc. Fed. Am. Soc. Exp. Biol., 12:414-15. With V. J. Brookes. The process by which the puparia of many species of flies become fixed to a substrate. Biol. Bull., 105:442-49. 1954 With E. W. French. Carnitine (vitamin BT) as a nutritional requirement for the confused flour beetle, Tribolium confusum Duval. Nature, 173:173. With H. E. Gray. The carbohydrate components of honeydew. Physiol. Zool., 27:56-65. With P. I. Chang. Manifestations of a vitamin BT (carnitine) deficiency in the larvae of the meal worm, Tenebrio molitor L. Physiol. Zool., 27:40-56. With G. E. Printy. The amino acid requirements of the confused flour beetle, Tribolium confusum Duval. Biol. Bull., 106:149-57. With H. H. Moorefield. The character and ultimate fate of the larval salivary secretion of Phormia regina Meigen (Diptera, Calliphoridae). Biol. Bull., 106:178-84. With H. Lipke and I. E. Liener. Effect of soybean inhibitors on growth of Tribolium confusum. Agric. Food Chem., 2:410-14. With P. I. Chang. Histopathology of vitamin BT (carnitine) deficiency in larvae of meal worm, Tenebrio molitor L. Physiol. Zool., 27:259-67. The distribution of vitamin BT (carnitine) throughout the animal kingdom. Arch. Biochem. Biophys., 50:486-95. With S. C. Rasso. The food requirements of the adult female blowfly, Phormia regina (Meigen), in relation to ovarian development. Ann. Entomol. Soc. Am., 47:636-45. With N. C. Pant. Studies on the symbiotic yeasts of two insect species, Lasioderma serricorne F. and Stegobium paniceum L. Biol. Bull., 107:420-32. With N. C. Pant. On the function of the intracellular symbionts of Oryzaephilus surinamensis L. (Cucujidae, Coleoptera). J. Zool. Soc. India, 6:173-77. 1955 Inhibitory effects of sugars on the growth of the mealworm, Tenebrio molitor L. J. Cell Comp. Physiol., 45:393-408.
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With P. K. Bhattacharyya and S. Friedman. The effect of some derivatives and structural analogues of carnitine on the nutrition of Tenebrio molitor. Arch. Biochem. Biophys., 54:424-31. With S. Friedman, T. Hinton, S. Laszlo, and J. L. Noland. The effect of substituting carnitine for choline in the nutrition of several organisms. Arch. Biochem. Biophys., 54:432-39. With H. Lipke. The toxicity of corn germ to the meal worm, Tenebrio molitor. J. Nutr., 55:165-78. With M. Brust. The nutritional requirements of the larvae of a blowfly, Phormia regina Meig. Physiol. Zool., 28:186-204. With S. Friedman. Reversible enzymatic acetylation of carnitine. Arch. Biochem. Biophys., 59:491-501. With S. Friedman, J. E. McFarlane, and P. K. Bhattacharyya. Quantitative separation and identification of quaternary ammonium bases. Arch. Biochem. Biophys., 59:484-90. 1956 With H. Lipke. Insect nutrition. Annu. Rev. Entomol., 1:17-44. Insects and plant biochemistry. The specificity of food plants for insects. Proc. 14th Int. Congr. Zool. Copenhagen, pp. 383-87. With J. Leclercq. Nouvelles recherches sur les besoins nutritifs de la larve du Tenebrio molitor L. (Insecte, Coleoptère) Arch. Int. Physiol. Biochim., 64:601-22. With K. Bloch, R. G. Langdon, and A. J. Clark. Impaired steroid biogenesis in insect larvae. Biochim. Biophys. Acta., 21:176. 1957 The Tenebrio assay for carnitine. In: Methods of Enzymology, ed. S. P. Colowick and N. O. Kaplan, New York: Academic Press, vol. 3, pp. 662-67. With S. Friedman and A. B. Galun. Isolation and physiological action of (+)-carnitine. Arch. Biochim. Biophys., 66:10-15. With T. Ito. γ-butyrobetaine as a specific antagonist for carnitine in the development of the early chick embryo. J. Gen. Physiol., 41:279-88. With S. Friedman. Carnitine. Vitam. Horm., 15:73-118. With R. Galun. Physiological effects of carbohydrates in the nutrition of a mosquito, Aedes aegypti and two flies, Sarcophaga bullata and Musca domestica. J. Cell. Comp. Physiol., 50:1-23.
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1958 With V. J. Brooks. The nutrition of the larva of the housefly Musca domestica L. Physiol. Zool., 31:208-23. The effect of zinc and potassium in the nutrition of Tenebrio molitor, with observations on the expression of a carnitine deficiency. J. Nutr., 65:361-96. The basis of food selection in insects which feed on leaves. Abstracts of Invitational Papers. 18th Annu. Meet. Entomol. Soc. of Japan. 5 pp. 1959 A historical and comparative survey of the dietary requirements of insects. Ann. N.Y. Acad. Sci., 77:267-74. The raison d'être of secondary plant substances. Science, 129:1466-70. The chemistry of host specificity of phytophagous insects. In: Biochemistry of Insects, 4th Int. Congr. Biochem., London: Pergamon Press, vol. 12, pp. 1-14. With T. Ito and Y. Horie. Feeding on cabbage and cherry leaves by maxillectomized silkworm larvae. J. Seric. Sci. Jpn., 28:107-13. With R. T. Yamamoto. Common attractant for the tobacco hornworm, Protoparce sexta (Johan.) and the Colorado potato beetle, Leptinotarsa decemlineata (Say). Nature, 184:206-7. 1960 With R. T. Yamamoto. The specificity of the tobacco hornworm, Protoparce sexta to solanaceous plants. Ann. Entomol. Soc. Am., 53:503-7. With R. T. Yamamoto. Assay of the principal gustatory stimulant for the tobacco hornworm, Protoparce sexta from solanaceous plants. Ann. Entomol. Soc. Am. , 53:499-503. With R. T. Yamamoto. The suitability of tobaccos for the growth of the cigarette beetle, Lasioderma serricorne. J. Econ. Entomol., 53:381-84. Lethal high temperatures for three marine invertebrates, Limulus polyphemus, Littorina littorea and Pagurus longicarpus. Oikos, 11:171-82. With S. Friedman, J. E. McFarlane, and P. K. Bhattacharyya.
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(-)-Carnitine chloride. In: Biochemical Preparations, 7:26-30. New York: John Wiley & Sons. 1961 A new type of negative phototropotaxis observed in a marine isopod, Eurydice. Physiol. Zool., 34:228-32. Resistance to high temperatures in a Mediterranean snail, Littorina neritoides. Ecology, 42:604-6. Quelques observations sur le comportement de Convoluta roscoffensis . Cah. Biol. Mar., 2:155-60. With G. P. Waldbauer. Feeding on normally rejected plants by maxillectomized larvae of the tobacco hornworm, Protoparce sexta (Lepidoptera, Sphingidae). Ann. Entomol. Soc. Am., 54:47785. With D. L. Gunn. The Orientation of Animals. Kineses, Taxes and Compass Reactions. New York: Dover Publications, Inc. 376 pp. With Galun, R. The effect of low atmospheric pressure on adult Aedes aegypti and on housefly pupae. J. Insect Physiol., 7:161-76. Die biologische Funktion der sekundären Pflänzenstoffe im Allgemeinen und solcher Stoffe in Solanaceen im Besonderen. In: Chemie und Biochemie der Solanum-Alkaloide, Tagungsberichte 27, Int. Symp. Deutsch. Akad. Landw., Berlin, pp. 297-307. 1962 The physiology of insect nutrition. (Atti del Simposio Internazionale di Biologi Sperimentale, Celebrazione Spallanzaniana, Reggio Emilia-Pavia, May 2-7, 1961.) Symp. Genet. Biol. Ital., 9:3-11. With J. Nayar, O. Nalbandov, and R. T. Yamamoto. Further investigations into the chemical basis of the insect-host plant relationship, XI. Int. Congr. Entomol. Vienna, Verhandlungen, 3:122-26. With R. T. Yamamoto. The physiological basis for the selection of plants for egg-laying in the tobacco hornworm, Protoparce sexta (Johan.). XI. Inter. Congr. Entomol. Vienna, Verhandlungen, 3:127-133. With J. K. Nayar. The chemical basis of host plant selection in the silkworm, Bombyx mori (L.). J. Insect Physiol., 8:505-25.
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With C. Hsiao. Hormonal and nervous control of tanning in the fly. Science, 138:27-29. 1963 With J. K. Nayar. The chemical basis of host selection in the Catalpa sphinx, Ceratomia castalpae (Lepidoptera, Sphingidae). Ann. Entomol. Soc. Am., 56:119-22. With J. K. Nayar. The chemical basis of the host selection in the Mexican bean beetle, Epilachna varivestis (Coleoptera, Coccinellidae). Ann. Entomol. Soc. Am., 56:174-78. With J. K. Nayar. Practical methods of year-round laboratory rearing of the silkworm, Bombyx mori (L.) (Lepidoptera, Bombycidae). Ann. Entomol. Soc. Am., 56:122-23. With C. Hsiao. Tanning in the adult fly: A new function of neurosecretion in the brain. Science, 141:1057-58. Berlioz, the princess and 'Les Troyens.' Mus. Let., 44:249-56. 1964 With O. Nalbandov and R. T. Yamamoto. Insecticides from plants. Nicandrenone, a new compound with insecticidal properties, isolated from Nicandra physalodes. Agric. Food Chem., 12:55-59. With C. F. Soo Hoo. The resistance of ferns to the feeding of Prodenia eridania larvae. Ann. Entomol. Soc. Am., 57:788-90. With C. F. Soo Hoo. A simplified laboratory method for rearing the Southern armyworm, Prodenia eridania for feeding experiments. Ann. Entomol. Soc. Am., 57:798-99. 1965 With C. Hsiao. Bursicon, a hormone which mediates tanning of the cuticle in the adult fly and other insects. J. Insect Physiol., 11:513-56. A brief survey of the recognition of carnitine as a substance of physiological importance. In: Recent Research on Carnitine. Its Relation to Lipid Metabolism, ed. G. Wolf, Cambridge: The MIT Press, pp. 1-3. 1966 With C. Hsiao and M. Seligman. Properties of bursicon: An insect hormone that controls cuticular tanning. Science, 151:91-93.
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With R. D. Pausch. The nutrition of the larva of the oriental rat flea, Xenopsylla cheopis (Rothschild). Physiol. Zool., 39:202-22. With C. Hsiao. Neurosecretory cells in the central nervous system of the adult blowfly, Phormia regina Meigen (Diptera, Caliphoridae). J. Morphol., 119:21-38. With C. F. Soo Hoo. The consumption, digestion and utilization of food plants by a polyphagous insect, Prodenia eridania (Cramer). J. Insect Physiol., 12:711-30. With C. F. Soo Hoo. The consumption, digestion and utilization of food plants by a polyphagous insect, Prodenia eridania (Cramer). J. Insect Physiol., 12:711-30. With T. Ito. The effect of nitrogen starvation on Tenebrio molitor L. J. Insect Physiol., 12:803-17. The heat resistance of intertidal snails at Shirahama, Wakyamaken, Japan. Publ. Seto Mar. Biol. Lab., 14:185-95. 1967 With C. Hsiao. Calcification, tanning, and the role of ecdysone in the formation of the puparium of the facefly, Musa autumnalis. J. Insect Physiol., 13:1387-94. 1968 With C. Hsiao. Manifestations of a pupal diapause in two species of flies, Sarcophaga argyrostoma and S. bullata. J. Insect Physiol., 14:689-705. With C. Hsiao. Morphological and endocrinological aspects of pupal diapause in a fleshfly, Sarcophaga argyrostoma. J. Insect Physiol., 14:707-18. The heat resistance of intertidal snails at Bimini, Bahamas; Ocean Springs, Mississippi; and Woods Hole, Massachusetts. Physiol. Zool., 41:1-13. With T. H. Hsiao. The influence of nutrient chemicals on the feeding behavior of the Colorado potato beetle, Leptinotarsa decemlineata (Coleoptera: Chrysomelidae). Ann. Entomol. Soc. Am., 61:44-54. With T. H. Hsiao. Isolation of phagostimulatory substances from the host plant of the Colorado potato beetle. Ann. Entomol. Soc. Am., 61:476-84. With T. H. Hsiao. The role of secondary plant substances in the
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food specificity of the Colorado potato beetle. Ann. Entomol. Soc. Am. , 61:485-93. With T. H. Hsiao. Selection and specificity of the Colorado potato beetle for solanaceous and nonsolanaceous plants. Ann. Entomol. Soc. Am., 61:493-503. Decorative Music Title Pages. New York: Dover Publications. 230 pp. 1969 With W. Fogal. Melanin in the puparium and adult integument of the fleshfly, Sarcophaga bullata. J. Insect Physiol., 15:1437-47. With W. H. Fogal. The role of bursicon in melanization and endocuticle formation in the adult fleshfly, Sarcophaga bullata. J. Insect Physiol., 15:1235-47. With M. Seligman and S. Friedman. Hormonal control of turnover of tyrosine and tyrosine phosphate during tanning of the adult cuticle in the fly, Sarcophaga bullata. J. Insect Physiol., 15:1085-101. With M. Seligman and S. Friedman. Bursicon mediation of tyrosine hydroxylation during tanning of the adult cuticle of the fly, Sarcophaga bullata. J. Insect Physiol., 15:553-62. With T. H. Hsiao. Properties of leptinotarsin: A toxic hemolymph protein from the Colorado potato beetle. Toxicon, 7:119-30. With P. Berreur. Puparium formation in flies: Contraction to puparium induced by ecdysone . Science, 164:1182-83. With J. Zdarek. Correlated effects of ecdysone and neurosecretion in puparium formation (pupariation) of flies. Proc. Natl. Acad. Sci. USA, 64:565-72. Evaluation of our thoughts on secondary plant substances. Entomol. Exp. Appl., 12:473-86. 1970 With E. Zlotkin. Acceleration of puparium formation in Sarcophaga argyrostoma by electrical stimulation or scorpion venom. J. Insect Physiol., 16:549-54. With J. Zdarek. The evaluation of the ''Calliphora test" as an assay for ecdysone. Biol. Bull., 139:138-50. With W. Fogal. Histogenesis of the cuticle of the adult flies, Sarcophaga bullata and S. argyrostoma. J. Morphol., 130:137-50. With J. Zdarek. Overt and covert effects of endogenous and
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exogenous ecdysone in puparium formation of flies. Proc. Natl. Acad. Sci. USA, 67:331-37. 1971 With J. Zdarek. Neurosecretory control of ecdysone release during puparium formation of flies. Gen. Comp. Endocrinol., 17:483-89. With E. Zlotkin, F. Miranda, and S. Lissitzky. The effect of scorpion venoms on blowfly larvae-a new method for the evaluation of scorpion venoms potency. Toxicon, 9:1-8. 1972 With D. L. Denlinger and J. H. Willis. Rates and cycles of oxygen consumption during pupal diapause in Sarcophaga flesh flies. J. Insect. Physiol., 18:871-82. With J. Zdarek. The mechanism of puparium formation in flies. J. Exp. Zool., 179:315-24. With S. Friedman. Carnitine. In: The Vitamins, ed. W. H. Sebrell and R. S. Harris, New York and London: Academic Press, vol. 5., pp. 329-55. 1973 With D. M. DeGuire. The meconium of Aedes aegypti (Diptera: Culicidae). Ann. Entomol. Soc. Am., 66:475-76. With G. Bhaskaran. Pupariation and pupation in cyclorrhaphous flies (Diptera): terminology and interpretation. Ann. Entomol. Soc. Am., 66:418-22. With N. P. Ratnasiri. Inhibition of purpariation in Sacrophaga bullata . Nature, 243:91-93. With N. Ratnasiri. Anterior inhibition of pupariation in ligated larvae of Sarcophaga bullata and other fly species: Incidence and expression. Ann. Entomol. Soc. Am., 67:195-203. 1974 With N. Ratnasiri. The physiological basis of anterior inhibition of puparium formation in ligated fly larvae. J. Insect Physiol., 20:105-19. With P. Sivasubramanian and H. S. Ducoff. Effect of X-irradiation on the formation of the puparium in the fleshfly, Sarcophaga bullata . J. Insect Physiol., 20:1303-17.
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With P. Sivasubramanian and S. Friedman. Nature and role of proteinaceous hormonal factors acting during puparium formation in flies. Biol. Bull., 147:163-85. 1975 Interactions between ecdysone, bursicon, and other endocrines during puparium formation and adult emergence in flies. Am. Zool. 15 Suppl. 1, 15:29-48. 1976 Molting and development in undersized fly larvae. In: The Insect Integument, ed. H. R. Hepburn, Amsterdam: Elsevier, pp. 323-38. 1977 With A. Blechl, J. Blechl, P. Herman, and M. Seligman. 3':5'-cyclic, AMP, and hormonal control of puparium formation in the fleshfly Sarcophaga bullata. Proc. Natl. Acad. Sci. USA, 74:2182-86. With C. Pappas. Nutritional aspects of oogenesis in the flies Phormia regina and Sarcophaga bullata. Physiol. Zool., 50:237-46. With C. Pappas. Hormonal aspects of oogenesis in the files Phormia regina and Sarcophaga bullata. J. Insect Physiol., 24:75-80. With M. Seligman, A. Blechl, J. Blechl, and P. Herman. Role of ecdysone, pupariation factors, and cyclic AMP in formation and tanning of the puparium of the fleshfly Sarcophaga bullata. Proc. Natl. Acad. Sci. USA, 74:4697-701. 1979 With M. Hollowell. Actions of the juvenile hormone, 20-hydroxyecdysone and the oostatic hormone during oogenesis in the flies Phormia regina and Sarcophaga bullata. J. Insect Physiol., 25:305-10. With J. Zdarek and K. Slama. Changes in internal pressure during puparium formation in flies. J. Exp. Zool., 207:187-95. 1980 The proposed vitamin role of carnitine. In: Carnitine Biosynthesis, Metabolism and Functions, ed. R. E. Fraenkel and J. D. McGarry, New York: Academic Press, pp. 1-6.
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Foreword and overview. In: Neurohormonal Techniques in Insects, ed. T. A. Miller, Heidelberg: Springer-Verlag, pp. IX-XV. 1981 Importance of allelochemics in plant insect relations. In: The Ecology of Breuchids Attacking Legumes (Pulses), ed. V. Labeyrie, Hingham, Mass.: Junk Publishers, pp. 57-60. With F. Fallil. The spinning (stitching) behavior of the rice leaf folder, Cnaphalocrocis medinalis. Entomol. Exp. Appl., 29:138-46. With F. Fallil and K. S. Kumarasinghe. The feeding behavior of the rice leaf folder, Cnaphalocrocis medinalis. Entomol. Exp. Appl. 29:147-61. With B. Bennetova. What determines the number of ovarioles in a fly ovary? J. Insect Physiol., 27:403-10. Food conversion efficiency by fleshfly larvae, Sarcophaga bullata. Physiol. Entomol., 6:157-60. With J. Zdarek, R. Rohlf, and J. Blechl. A hormone effecting immobilization in pupariating fly larvae. J. Exp. Biol., 93:51-63. 1984 This week's citation classic. The raison d'être of secondary plant substances. C. C. Life Sci., 11:18. With J. Zdarek, J. Zavidilova, and J. Su. Post-eclosion behaviour of flies after emergence from the puparium. Acta Entomol. Bohemoslov., 81:161-70. With J. Su. Hormonal control of eclosion of flies from the puparium. Proc. Natl. Acad. Sci. USA, 81:1457-59. With J. Su and J. Zdarek. Neuromuscular and hormonal control of post-eclosion processes in flies. Arch. Insect Biochem. Physiol., 1:345-66. 1986 With J. Zdarek and S. Reid. How does an eclosing fly deal with obstacles? Physiol. Entomol., 11:107-14. 1987 With J. Zdarek. Pupariation in flies: A tool for monitoring effects of drugs, venoms and other neurotoxic compounds. Arch. Insect Biochem. Physiol., 4:29-46.
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With S. N. M. Reid and S. Friedman. Extrication, the primary event in eclosion, and its relationship to digging, pumping and tanning in Sarcophaga bullata. J. Insect Physiol., 33:339-348. With S. N. M. Reid and S. Friedman. Extrication, the primary event in eclosion, and its neural control in Sarcophaga bullata. J. Insect Physiol., 33:481-486.
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Haldan Keffer Hartline December 22, 1903-March 18, 1983
By Floyd Ratliff For more than half a century Haldan Keffer Hartline, Keffer to friends and close colleagues, conducted biophysical research on vision and the retina. He studied retinas from arthropods, vertebrates, and molluscs—the three major phyla with well-developed eyes—and his investigations extended into many and diverse branches of the field. During this long career Hartline elucidated numerous fundamental principles of retinal physiology, laying the foundations for the present-day study of the neurophysiology of vision. Hartline's four major accomplishments were all "firsts" in their respective fields: With Clarence H. Graham he recorded the activity of single optic nerve fibers. He mapped the activity of the visual receptive field to reveal a system of many convergent pathways from many photoreceptors (the foundation for modern concepts of parallel processing by specialized channels). He recorded— with Wagner and MacNichol—intracellular generator potentials. And finally, he discovered lateral inhibition in the retina and described the integrative activity of neural networks with the Hartline-Ratliff equations. EDUCATION AND EARLY LIFE Keffer Hartline was born on December 22, 1903, in Bloomsburg, Pennsylvania, to Daniel Schollenberger
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Hartline and Harriet Franklin Keffer Hartline. His father taught science and his mother English at the Bloomsburg State Normal School (now Bloomsburg State College) where the young Hartline received his early formal education. Perhaps more significant to the young Keffer was the informal but intensive training he received at home as an only child. Both of his parents had a strong interest in the natural world around them, an interest that deeply affected the young Keffer. Indeed, he was later to refer to his father as "my first and best teacher," and the love of nature his parents instilled surely influenced his choice of experimental research in biology as his lifelong career. Upon completion of his studies at Bloomsburg in 1920, Hartline spent the summer at the marine laboratory in Cold Spring Harbor, Long Island, taking a six-week course in comparative anatomy. That fall he entered Lafayette College to study biology and was encouraged by Professor B. W Kunkel to do research. Hartline was much impressed by Jacques Loeb's quantitative work on tropisms, and his very first experiments—phototropic responses of land isopods—were along the same lines. At Woods Hole in the summer of 1923 he showed the results of his experiments to Loeb, who encouraged him to publish the work in the Journal of General Physiology. Loeb also introduced Hartline to the biophysicist Selig Hecht, who was just coming into prominence in the field of vision research. That fall, Hartline entered the Johns Hopkins University School of Medicine. Finding time at Hopkins to continue his research, he came under the influence of E. K. Marshall, head of the Department of Physiology, and, even more strongly, of Charles D. Snyder. Snyder taught Hartline how to use and replace the inevitable broken strings on a string galvanometer, then gave him free access to that delicate instrument. Hartline bore out his confidence and soon thereafter published pioneering
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work on the retinal action potential he had recorded from a variety of species, including humans. His early research helped lay the groundwork for modern electroretinography. In 1927, Hartline received the M.D. degree from Hopkins but—clearly more interested in research—never went on to practice medicine. Remaining at Hopkins for two years as a National Research Council Fellow, Hartline decided, after a brief exposure to quantitative experimental biology, to study mathematics and physics. Drawn to these disciplines, he went so far as to consider a career in either one or the other. On a Johnson Research Scholarship from the Eldridge Reeves Johnson Foundation, he went to Germany to study under Arnold Sommerfeld at Munich and under Werner Heisenberg at Leipzig. It soon became evident, however, that Hartline lacked the background for these advanced courses and lectures, and-disappointed with the outcome of this venture—he returned to the United States after one year to take up his first appointment in biology. Hartline's interest in mathematics and physics never waned, and his approach to experimental biology remained rigorously quantitative and based on sound physical principles. PROFESSIONAL CAREER Detlev W. Bronk, director of the Eldridge Reeves Johnson Foundation at the University of Pennsylvania from 1929, was quick to recognize genius and soon offered Hartline a position as a fellow in medical physics. This proved ideal for the frustrated theoretical physicist, and Hartline remained at the Johnson Foundation from 1931 until 1949 (except for a brief and unsuccessful move, with Bronk, to Cornell University Medical College from 1940 to 1941). While at the Johnson Foundation, Hartline met a number of investigators who later became prominent in vision re
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search. Among these were psychologists Clarence H. Graham and Lorrin A. Riggs, who became his research collaborators, and physiologist William A. H. Rushton, who turned to vision research in his later years. It was also at the Johnson Foundation that Hartline first met the neurophysiologist Ragnar Granit. While at Woods Hole he became acquainted with the biochemist George Wald. Hartline, Granit, and Wald each went his independent way in vision research, following the work of the other two closely and admiring it, but never working together in collaboration. They little dreamed that—a quarter of a century later— they would share the Nobel Prize. In 1949, Bronk accepted the presidency of the Johns Hopkins University on the condition (among others) that a biophysics department be established on the Homewood campus. He appointed Hartline the first professor of biophysics and chairman of the new Thomas C. Jenkins Laboratory of Biophysics. There Hartline continued his earlier close association with Henry G. Wagner and E. F. (Ted) MacNichol, Jr., while electronics engineer John P. Hervey and instrument maker Walter Biderlich provided valuable support services. I first met Hartline in 1950 when I joined his laboratory on a one-year National Research Council fellowship. We felt an instant rapport and would work together in close collaboration for the next twenty-five years. In September of 1953, Bronk became president of the Rockefeller Institute for Medical Research (later The Rockefeller University) and immediately appointed Hartline a member and head of the Institute's Laboratory of Biophysics. Within the year, Hartline invited me to leave Harvard for The Rockefeller, and I immediately accepted. Over the next few years we were joined by William H. Miller, Bruce W. Knight, Jr., Frederick A. Dodge, Jr., and electronics engineer Norman Milkman. When the Rockefeller Institute became
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The Rockefeller University, Hartline was appointed professor and head of laboratory. He never left The University thereafter for any extended period, except for a sabbatical leave as George C. Eccles Professor at the University of Utah in 1972. That same year, he was named Detlev W. Bronk Professor at The Rockefeller, the post he held until his retirement until 1974. MAJOR SCIENTIFIC CONTRIBUTIONS Single Optic Nerve Fibers In 1927, Edgar D. Adrian and Rachel Matthews successfully recorded electrical activity in an optic nerve, though—in this early work (on the eye of the eel)—they were only able to record the massive discharge of the whole nerve trunk. Adrian and Bronk later managed to dissect and isolate a single fiber of the phrenic nerve and record its activity. Inspired by their success, Hartline and Graham undertook similar studies on the optic nerve of the horseshoe ''crab," Limulus. The compound eye of this venerable animal, with its large photoreceptors and long optic nerve, was ideally suited for this study, and in 1932 they were able to record the activity of single optic nerve fibers for the first time. Their research showed that impulses transmitted by an optic nerve fiber are essentially identical and that information about the intensity of light incident on the photoreceptor is coded in terms of the rate of discharge of impulses rather than the shape or amplitude of individual impulses. Here began the direct, quantitative, experimental investigation of information-processing in the visual system. The techniques used by Hartline and Graham also provided an indirect but proximate method for studying the physical and chemical events in the photoreceptor that give rise to nerve impulses. In 1935, for example, the two re
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searchers used it to determine the spectral sensitivity of the Limulus photoreceptor. Later, with P. R. McDonald, they measured light and dark adaptation. Twenty-five years later Ruth Hubbard and George Wald confirmed the precision and reliability of Hartline's early spectral measurements by extracting the photopigment from the eye of Limulus to determine its spectral absorption by direct methods. The two curves agreed almost point for point. The Receptive Field In his early research at the Johnson Foundation and later at Johns Hopkins and Rockefeller, Hartline nearly always worked in collaboration with other investigators. In all of these collaborations, however, there was never a question in anyone's mind about who was the master and who the apprentice. Though unquestionably a brilliant collaborator, Hartline's extraordinary ability and unique talents produced the most startling results during the period of his thirties and forties when he worked alone. The single-handed investigations, mainly on the vertebrate retina, of those years are perhaps his most significant contribution to science. With his exquisite microdissection technique, Hartline was able to isolate single optic nerve fibers of the vertebrate retina and, for the first time, record their activity. He found that the response of the whole nerve resulted from the summated activity of fibers whose individual responses differed markedly. Some fibers discharged steadily in response to steady illumination, some in response to the onset and cessation of illumination, others only to its cessation. Many fibers showed extreme sensitivity to moving patterns of light and shade. Mapping the "receptive fields" of some of them in detail showed that a retinal ganglion cell can receive excitatory and inhibitory influences over many convergent pathways from many photoreceptors. The optic nerve fiber
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arising from the retinal ganglion cell is simply the final common pathway. Hartline found that the processing of visual information begins in the retina with the specialized activity of diverse types of ganglion cells, thereby laying the foundation for modern concepts of parallel processing by specialized channels. "The study of these retinal neurons has emphasized the necessity for considering patterns of activity in the nervous system," he remarked in his 1942 Harvey Lecture. "Individual nerve cells never act independently; it is the integrated action of all the units of the visual system that gives rise to vision" (1942,1). The Generator Potential As early as 1935 Hartline, using external electrodes, had recorded the local "action current" of a single photoreceptor unit in the compound eye of Limulus. Simultaneous records of the propagated impulses in the optic nerve suggested that this retinal action potential might be the generator of the impulses. When micropipette electrodes with tips small enough to penetrate cells were developed, opening the generator potential to direct study, Hartline's earlier interest in this hypothesis was rekindled. Using the new micropipettes, Hartline, Wagner, and MacNichol recorded intracellular generator potentials for the first time and were able to study the photoreceptor as a biological transducer—relating nerve impulses to a generator potential, and generator potential to the light incident on the photoreceptor. MacNichol, Wagner, and Hartline further observed that the rate of discharge of impulses was approximately linear with depolarization of the cell— whether induced by light or by current passed through the electrode—and that spontaneous activity was suppressed by hyperpolarizing current.
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Hartline's colleague, Tsuneo Tomita, soon demonstrated that the depolarization resulted from an increase in membrane conductance shortcircuiting the resting potential of the cell. The way was now open to a proper biophysical understanding of the generation of impulses by sensory receptor cells. The Hartline-Ratliff Equations One of Hartline's most important contributions to the physiology of vision was his discovery of lateral inhibition in the retina of the compound eye of Limulus. It is uncertain when the discovery of this "lateral effect" (as it was first called) was actually made, although—according to Hartline's best recollection— it was the late 1930s. The first published report (1949,1) on this pattern of central excitation and surround inhibition was long delayed, but even so it predated the discovery of the analogous center-surround organization of the vertebrate retina. In our first studies carried out at Rockefeller, Hartline and I focused on a quantitative account of the inhibitory interactions in the eye of Limulus. We were able—with a pair of simultaneous equations—to express the reciprocal interactions between two photoreceptor units in the steady state. Although these equations were strongly nonlinear overall, they were, as Hartline put it, "mercifully, piece-wise linear, to a good approximation." These so-called "Hartline-Ratliff equations"—actually based upon, and testable by, direct electrophysiological measurements—provided the first mathematical description of the integrative activity of a real neural network. Our subsequent discovery of the phenomenon of "inhibition of inhibition" enabled us to extend the mathematical description to any number of interacting units. This inhibition of inhibition—or disinhibition as we preferred (following
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Pavlov) to call it—confirmed the notion we had already expressed in our pair of simultaneous equations describing the interaction of two elements: the interaction was both mutual and recurrent. With this knowledge, Hartline and I could now express the interactions among not just two units, but any number n— either with a set of n simultaneous equations, or, if the number was large enough, with integral equations. The phenomenon of disinhibition first thought to be unique to the Limulus retina has since turned out to be a general principle of neural organization, widespread in the other species and neural systems. Our earliest studies of the dynamics of lateral inhibition—with William H. Miller and G. David Lange—were purely empirical, but quantitative, theoretical approaches to the dynamics of neural mechanisms were in the air. Attracted by the symmetry of responses of the Limulus eye to equal increments and decrements, Bruce W. Knight, Jr., a physicist and applied mathematician, joined the Laboratory in 1961. Knight realized that the Limulus eye appeared to be a "time-invariant linear system" that could be treated as a system of linear transducers, and that the several transductions could all be characterized by transfer functions. The transduction from light to generator potential, generator potential to impulses, and impulses to self-and lateral-inhibitory potentials were directly measured and characterized as transfer functions, enabling the Laboratory to make successful theoretical predictions of responses to a wide variety of stimuli. These experiments—performed mainly in collaboration with Bruce Knight, Jun-ichi Toyoda, and Fred Dodge—showed the appropriateness of treating the Limulus eye as a system of linear transducers over a wide range of experimental conditions. But Hartline remained wary. "The trouble with theories," he once said, "is that after a while one begins to believe them."
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A SENSE OF HUMOR Hartline's wry humor often produced unexpected and telling remarks. Capping a discussion of a new laboratory building on campus much criticized by the scientists who had to use it, he said drily that "it must have been designed by an architect." He was also given to telling tall tales with a straight face, many of which were taken for truth. His often repeated assertion that he was "awarded the M.D. on the condition that he never practice medicine," for instance, was widely believed. But Hartline's humor was a two-way street, and he often quoted my own description of his untidy laboratory as "a slightly disorganized, but extremely fertile, chaos." HONORS AND AWARDS While still in medical school Hartline received the William H. Howell Award in Physiology. Experimental and physiological psychologists were among the first to recognize the importance of his later work to an understanding of human visual perception, and the Society of Experimental Psychologists awarded him the Howard Crosby Warren Medal in 1948. That same year saw his election to the National Academy of Sciences. He was elected to the American Philosophical Society in 1962, received Case Institute of Technology's Albert A. Michelson Award in 1964, became a foreign member of the Royal Society in 1966, and, in 1969, received the Lighthouse Award for Distinguished Service. In 1967 the Nobel Prize in Physiology or Medicine was awarded jointly to Ragnar Granit (Karolinska Institute), Haldan Keffer Hartline (The Rockefeller University), and George Wald (Harvard University) "for their discoveries concerning the primary physiological and chemical visual processes in the eye." Ironically, the Nobel Prize for Hartline's contributions to
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vision research coincided with a decline in his direct participation in such research. Slowly failing eyesight, a result of senile macular degeneration, made it increasingly difficult for Hartline to read and write, to use a microscope, and to perform the highly skilled manual techniques for which he was noted. "The loss of central vision is bad enough in itself," he once remarked, "but to be prematurely labeled senile only adds insult to injury." HOME AND FAMILY In 1936 Hartline married Elizabeth Kraus, daughter of the eminent chemist C. A. Kraus, and, at that time, instructor in comparative psychology at Bryn Mawr College. Mrs. Hartline shared her husband's interest in nature and later became a dedicated conservationist. Their three sons Daniel Keffer, Peter Haldan, and Frederick Flanders—tutored by their father as he had been by his— all became biologists. When Hartline accepted a position at Johns Hopkins in 1949, the family purchased a house near Hydes, Maryland, about twenty miles from Baltimore. This country house, which they called Turtlewood, is still the family home. In 1953, Hartline became a member of the Rockefeller Institute and moved to an apartment in New York City. Leaving Mrs. Hartline and their three sons in Maryland, Hartline returned home for long weekends and holidays, viewing the New York apartment as little more than a "winter camp" in the city. The family's "summer camp" was the Kraus family place on Old Point, just across Frenchman Bay, northwest of Bar Harbor, Maine. CONCLUDING REMARKS Hartline enjoyed good health throughout most of his life and, despite his slight stature and rather frail appearance, was an active outdoorsman. When young he enjoyed moun
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tain climbing and had some first ascents to his credit in the Wyoming Rockies. He piloted his own open-cockpit plane around the country. He enjoyed sailing— with Bronk near their summer home in Maine and, on occasion, with Ragnar Granit in the Baltic. Continuing his outdoor activities even into old age, Hartline decided in his seventies to take a long-postponed rafting trip through the Grand Canyon. His cardiologist recommended against the trip, but Hartline decided that it was now or never, basing his decision (according to one of his apocryphal stories) on a favorable second opinion from a dermatologist. In any event, he and Mrs. Hartline took the trip and—except for being too cold and wet on the raft in the rapids and too hot and dry on the desert shore—both enjoyed it immensely. In his late seventies Hartline's chest pains became more frequent and severe, and on March 18, 1983—as he was entering his eightieth year—he died of a heart attack at the Fallston General Hospital in Maryland. Keffer Hartline achieved great distinction in every phase of his halfcentury of research on the physiology of vision and was awarded the highest of all honors in science. Yet he remained modest and unassuming throughout and was somewhat embarrassed by fame and public acclaim. He specifically requested that there be no official memorial service or organized tribute to him at The Rockefeller University, suggesting rather that one of the University concerts—which he had enjoyed so much over so many years—would be an appropriate memorial, bringing joy to others rather than sorrow. On March 7, 1984, the Stuttgart Chamber Orchestra, with Karl Münchinger conducting, played to a full house in a performance dedicated to Keffer Hartline's memory. Keffer Hartline and I worked together day after day, year after year, for more than a quarter of a century. The strong
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bond of friendship between us transcended all time and place, and all human frailty. To such a friend, the truest tribute is one enshrined in memory and thought, unspoken. Information about Hartline's life and work during the period of 1903-1950 came from his own reminiscences dictated during the last years of his life and transcribed by his long-time secretary, Maria Lipski. The period of 1950-1983 is based primarily on my own records and firsthand knowledge. For other accounts, see: John E. Dowling and Floyd Ratliff, "Nobel Prize, Three Named for Medicine, Physiology Award," Science, 158(1976):468-73; Ragnar Granit and Floyd Ratliff, "Haldan Keffer Hartline, 1903-1983," Biographical Memoirs of Fellows of the Royal Society, 31( 1985):262-92; and Floyd Ratliff, "Haldan Keffer Hartline (1903-1983)," Year Book 1984 (Philadelphia: American Philosophical Society), pp. 111-120.
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Selected Bibliography
1923 Influences of light of very low intensity on phototropic reactions of animals. J. Gen. Physiol., 6:137-52. 1925 The electrical response to illumination of the eye in intact animals, including the human subject; and in decerebrate preparations. Am. J. Physiol., 73:600-612. 1928 A quantitative and descriptive study of the electric response to illumination of the arthropod eye. Am. J. Physiol., 83:466-83. 1930 The dark adaptation of the eye of Limulus, as manifested by its electric response to illumination. J. Gen. Physiol., 13:379-89. With C. H. Graham. Nerve impulses from single receptors in the eye. J. Cell. Comp. Physiol., 1:277-95. 1934 Intensity and duration in the excitation of single photoreceptor units. J. Cell. Comp. Physiol., 5:229-47. With C. H. Graham. The response of single visual sense cells to lights of different wave lengths . J. Gen. Physiol., 18:917-31. 1938 The discharge of impulses in the optic nerve of Pecten in response to illumination of the eye. J. Cell. Comp. Physiol., 11:465-78. The response to single optic nerve fibers of the vertebrate eye to illumination of the retina. Am. J. Physiol., 121:400-15. 1940 The receptive fields of optic nerve fibers. Am. J. Physiol., 130:690-99. The effects of spatial summation in the retina on the excitation of the fibers of the optic nerve. Am. J. Physiol., 130:700-11.
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The nerve messages in the fibers of the visual pathway. J. Opt. Soc. Am., 30:229-47. 1941-1942 The neural mechanisms of vision. Harvey Lect., 37:39-68. 1947 With P. R. McDonald. Light and dark adaptation of single photoreceptor elements in the eye of Limulus. J. Cell. Comp. Physiol., 30:225-54. 1949 Inhibition of activity of visual receptors by illuminating nearby retinal areas in the Limulus eye. Fed. Proc., 8:69. With H. G. Wagner and E. F. MacNichol. The peripheral origin of nervous activity in the visual system . Cold Spring Harbor Symp. Quant. Biol., 17:125-41. 1954 With F. Ratliff. Spatial summation of inhibitory influences in the eye of Limulus (Abstr.). Science, 120:781. 1956 With H. G. Wagner and F. Ratliff. Inhibition in the eye of the Limulus . J. Gen. Physiol., 39:651-73. 1957 With F. Ratliff. Inhibitory interaction of receptor units in the eye of Limulus. J. Gen. Physiol., 40:357-76. 1958 With F. Ratliff. Spatial summation of inhibitory influences in the eye of Limulus, and the mutual interaction of receptor units. J. Gen. Physiol., 41:1049-66. With F. Ratliff and W. H. Miller. Neural interaction in the eye and the integration of receptor activity. Ann. N.Y. Acad. Sci., 74:210-22.
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1959 With F. Ratliff. The responses of Limulus optic nerve fibers to patterns of illumination on the receptor mosaic. J. Gen. Physiol., 42:1241-55. 1961 With F. Ratliff and W. H. Miller. Inhibitory interaction in the retina and its significance in vision. In: Nervous Inhibition, ed. E. Florey, New York: Pergamon Press, pp. 141-84. 1963 With F. Ratliff and W. H. Miller. Spatial and temporal aspects of retinal inhibitory interaction. J. Opt. Soc. Amer., 53:110-20. 1966 With D. Lange and F. Ratliff. Inhibitory interaction in the retina: Techniques of experimental and theoretical analysis. Ann. N.Y. Acad. Sci., 128:955-71. With F. Ratliff and D. Lange. The dynamics of lateral inhibition in the compound eye of Limulus I. In: Proceedings of an International Symposium on The Functional Organization of the Compound Eye, ed. C. G. Bernhard, Oxford and New York: Pergamon Press, pp. 399-424. With D. Lange and F. Ratliff. The dynamics of lateral inhibition in the compound eye of Limulus II. In: Proceedings of an International Symposium on The Functional Organization of the Compound Eye, ed. C. G. Bernhard, Oxford and New York: Pergamon Press, pp. 425-49. 1967 With F. Ratliff, B. W. Knight, Jr., and J. Toyoda. Enhancement of flicker by lateral inhibition. Science, 158:392-93. 1968 With F. Ratliff and D. Lange. Variability of interspike intervals in optic nerve fibers of Limulus: Effect of light and dark adaptation. Proc. Natl. Acad. Sci. USA, 60:464-69.
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1969 Visual receptors and retinal interaction. In: Les Prix Nobel en 1967 , The Nobel Foundation, Stockholm, pp. 242-59. Also in: Rockefeller Univ. Rev., 5(no. 5):9-11, and Science, 164:270-78. 1972 With F. Ratliff. Inhibitory interaction in the retina of Limulus. In: Handbook of Sensory Physiology, VII/2, Heidelberg: Springer-Verlag, pp. 381-447. 1973 With N. Graham and F. Ratliff. Facilitation of inhibition in the compound lateral eye of Limulus. Proc. Natl. Acad. Sci. USA, 70:894-98. 1974 With F. Ratliff, B. W. Knight, Jr., and F. A. Dodge, Jr. Fourier analysis of dynamics of excitation and inhibition in the eye of Limulus : Amplitude, phase, and distance. Vision Res., 14:1155-68. Studies on Excitation and Inhibition in the Retina—A Collection of Papers from the Laboratories of H. K. Hartline, ed. F. Ratliff, New York: The Rockefeller University Press, 668 pp.
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Mark Kac August 16, 1914-October 25, 1984
By H. P. Mckean Poland. Mark Kac was born ''to the sound of the guns of August on the 16th day of that month, 1914," in the town of Krzemieniec—then in Russia, later in Poland, now in the Soviet Ukraine (1985,1, p. 6). In this connection Kac liked to quote Hugo Steinhaus, who, when asked if he had crossed the border replied, "No, but the border crossed me." In the early days of the century Krzemieniec was a predominantly Jewish town surrounded by a Polish society generally hostile to Jews. Kac's mother's family had been merchants in the town for three centuries or more. His father was a highly educated person of Galician background, a teacher by profession, holding degrees in philosophy from Leipzig, and in history and philosophy from Moscow. As a boy Kac was educated at home and at the Lycee of Krzemieniec, a well-known Polish school of the day. At home he studied geometry with his father and discovered a new derivation of Cardano's formula for the solution of the cubic—a first bite of the mathematical bug that cost Kac père five Polish zlotys in prize money. At school, he obtained a splendid general education in science, literature, and history. He was grateful to his early teachers to the end of his life. In 1931 when he was seventeen, he entered the John
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Casimir University of Lwów, where he obtained the degrees M. Phil. in 1935 and Ph.D. in 1937. This was a period of awakening in Polish science. Marian Smoluchowski had spurred a new interest in physics, and mathematics was developing rapidly: in Warsaw, under Waclaw Sierpinski, and in Lwów, under Hugo Steinhaus. In his autobiography (1985,1, p. 29), Kac called this renaissance "wonderful." Most wonderful for him was the chance to study with Steinhaus, a mathematician of perfect taste, wide culture, and wit; his adored teacher who became his true friend and introduced him to the then undigested subject of probability. Kac would devote most of his scientific life to this field and to its cousin, statistical mechanics, beginning with a series of papers prepared jointly with Steinhaus on statistical independence (1936,1-4; and 1937,1-2). Kac's student days saw Hitler's rise and consolidation of power, and he began to think of quitting Poland. In 1938 the opportunity presented itself in the form of a Polish fellowship to Johns Hopkins in Baltimore. Kac was twentyfour. He left behind his whole family, most of whom perished in Krzemieniec in the mass executions of 1942-43. Years later he returned, not to Krzemieniec but to nearby Kiev. I remember him rapt, sniffing about him and saying he had not smelled such autumn air since he was a boy. On this trip he met with a surviving female cousin who asked him, at parting, "Would you like to know how it was in Krzemieniec?" then added, "No. It is better if you don't know" (1985,1, p. 106). These cruel memories and their attendant regrets surely stood behind Kac's devotion to the plight of Soviet refusniks and others in like distress. His own life adds poignancy to his selection of the following quote from his father's hero, Solomon Maimon: "In search of truth I left my people, my country and my family. It is not therefore to be assumed that I shall forsake the truth for any lesser motives" (1985,1, p. 9).
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AMERICA Kac came to Baltimore in 1938 and wrote of his reaction to his new-found land: "I find it difficult . . . to convey the feeling of decompression, of freedom, of being caught in the sweep of unimagined and unimaginable grandeur. It was life on a different scale with more of everything—more air to breathe, more things to see, more people to know. The friendliness and warmth from all sides, the ease and naturalness of social contacts. The contrast to Poland . . . defied description." (1985,1, p. 85)
After spending 1938-39 in Baltimore, Kac moved to Ithaca, where he would remain until 1961. Cornell was at that time a fine place for probability: Kai-Lai Chung, Feller, Hunt, and occasionally the peripatetic Paul Erdös formed, with Kac, a talented and productive group. His mathematics bloomed there. He also courted and married Katherine Mayberry, shortly finding himself the father of a family. So began, as he said, the healing of the past. From 1943 to 1947 Kac was associated off and on with the Radiation Lab at MIT, where he met and began to collaborate with George Uhlenbeck. This was an important event for him. It reawakened his interest in statistical mechanics and was a decisive factor in his moving to be with Uhlenbeck at The Rockefeller University in 1962. There Detlev Bronk, with his inimitable enthusiasm, was trying to build up a small, top-flight school. While this ideal was not fully realized either then or afterwards, it afforded Kac the opportunity to immerse himself in the statistical mechanics of phase transitions in the company of Ted Berlin and Uhlenbeck, among others. Retiring in 1981, Kac moved to the University of Southern California, where he stayed until his death on October 25, 1984, at the age of seventy. He is survived by his wife Kitty,
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his son Michael, his daughter Deborah, and his grandchildren. MATHEMATICAL WORK Independence and the Normal Law In the beginning was the notion of statistical independence to which Steinhaus introduced Kac. The basic idea is that the probability of the joint occurrence of independent events should be the product of their individual probabilities, as in 1/2 × 1/2 = 1/4 for a run of two heads in the tossing of an honest coin. The most famous consequence of this type of independence is the fact that, if #(n) is the number of heads in n tosses of such an honest coin, then the normal law of errors holds:
in which P signifies the probability of the event indicated between the brackets, the subtracted n/2 is the mean of #(n), and the approximation to the right-handed integral improves indefinitely as n gets large. The fact goes back to A. de Moivre (1667-1754) and was extended to a vague but much more inclusive statement by Gauss and Laplace. It was put on a better technical footing by P. L. Cebysev (1821-1890) and A. A. Markov (1856-1922), but as Poincaré complained, "Tout le monde y croit (la loi des erreurs) parce que les mathématiciens s'imaginent que c'est un fait d'observation et les observateurs que c'est un théorème de mathématiques." The missing ingredient, supplied by Steinhaus, was an unambiguous concept of independence. But that was only the start. All his life Kac delighted in extending the sway of the normal law over new and unforseen domains. I mention two instances:
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Let ω1, . . . , ωn be n(2) independent frequencies, meaning that no integral combination of them vanishes. Then
for large T and n, in which measure signifies the sum of the lengths of the several subintervals on which the indicated inequality takes place. In short, sinusoids of independent frequencies behave as if they were statistically independent, though strictly speaking they are not (1937,2; 1943,2). On another occasion, Kac looked to a vastly different domain: Let d(n) be the number of distinct prime divisors of the whole number n = 1,2,3, .... Then for large N,
in which # denotes the number of integers having the property indicated in the brackets and lg2n is the iterated logarithm lg (lgn). In short, there is some kind of statistical independence in number theory, too. Kac made this beautiful discovery jointly with P. Erdös (1940,4). These and other examples of statistical independence are explained in Kac's delightful Carus Monograph, Independence in Probability, Analysis and Number Theory (1951,1). Brownian Motion and Integration in Function Space The Brownian motion, typified by the incessant movement of dust motes in a beam of sunlight, was first discussed from a physical standpoint by M. Smoluchowski and A. Einstein (1905). N. Wiener later put the discussion on a solid
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mathematical footing. Kac was introduced to both developments during his association with MIT from 1943 to 1947. Now the statistical law of the Brownian motion is normal: if x(t) is the displacement of the Brownian traveler in some fixed direction, then
The fact is that Brownian motion is nothing but an approximation to honest coin-tossing:
in which T is the whole number nearest to tN and N is large. The normal law for coin-tossing cited before is the simplest version of this approximation. Kac, with the help of Uhlenbeck, perceived the general principal at work, of which the following is a pretty instance: Let p(n) be the number of times that heads outnumber tails in n tosses of an honest coin. Then the arcsine law holds:
the right-hand side being precisely the probability that the Brownian path x (t): 0 ≤ t ≤ 1, starting at x(0) = 0, spends a total time, T ≤ c, to the right of the origin (1947,2). Kac's next application of Brownian motion was suggested in a quantummechanical form by R. Feynman. It has to do with the so-called elementary solution e(t,x,y) of the Schrödinger equation:
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MARK KAC
The formula states that, with the left-hand imaginary unit
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removed :
in which the final factor is the free elementary solution (for V = 0) and Exy is the Brownian mean taken over the class of paths starting at x(0) = x and ending at x(t) = y. This is not really as explicit as it looks, as the mean is not readily expressible in closed form for any but the simplest cases, but it does exhibit just how e depends upon V in a transparent way. It can be used very effectively, as Kac illustrated by a beautiful derivation of the WKB approximation of classical quantum mechanics (1946,3). I will describe one more application of Brownian motion contained in Kac's Chauvenet Prize paper, Can One Hear the Shape of a Drum? (1976,1). The story goes back to H. Weyl's proof of a conjecture of H. A. Lorentz. Let D be a plane region bounded by one or more nice curves, holes being permitted, and let ω1, ω2, etc., be its fundamental tones, i.e., let etc., be the eigenvalues of Laplace's operator acting upon smooth functions that vanish at the boundary of D. Then Lorentz conjectured and Weyl proved:
for large ω. Kac found a remarkably simple proof of this fact based upon the self-evident principle that the Brownian traveller, starting inside D, does not feel the boundary of D until it gets there. He also speculated as to whether you could deduce the shape of D (up to rigid motions) if you could "hear" all of its fundamental tones and showed that, indeed, you can
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hear the length of the boundary, and the number of holes if any. The full question is still open. Statistical Mechanics As noted before, Kac's interest in this subject had been reawakened by Uhlenbeck at MIT. A famous conundrum of the field was the superficial incompatibility of the (obvious) irreversibility of natural processes and the reversibility of the underlying molecular mechanics. Boltzmann struggled continually with the problem, best epitomized by Uhlenbeck's teachers, P. and T. Ehrenfest, in what they called the "dogflea" model. Kac's debut in statistical mechanics was to provide its complete solution, put forth in his second Chauvenet Prize paper (1947,4). Next, Kac took up Boltzmann's equation describing the development, in time, of the distribution of velocities in a dilute gas of like molecules subject to streaming and to collisions (in pairs). I think this work was not wholly successful, but it did prompt Kac to produce a stimulating study of Boltzmann's idea of "molecular chaos" (Stosszahlansatz) and a typically elegant, Kac-type "caricature" of the Boltzmann equation itself. I pass on to the eminently successful papers on phase transitions. The basic question which Maxwell and Gibbs answered in principle is this: How does steam know it should be water if the pressure is high or the temperature is low, and how does that come out of the molecular model? There are as many variants of the question as there are substances. A famous one is the Ising model of a ferromagnet, brilliantly solved by L. Onsager in the two-dimensional case. Kac and J. C. Ward found a different and much simpler derivation (1952,2). The related "spherical model" invented by Kac was solved by T. Berlin (1952,1). But to my mind, Kac's most inspiring work in this line is
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contained in the three papers written jointly with P. C. Hemmer and Uhlenbeck (1963,1-3), in which they related the phase transition of a one-dimensional model of a gas to the splitting of the lowest eigenvalue of an allied integral equation and derived, for that model, the (previously ad hoc) van der Waal's equation of state, Maxwell's rule of equal areas included—a real tour de force. PERSONAL APPRECIATION I am sure I speak for all of Kac's friends when I remember him for his wit, his personal kindness, and his scientific style. One summer when I was quite young and at loose ends, I went to MIT to study mathematics, not really knowing what that was. I had the luck to have as my instructor one M. Kac and was enchanted not only by the content of the lectures but by the person of the lecturer. I had never seen mathematics like that nor anybody who could impart such (to me) difficult material with so much charm. As I understood more fully later, his attitude toward the subject was in itself special. Kac was fond of Poincaré's distinction between God-given and man-made problems. He was particularly skillful at pruning away superfluous details from problems he considered to be of the first kind, leaving the question in its simplest interesting form. He mistrusted as insufficiently digested anything that required fancy technical machinery—to the extent that he would sometimes insist on clumsy but elementary methods. I used to kid him that he had made a career of noting with mock surprise that ex = 1 + x + x2/2 + etc. when the whole thing could have been done without expanding anything. But he did wonders with these sometimes awkward tools. Indeed, he loved computation (Desperazionsmatematik included) and was a prodigious, if secret, calculator all his life. I cannot close this section without a Kac story to illustrate
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his wit and kindness. Such stories are innumerable, but I reproduce here a favorite Kac himself recorded in his autobiography: "The candidate [at an oral examination] was not terribly good—in mathematics at least. After he had failed a couple of questions, I asked him a really simple one . . . to describe the behavior of the function l/z in the complex plane. 'The function is analytic, sir, except at z = 0, where it has a singularity,' he answered, and it was perfectly correct. 'What is the singularity called?' I continued. The student stopped in his tracks. 'Look at me,' I said. 'What am I?' His face lit up. 'A simple Pole, sir,' which was the correct answer." (1985,1, p. 126)
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HONORS, PRIZES, AND SERVICE 1950 Chauvenet Prize, Mathematical Association of America 1959 American Academy of Sciences 1963 Lorentz Visiting Professor, Leiden 1965 National Academy of Sciences 1965-1966 Vice President, American Mathematical Society 1966-1967 Chairman, Division of Mathematical Sciences, National Research Council 1968 Chauvenet Prize, Mathematical Association of America 1968 Nordita Visitor, Trondheim 1969 Visiting Fellow, Brasenose College, Oxford 1969 American Philosophical Society 1969 Royal Norwegian Academy of Sciences 1971 Solvay Lecturer, Brussels 1976 Alfred Jurzykowski Award 1978 G. D. Birkhoff Prize, American Mathematical Society 1980 Kramers Professor, Utrecht 1980 Fermi Lecturer, Scuola Normale, Pisa
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Selected Bibliography
1934 A trigonometrical series. J. London Math. Soc., 9:116-18. 1935 Une remarque sur les series trigonométriques. Stud. Math., 5:99-102. 1936 Une remarque sur les équations fonctionnelles. Comment. Math Helv., 9:170-71. Sur les fonctions indépendantes I. Stud. Math., 6:46-58. Quelques remarques sur les fonctions indépendantes. Comptes Rendus Acad. Sci. (Paris), 202:1963-65. With H. Steinhaus. Sur les fonctions indépendantes II. Stud. Math., 6:89-97. With H. Steinhaus. Sur les fonctions indépendantes III. Stud. Math., 89-97. 1937 With H. Steinhaus. Sur les fonctions indépendantes IV. Stud. Math., 7:1-15. Sur les fonctions indépendantes V. Stud. Math., 7:96-100. Une remarque sur les polynomes de M. S. Bernstein. Stud. Math., 7:49-51. On the stochastical independence of functions (Doctoral dissertation, in Polish). Wiadomosci Matematyczne, 44:83-112. 1938 Quelques remarques sur les zéros des intégrales de Fourier. J. London Math. Soc., 13:128-30. Sur les fonctions 2nt - [2nt] - 1/2. J. London Math. Soc., 13: 131-34. 1939 Note on power series with big gaps. Am. J. Math., 61:473-76. On a characterization of the normal distribution. Am. J. Math., 61:726-28.
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With E. R. van Kampen. Circular equidistribution and statistical independence. Am. J. Math., 61:677-82. With E. R. van Kampen and A. Wintner. On Buffon's needle problem and its generalizations. Am. J. Math., 61:672-76. With E. R. van Kampen and A. Wintner. On the distribution of the values of real almost periodic functions. Am. J. Math., 61:985-91. 1940 On a problem concerning probability and its connection with the theory of diffusion. Bull. Am. Math. Soc., 46:534-37. With P. Erdös, E. R. van Kampen, and A. Wintner. Ramanujan sums and almost periodic functions. Stud. Math., 9:43-53. Also in: Am. J. Math., 62:107-14. Almost periodicity and the representation of integers as sums of squares. Am. J. Math., 62:122-26. With P. Erdös. The Gaussian law of errors in the theory of additive number-theoretic functions. Am. J. Math., 62:738-42. 1941 With R. P. Agnew. Translated functions and statistical independence. Bull. Am. Math. Soc., 47:148-54. Convergence and divergence of non-harmonic gap series. Duke Math. J., 8:541-45. Note on the distribution of values of the arithmetic function d(m). Bull. Am. Math. Soc., 47:815-17. Two number theoretic remarks. Rev. Ciencias, 43:177-82. 1942 Note on the partial sums of the exponential series. Rev. Univ. Nac. Tucumn Ser. A., 3:151-53. 1943 On the average number of real roots of a random algebraic equation. Bull. Am. Math. Soc., 49:314-20. On the distribution of values of trigonometric sums with linearly independent frequencies. Am. J. Math., 65:609-15. Convergence of certain gap series. Ann. Math., 44(2):411-15.
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1944 With Henry Hurwitz, Jr. Statistical analysis of certain types of random functions. Ann. Math. Stat., 15:173-81. 1945 Random walk in the presence of absorbing barriers. Ann. Math. Stat., 16:62-67. With R. P. Boas, Jr. Inequalities for Fourier transforms of positive functions. Duke Math. J., 12:189-206. A remark on independence of linear and quadratic forms involving independent Gaussian variables. Ann. Math. Stat., 16:400-1. 1946 On the distribution of values of sums of the type Σf(2kt). Ann. Math., 47(2):33-49. With P. Erdös. On certain limit theorems of the theory of probability. Bull. Am. Math. Soc., 52:292-302. On the average of a certain Weiner functional and a related limit theorem in calculus of probability. Trans. Am. Math. Soc., 59:401-14. 1947 With A. J. F. Siegert. On the theory of random noise in radio receivers with square law detectors. J. Appl. Phys., 18:383-97. With P. Erdös. On the number of positive sums of independent random variables. Bull. Am. Math. Soc., 53:1011-20. On the notion of recurrence in discrete stochastic processes. Bull. Am. Math. Soc., 53:1002-10. Random walk and the theory of Brownian motion. Am. Math. Month., 54:369-91. With A. J. F. Siegert. An explicit representation of a stationary Gaussian process. Ann. Math. Stat., 18:438-42. 1948 With R. Salem and A. Zygmund. A gap theorem. Trans. Am. Math. Soc., 63:235-48. On the characteristic functions of the distributions of estimates of
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various deviations in samples from a normal population. Ann. Math. Stat., 19:257-61. 1949 On distributions of certain Wiener functionals. Trans. Am. Math. Soc., 65:1-13. On deviations between theoretical and empirical distributions. Proc. Natl. Acad. Sci., 35:252-57. On the average number of real roots of a random algebraic equation (II). Proc. London Math. Soc., 50:390-408. Probability methods in some problems of analysis and number theory. Bull. Am. Math. Soc., 55:641-65. 1950 Distribution problems in the theory of random noise. Proc. Symp. Appl. Math., 2:87-88. With H. Pollard. The distribution of the maximum of partial sums of independent random variables . Canad. J. Math., 2:375-84. 1951 Independence in Probability, Analysis, and Number Theory. New York: John Wiley and Sons. With K. L. Chung. Remarks on fluctuations of sums of independent random variables. Mem. Am. Math. Soc., no. 6. (See also: corrections in Proc. Am. Math. Soc., 4:560-63.) On some connections between probability theory and differential and integral equations. Proc. 2d Berkeley Symp. Math. Stat. Prob., ed. J. Neyman, Berkeley: University of California Press, pp. 189-215. On a theorem of Zygmund. Proc. Camb. Philos. Soc., 47:475-76. With M. D. Donsker. A sampling method for determining the lowest eigenvalue and the principal eigenfunction of Schrodinger's equation. J. Res. Nat. Bur. Standards, 44:551-57. 1952 With T. H. Berlin. The spherical model of a ferromagnet. Phys. Rev., 86(2):821-35. With J. C. Ward. A combinatorial solution of the two-dimensional Ising model. Phys. Rev., 88:1332-37.
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1953 An application of probability theory to the study of Laplace's equation. Ann. Soc. Math. Pol., 25:122-30. With W. L. Murdock and G. Szego. On the eigenvalues of certain Hermitian forms. J. Ration. Mech. Anal., 2:767-800. 1954 Signal and noise problems. Am. Math. Month., 61:23-26. Toeplitz matrices, translation kernels and a related problem in probability theory. Duke Math. J., 21:501-9. 1955 Foundations of kinetic theory. Proc. 3d Berkeley Symp. Math. Stat. Prob., ed. J. Neyman, Berkeley: University of California Press, pp. 171-97. A remark on the preceding paper by A. Rényi, Acad. Serbes des Sci., Belgrade: Extrait Publ. de l'Inst. Math., 8:163-65. With J. Kiefer and J. Wolfowitz. On tests of normality and other tests of goodness of fit based on distance methods. Ann. Math. Stat., 26:189-211. Distribution of eigenvalues of certain integral operators. Mich. Math. J., 3:141-48. 1956 Some remarks on the use of probability in classical statistical mechanics. Bull. Acad. Roy. Belg. Cl. Sci., 42(5):356-61. Some Stochastic Problems in Physics and Mathematics: Collected Lectures in Pure and Applied Science, no 2. (Hectographed). Magnolia Petroleum Co. 1957 With D. A. Darling. On occupation times for Markoff processes. Trans. Am. Math. Soc., 84:444-58. A class of limit theorems. Trans. Am. Math. Soc., 84:459-71. Probability in classical physics. In: Proc. Symp. Appl. Math., ed. L. A. MacColl, New York: McGraw Hill Book Co., vol. 7., pp. 73-85. Uniform distribution on a sphere. Bull. Acad. Pol. Sci., 5:485-86. With R. Salem. On a series of cosecants. Indag. Math., 19:265-67.
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Some remarks on stable processes. Publ. Inst. Statist. Univ. Paris, 6:303-6. 1958 With H. Kesten. On rapidly mixing transformations and an application to continued fractions. Bull. Am. Math. Soc., 64:283-87. (See also correction, 65:67.) 1959 Remark on recurrence times. Phys. Rev., 115(2):1. Some remarks on stable processes with independent increments. In: Probability and Statistics: The Harald Cramér Volume, ed. Ulf Grenander, Stockholm: Almqvist & Wiksell, pp. 130-38; and New York: John Wiley & Sons. On the partition function of a one-dimensional gas. Phys. Fluids, 2:8-12. With D. Slepian. Large excursions of Gaussian processes. Ann. Math. Stat., 30:1215-28. Probability and Related Topics in Physical Sciences. New York: Interscience. Statistical Independence in Probability, Analysis, and Number Theory . New York: John Wiley & Sons. 1960 Some remarks on oscillators driven by a random force. IRE Trans. Circuit Theory. August, pp. 476-79. 1962 A note on learning signal detection. IRE Trans. Prof. Group Inform. Theory, 8:127-28. With P. E. Boudreau and J. S. Griffin. An elementary queueing problem. Am. Math. Month., 69:713-24. Probability theory: Its role and its impact. SIAM Rev., 4:1-11. Statistical mechanics of some one-dimensional systems. In: Studies in Mathematical Analysis and Related Topics , ed. G. Szego, Stanford: Stanford University Press, pp. 165-69. 1963 Probability theory as a mathematical discipline and as a tool in engineering and science. In: Proceedings of the 1st Symposium on
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Engineering, eds. J. L. Bogdanoff and F. Kozin, New York: John Wiley & Sons, pp. 31-68. With E. Helfand. Study of several lattice systems with long-range forces. J. Math. Phys., 4:1078-88. With G. E. Uhlenbeck and P. C. Hemmer. On the van der Waals theory of the vapor-liquid equilibrium. I. Discussion of a one-dimensional model. J. Math. Phys., 4: 216-28. With G. E. Uhlenbeck and P. C. Hemmer. On the van der Waals theory of the vapor-liquid equilibrium. II. Discussion of the distribution functions. J. Math. Phys., 4:229-47. 1964 With G. E. Uhlenbeck and P. C. Hemmer. On the van der Waals theory of the vapor-liquid equilibrium. III. Discussion of the critical region. J. Math. Phys., 5:60-74. Probability. Sci. Am., 211:92-106. The work of T. H. Berlin in statistical mechanics: A personal reminiscence. Phys. Today, 17:40-42. Some combinatorial aspects of the theory of Toeplitz matrices. Proceedings of the IBM Scientific Computing Symposium on Combinatorial Problems, 12:199-208. 1965 With G. W. Ford and P. Mazur. Statistical mechanics of assemblies of coupled oscillators. J. Math. Phys., 6:504-15. A remark on Wiener's Tauberian theorem. Proc. Am. Math. Soc., 16:1155-57. 1966 Can liberal arts colleges abandon science? Proceedings of the 52nd Annual Meeting of the Association of American Colleges. Bull. Assoc. Am. Coll., 52:41-49. Wiener and integration in function spaces. Bull. Am. Math. Soc., 72:52-68. Can one hear the shape of a drum? Am. Math. Month., 73:1-23. Mathematical mechanisms of phase transitions. In: 1966 Brandeis Summer Inst. Theor. Phys., ed. M. Chrétien, E. P. Gross, and S. Deser, New York: Gordon and Breach. 1:243-305. With C. J. Thompson. On the mathematical mechanism of phase transition. Proc. Natl. Acad. Sci., 55:676-83. (See also correction in 56:1625.)
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1967 The physical background of Langevin's equation. In: Stochastic Differential Equations. Lecture Series in Differential Equations, session 7, vol. 2, ed. A. K. Ariz, Van Nostrand Math. Studies, 19:147-66. New York: Van Nostrand. 1968 With S. Ulam. Mathematics and Logic: Retrospect and Prospects. New York: Frederick A. Praeger. On certain Toeplitz-like matrices and their relation to the problem of lattice vibrations. Arkiv for det Fysiske Seminar i. Trondheim, 11:1-22. 1969 With Z. Cielieski. Some analytic aspects of probabilistic potential theory. Zastosowania Mat., 10:75-83. Asymptotic behavior of a class of determinants. Enseignement Math., 15(2):177-83. With C. J. Thompson. Critical behavior of several lattice models with long-range interaction. J. Math. Phys., 10:1373-86. Some mathematical models in science. Science, 166:695-99. With C. J. Thompson. One-dimensional gas in gravity. Norske Videnskabers Selskabs Forhandl., 42:63-73. With C. J. Thompson. Phase transition and eigenvalue degeneracy of a one-dimensional anharmonic oscillator. Stud. Appl. Math., 48:257-64. 1970 Aspects probabilistes de la théorie du potential: Séminaire de mathématiques supérieures, Été 1968. Montréal: Les Presses de L'Université de Montréal. On some probabilistic aspects of classical analysis. Am. Math. Month., 77:586-97. 1971 The role of models in understanding phase transitions. In: Critical Phenomena in Alloys, Magnets and Superconductors, ed. R. E. Mills, New York: McGraw-Hill, pp. 23-39. With C. J. Thompson. Spherical model and the infinite spin dimensionality limit. Phys. Norveg., 5:163-68.
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1972 On applying mathematics: Reflections and examples. Q. Appl. Math., 30:17-29. With S. J. Putterman and G. E. Uhlenbeck. Possible origin of the quantized vortices in He, II. Phys. Rev. Lett., 29:546-49. William Feller, In memoriam. In: Proceedings of the 6th Berkeley Symposium, University of California, June/July 1970, ed. J. Neyman, Berkeley: University of California Press, vol. 2, pp. xxi-xxiii. 1973 With J. M. Luttinger. A formula for the pressure in statistical mechanics. J. Math. Phys., 14:583-85. With K. M. Case. A discrete version of the inverse scattering problem. J. Math. Phys., 14:594-603. Lectures on mathematical aspects of phase transitions: NATO Summer School in Math. Phys., Istanbul, 1970. NATO Adv. Stud. Inst. Ser. C, 1970, pp. 51-79. With D. J. Gates and I. Gerst. Non-Markovian diffusion in idealized Lorentz gases. Arch. Rational Mech. Anal., 51:106-35. Some probabilistic aspects of the Boltzmann equation. Acta Phys. Austriaca, suppl. 10:379-400. With J. M. Luttinger. Bose-Einstein condensation in the presence of impurities. J. Math. Phys., 14:1626-28. Phase transitions. In: The Physicists Conception of Nature, ed. J. Mehra, Dordrecht, Netherlands: Reidel Pub., pp. 514-26. 1974 Will computers replace humans? In: The Greatest Adventure, ed. E. H. Kone and H. J. Jordan, New York: Rockefeller University Press, pp. 193-206. With J. M. Luttinger. Bose-Einstein condensation in the presence of impurities II. J. Math. Phys., 15:183-86. Hugo Steinhaus-A reminiscence and a tribute. Am. Math. Month., 81:572-81. With P. van Moerbeke. On some isospectral second order differential operators. Proc. Natl. Acad. Sci. USA, 71:2350-51. Probabilistic methods in some problems of scattering theory. Rocky Mountain J. Math., 4:511-37.
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A stochastic model related to the telegrapher's equation. Rocky Mountain J. Math., 4:497-509. The emergence of statistical thought in exact sciences. In: The Heritage of Copernicus, ed. J. Neyman, Cambridge: MIT Press, pp. 433-44. Quelques problèmes mathématiques en physique statistique. Collection de la chaire Aisenstadt1. Montréal: Les Presses de l'Université de Montréal. 1975 With P. van Moerbeke. On some periodic Toda lattices. Proc. Natl. Acad. Sci. USA, 72:1627-29. With P. van Moerbeke. On an explicitly soluble system of nonlinear differential equations related to certain Toda lattices. Adv. Math., 16:160-69. An example of ''counting without counting" (to honor C. J. Bouwkamp). Philips Res. Rep., 30:20-22. With P. van Moerbeke. A complete solution of the periodic Toda problem. Proc. Natl. Acad. Sci. USA, 72:2879-80. The social responsibility of the academic community. Franklin Inst., 300:225-27. Some reflections of a mathematician on the nature and the role of statistics. Suppl. Advances Appl. Probab., 7:5-11. With W. W. Barrett. On some modified spherical models. Proc. Natl. Acad. Sci. USA, 72:4723-24. Henri Lebesgue et l'école mathématique Polonaise: Aperçus et souvenirs. Enseignement Math., 21:111-14. With P. van Moerbeke. Some probabilistic aspects of scattering theory. In: Functional Integration and Its Applications, eds. A. M. Arthur's and M. R. Bhagavan, Oxford: Oxford University Press, pp. 87-96. With J. M. Luttinger. Scattering length and capacity. Ann. Inst. Fourier Univ. Grenoble, 25:317-21. 1976 With J. Logan. Fluctuations and the Boltzmann equation, I. Phys. Rev. A., 13:458-70. 1985 Enigmas of Chance. (Autobiography). New York: Harper and Row.
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Aldo Starker Leopold October 22, 1913-August 23, 1983
By Robert A. Mccabe When a creative, innovative, talented, and intelligent colleague dies, we mourn his loss and honor his accomplishments in print, and doing so honor him no less than did the ancient Egyptians who carved pictures of their noble dead on the walls of tombs. Such a colleague was A. Starker Leopold, who died of a heart attack in his home in Berkeley, California, on August 23, 1983. A. Starker Leopold was born in Burlington, Iowa, on October 22, 1913, the oldest son of Aldo Leopold and Estella Bergere Leopold. Both his father and grandfather were outdoorsmen in the tradition of the early Midwest, and Starker in his turn was schooled in natural history and imbued with a sense of responsibility for the wild and free. While he was still a young boy, the family moved to Madison, Wisconsin, where Starker grew up. In 1936 he graduated from the University of Wisconsin with a B.S. degree in agriculture and went on to Yale, then to the University of California at Berkeley for graduate study. In 1944 he received his Ph.D. from Berkeley, where the eminent ornithologist Alden H. Miller guided his zoological studies. His doctoral thesis, The Nature of Heritable Wildness in Turkeys, was perhaps the first attempt to address the subject of wildness in birds.
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"The objectives of the study have been to determine insofar as possible the fundamental, heritable differences between wild and domestic turkeys and to compare the ecological relationships and general productivity of existing turkey populations which differ in degree of 'wildness.' The problem is of practical importance in wild turkey management because the intermixing of the domestic strain with wild populations has had certain adverse effects upon the hardiness of the native turkeys of Missouri. It is of theoretical importance in offering an opportunity better to understand the nature of wildness in a locally adapted, indigenous race of birds." (1945,1, p.133)
Leopold's results were commensurate with these stated objectives, and his paper, with its insights into the biology and behavior of turkeys, stands as a major contribution to the understanding of avian wildness. Though Starker Leopold functioned well as a lone scientist dealing with an ecological problem, he was also an excellent team worker. He listened to and understood the opinions of others, appreciated skills he himself did not possess, and was tolerant of the shortcomings of his associates. In 1952 he teamed with an ecologist who had few (if any) shortcomings: F. Fraser (Frank) Darling, then of the University of Edinburgh. The two undertook an ecological reconnaissance of Alaska to assess the current and potential impact of economic growth and technology on the natural resources of that territory, with particular reference to big game. Together they spent four months traveling, observing, and conducting interviews sponsored by the New York Zoological Society and the Conservation Foundation. Their efforts resulted in a clear, concise book unencumbered by jargon: "At the outset we stated that ideally a program of conservation and of land use should be devised before a new country is developed. Unfortunately the motive for conservation usually is impending shortage, which leads us to trim the resource boat after it is half full of water. But in Alaska, despite some buffeting about, the land resources are still largely intact, and what is more, they are still in government rather than private hands. The prob
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lem of planning and executing the best possible development of the Territory is therefore squarely up to the government. " . . . [if] mechanical and administrative difficulties can be overcome, we visualize an unusual opportunity for application of the principles of conservation to a fascinating and magnificent stretch of country." (1953,7, pp. 114-115)
It is difficult to evaluate the impact of that report on a state that has had more reports on its welfare and its resources than any other, but what could be said was perhaps best stated by Fairfield Osborn: "We could not have been more fortunate in the selection of the reconnaissance team for this study. Two eminent naturalists, one from the Old World and one from the New, have pooled their knowledge and experience to produce this report. On behalf of the two sponsoring organizations, it is a deep pleasure to commend and thank Dr. A. Starker Leopold and Dr. F. Fraser Darling for their accomplishment." (1953,7, Foreword)
Realizing the plight of our natural resources, S. Udall sought to achieve adequate stewardship of the land through science and education. He called on Starker Leopold to chair the Department of Interior Advisory Board on Wildlife Management.1 Leopold's Board first addressed the problem of wildlife management in the national parks, examining goals, policies, and methods of national wildlife management: "The goal of managing the national parks and monuments should be to preserve, or where necessary to recreate, the ecological scene as viewed by the first European visitors. As part of this scene, native species of wild animals should be present in maximum variety and reasonable abundance. Protection alone, which has been the core of Park Service wildlife policy, is not adequate to achieve this goal. Habitat manipulation is helpful and often essential to restore or maintain animal numbers. Likewise, populations of the animals themselves must sometimes be regulated to prevent habitat damage; this is especially true of ungulates." (1963,1, p. 43)
1 Stewart L. Udall, The Quiet Crisis (New York: Holt, Rinehart and Winston, 1953), p. 209.
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Ungulate excess within the National Parks became a core issue, exciting the hunting public, but the Committee concluded that: "Direct removal by killing is the most economical and effective way of regulating ungulates within a park. Game removal by shooting should be conducted under the complete jurisdiction of qualified park personnel and solely for the purpose of reducing animals to preserve park values. Recreational hunting is an inappropriate and nonconforming use of the national parks and monuments." (1963,1, p. 43)
This forthright position in the face of opposition was a cornerstone in National Park programs for wildlife management. The Advisory Board then investigated unnecessary destruction of animals by the Branch of Predator and Rodent Control of the United States Fish and Wildlife Service ". . . augmented by state, county, and individual endeavor," and recommended: ". . . a complete reassessment of the goals, policies, and field operations of the Branch of Predator and Rodent Control with a view to limiting the killing program strictly to cases of proven need, as determined by rigidly prescribed criteria." (1964,1, p. 47)
The Board's report was—and still is—the most penetrating assessment of United States government control of animals, and it put the responsibility for correcting the unwarranted destruction of animals on the Fish and Wildlife Service. Its appearance was followed by a series of rebuttals and explanations in defense of existing programs, but changes also resulted. Finally, the Board Leopold chaired evaluated the National Wildlife Refuge System to "appraise the significance of the national refuges in migratory bird conservation, with emphasis on waterfowl." Their report recommended the establishment of eleven more refuges, better financial support for
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existing refuges, and detailed long-range and multiple-use planning. Perhaps the most significant recommendation was that: "National wildlife refuges should be extensively used for research and teaching by qualified scientists and naturalists. In many localities refuges are the only land units devoted solely to wildlife preservation, and thus offer unique possibilities for continuous research and ecologic education." (1968,4, p. 52)
The Advisory Board's evaluations of wildlife management—or, as they are universally known, the "Leopold Reports"—are outstanding for their concision and depth of understanding. Though not everything they recommended came to fruition, the reports themselves are benchmarks in national conservation. Written with Riney, McCain, and Tevis, Leopold's ecological evaluation of the California jawbone deer herd (1951,2) was another significant contribution to the assessment of our natural resources. Though now nearly forty years old, both the data and narrative portions of this bulletin could serve as patterns for modern big game investigations. In 1961 Leopold produced a book on the desert for TIMELIFE'S Life Nature Library series (1961,1), a testimony to his intellectual versatility. In keeping with the format of that series he traced the work of wind and water as well as the ecology of men and animals living in the arid environments of the world. His chapters six, "Life Patterns in Arid Lands," and seven, "Man Against Desert," are particularly enlightening. But Starker Leopold's magnum opus was his survey, Wildlife of Mexico: The Game Birds and Mammals (1959,3). A skilled and astute field scientist, he began fieldwork for this impressive work in 1944 and ended it only with the book's publication in 1959. He followed up an initial two years in the field with
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a variety of short trips, and in the summer of 1948 I accompanied him on one of these expeditions. Little escaped Starker's attention, as he recorded all facets of the ecology and natural history of his fifty-one camp study sites extending from the northern Sonorán border to the Yucatán. His fluent Spanish helped him in getting both official sanction from comisarios (officials) and guidance and information from landowners and campesinos (farmers). Well written, easy to understand, and vital to Latin American conservationists, Wildlife of Mexico won the Wildlife Society's 1959 publication of the year award. As one reviewer aptly put it: "This publication is not only indispensable to any serious student of Mexican game birds and mammals, but it is also a guide to all thinking Mexican citizens who are interested in managing a valuable resource through wise use. It sets a pattern that other Latin American countries might well strive to emulate."2
In order that it could be used in Latin America, Leopold's book was translated into Spanish in 1965 by Luis Macias Arellano and Ambrosio Gonzales Cortes. It is a landmark publication for conservation in Mexico and Latin America. In 1979, Leopold again won the Wildlife Society's publication award for his book on the California quail (1977,1). One of the finest monographs on single species in the field of wildlife ecology, it contains not only insights into the ecology and life history of the species but also exemplary suggestions for the management of western quails. On his last hardcover book, Leopold collaborated with Gutierrez and Bronson to provide information on the life histories of 135 game species of the United States, Canada, and northern Mexico. An encyclopedic assessment of species
2 William B. Davis, review of Wildlife in Mexico, Journal of Wildlife Management 24,4(1960):446.
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that are hunted or trapped, North American Game Birds and Mammals (1981,1) is a valuable and accessible source of information for wildlife students and administrators. Choosing the right hypothesis to test and the tool most likely to solve a problem is an art. Starker Leopold's investigative choices were inspired, and he applied himself untiringly to follow them through to make worthy contributions to science. Excelling as a field ecologist, he was not parochial and in the field often found time to collect and prepare museum specimens for colleagues interested in classification and evolution. Nor did he limit himself to any particular species or group, as his many and varied published papers amply testify. Though dedicated, he did not sacrifice everything to his science. Throughout his life he divided his time among work, family, and hobbies (particularly hunting and fishing) and managed to do justice to all. Starker was a quiet and dignified man who was always neat and well groomed. He was jovial and fun loving without being boisterous. He was at ease among friends, with strangers, or on a lecture platform. Polite and well mannered, he gave special consideration to others. He had friends in all walks of life—from a member of the President's cabinet to a Mexican farmer eking out a living on the mountain slopes of Hidalgo and a sheepherder in the Australian outback. He also came from a remarkable family, and both his brother, Luna, and his sister, Estella, were elected to membership in the Academy—a unique occurrence in the Academy's history. Although his father, Aldo Leopold, was a leader of considerable prominence in the field of wildlife ecology, Starker did not seek to trade on his father's name. Earning his own achievements and honors, he yet benefitted considerably from the education he received from his father,
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and both men held to the credo that "good land use is good wildlife management." Today we know that good land use is imperative for the salvation of civilization itself. Starker's wife, Elizabeth Weiskotten Leopold, and his children, Frederic S. and Sarah Leopold, survive him. Ecologists and wildlife scientists universally—and particularly his fellow members of the National Academy—honored Starker Leopold, the kind of scientist who enhances the credibility of science. We all share in the loss of this outstanding colleague.
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PROFESSIONAL AND PUBLIC SERVICE 1972-1975 Marine Mammal Commission, appointed by the President 1970 Board of Ecology Team Consultant for U.S. Plywood-Champion Papers, Inc. 1970 Consultant on Research Policy, Tanzania National Parks 1969-1970 Chairman, Committee to Appraise the Program of the Missouri Conservation Commission 1969 Advisory Committee, Lawrence Hall of Science 1968-1972 Chief Scientist and Chairman, Advisory Committee, National Park Service 1968 Knapp Professorship, University of Wisconsin 1967-1983 Board of Advisors, National Wildlife Federation 1965-1969 Consultant, California Water Quality Control Board 1964 President, Board of Governors, Cooper Ornithological Society 1964 Advisory Trustee, Alta Bates Hospital Association 1962-1968 Chairman, Wildlife Management Advisory Committee, appointed by Secretary of the Interior Stewart L. Udall 1960 President, Northern Division, Cooper Ornithological Society 1959-1966 President, California Academy of Sciences 1957-1958 President, Wildlife Society 1956-1983 Member of Science Council and Board of Trustees, California Academy of Sciences 1955-1960 Vice President and Member of the Board of Directors, Sierra Club 1955-1959 Editorial Board, Sierra Club Bulletin 1954-1957 Council Member, Wilderness Society 1954-1956 Board of Governors, Nature Conservancy 1948-1966 Editorial Board, Pacific Discovery
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HONORS AND DISTINCTIONS 1947 Guggenheim Fellow 1959 Fellow, American Ornithologists' Union 1959 Wildlife Society Publication Award 1964 Department of Interior Conservation Award 1965 Aldo Leopold Medal of the Wildlife Society 1966 Audubon Society Medal 1969 Honorary Member, the Wildlife Society 1970 Member, National Academy of Sciences 1970 California Academy of Sciences Fellows Medal 1974 Winchester Award for Outstanding Accomplishment in Professional Wildlife Management 1978 Berkeley Citation, University of California 1979 Wildlife Society Publication Award 1980 American Institute of Biological Sciences, Distinguished Service Award 1980 Occidental College, Honorary Doctoral Degree 1980 Edward W. Browning Award for Conserving the Environment, Smithsonian Institution and the New York Community Trust
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Selected Bibliography
1939 Age determination in quail. J. Wildl. Manage., 3:261-65. 1941 Woven wire and the wild turkey. Missouri Conserv., 3:5. Report on the management of the Caney Mountain Turkey Refuge. Jefferson City: Missouri Conserv. Commiss. (mimeographed report). 19 pp. 1943 Results of wild turkey management at Caney Mountain Refuge, 1940 to 1943. Jefferson City: Missouri Conserv. Commiss. (mimeographed report). 13 pp. With P. D. Dalke. The 1942 status of wild turkeys in Missouri. J. Forest., 41:428-35. The molts of young wild and domestic turkeys. Condor, 45:133-45. Conservation of game. Address to Symposium on Science in Conservation During War Times. Trans. Acad. Sci. St. Louis, 31:63-67. Autumn feeding and flocking habits of the mourning dove in southern Missouri. Wilson Bull., 55:151-54. 1944 Cooper's hawk observed catching a bat. Wilson Bull., 56:116. The nature of heritable wildness in turkeys. Condor, 46:133-97. With M. Leopoldo Hernandez. Los recursos biologicos de guerrero con referencia especial a los mamiferos y aves de caza. Anuario comisión impulsora y coordinadora de la investigación cientifica (año 1944) , Mexico, D. F., pp. 361-90. 1945 Sex and age ratios among bobwhite quail in southern Missouri. J. Wildl. Manage., 9:30-34. With E. R. Hall. Some mammals of Ozark County, Missouri. J. Mammal., 26:142-45.
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1946 Clark's Nutcracker in Nuevo León, Mexico. Condor, 48:278. 1947 With David L. Spencer and Paul D. Dalke. The ecology and management of the wild turkey in Missouri. Tech. Bull. 1 (1946). Jefferson City: Conservation Commission, Federal Aid to Wildlife Program, State of Missouri, pp. 1-86. Status of Mexican big-game herds. Trans. 12th N. Am. Wildl. Conf., Washington, D.C.: Wildl. Mngmt. Inst., pp. 437-48. 1948 The threat to our western ranges. Pac. Discovery, 1:28-29. With William Longhurst. Deer damage in the Capay Valley. Report to the California Fish and Game Commission (mimeographed). 4 pp. Clear Water. Pac. Discovery, 1:21-23. Reviews of William H. Carr, Desert Parade: A Guide to South-Western Desert Plants and Wildlife; and E. F. Adolph et al., Physiology of Man in the Desert. Living Wilderness, 26:21-22. The wild turkeys of Mexico. Trans. 13th N. Am. Wildl. Conf., Washington, D.C.: Wildl. Mngmt. Inst., pp. 393-400. With Randal McCain and William M. Longhurst. Preliminary report on the problems of deer management in California. Rep. to the Calif. Fish and Game Commission (mimeographed). 16 PP. Of time and survival. Pac. Discovery, 1:28-29. 1949 Adiós, Gavilán. Pac. Discovery, 2:4-13. Review of Trippensee, Wildlife Management of Upland Game and General Principles. Calif. Fish Game, 35:205-6. 1950 The pheasant kill on the Conaway Ranch-1947-48, Univ. of Calif. Berkeley Mus. Vert. Zool. (mimeographed), 14 pp. Reviews of Henry E. Davis, The American Wild Turkey; and Robert J. Wheeler, The Wild Turkey in Alabama. Bird-Banding, 21:83-84.
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Deer in relation to plant succession. Trans. 15th N. Am. Wildl. Conf., Washington, D.C.: Wildl. Mngmt. Inst., pp. 571-80. Vegetation zones in Mexico. J. Ecol. Soc. Am., 31:507-18. 1951 Review of Richard H. Pough, Audubon Water Bird Guide. Pac. Discovery, 4:32. With T. Riney, R. McCain, and L. Tevis, Jr. The jawbone deer herd. California Division of Fish and Game, Dept. of Natl. Res. and Mus. Vert. Zool., Univ. of Calif., Berkeley. Game Bull. 4, 139 PP. Game Birds and Mammals of California: A Laboratory Syllabus. Berkeley: California Book Co., 125 pp. Review of Helmut K. Buechner, Life History, Ecology, and Range Use of the Pronghorn Antelope in Trans-Pecos, Texas. J. Wildl. Manage., 15:322-23. With R. A. McCabe. Breeding season of the Sonora white-tailed deer. J. Wildl. Manage., 15:433-34. Review of Ira N. Gabrielson, Wildlife Management. J. Wildl. Manage., 15:422-23. 1952 With W. M. Longhurst and R. F. Dasmann. A Survey of California Deer Herds, Their Ranges and Management Problems. State of California, Division of Fish and Game. Game Bull. 6, 136 pp. Ecological aspects of deer production on forest lands. In: Proc. 1949 U. N. Sci. Conf. Conserv. and Utiliza. Resour. U. N. Dept. Economic Affairs, Wildlife and Fish Resources, 7:205-7. With F. F. Darling. What's happening in Alaska. Anim. Kingdom, 55:170-74. 1953 With R. H. Smith. Numbers and winter distribution of Pacific black brant in North America. Calif. Fish and Game, 29:95-101. Intestinal morphology of gallinaceous birds in relation to food habits. J. Wildl. Manage., 17:197-203. Zonas de vegetación en Mexico. Bol. Soc. Mex. Geog. Estadist., 78:55-74. Report of the Committee on Research Needs. J. Wildl. Manage., 17:361-65.
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With F. F. Darling. Effects of land use on moose and caribou in Alaska. Trans. 18th N. Am. Wildl. Conf., Washington, D.C.: Wildl. Mngmt. Inst., pp. 553-62. Too many deer. Sierra Club Bull., 38:51-57. With F. F. Darling. Wildlife in Alaska: An Ecological Reconnaissance . New York: Ronald Press Co. 129 pp. What does conservation mean today? Pac. Discovery, 7:1-2. 1954 Review of Durward L. Allen, Our Wildlife Legacy. Sat. Rev., 23:55-56. Can we keep our outdoor areas? Audubon, 56:148-51, 179. Review of William F. Schulz, Jr., Conservation Law and Administration . Pac. Discovery, 7:29. The predator in wildlife management. Sierra Club. Bull., 39:34-38. Dichotomous forking in the antlers of white-tailed deer. J. Mammal., 35:599-600. Natural resources-whose responsibility? Trans. 19th N. Am. Wildl. Conf., pp. 589-98. Preserving the qualitative aspects of hunting and fishing. Conserv. News, 19:1-5. 1955 The conservation of wildlife. In: A Century of Progress in the Natural Sciences, San Francisco: California Academy of Sciences. Centennial volume, pp. 795-806. 1956 Foreword. In: Arctic Wilderness, by Robert Marshall. Berkeley: University of California Press. 171 pp. 1957 Public and private game management-we need both. Calif. Farmer, 206:12-13. With R. A. McCabe. Natural history of the Montezuma quail in Mexico. Condor, 59:3-26. Deer management or deer politics? Cent. Calif. Sportsman, 17:24-26. Arctic spring. Sierra Club Bull., 42:17-18.
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Wilderness and culture. Sierra Club Bull., 42:33-37. 1958 Review, ed. W. L. Thomas, Jr., Man's Role in Changing the Face of the Earth. Calif. Vector Views, 5:48-49. Situación del oso plateado en Chihuahua. Rev. Soc. Mex. Hist. Nat., 19:1-4. 1959 The range of the jaguar in Mexico. Excavation at La Venta Tabasco. Appendix 5, pp. 290-91. Big game management. Survey of Fish and Game Problems in Nevada, Bull. 36, pp. 85-99. Wildlife of Mexico: The Game Birds and Mammals. Berkeley: University of California Press. 568 pp. 1960 Save our remaining wilderness. Pac. Discovery, 13:1-2. Lois Crisler, chasseur d'images en Alaska. Flammes, 95:10-12. Biogeography. In: McGraw-Hill Encyclopedia of Science and Technology. New York: McGrawHill Book Co., pp. 204-7. 1961 The Desert. New York: TIME, Inc. 192 pp. 1963 With S. A. Cain, C. Cottam, I. N. Gabrielson, and T. L. Kimball. Wildlife management in the national parks. Report of the Advisory Board on Wildlife Management. Trans. 28th N. Am. Wildl. Nat. Resour. Conf., Washington, D.C.: Wildl. Mngmt. Inst., pp. 28-45. 1964 With S. A. Cain, C. M. Cottam, I. N. Gabrielson, and T. L. Kimball. Predator and rodent control in the United States. Trans. 29th N. Am. Wildl. Nat. Resour. Conf., Washington, D.C.: Wildl. Mngmt. Inst., pp. 27-49. Mexico and migratory waterfowl conservation. In: Waterfowl Tomorrow , ed. Joseph P. Linduska. (Translated from Spanish
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through the courtesy of the Mexican Embassy, Washington, D. C.) Washington, D.C.: U.S. Dept. Int., pp. 729-36. 1965 Harrier observed catching a fairy tern in Tahiti. Condor, 67:91. Wildlands in our civilization. In: Wilderness and Culture. San Francisco: Sierra Club Publ., pp. 81-85. Fauna Silvestre de Mexico. Mexico City: Ediciones del Instituto Mexicano de Recursos Naturales Renovables. 608 pp. 1966 Effects of Rampart Dam on wildlife resources. In: Rampart Dam and the Economic Development of Alaska, Ann Arbor: Univ. of Michigan School of Natural Resources, p. 12. With J. W. Leonard. Alaska Dam would be resources disaster. Audubon, 68:176-79. With J. W. Leonard. Effects of the proposed Rampart Dam on wildlife and fisheries (Alaska's economic Rampart). Trans. 31st N. Am. Wildl. Nat. Resour. Conf., Washington, D.C.: Wildl. Mngmt. Inst., pp. 454-59. Adaptability of animals to habitat change. In: Future Environments of North America, eds. F. F. Darling and J. P. Milton, Garden City, N.Y.: Natural History Press, pp. 65-75. 1967 With R. E. Jones. Nesting interference in a dense population of wood ducks . J. Wildl. Manage., 31:221-28. Quantitative and qualitative values in wildlife management. In: Natural Resources: Quality and Quantity, eds. S. V. Ciriacy-Wantrup and J. J. Parsons, Berkeley: University of California Press, pp. 127-36. Grizzlies of the Sierra del Nido. Pac. Discovery, 20:30-32. 1968 Electric power for Alaska-A problem in land-use planning. East Afr. Agric. For. J., 33:23-26. Ecologic objectives in park management. East Afr. Agric. For. J., 33:168-72. Optimum utilization of East African range resources. In: Report of a Symposium on East African Range Problems, eds. W. M. Long
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hurst and H. F. Heady, Villa Serbelloni, Lake Como, Italy. (Leopold Abstract, p. 81.) With C. C. Cottam, I. M. Cowan, I. N. Gabrielson, and T. L. Kimball. The National Wildlife Refuge System. Trans. 33rd N. Am. Wildl. Nat. Resour. Conf., Washington, D.C.: Wildl. Mngmt. Inst., pp. 30-54. The National Wildlife Refuge System. Natl. Wildl., 6:4-9. 1970 Weaning grizzly bears: A report on Ursus arctos horribilis. Nat. Hist., 79:94-101. With I. K. Fox and C. H. Callison. Missouri Conservation Program: An appraisal and some suggestions. Mo. Conserv., 31:3-31. With Herbert L. Mason, et al. The Scenic, Scientific and Educational Values of the Natural Landscape of California. Sacramento: California Department of Parks and Recreation. 36 pp. With T. O. Wolfe. Food habits of wedge-tailed eagles, Aquila audax , in south-eastern Australia. CSIRO Wildl. Res., 15:1-17. Research policy in the Tanzania National Parks. Arusha: Tanzania National Parks, 15 pp. What lies ahead in wildlife conservation. Ed. J. Yoakum, Trans. Calif.-N.W. Sect. Wildl. Soc., Fresno, Jan. 30-31, 1970, pp. 156-60. 1971 Editor's foreword. In: Environmental: Essays on the Planet as a Home , P. Shepard and D. McKinley. Boston: Houghton Mifflin Co. 308 PP. Introduction. In: The Environment, the Establishment, and the Law, H. Henkin, M. Merta, and J. Staples. Boston: Houghton Mifflin Co. 223 pp. Biogeography. In: McGraw-Hill Encyclopedia of Science and Technology , New York: McGrawHill Book Co., pp. 213-16. Sagehen Creek Field Station: The First Twenty Years. Berkeley: University of California Press. 27 pp. 1972 Symposium on predator control: Remarks by A. Starker Leopold. Trans. 37th N. Am. Wildl. Nat. Resour. Conf., Washington, D.C.: Wildl. Mngmt Inst., pp. 200-2.
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With S. A. Cain, J. A. Kadlec, D. L. Allen, R. A. Cooley, M. G. Hornocker, and F. H. Wagner. Predator Control-1971. Report of Advisory Committee on Predator Control to Secretary of Interior and Council on Environmental Quality. Ann Arbor: University of Michigan Institute for Environmental Quality. 207 pp. The essence of hunting. Nat. Wild., 10:38-40. With R. H. Barrett. Implications for Wildlife of the 1968 Juneau Unit Timber Sale. Berkeley: University of California Press, Department of Forestry and Conservation. 109 pp. 1973 The hunter's role in wildlife conservation. 4th Int. Big Game Hunters' and Fishermen's Conf., San Antonio, Texas, pp. 5-6. Reprinted in Penn. Game News, 45(4):16-21. 1974 Needed-A broader base for wildlife administration. Ed., J. Yoakum, Monterey: Trans. Calif.Nevada Sec., Wildl. Soc., pp. 90-95. Hunting versus protectionism-The current dilemma. Address to the 1974 National Wildlife Federation Annual Meeting in Denver, pp. 5-10. Reprinted in: Gun World, 14(6):50-53. 1975 Ecosystem Deterioration Under Multiple Use. Wild Trout Management Symposium, Yellowstone National Park. Denver, Colorado: Trout Unlimited. 103 pp. 1976 With M. Erwin, J. Oh, B. Browning. Phytoestrogens: Adverse effects on reproduction in California quail. Science, 191:98-100. 1977 The California Quail. Berkeley: University of California Press. 281 pp. Meditations in a duck blind. Gray's Sporting J., 2:6-10. 1978 Wildlife in a prodigal society. Trans. 43rd N. Am. Wildl. Nat. Resour. Conf., Washington, D.C.: Wildl. Mngmt. Inst., pp. 5-10.
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Wildlife and forest practice. In: Wildlife and America, ed. H. P. Brokaw , Washington, D.C.: Council on Environmental Quality. 532 pp. 1979 Search for an environmental ethic. Review of Robert Cahn, Footprints on the Planet. Sierra Club Bull., 64:58. 1981 With R. J. Gutierrez and M. T. Bronson. North American Game Birds and Mammals. New York: Charles Scribner's Sons.
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Manfred Martin Mayer June 15, 1916-September 18, 1984
By K. Frank Austen What is the legacy of a scientist? A pioneer in the field of immunochemistry, Manfred Mayer almost singlehandedly established the discipline of complement. He contributed the one-hit theory of immune hemolysis. He uncovered the first indications of the enzymatic cleavage of one complement protein by another, leading to our eventual understanding of the sequential interaction and function of the eighteen proteins of the complement system. He appreciated that cytolysis by complement is due to the insertion of hydrophobic complement peptides into the lipid bilayer of biomembranes and formation of transmembrane channels. Finally, on a different tack, he and Robert Nelson developed the Triponema pallidum immobilization test for syphilis. As a teacher and mentor, his impeccable methodology and the care he lavished on the members of his laboratory produced many distinguished intellectual descendants. Finally, Manfred Mayer will always remain the model of a life lived by the highest values, scientific and personal. EDUCATION AND EARLY LIFE Manfred was born in Frankfurt-am-Main, Germany, on June 15, 1916, and died in Baltimore, Maryland, on September 18, 1984. He received his primary and secondary school
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ing in Germany but was forced to leave that country in 1933, at the age of seventeen, because of political events. He worked his way through the City College of New York, receiving a B.S. in 1938, then entered a doctoral program at Columbia University. His doctoral thesis was on the chemical and immunologic properties of phosphorylated serum albumin. He received the Ph.D. degree in 1946. From 1938 through 1942, Manfred supported himself working as a laboratory assistant to Dr. Michael Heidelberger—a founder of the discipline of immunochemistry—at Columbia University. His background in physical chemistry fit well with Heidelberger's organic chemical background and approach, and he was very comfortable in this laboratory that also contained Forest Kendall and had just trained Elvin Kabat. During his four years there, Manfred progressed from laboratory assistant to the role of distinguished graduate student. He worked on both the cross-reactions to Type III pneumococcal capsular polysaccharides and the fixation of the activity in immune complex reactions known as ''complement." By 1946 Manfred was an accomplished immunochemist with two unique interests of his own that would occupy his subsequent scientific career: quantitative assessment of the complement system and its components, and the elucidation—in biochemical terms—of the reaction sequence. The same year that Manfred received his Ph.D., Thomas B. Turner, chairman of the Department of Bacteriology at the Johns Hopkins School of Medicine, asked Michael Heidelberger to recommend someone in immunology. Heidelberger praised Mayer highly, and he was offered the position of assistant professor. With his wife, Elinor, Manfred proceeded to Baltimore, and within two years his contributions as a teacher and investigator had earned him promotion to associate professor. In recognition of the quality of his scholarship and his balanced approach to departmental
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issues, he was chosen acting head of the Department (though not yet a full professor) when Thomas Turner left to become dean. He served throughout 1957, when Barry Wood arrived to take over the chairmanship, and was appointed full professor in 1960. SCIENTIFIC CONTRIBUTION Working with Elvin Kabat from 1942 to 1945, long before his arrival at Hopkins, Mayer had completed Experimental Immunochemistry (1948,1), though this most important volume did not appear in print until 1948. During that era, everyone in the field of immunochemistry had been instructed by Michael Heidelberger, either personally or through his distinguished disciples, Elvin Kabat and Manfred Mayer. The Heidelberger school had developed techniques for conducting quantitative precipitin reactions and agglutination determinations, and Kabat and Mayer decided it was critical for the future of research in the field to produce a textbook of quantitative immunochemistry that was both conceptual and practical in content. For a number of years, Elvin Kabat and Manfred Mayer met virtually every weekend in one another's apartments to read aloud and revise every word of the proposed text. Heidelberger also read it and ultimately prepared the introduction to the volume. These were difficult times for the wives of immunochemists, but Elinor supported Manfred throughout while at the same time proceeding with her own substantial interests. By 1945 the unique and historically critical volume was complete, only to be delayed three years by the publishers—allegedly because of a paper shortage. The authors, however, used the delay to revise the manuscript extensively and produced a volume that went through three printings without revision over the next ten years. In 1958 the authors began work on a second edition in which Mayer's contribution on
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the complement system was greatly expanded, largely due to the findings of his laboratory. This edition (1961,5) went through four printings before going out of print in 1984. During the late 1940s and early 1950s Manfred had begun to assemble an outstanding group committed to immunochemistry in general and to complement research in specific—with startling results. During those years Lawrence Levine demonstrated that the introduction of diisopropyl fluorophosphate (DFP) would block enzymatic activity in the first component of complement—a critical step in the recognition of the biochemical events in the complement cascade. Keith Cowan was studying how carbowax acted as a substitute for specific antibody in mediating the hemolytic action of complement. Al Marucci, with Manfred's guidance, had begun to evaluate the use of radiolabeled antibody as an analytic tool in defining immunochemical events. Finally, Herbert Rapp was analyzing the different functions of rabbit antibodies of different immunoglobulin classes and, with Manfred, was beginning to develop a mathematical basis for the analysis of the reaction sequence. Their work resulted in the conclusion that the "third component of complement" was not a single substance but, based upon its behavioral characteristics as defined in mathematical terms, represented multiple substances —a conclusion subsequently substantiated by the identification of five component proteins. Manfred's definition of the cofactor functions of calcium and magnesium made possible the singularly important demonstration that the functional interactions of the components of the complement system met the "one-hit" model of interactions. By preparing, with his students, specific intermediates, he broke the reaction down into sequential events and initially purified the components being analyzed. This "onehit" analysis permitted the measurement of complement components—or proteins—in molecular terms with a level
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of sensitivity that enabled the researchers, working with both guinea pig and human sources, to isolate each individual protein. Effective molecule titration proved useful again some years later when the alternative complement pathway—or properdin system—was rediscovered as a non-antibody-dependent mechanism for recruiting the terminal capabilities of the complement system. On this occasion, the method's sensitivity and specificity enabled the researchers to isolate and characterize the activating proteins rapidly. The work of Mayer's laboratory on effective molecule titration of the components of the complement system also led to the initial recognition that certain of the components had multiple biologically-active sites. In the case of the second complement component, these studies showed, the binding site to the fourth component was clearly distinguished from the catalytic site, resulting in the cleavage activation of the third component. Mayer later turned his attention to the mechanism by which the sequentially reacting proteins (at one time termed "C3") produced "holes" in the membrane of a target cell destined to undergo lysis. He established that lysis was caused by a pentamolecular complex of the terminal five components, C5-9, which formed a transmembrane channel identified (in earlier studies by English electron microscopists) as discontinuities with an elevated border. In addition to his unique contributions to the understanding of the sequential interaction and function of the eighteen proteins of the complement system, Mayer and his colleague Robert Nelson developed the Triponema pallidum immobilization test for syphilis—an important contribution to clinical medicine capable of eliminating false-positive reactions. At that time the conventional test for syphilis yielded false-positive results in individuals with gamma globulin abnor
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malities, including those with autoimmune diseases who did not have the antibody specific to the spirochete. Dr. Mayer's contributions to the immunochemistry and biochemistry of the complement field were recognized in 1969 by an honorary degree in medical science from the Johannes Gutenberg University of Mainz, Germany; in 1974 by the Karl Landsteiner Award of the American Association of Blood Banks; in 1979 by election to the presidency of the American Association of Immunologists; and in 1982 by the Gairdner Foundation International Award. In 1953 he shared with Robert Nelson the Kimble Award for Methodology for the development of the T. pallidum immobilization test. He was elected to the National Academy of Sciences in 1979. TEACHER AND MENTOR Most teachers of science provide their students with basic skills and knowledge, but few can instill that additional ingredient: confidence to meet the challenges of independent research. Manfred Mayer was an inspiring scholar who —by example, instruction, and wisdom—made independent researchers of many of his students. Well aware that Mayer's own vision had uncovered the immunochemistry of complement (today a significant portion of the discipline of immunology), they used his laboratory as the reference point for all aspects of the field of complement research and the model for addressing—with technical resourcefulness and appropriate critical analysis—all difficult scientific questions. Dr. Mayer, politely but firmly, demanded technical mastery of all the relevant immunologic methodologies before he would trust a member of his laboratory to deal with critical research questions. Technical competence, he maintained, was the essential prerequisite for personal creativity. He examined each experiment with an open mind, exploring the
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established results and their implications. Several times a day he would go with colleagues to the blackboard to discuss which data were secure and which required more work. He frequently suggested an alternative hypothesis that required the development of a new methodology. If the new methodology took months but was the only way to obtain an answer, that was the direction the research took. Mayer's committed belief that correct methodology was the prerequisite for meaningful research meant that his laboratory's methodologic development was continually in flux. His science was state-of-the-art. After a piece of work had been completed, the researchers had the remarkable experience of putting their results down on paper for critique by other members of Mayer's laboratory. Manfred always treated the literature of his field with integrity, while discussing his own data with great imagination and insight. What more can be said of a giant who developed a whole scientific field not only in his personal research, but also through the training he so generously gave to others? His rocklike personal integrity became a part of his students' educational environment. Never forgetting his own early years as a refugee from Nazi oppression, he did all in his power for the displaced of any background. Truth—not politics—was his only goal, and in the search for truth he generously shared new hypotheses to be tested with every student, making sure that each had a part in the joy of discovery. His hypotheses further stimulated those about him, generating ever more definitive experiments. Not surprisingly, Mayer's laboratory produced a number of distinguished colleagues and students who carry on his own high standards in a variety of fields (immunochemistry, complement biology, cellular immunology), among them Teruko Ishizaka, Moon Shin, and Hyun Shin.
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Manfred was equally committed to the development of new knowledge and to the education of those of us who interacted with him. He had no sense of status or rank, and the friendships he formed with colleagues and students were lifelong and meaningful. He felt the opportunity for a life in research a rare privilege that obliged the researcher to strive for the highest possible level of technical competence, resourcefulness, integrity, and commitment, both to research and to education—and he transferred these values to his students. Manfred was conspicuously more concerned about the development of the discipline of immunology and of complement immunochemistry than about his own personal fame. Manfred's nonprofessional interests centered on his wife and four children. Born into a musical family, he maintained interests in music, languages, and art throughout his life. Both he and Elinor were accomplished amateur pianists, as well as collectors of art and archaeology. An admirer of beauty in art, music, and science, Manfred Mayer was a true role model of the scientist-teacher. He developed a major area of immunology and, with the aid of his concepts and technologies, prepared those individuals who now pursue it. He is sorely missed by everyone who trained with him and or was influenced by his work. He will be remembered always as a scientist, a teacher, and the founder of the discipline of complement immunochemistry.
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HONORS AND DISTINCTIONS Professional And Academic Positions 1938-1942 Laboratory Assistant in Immunochemistry, Columbia University 1942-1945 Member of the Scientific Staff, Project of the Office of Scientific Research and Development, Columbia University 1946 Instructor in Biochemistry, Columbia University 1946-1947 Assistant Professor of Bacteriology, Johns Hopkins University School of Hygiene and Public Health 1948 Associate Professor of Microbiology, Johns Hopkins University School of Hygiene and Public Health 1957 Acting Chairman, Department of Microbiology, Johns Hopkins University School of Hygiene and Public Health 1960 Professor of Microbiology, Department of Microbiology, Johns Hopkins University of Medicine Learned Societies American Association for the Advancement of Science American Association of Immunologists American Chemical Society American Society of Biological Chemists Biochemical Society National Academy of Sciences Society for Experimental Biology and Medicine Honorary Memberships Phi Beta Kappa Sigma Xi Collegium Internationale Allergologicum Other Professional Activities Consultant, United States Public Health Service Consultant, National Science Foundation Consultant, Office of Naval Research Consultant, Plum Island Animal Disease Laboratory, Department of Agriculture Associate Editor, Biological Abstracts
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Associate Editor, Journal of Immunology Associate Editor, Analytical Biochemistry Associate Editor, Immunochemistry President, American Association of Immunologists Editorial Board, Journal of Immunology Prizes And Awards 1945 Citation, Columbia University, for work in the Division of War Research during World War II 1953 Kimble Award for Methodology 1957 Selman Waksman Lectureship Award 1969 Honorary Doctor of Medical Science, Johannes Gutenberg University, Mainz, Germany 1974 Karl Landsteiner Award, American Association of Blood Banks 1976 Albion O. Bernstein Award, Medical Society of the State of New York 1982 Gairdner Foundation International Award, Toronto, Canada
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Selected Bibliography
1940 With M. Heidelberger and H. P. Treffers. A quantitative theory of the precipitin reaction. VII. The egg albumin-antibody reaction in antisera from the rabbit and horse. J. Exp. Med., 71:271. 1941 With M. Heidelberger and M. Rocha e Silva. Quantitative chemical studies on complement or alexin. III. Uptake of complement nitrogen under varying experimental conditions. J. Exp. Med., 74:359. 1942 With M. Heidelberger and E. A. Kabat. A further study of the cross reaction between the specific polysaccharides of Types III and VIII pneumococci in horse antisera. J. Exp. Med., 75:35. With M. Heidelberger. Quantitative chemical studies on complement or alexin. IV. Addition of human complement to specific precipitates. J. Exp. Med., 75:285. With M. Heidelberger. Velocity of combination of antibody with specific polysaccharides of pneumococcus. J. Biol. Chem., 143:567. 1944 With D. H. Moore. Note on changes in horse serum albumin on aging. J. Biol. Chem., 156:777. With M. Heidelberger. Normal human stromata as antigens for complement fixation in the sera of patients with relapsing vivax malaria. Science, 100:359. 1945 With M. Heidelberger, O. G. Bier, and G. Leyton. Complement titrations in human sera. II. J. Mt. Sinai Hosp., 12:285. With O. G. Bier, G. Leyton, and M. Heidelberger. A comparison of human and guinea pig complements and their component fractions. J. Exp. Med., 81:449.
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1946 With M. Heidelberger. Physical, chemical and immunological properties of phosphorylated crystalline horse serum albumin. J. Am. Chem. Soc., 68:18. With M. Heidelberger and C. R. Demarest. Studies in human malaria. I. The preparation of vaccines and suspensions containing plasmodia. J. Immunol., 52:325. With B. B. Eaton and M. Heidelberger. Spectrophotometric standardization of complement for fixation tests. J. Immunol., 53:31. With M. Heidelberger and W. A. Coates. Studies in human malaria. II. Attempts to influence relapsing vivax malaria by treatment of patients with vaccine (Pl. vivax). J. Immunol., 53:101. With M. Heidelberger, A. A. Alving, B. Craige, Jr., R. Jones, Jr., T. N. Pullman, and M. Whorton. Studies in human malaria. IV. An attempt at vaccination of volunteers against mosquitoborne infection with Pl. vivax. J. Immunol., 53:113. With M. Heidelberger. Studies in human malaria. V. Complement fixation reactions. J. Immunol., 54:89. With A. G. Osler, O. G. Bier, and M. Heidelberger. The activating effect of magnesium and other cations on the hemolytic function of complement. J. Exp. Med., 185:535. 1947 With A. G. Osler, O. G. Bier, and M. Heidelberger. Quantitative studies of complement fixation. Proc. Soc. Exp. Biol. Med., 65:66. With H. N. Eisen, D. H. Moore, R. Tarr, and H. C. Stoerck. Failure of adrenal cortical activity to influence circulating antibodies and gamma globulin. Proc. Soc. Exp. Biol. Med., 65:301. 1948 With E. A. Kabat. Experimental Immunochemistry. Springfield, Ill.: C. C. Thomas. With A. G. Osler, O. G. Bier, and M. Heidelberger. Quantitative studies on complement fixation. I. A method. J. Immunol., 59:195. With A. G. Osler, O. G. Bier, and M. Heidelberger. Quantitative studies on complement fixation. II. Fixation of complement in
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the reaction between Type III pneumococcus specific polysaccharide and homologous antibody. J. Immunol., 60:205. With M. Heidelberger. Review on complement. Adv. Enzymol., 3:71. With C. C. Croft and M. Gray. Kinetic studies on immune hemolysis. J. Exp. Med., 88:427. 1949 With R. A. Nelson. Immobilization of T. pallidum in vitro by antibody produced in syphilis infection. J. Exp. Med., 89:369. 1950 With A. L. Wallace and A. G. Osler. Quantitative studies of complement fixation. V. Estimation of complement. Fixing potency of immune sera and its relation to antibody nitrogen content. J. Immunol., 65:661. 1951 With W. Bowman and H. J. Rapp. Kinetic studies on immune hemolysis. II. The reversibility of red cell-antibody combination and the resultant transfer of antibody from cell to cell during hemolysis. J. Exp. Med., 94:87. Immunochemistry. Annu. Rev. Biochem., 20:415. 1952 With A. G. Osler and J. H. Strauss. Diagnostic complement fixation. I. A method. Am. J. Syph. Gonorrhea Vener. Dis., 36:140. 1953 With L. Levine, K. M. Cowan, and A. G. Osler. Studies on the role of Ca++ and Mg++ in complement fixation and immune hemolysis. I. Uptake of complement nitrogen by specific precipitates and its inhibition by ethylene diamine tetraacetate. J. Immunol., 71:359. The mechanism of hemolysis by antibody and complement. Is immune hemolysis a single or multiple-hit process? Atta VI. Congr. Int. Microbio., Rome, September 6-12, 1953, vol. 2, pp. 151-57. With L. Levine, K. M. Cowan, and A. G. Osler. Studies on the role of Ca++ and Mg++ in complement fixation and immune he
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molysis. II. The essential role of calcium in complement fixation. J. Immunol., 71:367. With L. Levine and A. G. Osler. Studies on the role of Ca++ and Mg ++ in complement fixation and immune hemolysis. III. The respective roles of Ca++ and Mg++ in immune lysis. J. Immunol., 71:374. 1954 With L. Levine. Kinetic studies on immune hemolysis. III. Description of a terminal process which follows the Ca++ and Mg++ steps in the action of complement on sensitized erythrocytes. J. Immunol., 72:511. With L. Levine. Kinetic studies on immune hemolysis. IV. Rate determination of the Mg++ and terminal reaction steps . J. Immunol., 72:516. Studies on the nature of C'y and its hemolytic action. Baskerville Chem. J., (May): 12. With L. Levine. Kinetic studies on immune hemolysis. V. Formation of the complex EAC'X and its reactions with C'y. J. Immunol., 73:426. With L. Levine and H. J. Rapp. Kinetic studies on immune hemolysis. VI. Resolution of the C'y step into two successive processes involving C'2 and C'3. J. Immunol., 73:435. With L. Levine, H. J. Rapp, and A. A. Marucci. Kinetic studies on immune hemolysis. VII. Decay of EAC'1,4,1, fixation of C'3, and other factors influencing the hemolytic action of complement. J. Immunol., 73:443. 1955 With A. A. Marucci. Quantitative studies on the inhibition of crystalline urease by rabbit antiurease. Arch. Biochem. Biophys., 54:330. 1957 With H. J. Rapp, B. Roizman, S. W. Klein, K. M. Cowan, D. Lukens, et al. The purification of poliomyelitis virus as studied by complement fixation. J. Immunol., 78:435.
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1958 Studies on the mechanism of hemolysis by antibody and complement. Prog. Allergy, 5:215. With B. Roizman and W. Hopken. Immunochemical studies of poliovirus. II. Kinetics of the formation of infectious and noninfectious type I poliovirus in three cell strains of human derivation. J. Immunol., 80:386. With B. Roizman and H. J. Rapp. Immunochemical studies of poliovirus. III. Further studies on the immunological and physical properties of poliovirus particles produced in tissue culture. J. Immunol., 81:419. 1959 With B. Roizman and P. R. Roane, Jr. Immunochemical studies of poliovirus. IV. Alteration of the immunological specificity of purified poliovirus by heat and ultraviolet light. J. Immunol., 82:119. With L. G. Hoffmann, H. J. Rapp, and J. R. Vinas. A kinetic flow technique for study of immune hemolysis. Proc. Soc. Exp. Biol. Med., 100:211. 1961 Development of the one-hit theory of immune hemolysis. In: Immunochemical Approaches to Problems in Microbiology. New Brunswick, N. J.: Rutgers University Press. With T. Borsos and H. J. Rapp. Studies on the second component of complement. II. The reaction between EAC'1,4 and C'2: Evidence on the single site mechanism of immune hemolysis and determination of C'2 on an absolute molecular basis. J. Immunol., 87:310. With T. Borsos and H. J. Rapp. Studies on the second component of complement. II. The nature of the decay of EAC'1,4,2. J. Immunol., 87:326. On the destruction of erythrocytes and other cells by antibody and complement. Cancer Res., 21:1262. With E. A. Kabat. Experimental Immunochemistry, 2d ed. Springfield, Ill.: C. C. Thomas.
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1962 With T. Borsos. Mechanism of action of guinea pig complement. In: Mechanism of Cell and Tissue Damage Produced by Immune Reactions (Second International Symposium on Immunopathology, Brook Lodge, Michigan). Basel: Benno Schwabe & Co. 1963 Enzymatic cleavage of C'2 by EAC'la,4: Fixation of C'2a on the cell and release of C'2i. Science, 141:738. With H. J. Rapp and T. Borsos. Complement. National Cancer Institute Workshop, February 28March 1, 1963. Bethesda, Maryland. 1965 With W. F. Willoughby. Antibody-complement complexes. Science, 150:907. Mechanism of hemolysis by complement. In: CIBA Foundation Symposium on Complement, eds. G. E. W. Woltstenholme and J. Knight, London: J. & A. Churchill, Ltd., p. 4. With L. G. Hoffmann and A. T. McKenzie. The steady state system in immune hemolysis. Description and analysis; Application to the enumeration of SAC'4. Immunochemistry, 2:13. With J. A. Miller. Inhibition of guinea pig C'2 by rabbit antibody, quantitative measurement of inhibition, discrimination between immune inhibition and complement fixation, specificity of inhibition and demonstration of uptake of C'2 by EAC'la,4. Immunochemistry, 2:71. With R. M. Stroud and K. F. Austen. Catalysis of C'2 fixation by C'la. Reaction kinetics, competitive inhibition by TAMe, and transferase hypothesis of the enzymatic action of C' la on C'2, one of its natural substrates. Immunochemistry, 2:219. 1966 With G. Sitomer and R. M. Stroud. Reversible adsorption of C'2 by EAC'4: role of Mg++, enumeration of competent SAC'4, two-step nature of C'2a fixation and estimation of its efficiency. Immunochemistry, 3:57. With R. M. Stroud, J. A. Miller, and A. T. McKenzie. C'2ad, an inactive derivative of C'2 released during decay of EAC'4,2a. Immunochemistry, 3:163.
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1967 With H. S. Shin and J. A. Miller. Fragmentation of guinea pig complement components C'2 and C'3c. In: Protides of the Biological Fluids (15th Annual Colloquium), Amsterdam: Elsevier Publishing Co., p. 411. 1968 With H. S. Shin. The third component of the guinea pig complement system. I. Purification and characterization. Biochemistry, 7:2991. With H. S. Shin. The third component of the guinea pig complement system. II. Kinetic study of the reaction of EAC'4,2a with guinea pig C'3. Enzymatic nature of C'3 consumption, multiphasic character of fixation, and hemolytic titration of C'3. Biochemistry, 7:2997. With H. S. Shin. The third component of the guinea pig complement system. III. Effect of inhibitors. Biochemistry, 7:3003. With J. A. Miller. On the cleavage of C'2 by Cla: Immunological and physical comparisons of C'2ad and C'2a/i. Proc. Soc. Exp. Biol. Med., 129:127. With H. S. Shin, R. Snyderman, E. B. Friedman, and A. J. Mellors. A chemotactic and anaphylatoxic fragment cleaved from the fifth component of guinea pig complement. Science, 162:361. 1969 With D. J. Hingson and R. K. Massengill. The kinetics of release of 86rubidium and hemoglobin from erythrocytes damaged by antibody and complement. Immunochemistry, 6:295. 1970 With J. A. Miller. Photometric analysis of proteins and peptides at 191-194 m. Analyt. Biochem., 36:91. With F. A. Rommel, M. B. Goldlust, F. C. Bancroft, and A. H. Tashjian, Jr. Synthesis of the ninth component of complement by a clonal strain of rat hepatoma cells. J. Immunol., 105:396. With J. A. Miller and H. S. Shin. A specific method for purification of the second component of guinea pig complement and a chemical evaluation of the one-hit theory. J. Immunol., 105:327.
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Highlights of complement research during the past twenty-five years. Immunochemistry, 7:485. 1971 With C. T. Cook, H. S. Shin, and K. Laudenslayer. The fifth component of the guinea pig complement system. I. Purification and characterization. J. Immunol., 106:467. With H. S. Shin and R. J. Pickering. The fifth component of the guinea pig complement system. II. Reaction of C5 with EAC' 1,4,2,3. J. Immunol., 106:473. With H. S. Shin and R. J. Pickering. The fifth component of the guinea pig complement system. III. The properties of EAC1,4,2,3,5b. J. Immunol., 106:480. With M. B. Goldlust and H. S. Shin. Elution of guinea pig ''C5b/6" activity from EAC1,4,2a,3b,5b,6. J. Immunol (abstract)., 107:318. With R. L. Marcus and H. S. Shin. An alternate pathway: Demonstration of C3 cleaving activity, other than C4,2a, on endotoxic lipopolysaccharide after treatment with guinea pig serum. Relation to the properdin system. Proc. Natl. Acad. Sci. USA, 68:1351. With C. S. Henney. Specific cytolytic activity of lymphocytes: Effect of antibodies against complement components C2, C3, and C5. Cell. Immunol., 2:702. 1972 Mechanism of cytolysis by complement. Proc. Natl. Acad. Sci. USA, 69:2954. With M. K. Gately. The effect of antibodies to complement components C2, C3, and C5 on the production and action of lymphotoxin. J. Immunol., 109:728. With V. Brade, C. T. Cook, and H. S. Shin. Studies on the properdin system: Isolation of a heatlabile factor from guinea pig serum related to a human glycine-rich beta-glycoprotein (GBC or factor B). J. Immunol., 109:1174. 1973 With F. A. Rommel. Studies of guinea pig complement component C9: Reaction kinetics and evidence that lysis of EACI-8 results from a single membrane lesion caused by one molecule of C9. J. Immunol., 110:637.
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With A. Eden, C. Bianco, and V. Nussenzweig. C3 split products inhibit the binding of antigenantibody-complement complexes to B lymphocytes. J. Immunol., 110:1452. The complement system. Sci. Am., 229:54. With V. Brade, G. D. Lee, A. Nicholson, and H. S. Shin. The reaction of zymosan with the properdin system in normal and C4 deficient guinea pig serum: Demonstration of C3- and C5 cleaving and multi-unit enzymes, both containing factor B, and acceleration of their formation by the classical complement pathway. J. Immunol., 111:1389. 1974 With A. Nicholson, V. Brade, G. D. Lee, and H. S. Shin. Kinetic studies of the formation of the properdin system enzymes on zymosan. Evidence that nascent C3b controls the rate of assembly. J. Immunol., 112:1115. With M. K. Gately. The molecular dimensions of guinea pig lymphotoxin. J. Immunol., 112:168. With V. Brade, A. Nicholson, and G. D. Lee. The reaction of zymosan with the properdin system. Isolation of purified factor D from guinea pig serum and study of its reaction characteristics. J. Immunol., 112:1845. With M. B. Goldlust, H. S. Shin, and C. H. Hammer. Studies of complement complex C5b,6 eluted from EAC-6: Reaction of C5b,6 with EAC4b,3b and evidence on the role of C2a and C3b in the activation of C5. J. Immunol., 113:998. 1975 Complement. An immunological and pathological mediator system. Medizin. Prisma, (May):2. With C. L. Gately and M. K. Gately. The molecular dimensions of mitogenic factor from guinea pig lymph node cells. J. Immunol., 114:10. With C. H. Hammer and A. Nicholson. On the mechanism of cytolysis by complement. Evidence on insertion of the C5b and C7 subunits of the C5b,6,7 complex into the phospholipid bilayer of the erythrocyte membrane. Proc. Natl. Acad. Sci. USA, 72:5076. The complex complement system. Inflo, 8:1.
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1976 With C. L. Gately and M. K. Gately. Separation of lymphocyte mitogen from lymphotoxin and experiments on the production of lymphotoxin by lymphoid cells stimulated with the partially purified mitogen: A possible amplification mechanism of cellular immunity and allergy. J. Immunol., 116:669. With D. W. Michaels and A. S. Abramovitz. Increased ion permeability of planar lipid bilayer membranes after treatment with the C5b-9 cytolytic attack mechanism of complement. Proc. Natl. Acad. Sci. USA, 73:2852. With C. H. Hammer and A. S. Abramovitz. A new activity of complement component C3: Cellbound C3b potentiates lysis of erythrocytes by C5b,6 and terminal components. J. Immunol., 117:830. On the mechanism of cytolysis by complement: Experimental studies of the transmembrane channel hypothesis. In: The Nature and Significance of Complement Activation (An international symposium sponsored by Ortho Research Institute of Medical Science September, 1976, in Raritan, New Jersey). With M. K. Gately and C. S. Henney. Effect of anti-lymphotoxin on cell-mediated cytotoxicity. Evidence for two pathways, one involving lymphotoxin and the other requiring intimate contact between the plasma membranes of killer and target cells. Cell. Immunol., 27:82. 1977 The cytolytic attack mechanism of complement. In: Mediators of the Immediate Type Inflammatory Reaction. Mono. Allergy 12, Basel: S. Karger. With C. H. Hammer, M. L. Shin, and A. S. Abramovitz. On the mechanism of cell membrane damage by complement: Evidence on insertion on polypeptide chains from C8 and C9 into the lipid of erythrocytes . J. Immunol., 119:1. With M. L. Shin, W A. Paznekas, and A. S. Abramovitz. On the mechanism of cell membrane damage by complement: Exposure of hydrophobic sites on activated complement protein. J. Immunol., 119:1358. On the mechanism of cytolysis by lymphocytes: A comparison with
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complement. (Presidential address to the American Association of Immunologists, April, 1977.) J. Immunol., 119:1195. With H. J. Müller-Eberhard and L. G. Hoffmann. Complement. In: Methods in Immunology and Immunochemistry, eds. A. W. Curtis and M. W Chase, vol. 4, p. 127. With S. Cohen, J. David, M. Feldmann, P. R. Glade, J. J. Oppenheim, et al. Current state of studies of mediators of cellular immunity: A progress report. Cell. Immunol., 33:233. 1978 Complement, past and present. In: The Harvey Lect., 72, 1976-1977. New York: Academic Press. With M. Okamoto. Studies on the mechanism of action of guinea pig lymphotoxin. I. Membrane active substances prevent target cell lysis by lymphotoxin. J. Immunol., 120:272. With M. Okamoto. Studies on the mechanism of action of guinea pig lymphotoxin. II. Increase of calcium uptake rate in LT-damaged target cells. J. Immunol., 120:279. With M. L. Shin and W A. Paznekas. On the mechanism of membrane damage by complement: The effect of length and unsaturation of the acyl chains in liposomal bilayers and the effect of cholesterol concentration in sheep erythrocytes and liposomal membranes. J. Immunol., 120:1996. With M. K. Gately. Purification and characterization of lymphokines: An approach to the study of molecular mechanisms of cell-mediated immunity. Prog. Allergy, 25:106. With C. H. Hammer, D. W Michaels, and M. L. Shin. Immunologically mediated membrane damage: The mechanism of complement action and the similarity of lymphocyte-mediated cytotoxicity. Transplant. Prox., 10:707. 1979 With C. H. Hammer, D. W. Michaels, and M. L. Shin. Immunologically mediated membrane damage: The mechanism of complement action and the similarity of lymphocyte-mediated cytotoxicity. Immunochemistry, 15:813. Complement and lysis. In: Principles of Immunology, eds. N. R. Rose, F. Milgrom, and C. J. Van Oss, New York: Macmillan. With M. K. Gately and M. Okamoto. Biochemical studies of guinea
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pig lymphotoxin. In: Immunopathology, eds. F. Milgrom and B. Albini, Basel: S. Karger, p. 301. With M. L. Shin and D. W. Michaels. Membrane damage by a toxin from the sea anemone Stoichactis helianthus. II. Effect of membrane lipid composition in a liposome system. Biochim. Biophys. Acta, 55:79. With M. K. Gately, M. Okamoto, M. L. Shin, and J. B. Willoughby. Two mechanisms of cellmediated cytotoxicity: (1) Ca++ transport modulation by lymphotoxin, and (2) transmembrane channel formation by antibody and non-adherent spleen cells. Ann. N.Y. Acad. Sci., 332:395. 1980 With L. E. Ramm. Life-span and size of the trans-membrane channels formed by large doses of complement. J. Immunol., 124:2281. With J. B. Willoughby. On the channel hypothesis of antibody-dependent cell-mediated cytotoxicity (ADCC): Evaluation of a liposome model system. In: Biochemical Characterization of Lymphokines, ed. F. Kristensen, M. Landy, and A. deWeck, New York: Academic Press. Trans-membrane channels produced by complement proteins. Ann. N.Y. Acad. Sci., 358:43. 1981 With M. L. Shin and G. M. Hänsch. Effect of agents that produce membrane disorder on the lysis of erythrocytes by complement. Proc. Natl. Acad. Sci. USA, 78:2522. Membrane damage by complement. (The Dean's Lecture.) Johns Hopkins Med. J., 148:243. With G. M. Hänsch, C. H. Hammer, and M. L. Shin. Activation of the fifth and sixth component of the complement system: Similarities between C5b6 and C(56)a with respect to lytic enhancement by cell-bound C3b or A2C, and species preferences of target cell. J. Immunol., 127:999. With D. W. Michaels, L. E. Ramm, M. B. Whitlow, J. B. Willoughby, and M. L. Shin. Membrane damage by complement. Crit. Rev. Immunol., 2:133.
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1982 Membrane attack by complement (with comments on cell-mediated cytotoxicity). In: Mechanisms of Cell-Mediated Cytotoxicity, eds. W. R. Clark and P. Golstein, Adv. Exp. Biol. Med., 146:193. With L. E. Ramm and M. B. Whitlow. Size of the trans-membrane channels produced by complement proteins C5b-8. J. Immunol., 129:1143. With L. E. Ramm and M. B. Whitlow. Trans-membrane channel formation by complement. Functional analysis of the number of C5b6, C7, C8, and C9 molecules required for a single channel. Proc. Natl. Acad. Sci. USA, 79:4751. 1983 With L. E. Ramm and M. B. Whitlow. Size distribution and stability of the trans-membrane channels formed by complement complex C5b-9. Mol. Immunol., 20:155. With L. E. Ramm, D. W. Michaels, and M. B. Whitlow. On the size, heterogeneity and molecular composition of the transmembrane channels produced by complement. In: Biological Response Mediators and Modulators , ed. J. T. August, New York: Academic Press. With C. L. Koski, L. E. Ramm, C. H. Hammer, and M. L. Shin. Cytolysis of nucleated cells by complement: Cell death displays multi-hit characteristics. Proc. Natl. Acad. Sci. USA, 80:3816. With L. E. Ramm, M. B. Whitlow, C. L. Koski, and M. L. Shin. Elimination of complement channels from the plasma membranes of U937, a nucleated mammalian cell line: Temperature dependence of the elimination rate. J. Immunol., 121:1411. With D. K. Imagawa, L. E. Ramm, and M. B. Whitlow. Membrane attack by complement and its consequences. In: Progress in Immunology, eds. Y. Yamamura and T. Tada, Tokyo: Academic Press Japan, p. 427. With D. K. Imagawa, N. E. Osifchin, W. A. Paznekas, and M. L. Shin. Consequences of cell membrane attack by complement: Release of arachidonate and formation of inflammatory derivatives. Proc. Natl. Acad. Sci. USA, 80:6647. Complement. Historical perspectives and some current issues. Complement, 1:2.
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With L. E. Ramm and M. B. Whitlow. Complement lysis of nucleated cells: Effect of temperature and puromycin on the number of channels required from cytolysis. Mol. Immunol., 21:1015. With M. B. Whitlow and L. E. Ramm. Penetration of C8 and C9 in the C5b-9 complex across the erythrocyte membrane into the cytoplasmic space. J. Biol. Chem., 260:998. With L. E. Ramm and M. B. Whitlow. The relationship between channel size and the number of C9 molecules in the C5b-9 complex. J. Immunol., 134:2594. 1987 With D. K. Imagawa, N. E. Osifchin, L. E. Ramm, P. G. Koga, C. H. Hammer, and H. S. Shin. Release of arachidonic acid and formation of oxygenated derivatives following complement attack on macrophages: Role of channel formation. J. Immunol.
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Walsh McDermott October 24, 1909-October 17, 1981
By Paul B. Beeson Walsh McDermott's professional life divides into two phases. Until his mid-forties he followed a highly productive career in academic clinical medicine and laboratory investigation. He then decided to shift emphasis and work in the field of public health at the local, national, and international levels. From this vantage point he played an influential role in the development of national health policy and the reorganization of U.S. medical research, earning recognition as a leading statesman in American medicine. EDUCATION AND EARLY LIFE McDermott was born on October 24, 1909, in New Haven, Connecticut, where his father was a family doctor. His mother, the former Rosella Walsh, came from Massachusetts. After attending New Haven public schools and Andover, he went to Princeton for premedical studies, receiving the B.A. degree in 1930. He then entered Columbia University's College of Physicians and Surgeons, earning the M.D. degree in 1934. In college and medical school he had little financial support and had to obtain scholarships as well as part-time jobs. For residency he moved across Manhattan Island to the New York Hospital. Thus began a long association with that hospital and with the Cornell University College of Medicine.
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After his death, Cornell created an endowed chair of medicine in his name in recognition of this association. During the second year of residency training, in August, 1935, Walsh McDermott was diagnosed as having tuberculosis. He was transferred to the Trudeau Sanitarium, Saranac Lake. Over the next nineteen years he would be admitted to the New York Hospital nine times for treatment of the disease. At Saranac, he seemed to make good progress and after seven months returned to take a part-time appointment in an outpatient clinic of the New York Hospital devoted to the treatment of syphilis, though—priding themselves on the practice of general internal medicine—the clinic physicians seldom referred patients elsewhere for the treatment of nonsyphilitic problems. At that time penicillin had not been introduced into clinical practice, and the mainstay of antisyphilitic treatment was injection of arsenical compounds at weekly intervals over periods of months or years. In the New York Hospital syphilis clinic, McDermott demonstrated his capabilities as physician, teacher, and humane care-giver. It is also reasonable to assume that his own protracted illness and the long-term care for patients with syphilis influenced the nature of his work during both phases of his medical career. First, it brought home the fact that the etiologic agent of a disease can remain in the body for long periods without causing discernible evidence of disease. Second, it underscored the importance of the samaritan role of the physician and the need to treat the whole person rather than focusing on a single process or etiologic agent. Another dividend of incalculable importance came out of his work in the syphilis clinic, for it was there that he met Marian MacPhail—of the MacPhail baseball dynasty—who was serving as a volunteer clinic worker. They married in 1940 and their home was always in Manhattan, though they
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used a vacation house in Pawling, New York, on weekends and summer holidays. Marian's support during McDermott's illnesses and her influence on his style of living were of greatest significance to the successful pursuit of his professional life. In 1941, Marian joined the staff of Time Magazine as a researcher and in 1947 transferred to Life Magazine, where she eventually became senior research editor and a member of the Board of Editors. Both McDermott's thus enjoyed productive individual careers, and their friends included not only colleagues from the field of medicine, but also writers, political figures, photographers, and sports executives. THE MCDERMOTT LABORATORY Penicillin In 1942 David Barr, chief of medicine at the Cornell-New York Hospital, appointed McDermott head of the Division of Infectious Diseases. By that time penicillin was being made available for the treatment of certain diseases. McDermott was chosen as one of the clinicians responsible for using the limited supplies of the drug in trials against certain defined clinical infections. It was soon discovered that penicillin was far more effective than the arsenicals in treatment of syphilis and the management of that disease was so simplified that the special out-patient clinic could be closed. McDermott's scene of operations then moved to the infectious disease floor of the hospital, and his investigations broadened to include many other infections produced by staphylococci, pneumococci, the typhoid bacillus, and brucella. In the next few years, several other effective antimicrobial drugs became available: streptomycin, the tetracyclines, and chloramphenicol. McDermott's infectious disease service at New York Hospital became an exciting training area where members of the
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resident staff as well as research fellows were eager to be assigned. McDermott studied the pharmacological behavior of the new antimicrobial agents in a variety of clinical situations. He showed, for example, that in some circumstances penicillin could exert its beneficial effect when given orally, though it was originally thought that the drug had to be administered by injection to avoid the destructive effect of gastric acid. Travelling to Mexico, he also collaborated with health authorities in Guadalajara in devising therapy for such diseases as typhoid fever and brucellosis, comparing the relative effectiveness of the tetracyclines, streptomycin, and chloramphenicol. Yet flareups of his own tuberculosis kept intervening during these years, necessitating periods of bed rest either in the hospital or at home. McDermott was treated with several new drugs thought to be active against the tubercle bacillus, but the disease continued to manifest itself from time to time with pulmonary spread, cervical adenitis, and uveitis. Despite periods of incapacity, he continued to direct the work of the Infectious Disease Ward and—through his team of colleagues—of his research laboratories. Even when forced to give advice and directions from his bed, his voracious reading of the medical literature and remarkable memory enabled him to retain the respect and leadership of his team. McDermott's most serious episode of tuberculosis occurred in 1950 when he developed a bronchopleural fistula. After a series of consultations and at his own urging, an attempt was made to close the fistula surgically. This was accomplished by a high-risk lobectomy and thoracoplasty, fortunately with supplementary treatment by the newly introduced drug isoniazid. After that, the disease began to abate, although there was some radiologic evidence of active progression in the left lung, for which he received further chemotherapy.
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Antimicrobial Therapy for Infections in Animals Along with an extensive program of clinical investigations into the treatment of several infectious diseases, McDermott and his team of associates undertook laboratory investigations involving antimicrobial therapy of infections in animals. His able young associates included Paul Bunn, Ralph Tompsett, David Rogers, Vernon Knight, Robert McCune, Floyd Feldman, Charles LeMaistre, Edwin Kilbourne, Roger DesPrez, Harold Lambert, and John Batten. Their special focus of attention was the interaction of microbes and drugs in living tissues, with particular emphasis on the phenomenon of microbial persistence. In such cases a microbe susceptible to a drug in vitro can, nevertheless, survive long-term exposure to that drug in the living animal host. For nearly two decades McDermott explored this phenomenon—which he had observed clinically in syphilis, tuberculosis, typhoid fever, typhus fever, brucellosis, and more rarely in staphylococcal infections. In certain circumstances a latent microbe can again acquire the ability to reproduce and cause disease within the host. McDermott and his team studied mice inoculated with human tubercle bacilli most intensively and subsequently determined the number of living organisms recoverable from these animals' spleens. The experiments were timeconsuming and tedious, requiring months for completion. Mice that received no therapy were found to have fairly constant numbers of organisms in their spleens during succeeding months. Certain drugs caused a rapid decline in the number of organisms during the first three weeks but no further reduction in the number of culturable units when therapy was continued for as long as seventeen weeks. The bacteria recovered from treated animals showed the same susceptibility to the antimicrobial drugs as at the beginning of the experiment, i.e., microbial persistence.
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When the researchers used the potent drugs isoniazid and pyrazinamide, what appeared to be complete sterilization came about within twelve weeks: no living organisms could be demonstrated by culture of spleens. Yet after a rest period of three months, viable organisms were once again found in about onethird of the animals. Treatment with cortisone seemed to favor the infecting agent, so that viable organisms could be demonstrated earlier and in a higher population of treated mice. After investigating this phenomenon for many years, McDermott concluded that antimicrobial therapy induced a kind of temporary ''adaptive plasticity" in a certain proportion of the infecting inoculum, or change that could undergo spontaneous reversal. This long quest is recounted in his 1959 Dyer Lecture and his 1967 Harvey Lecture (1959,1). On the basis of many lines of reasoning, McDermott and his colleagues concluded that the phenomenon of microbial persistence could not be explained on the basis of survival in certain "sanctuaries," e.g., within cells. They also showed that microorganisms were not protected from the effect of drugs by the chemical milieu of an inflammatory reaction. More Penicillin and the Role of Drugs in Combination McDermott's laboratory tested the bactericidal effect of penicillin against the staphylococcus, and a series of imaginative experiments produced evidence that here, too, pointed to an effect on the microbe. McDermott suggested that microorganisms became "indifferent" to the drug by some change in form (possibly analogous to protoplasts), a transformation he often described as "adaptive plasticity." McDermott also investigated the mechanisms of action of drug combinations. Clinical and experimental evidence showed that two different antimicrobial drugs can sometimes sterilize a bacterial population, either in vitro or in vivo, more
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effectively than either one alone. The McDermott team came to conclusions not in accord with conventional thinking—that each drug kills those bacterial cells susceptible to it. Their findings favored an enhanced antimicrobial effect greater than a simple additive action in which each drug exerts its effect by its own mechanism. It is interesting to note that McDermott never had any formal research training in college, medical school, or in his postgraduate years. He was able, nevertheless, to organize a microbiology and experimental pathology research laboratory and to attract talented younger people to work with him. He taught himself much by extensive reading and in conversations with his colleagues. In this connection he was particularly fortunate to form a lasting friendship with René Dubos of The Rockefeller Institute (later University). They were frequently in touch and were both superb communicators. Some of McDermott's scientific success is surely attributable to the close association he maintained with Dubos and other Rockefeller scientists. EDITORIAL WORK McDermott became managing editor of the American Review of Tuberculosis in 1948 and editor in 1952, when Esmond Long retired. He held the position for twenty years. During that time tuberculosis diminished as a cause of morbidity and mortality, the interest of pulmonary physicians shifted to other diseases and problems, and the name of the journal was changed to the American Review of Respiratory Disease . McDermott managed the transition smoothly and, under his editorship, the journal's importance in the biomedical world grew. He was known to be a conscientious editor who often revised the manuscripts submitted to him extensively. He also played a leading role in the custodianship of the Cecil Textbook of Medicine (first edition, 1928). In the early
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1950s, editors Russell Cecil and Robert Loeb invited McDermott to become associate editor with special responsibility for the infectious diseases section of the textbook. Cecil and Loeb retired after the 10th edition in 1959 and were succeeded by McDermott and this author as coeditors. We collaborated in that work through the next five editions of the textbook, until 1979. For me this joint effort was both enjoyable and instructive. Our function was mainly to add new subjects to the contents, to select contributors (more than 200 in each edition), and to ensure that manuscripts were ready by the deadline. We were in touch constantly—by meetings, by telephone, and by letters. Because this relationship exposed me to McDermott's broad concepts of man, disease, and society, I came to enjoy it more and more. I was, therefore, especially interested to read something he said about this textbook work in 1973, when being interviewed as "Medicine's Man of the Year." The greatest compensation for such work, said McDermott, was "knowing that the volume goes to the remotest parts of the world—that someplace, perhaps in an African jungle, some human being is getting correct treatment because a doctor or a nurse has our book." This statement illustrates his sincere concern for the delivery of medical care in underserved segments of the population, at home and abroad. CHANGE IN FOCUS: PUBLIC HEALTH The necessity to carry out some field trials of antimicrobial therapy, plus an interest in the social and political problems in his own metropolitan area, caused McDermott to change the character of his medical work. In the course of his long-term studies on streptomycin therapy he had observed clinical relapses caused by the emergence of resistant microbes during a long course of therapy. When isoniazid became available there was reason to hope that more effective
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treatment was at hand. But the matter of testing a new agent in a life-threatening disease presented a grave ethical dilemma. Was it justifiable to try a new agent, isoniazid, while withholding streptomycin—an agent which indubitably had some therapeutic value? His concern about this ethical problem was resolved when one of his fellows, serving at the Communicable Disease Center, learned that Navajo Indians with serious and uniformly fatal forms of tuberculosis—i.e., meningitis and military tuberculosis—were dying on their reservations in Arizona and New Mexico because conditions did not permit the required daily injections of streptomycin over long periods of time. It was, therefore, justifiable to test isoniazid alone. McDermott then arranged a program, the Many Farms Project, to use isoniazid therapy in that population. Physicians and nurses manned aid stations and a mobile visiting service reached wide territorial areas. McDermott made many visits there, negotiated with tribal leaders, and secured agreements for the drug trials to be carried out. The Many Farms Project provided unequivocal evidence of the superiority of isoniazid, which has largely supplanted streptomycin, although other antituberculous drugs of unquestioned value later became available. The success of isoniazid in curing an otherwise lethal infection among the Navajo suggested the possible benefit of bringing other sophisticated medical service to that underserved population, and the Many Farms Project was expanded to include many other forms of modern medical care. The experiment continued for six years, and some parts of the program were continued beyond that time with benefit to the Navajo population. But—as McDermott and his colleague Kurt Deuschle reported in 1972—even the best medical care could not bring about a general improvement in the health of people who had inadequate food, insufficient
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drinking water, lived in extreme poverty, and lacked modern sanitary services. By 1955 McDermott decided that he could make his most important contribution to medicine in the area of public health. Maintaining his appointment in the Department of Medicine at Cornell, he became professor of Public Health and chairman of that Department, a position he held until 1972. During that period, he and his Department focused much attention on the public health problems to be found in a modern city: air pollution, poverty, malnutrition, drug addiction, alcoholism, tobacco usage, etc. A pilot project was set up with Kenneth Johnson in the Bedford-Stuyvesant area of Brooklyn, including day clinics, visiting nurses, and social work services. McDermott used this project in his teaching of public health and arranged for dozens of Cornell medical students to observe and participate. In addition to the work at home, he served on committees dealing with international health problems and traveled widely in Central America, South America, Europe, and Asia. He spoke of this kind of work as "statistical compassion," i.e., a kind of activity that allows members of the medical profession to help people they never get to see. WORK IN THE JOHNSON FOUNDATION The early 1970s saw the creation of The Robert Wood Johnson Foundation, headquartered in Princeton, New Jersey. The income from a very large endowment was to be used in support of projects testing ways to provide better access to medical care. The creation of this Foundation provided an ideal opportunity for McDermott to work in health care delivery, a field for which he was so superbly prepared. David Rogers, the first president of the Johnson Foundation, who had some years earlier collaborated on research
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with McDermott at Cornell, persuaded his old colleague to accept a unique appointment as special advisor and to commute to Princeton. McDermott's academic title at Cornell was appropriately changed to professor of public affairs in medicine. McDermott was especially interested in ways to provide better care for the most vulnerable members of the population—the elderly and the newborn. This involved setting up visiting nurse services, social services, welfare programs, prenatal care, and perinatal care. McDermott's interests and experience made him ideally suited for the task. He wrote position papers, took an active part in staff discussions, counseled other staff members, and made site visits— continuing actively in this work until his sudden death in 1981. After his death, Rogers wrote several moving tributes to this friend and colleague, detailing how very great his contribution to the work of the Foundation had been. PROFESSIONAL MEMBERSHIPS AND OTHER ACTIVITIES McDermott was elected to many learned societies, including the American Academy of Arts and Sciences, American College of Physicians, American Public Health Association, American Society for Clinical Investigation, American Thoracic Society, Association of American Physicians, Infectious Diseases Society of America, the National Academy of Sciences, and Britain's Royal College of Physicians. Of nonmedical associations, he belonged to the Century Association and the Council on Foreign Relations in New York City. He was also a member of the honorific Cosmos Club in Washington, D. C. From the late 1940s to the late 1960s, McDermott was much in demand as a consultant to the National Institutes of Health, particularly in the fields of tuberculosis and anti
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microbial therapy. During that time he was appointed to numerous advisory councils, study sections, and special advisory councils within the United States Public Health Service. These appointments included: chairman, the Experimental Therapeutics Study Section, NIH, 1947-1953; chairman, Cancer Chemotherapy Committee, NIH, 1953-1954; member, National Advisory Health Council, NIH, 1955-1959; member, National Advisory Council, Allergy and Infectious Diseases, NIH, 1960-1963; member, Board of Regents, National Library of Medicine, 1964-1968; consultant, Division of Indian Health, 1965-1968. In the 1960s he chaired several boards and panels concerned with involving American academia and industry in health projects administered under United States foreign aid programs. These included the Development Assistance Panel of the President's Advisory Committee on Science and Technology, the Public Advisory Board in the Department of State, and the U.S. delegation to the United Nations Conference on the Application of Science and Technology for the Benefit of the Less Developed Areas, made up of nearly a hundred American scientists representing many fields. He also chaired the Research Advisory Committee of the Agency for International Development. In the World Health Organization he was a member of the Expert Advisory Panel on Tuberculosis (1958-1973) and the Advisory Committee on Medical Research (1964-1967). He also served on the Pan American Health Organization's Advisory Committee on Medical Research (1962-1970). In New York City, under Mayors Wagner and Lindsay, he was one of four members of the Board of Health and, with Leona Baumgartner and Colin MacLeod, played a key role in establishing the New York Health Research Council—for a number of years the major financial supporter of the City's various medical schools.
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FRIEND AND COLLEAGUE I became acquainted with Walsh McDermott in 1949-50, when I was appointed to serve, under his chairmanship, on the Experimental Therapeutics Study Section of the National Institutes of Health. He had a light touch and often injected a bit of humor into the discussions while he kept things moving. I was impressed by the way he made our business go. By the end of the day our work was done, and we were satisfied with it. As I look back on his performance, I am convinced that the reason he guided us so well was that he always did his "homework." He studied carefully every grant request that was to come before us with skill and dedication—accounting, doubtless, for the many invitations he received to serve on committees and advisory boards. During the last eight years of his life, Walsh McDermott was a trustee of Columbia University. At a memorial service after his death, Columbia's President Sovern said of him: "What Walsh communicated was warmth, good sense, and wonderful humor. He brightened the deliberations of our Board of Trustees even as he made them wiser. Though it strain credulity, even committee meetings could be fun if Walsh was there. . . ." CREATION OF THE INSTITUTE OF MEDICINE From time to time throughout the 1960s there had been suggestions that a National Academy of Medicine, related to the National Academy of Sciences, should be formed. McDermott was elected to the NAS in 1967, undoubtedly because of his studies of chemotherapy and his work on the phenomenon of microbial persistence. Soon thereafter he was asked to chair a new planning committee called the
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Board on Medicine, whose deliberations have been described by Irving M. London: "As you know, the president of the National Academy of Sciences, Fred Seitz, appointed a Board on Medicine with Walsh as chairman in the late 1960s. A major function of the Board was to speak to important issues in medicine, to provide informed advice, and to avoid the lobbying posture of organizations such as the American Medical Association. "An additional important function of the Board was to consider the form that such an organization should develop. There were various currents of thought concerning this organizational form. Some individuals advocated the establishment of a National Academy of Medicine which would be largely honorific, free-standing, and not associated with the National Academy of Sciences. Those who held this position argued that association with the National Academy of Sciences would be too restrictive. Others—particularly Walsh and I—favored close association with the National Academy of Sciences because we felt that the NAS would lend its prestige to our new organization and at the same time would help to exercise a kind of desirable quality control."
McDermott and others argued successfully that what was needed was a prestigious organization affiliated with the NAS but with a diverse membership, to include not only members of the medical profession but also people with expertise in related fields of economics, law, social sciences, and other health care professions such as nursing. This notion was accepted by the new president of the NAS, Philip Handler. The result was a unique organization—the Institute of Medicine of the National Academy of Sciences. In the two decades of its existence, the Institute of Medicine has served a variety of important functions and come to be regarded as an influential force in American medicine. It has conducted many excellent studies and fulfills a function not appropriate to other societies or organizations in the health care field. Summarizing Walsh McDermott's contribution to the establishment of the IOM, Irving London wrote:
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"In the creation of the Board of Medicine and its evolution [in]to the Institute of Medicine, Walsh was absolutely critical to the success of these developments. He had a deft touch, he was politically sensitive and astute, and he spoke with the authority of one who had achieved scientific distinction, was a recognized authority in his field, and enjoyed the respect of physicians in the practice of medicine. He deserves to be regarded as the Founding Father of the Institute of Medicine."
AWARDS Walsh McDermott's first major recognition came in 1955 when, with Carl Muschenheim and two other clinicians, he received the Albert Lasker Award for "contribution of the first order to our knowledge of the principles of the treatment and control of tuberculosis. . . ." In 1963, the National Tuberculosis Association gave him its Trudeau Medal. In 1968, he won the James D. Bruce Memorial Award of the American College of Physicians, and in 1969, received the Woodrow Wilson Award of Princeton University "to a Princeton alumnus in recognition of distinguished achievement in the nation's service .. ." In 1970, the College of Physicians and Surgeons' Alumni Association gave him its Alumni Gold Medal Award "for distinguished achievement in medicine . . . ." In 1975, the Association of American Physicians gave him the Kober Medal in "full realization of the commanding knowledge in medicine . . ." In 1979 he received the Blue CrossBlue Shield Association's National Health Achievement Award "for his monumental contribution to the education of generations of physicians . . . [and] for playing a major role in shaping the health policy in the United States." Princeton and Columbia universities awarded him honorary degrees. He gave dozens of special lectures in American medical schools and other institutions. Among these may be mentioned the William Allen Pusey Memorial Lecture at the Chicago Institute of Medicine, 1949; the Jenner Lecture at
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St. George's Hospital Medical School, London, 1958; the R. E. Dyer Lectureship of the National Institutes of Health, 1959; the J. Burns Amberson Lecture of the National Tuberculosis Association, 1962; the Holme Lecture, University College Hospital, University of London, 1967; the Barnwell Memorial Lecture of the National Tuberculosis Association, 1969; the Heath Clark Lecture, London School of Preventive Medicine and Tropical Hygiene, London, 1971; the William S. Paley Lecture, Cornell Medical College, 1967. ETHICS, THE MEDICAL PROFESSION, AND MODERN SCIENCE It seems appropriate to conclude this memoir with something of McDermott's philosophy expressed in his own carefully chosen words. In an introductory chapter to the Textbook of Medicine, of which he was co-editor, he explained the expression ''statistical compassion:" "The physician who treats one patient at a time and the physician who deals with a community as a whole both exert compassion, but it is of two quite different sorts. The compassion exercised by the physician who treats individuals takes the form of a cultivated instinct to lend support and comfort to a particular fellow human being. By contrast, the 'group' compassion of the public health or community physician necessarily takes the form of what the writer has previously termed 'statistical compassion.' By this is meant an imaginative compassion for people whom one never gets to see as individuals and, indeed, can know only as data on a graph."
In 1978, in an article entitled "Medicine: The Public Good and One's Own," he wrote further: "Medicine itself is deeply rooted in a number of sciences, but it is also deeply rooted in the samaritan tradition. The science and the samaritanism are both directed toward the same goal of tempering the harshness of illness and disease. Medicine is thus not a science but a learned profession that attempts to blend affairs of the spirit and the cold objectivity of science
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. . . These two functions, the technologic and the samaritan, are separable in the world of analysis but not in the world of real life . . ."
Accepting the Kober Medal in 1975, McDermott spoke of the explosion of medical science and technology over the preceding fifty years: "The importance today of these developments that started fifty years ago can hardly be exaggerated. What was substantially a whole new technology was born. Had this new technology, like atomic energy, been ushered in with one big bang on a single day, the implications would have been so obvious that medicine would have been forced to create a comprehensive institutional framework for the new science [like] . . . the Atomic Energy Commission. But the rate of change, although rapid, was just slow enough that it was easy to miss that something quite different was going on from just the logical extension of what had gone on before. The scene was now occupied by a new, powerful, and unruly force which on the one hand could lift our profession into the heights of much greater usefulness, but on the other could destroy it as a profession. . . . "The piecemeal nature of our institutional approach was greatly furthered by the fact that, with medicine, the coming of the new technology was not followed by a delivery system shaped to fit it. Instead, the new technology was simply engrafted on a centuries-old delivery system—the personal-encounter physician. As a result the profession was stressed almost to the bursting point by the new science—a stress that still continues. This turmoil is not the fault of our science and technology; it results from the relative failure of the institutions for their management."
Regarding the social consequences of modernization, he added: " . . . Something quite new has been added to the social contract—namely the idea that each of us as an individual bears a moral responsibility for the collective acts of our particular society. No longer are we allowed to cling either to [the excuse of] 'orders from above' or to the personal hypocrisies that enabled us to avoid looking at what was morally outrageous. Thanks to communication technology, we cannot escape a virtually daily awareness of the extended consequences of our acts or of our failures to act. There are now very few places to hide."
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Selected Bibliography
1941 With W. G. Downs and B. Webster. Reactions to tryparsamide therapy. Am. J. Syph. Gonorrhea Vener. Dis., 25:16. With B. Webster and D. Macrae. The effect of arsphenamine on tuberculosis in syphilitic animals. Am. Rev. Tuberc., 44:3. With R. Tompsett, W. G. Downs, and B. Webster. The use of clorarsen in the treatment of syphilis. J. Pharmacol. Exp. Ther., 73:412. 1942 With R. Tompsett and B. Webster. Syphilitic aortic insufficiency: The asymptomatic phase. Am. J. Med. Sci., 2:203. 1943 With B. Webster, R. Baker, J. Lockhart, and R. Tompsett. Nutritional degeneration of the optic nerve in rats: Its relation to tryparasamide amblyopia. J. Pharmacol. Exp. Ther., 77:24. 1944 With D. R. Gilligan and J. A. Dingwall. The parenteral use of sodium lactate solution in the prevention of renal complications from parenterally administered sodium sulfadiazine. Ann. Int. Med., 20:604. With D. R. Gilligan, C. Wheeler, and N. Plummer. Clinical studies of sulfamethazine. N. Y. State J. Med., 44:394. Recent advances in the treatment of syphilis. Med. Clin. N. Am., 293:308. 1945 With P. A. Bunn, M. Benoit, R. Dubois, and W. Haynes. Oral penicillin. Science, 101:2618, 228-29. With M. Benoit and R. Dubois. Time-dose relationships of penicillin therapy. Regimens used in early syphilis. Am. J. Syph. Gonorrhea Vener. Dis., 29:345. With R. A. Nelson. The transfer of penicillin into the cerebrospinal fluid following parenteral administration. Am. J. Syph. Gonorrhea Vener. Dis., 29:403. With M. M. Leask and M. Benoit. Streptobacillus moniliformis as
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a cause of subacute bacterial endocarditis. Ann. Int. Med., 22:414. With P. A. Bunn, S. Hadley, and A. Carter. The treatment of pneumococcic pneumonia with orally administered penicillin. J. Am. Med. Assoc., 129:320. 1946 With P. A. Bunn, M. Benoit, R. Dubois, and M. Reynolds. The absorption of orally administered penicillin. Science, 103:2673, 359-61. With P. A. Bunn, M. Benoit, R. Dubois, and M. Reynolds. The absorption, excretion, and destruction of orally administered penicillin. J. Clin. Invest., 25:2, 190-210. 1947 With R. Tompsett and S. Schultz. Influence of protein-binding on the interpretation of penicillin activity in vivo. Proc. Soc. Exp. Biol. Med., 65:163. With H. Koteen, E. J. Doty, and B. Webster. Penicillin therapy in neurosyphilis. Am. J. Syph. Gonorrhea Vener. Dis., 31:1. With R. Tompsett and S. Schultz. The relation of protein-binding to the pharmacology and antibacterial activity of penicillins X, G, Dihydro F, and K. J. Bacteriol., 53:581. Toxicity of streptomycin. Am. J. Med., 2:491. With G. G. Reader, B. J. Romeo, and B. Webster. The prognosis of syphilitic aortic insufficiency. Ann. Int. Med., 27:584. With H. Koprowski and T. W. Norton. Isolation of poliomyelitis virus from human serum by direct inoculation into a laboratory mouse. Publ. Hea. Rep., 62:1467. With C. Muschenheim, S. J. Hadley, P. A. Bunn, and R. V. Gorman. Streptomycin in the treatment of tuberculosis in humans. I. Meningitis and generalized hematogenous tuberculosis. Ann. Int. Med., 27:769. With C. Muschenheim, S. J. Hadley, H. Hull-Smith, and A. Tracy. Streptomycin in the treatment of tuberculosis in humans. Ann. Int. Med., 27: 769. 1948 With H. Gold and H. Koteen. Conference on streptomycin. Am. J. Med., 4:130. With C. M. Flory, J. W. Correll, J. G. Kidd, L. D. Stevenson, E. C.
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Alvord, et al. Modifications of tuberculous lesions in patients treated with streptomycin. Am Rev. Tuberc., 58:4. With L. B. Hobson, R. Tompsett, and C. Muschenheim. A laboratory and clinical investigation of dihydrostreptomycin. Am. Rev. Tuberc., 58:5. 1949 With R. Tompsett, A. Timpanelli, and O. Goldstein. Discontinuous therapy with penicillin. J. Am. Med. Assoc., 139:555. With V. Knight and F. Ruiz-Sanchez. Antimicrobial therapy in typhoid fever. Trans. Assoc. Am. Phys., 62:46. With V. Knight, F. Ruiz-Sanchez, and A. Ruiz-Sanchez. Aureomycin in typhus and brucellosis. Am. J. Med., 6:407. Streptomycin in the treatment of tuberculosis. J. Natl. Med. Assoc., 41:167. With R. Tompsett. Recent advances in streptomycin therapy. Am. J. Med., 7:371. With L. B. Hobson. Criteria for the clinical evaluation of antituberculous agents. Ann. N. Y. Acad. Sci., 52:782. 1950 With H. C. Hinshaw. Thiosemicarbazone therapy of tuberculosis in humans. Am. Rev. Tuberc., 61:145. With V. Knight, F. Ruiz-Sanchez, A. Ruiz-Sanchez, and S. Schultz. Antimicrobial therapy in typhoid. Arch. Int. Med., 85:44. With V. Knight and F. Ruiz-Sanchez. Chloramphenicol in the treatment of the acute manifestations of brucellosis. Am. J. Med. Sci., 219:627. With C. A. Werner and V. Knight. Absorption and excretion of terramycin in humans; comparison with aureomycin and chloramphenicol. Proc. Soc. Exp. Biol. Med., 74:261. With R. Tompsett and J. G. Kidd. Tuberculostatic activity of blood and urine from animals given gliotoxin. J. Immunol., 65:59. With A. Timpanelli and R. D. Huebner. Terramycin in the treatment of pneumococcal and mixed bacterial pneumonias. Ann. N. Y. Acad. Sci., 53:440. 1951 With C. A. Werner, R. Tompsett, and C. Muschenheim. The toxicity of viomycin in humans. Am. Rev. Tuberc., 63:49.
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With C. A. LeMaistre, R. Tompsett, C. Muschenheim, and J. A. Moore. Effects of adrenocorticotropic hormone and cortisone in patients with tuberculosis. J. Clin. Invest., 30:445. With C. Muschenheim and R. Maxwell. The therapy of miliary and meningeal tuberculosis: Review of a five-year experience. Trans. Am. Clin. Climatol. Assoc., 63:257. 1952 With DuM. F. Elmendorf, Jr., W. U. Cawthon, and C. Muschenheim. The absorption, distribution, excretion, and short-term toxicity of isonicotinic acid hydrazide (Nydrazid) in man. Am. Rev. Tuberc., 65:429. With C. M. Clark, DuM. F. Elmendorf, Jr., W. U. Cawthon, and C. Muschenheim. Isoniazid (isonicotinic acid hydrazide) in the treatment of miliary and meningeal tuberculosis. Am. Rev. Tuberc., 66:391. With C. Muschenheim, C. M. Clark, DuM. F. Elmendorf, Jr., and W. U. Cawthon. Isonicotinic acid hydrazide in tuberculosis in man. Trans. Assoc. Am. Phys., 65:191. 1953 Antimicrobial therapy in tuberculosis. Bull. St. Louis Med. Soc., 47:472. With C. A. LeMaistre and R. Tompsett. The effects of corticosteroids upon tuberculosis and pseudotuberculosis. Ann. N. Y. Acad. Sci., 56:772. With C. Muschenheim, DuM. F. Elmendorf, Jr., and W. U. Cawthon. Failure of paraisobutoxybenzaldehyde thiosemicarbazone as an antituberculous drug in man. Am. Rev. Tuberc., 68:791. The antimicrobial therapy of tuberculosis. Bull. Quezon Inst., 2:169. 1954 With L. Ormond, C. Muschenheim, K. Deuschle, R. M. McCune, Jr., and R. Tompsett. Pyrazinamide-isoniazid in tuberculosis. Am. Rev. Tuberc., 69:319. With C. A. Werner and V. Knight. Studies of microbial populations artificially localized in vivo. I. Multiplication of bacteria and distribution of drugs in agar loci. J. Clin. Invest., 33:742.
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With C. A. Werner. Studies of microbial populations artificially localized in vivo. II. Differences in antityphoidal activities of chloramphenicol and chlortetracycline. J. Clin. Invest., 33:753. With D. E. Rogers. Neoplastic involvement of the meninges with low cerebrospinal fluid glucose concentrations simulating tuberculous meningitis. Am. Rev. Tuberc., 69:1029. With R. Tompsett, R. M. McCune, Jr., L. Ormond, K. Deuschle, and C. Muschenheim. The influence of pyrazinamide-isoniazid on M. tuberculosis in animals and man. Trans. Assoc. Am. Phys., 67:224. With K. Deuschle, L. Ormond, Du M. F. Elmendorf, Jr., and C. Muschenheim. The course of pulmonary tuberculosis during long-term single-drug (isoniazid) therapy. Am. Rev. Tuberc., 70:228. With R. Tompsett. Activation of pyrazinamide and nicotinamide in acidic environments in vitro. Am. Rev. Tuberc., 70:748. With C. Muschenheim, R. McCune, K. Deuschle, L. Ormond, and R. Tompsett. Pyrazinamideisoniazid in tuberculosis. II. Results in fifty-eight patients with pulmonary lesions one year after the start of therapy (notes). Am. Rev. Tuberc., 70:743. 1955 The enlarging role of the general practitioner in tuberculosis therapy (editorial). J. Chron. Dis., 2:234. With Y. Kneeland, Jr., A. L. Barach, D. V. Habif, and H. M. Rose. Current concepts in the use of antibiotics. Panel meeting on therapeutics. Bull. N. Y. Acad. Med., 31:639. 1956 The problem of staphylococcal infections. Ann. N. Y. Acad. Sci., 65:58. With O. Wasz-Hoeckert, R. M. McCune, Jr., S. H. Lee, and R. Tompsett. Resistance of tubercle bacilli to pyrazinamide in vivo. Am. Rev. Tuberc. Pulm. Dis., 74:572. With R. M. McCune, Jr., and R. Tompsett. The fate of mycobacterium tuberculosis in mouse tissues as determined by the microbial enumeration technique. II. The conversion of tuberculous infection to the latent state by the administration of pyrazinamide and a companion drug. J. Exp. Med., 104:763.
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1957 With J. Adair and K. Deuschle. Patterns of health and disease among the Navajos. Ann. Am. Acad. Polit. Soc. Sci., 311:80. 1958 With C. Jordahl, R. Des Prez, K. Deuschle, and C. Muschenheim. Further experience with single-drug (isoniazid) therapy in chronic pulmonary tuberculosis. Am. Rev. Tuberc. Pulm. Dis., 77:539. 1959 Inapparent infection. The R. E. Dyer Lecture (delivered at the National Institutes of Health). Publ. Hea. Rep., 74:485. With R. Des Prez, C. Jordahl, K. Deuschle, and C. Muschenheim. Streptovaricin and isoniazid in the treatment of pulmonary tuberculosis (notes). Am. Rev. Respir. Dis., 80:431. Drug-microbe-host mechanisms involved in a consideration of chemoprophylaxis. 15th International Tuberculosis Conference, Istanbul, Sept., 1959. Bull. Int. Union Tuberc., 29:243. 1960 With E. D. Kilbourne, D. E. Rogers, and H. M. Rose. Influenza upper respiratory infections (a panel meeting). Bull. N. Y. Acad. Med., 36:22. With K. Deuschle, J. Adair, H. Fulmer, and B. Loughlin. Introducing modern medicine in a Navajo community. Science, 131:197. With C. A. Berntsen. Increased transmissibility of staphylococci to patients receiving an antimicrobial drug. N. Engl. J. Med., 262:637. The community's stake in medical research. Am. Rev. Respir. Dis., 81:279. Antimicrobial therapy of pulmonary tuberculosis. Bull. WHO, 23:427-61. 1961 Air pollution and public health. Sci. Am., 205:49-57.
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1962 The chemotherapy of tuberculosis. The J. Burns Amberson Lecture. Am. Rev. Respir. Dis., 86:323. 1963 Science for the individual—the university medical center. J. Chron. Dis., 16:105-10. 1964 The role of biomedical research in international development. J. Med. Ed., 39:655. 1965 Summary remarks. Dedication symposium of the Institute for Biomedical Research of the American Medical Association. J. Am. Med. Assoc., 194:1374. 1966 With R. McCune, F. Feldman, and H. Lambert. Microbial persistence. I. The capacity of tubercle bacilli to survive sterilization in mouse tissues. J. Exp. Med. With R. McCune and F. Feldman. Microbial persistence. II. Characteristics of the sterile state of tubercle bacilli. J. Exp. Med. Modern medicine and the demographic disease pattern of overly traditional societies: A technologic misfit. J. Med. Ed., 41:9. 1967 Ed. W. McDermott and P. B. Beeson. Cecil-Loeb Textbook of Medicine , 12th ed. Philadelphia: W. B. Saunders Company. The changing mores of biomedical research. A Colloquium on Ethical Dilemmas from Medical Advances (opening comments). Ann. Int. Med., 67:39. 1969 Early days of antimicrobial therapy. Presidential address delivered at the meeting of the Infectious Diseases Society of America. In: Antimicrobial Agents & Chemotherapy, 1968, pp. 1-6. Washington, D.C.: American Society for Microbiology.
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Microbial persistence. The Harvey Lectures, Series 63, delivered September 21, 1967. New York: Academic Press. 1970 Microbial drug resistance. The John Barnwell Lecture. Am. Rev. Respir. Dis., 102:857-76. 1972 With K. W. Deuschle and C. R. Barnett. Health care experiment at Many Farms. Science, 175:23. 1974 General medical care: Identification and analysis of alternative approaches. Johns Hopkins Med. J., 135:5, 292-321. 1977 Evaluating the physician and his technology. Daedalus, 106:135. 1978 Medicine: The public good and one's own. The Paley Lecture. Perspect. Biol. Med., 21:167. Health impact of the physician. Am. J. Med., 65:569. 1980 Pharmaceuticals: Their role in developing societies. Science, 209:240. 1981 Absence of indicators of the influence of its physicians on a society's health. Am. J. Med., 70:833-43. 1982 Education and general medical care. Ann. Int. Med., 96:512. 1983 With D. Rogers. Technology's consort. Am. J. Med., 74:353.
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Theophilus Shickel Painter August 22, 1889-October 5, 1969
By Bentley Glass In the stellar days of Drosophila genetics during the 1920s and 1930s, only two principal centers of such research existed in the United States. The California Institute of Technology attracted Thomas Hunt Morgan from Columbia University in 1929, and he brought with him his two students, Alfred H. Sturtevant and Calvin B. Bridges, who a decade earlier had contributed to the establishment of the chromosome theory of heredity. The CalTech group also included Theodosius Dobzhansky, Jack Schultz, and a constellation of notable visiting fellows, present for a year or two, such as George Beadle and Curt Stern. During the same period a second stellar group formed at the University of Texas in Austin. H. J. Muller, one of the original trio of Morgan's graduate students, had created a great stir in genetics with his 1927 discovery that X-rays will induce mutations at frequencies hundreds, even thousands, of times higher than rates of spontaneous mutation. A generous grant from the Rockefeller Foundation made it possible for Muller, joined by J. T. Patterson and T. S. Painter of the Department of Zoology at Austin, to establish a cytogenetical program for exploiting the new discovery. Graduate students were recruited and given fellowships, the earliest of which went to C. P. Oliver, Wilson S. Stone, and the writer of
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this memoir. Muller soon found that X-rays produce chromosomal breaks and rearrangements in addition to gene mutations, Oliver worked out the relation of point mutations to radiation dosage, Painter collaborated with Muller in analyzing chromosomal rearrangements, and Patterson explored an exciting new field—mosaic types of mutation produced by X-rays. Bursts of exciting new findings made the rivalry with CalTech as hectic as a close basketball game, and Painter was a central figure in all of it. EARLY LIFE AND EDUCATION T. S. Painter was born in Salem, Virginia, the son of Franklin V. N. Painter and Laura T. Shickel Painter. T. S.'s father was an esteemed educator, a professor of modern languages and English literature at Roanoke College. Both parents were very religious, and their son was brought up in an atmosphere of culture and religious faith that marked him deeply. His middle name was that of his mother's family; his given name reflects his parents' Christian orientation. As a boy, T. S. was sickly and obtained most of his elementary and secondary education by home tutoring. He entered Roanoke College in 1904 and graduated with a B.A. degree in 1908. The college was a small one and did not provide a diversity of scientific courses. Painter was attracted to chemistry and physics but had no opportunity to acquaint himself with biology. Having received a scholarship in chemistry, he entered Yale University as a graduate student in 1908. Here he met Professor L. L. Woodruff of the Biology Department and asked to be permitted to sit in a corner of the laboratory and look at objects under a microscope, which he had never had an opportunity to use before. Professor L. L. Woodruff assigned Painter a microscope and provided him with a hay
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infusion full of active bacteria, protozoans, and algae. Painter was fascinated and soon decided that he wanted to change his field from chemistry to biology. He received an M.A. degree in 1909 and a Ph.D. in 1913, under the direction of the famed authority on spiders Alexander Petrunkevitch. Painter learned the techniques of cytology as practiced at that time and for his thesis explored the process of spermatogenesis in a species of spider. His first scientific publication (1913,1) was a paper on dimorphism in males of the jumping spider, Maevia vittata. His second (1914,1) was his thesis research. Painter then went to Europe for a year of postdoctoral study, partly in the laboratory of Theodor Boveri, in Würzburg, and partly at the famed Marine Zoological Station at Naples. At that time Boveri was among the foremost cytologists in the world. More than a decade earlier he had established, in studies of the fertilization and development of Ascaris eggs, that each chromosome controls development individually. Chromosomes, furthermore— although they seem to disappear after the close of each mitotic cell division— have a persistent continuity and reappear in the next mitosis in the same place they occupied before their disappearance. Most surprisingly, they continue to bear whatever aberrant distinctions they might previously have acquired by accident. Boveri was a stout supporter of the chromosome theory of heredity— which he had enunciated independently of W. S. Sutton, a student of E. B. Wilson at Columbia. Later, when I was taking a graduate course with Painter at Austin, it was a matter of astonishment to me that I never heard him reminisce about those exciting times or make any reference to Boveri or to what he learned from him. The experience at Naples, with its marvels of marine life for a cytologist to explore, seemed to affect Painter more. His next publications dealt with problems of the forces involved
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in the cleavage of the fertilized egg into a multiplicity of cells by means of repeated mitotic cell divisions. Back in the United States from a war-torn Europe, Painter received an appointment as an instructor in zoology at Yale for two years. He was also asked to teach marine invertebrate zoology at the Woods Hole Laboratory in the summers of 1914 and 1915. There he met two persons who were to be exceedingly important in his life. The first, Mary Anna Thomas, was a young student in his course who would later become his devoted wife. The second, John Thomas Patterson, was the young head of the Zoology Department at the University of Texas in Austin. Patterson offered Painter the academic post that brought him to the institution where he would spend the remainder of his life. In his Biographical Memoir of J. T. Patterson (1965,1), Painter told of the warm and friendly way in which the two first met while playing baseball with other teachers and researchers at Woods Hole. Painter's research at this period greatly resembled the type of experimentation on developing invertebrate embryos favored by E. B. Wilson and E. G. Conklin. He first studied the effects of carbon dioxide on the developing eggs of Ascaris, the material for which had been obtained at Würzburg. His next study also took its origin from work begun in Europe, this time at Naples, where Painter had discovered spiral asters in developing eggs of sea urchins and become curious about their participation in the process of embryonic cleavage. He investigated the occurrence of monaster eggs, the light they threw on cell mechanics during division, and the influence of narcotics on cell division. Painter demonstrated that eggs may divide in the absence of asters, that a factor derived from the nucleus is required for division, and that the asters presumably play a regulatory role in the distribution of the nuclear factor. In May 1916, Painter enlisted in the National Guard at
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New Haven and became a sergeant of the Headquarters Company of the Tenth Regiment of Field Artillery. Discharged in September 1916, he married Anna Thomas on December 19, 1917. Their children—two boys and two girls—and, eventually, their grandchildren made a warm, closely knit family. With the advent of World War I in 1917, Painter was commissioned a first lieutenant of the U.S. Army Signal Corps and was sent to Toronto's Imperial Flying School to find out what measures were needed to establish a ground school of aviation in Austin. After the school was established, he served as a member of its academic board and was promoted in 1918 to captain in the U.S. Army Air Service. In April 1919 he retired as a captain of the Reserve Corps. Though Painter went to Austin in 1916 as an adjunct professor of zoology, military service interrupted his research for several years, and he was not promoted to associate professor until 1921. Four years later, in 1925, he was appointed full professor with membership in the graduate faculty. Painter was a man of broad interests and cheerful disposition. He often visited his students in the laboratory to exchange ideas, giving them encouragement as well as direction. He taught undergraduate courses in addition to graduate cytology, and—for many years—a popular premedical course in comparative anatomy. He played tennis and golf and loved swimming, fishing, and crabbing. He was also an inveterate hunter, liking nothing more than to take down his rifle to hunt deer or antelope. He was a fine gardener, and his flower displays were a marvel to all visitors. He particularly enjoyed hybridizing irises to produce new patterns of remarkable color. He was an expert with tools and made furniture for his home. In later years he turned to jewelry-making and again developed great skill at producing objects that reflected his fine taste. He took a strong part
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in his church's activities and in various clubs. In many ways the antithesis of the stereotypical Texan, he was both reserved and self-controlled. CHROMOSOME CYTOLOGY AND SEX CHROMOSOMES Back at the University of Texas after his military service, Painter resumed his cytological studies of spermatogenesis in a common small lizard, Anolis carolinensis. But he quickly turned to a new problem: the number of mammalian chromosomes and their morphology, with particular emphasis on the nature of sex determination. In the zoology laboratories of the Department, embryologist Carl G. Hartmann was engaged in studying the reproduction of the opossum. ''There was 'possum meat all over the lab," Painter remarked, a fine opportunity for him to switch from spiders, marine organisms, and lizards to the enticing field of mammalian cytology. Almost nothing was known about mammalian chromosomes at the time, although it was supposed that mammals must have sex chromosomes corresponding to those of insects and that an XX(female)-XY(male) distinction would exist. It proved quite easy, in fact, to find the sex chromosomes of the opossum, for they were the smallest pair of chromosomes in the cell, and during spermatogenesis they always lay in the center of a ring of the other, larger chromosomes during the metaphase of mitosis. In those days all tissues used for cytological examination were successively fixed, embedded in paraffin, sectioned, and stained. It was of prime importance to get the tissues fresh from dissection into the fixing fluid. Painter invented a sort of multibladed knife by mounting a number of safety razor blades in parallel, close together, which he used to cut up the spermatogenic tubules of the testis immediately after the organ was excised. Painter demonstrated that the male opossums sex is de
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termined by a tiny Y-chromosome in place of one of the female's larger Xchromosomes. He showed that in meiosis of the male's spermatocytes prior to formation of spermatozoa, the X and Y chromosomes pair and then segregate, so that each male reproductive cell carries either an X- or a Y-chromosome, but not both. As in insects, then, if all egg cells carry a single X-chromosome and if fertilization by the two sorts of spermatozoa is random, the X-bearing sperm would produce female offspring; the Y-bearing sperm would produce males. Having thus shown that sex determination in a marsupial mammal corresponds to the process already known from invertebrates, Painter set his sights on placental, or eutherian, mammals, and—through a fortunate circumstance—was able to obtain fresh human testicular tissue. One of his former premedical students was practicing medicine in a state mental institution in Austin where, "for therapeutic reasons," Painter wrote, "they occasionally castrated male individuals." Painter's former student made it possible for him to obtain and preserve, "within thirty seconds or less after the blood supply was cut off, a human testis" (1971,1). We students in the Austin laboratory speculated widely that such tissue was also obtained from criminals executed at the nearby Huntsville prison, but this was probably just idle gossip. Painter himself never confirmed such a source. Painter's first work on human chromosomes, therefore, preceded his study of primates, though their order of publication was reversed. A year before he published his fuller account of human spermatogenesis and human sex chromosomes (1923,1), a short announcement on the sex chromosomes of "the monkey" appeared in Science. To solve the enigma of sex determination in humans, Painter turned to two species of monkey—the New World Brown Cebus and the Old World Rhesus (Rhesus macacus). As
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he pointed out in this pioneering work (1924,3), it was highly desirable and perhaps necessary to establish four matters for each species examined: (1) the morphology of the diploid chromosome complex and the chromosome number of the male; (2) the haploid number revealed in the second spermatocytes; (3) the morphology and behavior of the sex chromosomes (X and Y) during meiosis; and (4) the morphology and chromosome number of the female complex. Cross-checks among these observations should bar all possibility of error, even though many species of mammals—including the primates Painter was investigating—have many more and much smaller chromosomes in their karyotypes than do opossums or the insect species in which the chromosomal determination of sex was first established. (A "karyotype" is the term used to designate the entire group of chromosomes characteristic of a cell of a particular species. This could be a diploid cell with two complete sets of chromosomes or, more frequently, the chromosome complement of a haploid cell with a single set of chromosomes—one of each distinctive kind characterizing the species.) Painter's demonstration of the X-Y type of sex determination in these mammals and in the human species was compelling. His drawings of the larger X-chromosome and the much smaller Y-chromosome, connected to each other by a thin strand while segregating in the first prophase of meiosis, left no doubt. The number of chromosomes was less certain. Some human cells seemed to show a count of forty-eight chromosomes in the diploid primary spermatocyte, others only forty-six. Previous investigators of human chromosome number also varied in their counts, though most settled for fortyeight. Painter himself took the evidence of his "best cell" and reported the number as forty-eight, confirming an error that
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would be perpetuated in dozens of textbooks (including one of my own) until a new set of techniques for counting chromosomes was introduced in the mid-1950s. In 1956, using new stains (such as acetocarmine and Feulgen's stain specific for DNA) and soft somatic tissues (especially embryonic tissues) that could be smeared; using colchicine to halt dividing cells in metaphase and hence greatly increase the number of such cells observable; and using hypotonic salt solutions to spread the chromosomes of dividing cells apart to eliminate their clumping into uncountable masses, J. H. Tjio and A. Levan made a definitive determination that the human diploid chromosome number is fortysix, i.e., twenty-three pairs of homologous chromosomes in human diploid cells. Painter experienced deep chagrin over this error in what had long been regarded as a primary discovery for which he was known and universally cited. Yet—given the source of his material and the procedures available to him in the early 1920s—he may not have been entirely wrong. Individuals with mental disorders are not prime material for determining normal chromosome number and morphology, for they sometimes have forty-seven, forty-eight, or even more chromosomes and exhibit more frequently than normal persons translocations and deletions of chromosomes that would appear to alter their number. Recently T. C. Hsu, a well-known cytogeneticist, reexamined some of the original preparations on which Painter based his erroneous chromosome count and found that the chromosomes were so badly clumped and cut into segments by the microtome knife, it was a marvel Painter was able to find any cells at all that seemed to give a clear chromosome count. Given that human chromosomes are exceedingly small, that the dyes used in the 1920s darkly stained other matter in addition to chromosomes, and that microtome slices rarely produced whole, undamaged cells for examina
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tion, Painter's error was wholly natural and forgivable. In any case, it in no way diminishes the importance of his discovery of the XX-XY mechanism for determining sex in mammals (including humans), a significant contribution to science. Painter subsequently examined and recorded the chromosome number of the horse (probably 60; XX-XY sex determination), the bat Nyctinomous mexicanus (2N = 48), the European hedgehog (2N=48), the armadillo (2N = 60), the rabbit (2N = 44), and the dog (2N prob. 52). Additional marsupials examined included—besides the opossum (2N = 22)—Phascolarctus (2N = 16), Sarcophilus (2N = 14), Dasyurus (2N = 14), and the kangaroo Macropus (2N = 12). Painter identified an XY pair of sex chromosomes in all of these marsupial and placental mammals except the hedgehog, armadillo, and dog—species he did not investigate extensively enough to judge—though an XY male type was not excluded in them either. In summary, Painter showed that marsupial mammals in general have a lower chromosome number than placental mammals; that all, or almost all, placentals (including humans) have a high chromosome number ranging from forty-four to sixty; and that all of them have, or probably have, an XX-XY type of sex determination depending upon a particular pair of sex chromosomes in which the Y-chromosome (carried by the male) is far smaller in size than the Xchromosome. If these studies placed Painter in the first rank of cytogeneticists, the focus of his next research project established him firmly in the forefront of classical genetics. One of Painter's students, E. K. Cox, had determined that the chromosome number of the common house mouse, Mus musculus, is forty. Yet W. H. Gates reported that a Japanese waltzing mouse found in the Fl offspring of a cross between normal (dominant) and Japanese waltzer (recessive) parents seemed
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to owe its phenotype to the loss of the chromosome carrying the normal dominant allele. Carefully examining descendants of this mouse, Painter found that all of them had the full complement of forty diploid chromosomes. He also determined that the males carried a typical XY chromosome pair and concluded, therefore, that the original mouse found to be exceptional by Gates could not have suffered the nondisjunctional loss of an entire chromosome—the one carrying the normal allele of the waltzing gene. He hypothesized instead that there had been a deletion of the part of that chromosome that normally carries the allele in question—a hypothesis he subsequently verified by observing that these mice carried two heteromorphic pairs of chromosomes, the sex chromosome pair, plus another pair in which one homologue was very much smaller than its partner. Painter's study of the Japanese waltzing mouse appears to have been the first cytological identification of a deletion producing a specific genetic effect (1927,1). DROSOPHILA CYTOGENETICS "One day," Painter wrote, " . . . I found [H. J.] Muller down on the floor with a pipette trying to recover some ovaries which he had spilled from a dish. As skillful as he was in genetic analysis, he didn't have great skill in handling such small material. So I suggested to him—I think I caught him just at the right time-'Why don't you let me study those ovaries and tell you where the oogonial chromosomes have actually been broken?' Again, it was a case of being in the right place at the right time! Muller furnished me with female Drosophila carrying a translocation and by examining oogonial metaphases I would determine how much of an exchange had taken place." (1971,1, pp. 34-35.) So began a collaboration that eventually led to groundbreaking, parallel investigations of genetic and cytological
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variations induced by the action of X-rays on genes and chromosomes and to Painter and Muller's paper on the parallel cytology and genetics of induced translocations and deletions in Drosophila—a genetics classic (1929,1). Though translocations investigated (III-Y and III-II) did not at that time reveal the fact that all translocations are actually reciprocal exchanges, they did show that the size of the cytological piece taken from one chromosome and attached to another did not correspond precisely in size to the portion of the genetic map that was translocated. The importance of this observation was greatly enhanced by the finding that—in the case of deletions of a coherent portion of the genetic map of the X-chromosome—the cytological loss was much greater than would be expected from the ratio of the lost portion to the total genetic length of the chromosome. This finding led, furthermore, to the discovery that there is a large portion of "heterochromatin" at the base of the Xchromosome—a segment that appears to carry few, if any, genes. Most of the deletions excised a considerable part of this heterochromatin. The two authors went on to find a case of a new linkage-group established by the translocation of a fragment carrying certain genes to an independent spindle fiber attachment. Only much later was it learned that this case represented a translocation of a portion of an autosome to the basal portion of a Chromosome IV that—having lost most of the regular fourth chromosome genes —could freely undergo nondisjunction, eventually to become a new pair of chromosomes. Painter published a cytological "map" of the X-chromosome that reflected this discovery, and Muller reported on their joint studies at the Sixth International Congress of Genetics in 1932. What is generally regarded as Painter's most notable discovery in cytogenetics occurred in 1932, while the writer of
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this memoir was still a graduate student in his Department. Quite independently, but simultaneously with E. Heitz and Hans Bauer in Switzerland, Painter identified the strange-looking tangled balls of thick strands to be seen in the nuclei of the salivary glands of all Diptera (first described by E. G. Balbiani in 1881) as being closely paired homologous chromosomes. Aided by the wealth of established genetical information then available on Drosophila melanogaster, he then carried the genetic analysis considerably further than his codiscoverers in Europe. Painter also introduced a new cytological method for making salivary gland preparations, mentioned casually in his first paper announcing the new kind of chromosomes (1933,2). It was an application of the acetocarmine smear method, long used by cytologists who worked on maize chromosomes. Painter adapted the method to the fruitfly. He simply dissected out the salivary glands from a third instar Drosophila larva in a drop of physiological saline solution, transferred the glands to a drop of acetocarmine stain, placed a coverglass over them, and—under the dissecting microscope—pressed with the point of a dissecting needle on each nucleus within the gland. When an appropriate amount of pressure was exerted, the nuclear membrane burst and the released chromosomes took up the stain in their numerous crossbands. Painter saw that there were six strands, one short and five long. Each strand remained attached at one end to a mass identified as a "chromocenter," the fused heterochromatin of each chromosome. Painter identified each chromosome by using Drosophila stocks that had a deletion of a portion of one chromosome that would enable that particular chromosome to be picked out. One strand was identified as the X-chromosome; two as the respective left and right arms of Chromosome II; and two as the left and right arms of Chro
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mosome III. The short strand, by process of elimination, was Chromosome IV. Painter recognized, again from the study of the giant Drosophila chromosomes in individuals that were heterozygous for a deletion, that each strand consists of two closely-paired, homologous chromosomes. By using a variety of genetically known stocks containing deletions of short portions of the sequence of genes in the X-chromosome (the supply of which was expertly furnished to Painter by Wilson S. Stone), Painter quickly made a cytological salivary chromosome map of the X-chromosome of D. melanogaster. The cytological sequence of genes was in the same order as the known genetic map of X-chromosome loci based on crossover frequencies, but the distances between genetic loci did not correspond exactly to the cytological map. While certain regions were expanded somewhat, others were contracted. In general, however, the agreement was very good—better than for the agreement between crossover linkage maps and the cytological map derived from ordinary somatic or germ cells that did not develop giant chromosomes. In a second paper published in 1934, Painter continued his analysis of giant salivary gland chromosomes in stocks carrying deletions, inversions, or translocations. When one chromosome of a homologous pair carried a deletion, the longer mate formed a loop or buckle at the region, so that the exact points of breakage of the deletion could be determined at the level of individual crossbands. In the case of a heterozygous inversion, a large loop was formed with the two homologues passing around the loop in opposed directions, so that every band could still find and pair precisely with its mate in the other chromosome. In translocations a crossshaped figure would result, for at the point of the exchanged strands, the chromosomes would switch partners. From these studies it became apparent that all translocations are in fact mutual—or reciprocal—exchanges, even
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though the fragment from one chromosome may be large and that from the other very small. It also became established, as Muller and others had previously conjectured, that the reattachments of fragments of broken chromosomes take place only between two broken ends, as though they were in some way "sticky," or as we would now say, through the reunion of broken chemical bonds. These studies showed conclusively, as the genetic studies had intimated, that the attraction between homologous chromosomes is point by point, locus by locus, band by band, and not a synapsis caused in some vague way by chromosomes as entire units. From the standpoint of physics and chemistry, this conclusion is one of the most interesting findings of cytogenetics. At this stage of his career, honors came rapidly to T. S. Painter. Yale University conferred on him the honorary degree of D.Sc. in 1936. He was awarded the Daniel Giraud Elliot Medal of the National Academy of Sciences in 1933 and was elected a member of the Academy in that same year. He was elected a member of the American Philosophical Society in 1939. Painter was greatly interested in the nature and function of the heterochromatin. From the comparison of salivary chromosomes with those of regular somatic cells or cells of the germ line, he concluded that about threeeighths of the X-chromosome of Drosophila is missing in the salivary gland chromosomes, and that the Y-chromosome of the male is missing almost entirely, although in the usual somatic cells the Y-chromosome—unlike the Y of a mammal—is very large, almost as large as the X-chromosome. The apparent disappearance in the salivary gland cells of the heterochromatin must, he thought, be related in some way to difference in function. The salivary gland cells did not seem to carry the usual kind of genes that become evident from their mutation.
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Musing over this problem, he was led away from the detailed task of chromosome mapping, which he willingly left to Calvin Bridges' sharp eyes and unending appreciation of detail. Painter resolved to seek out the functions of different kinds of genetic material, especially the heterochromatin. How, he wondered, does the altered nature of chromosomes in particular organs, such as salivary glands, relate to specialized cellular function? Except for a joint paper with Wilson Stone on the relation of chromosome fusion to speciation in the Drosophilidae (1935,3), and two papers (1935,2 and 4) —one written jointly with J. T. Patterson—on the salivary gland chromosome map of Chromosome III, Painter concentrated on this new direction until his research was interrupted in 1944. With his student Allen Griffen, he examined the course of development of the salivary gland nucleus in the fly Simulium virgatum in order to see just how the giant paired salivary gland chromosomes arose and what their structure might be in comparison with simpler, single-stranded chromatids of more ordinary cells. With another student, Elizabeth Reindorp, he traced the development of endomitosis in the nurse cells of the Drosophila ovary, a process that gives rise to multistranded chromosomes that do not aggregate and consolidate into giant chromosomes of the salivary gland type. He studied the synthesis of cleavage chromosomes and demonstrated that the rapid series of cleavage divisions, involving the synthesis of great numbers of new chromosomes from the original new sets in the zygote, or fertilized egg, would be impossible were it not for the abundant feeding of amino acids and nucleotides derived from previously synthesized proteins and nucleic acids in the nurse cells into the oocyte during its period of maturation. Cases of cytoplasmic or matroclinous inheritance might also be explained by the
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accumulation of such materials in the cytoplasm of the egg cell. Painter summarized this work at a Cold Spring Harbor Symposium in 1940 (1941,2). With A. N. Taylor he continued working on nucleic acid storage in the toad's egg, while with J. J. Biesele he examined the alterations in the nature of chromosomes in cancerous cells of the mouse, where much endomitosis and polyploidy were found. Painter even undertook to assay the relation of cell growth in the pollen grains of a flowering plant, Rhoeo discolor, to the amounts of nucleic acid they possessed—an investigation he initiated prior to Avery, MacLeod, and McCarty's demonstration that, in pneumococcus transformations of genetic type, it is the nucleic acid, not protein, that acts as the genetic material. In light of this research, Painter also seems to have suspected that nucleic acid was the material responsible for the hereditary transmission of characters. UNIVERSITY ADMINISTRATION In 1944 T. S. Painter's professional life changed abruptly: he became a university administrator. The president of the University of Texas at that time had defended the academic freedom of two faculty members who had engaged in liberal political activities and spoken at meetings of labor organizations. The Regents of the University forced the president to resign and looked hastily for a caretaker who could be expected to refrain from political action and at the same time would be of high academic reputation. A committee of three members of the faculty met with the Regents in order to make suggestions for a resolution of the difficulties, and Painter was one of the three. According to the minutes of the Special Committee of the Faculty that was delegated the task of preparing a memorial resolution following Painter's death,
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the committee of which Painter was a member met with the Regents and then retired for the night. After Dr. Painter was asleep, he was called and asked to return to the meeting. He was told that the president had been dismissed. The Board of Regents asked Dr. Painter to become the acting president. He faced a dilemma. His research program was at a critical stage. He received many pro and con opinions from the faculty and other friends of the University. Finally he decided to accept the temporary appointment because that seemed to be the best way to keep faculty control over the destiny of the University of Texas. He and the Regents asked the faculty to form a committee to suggest nominees for permanent president. When no satisfactory nominee was named, the Board of Regents appointed Dr. Painter to be president so that he could have full authority to carry out the needs of the University. The appointment was accepted with the stipulation that the term would last only until a satisfactory president could be found. Twice Dr. Painter wanted to resign from the presidency but each time he was persuaded to continue in the position. In 1952, his resignation was accepted and he returned to his duties as a teacher.
Without a doubt Painter served his university effectively during a most trying period. He played the role of conservative in the best sense. Although some members of the faculty protested when he accepted the change from acting president to president, because they felt that this was a repudiation of his promise not to accept an offer for the full presidency, it may have been the only reasonable solution at the time to an irreconcilable conflict between the state— represented by the Board of Regents and the governor—and the faculty of the University. Today, after decades have passed, the entire academic community can be grateful for Painter's skill at mediation and compromise. He retained the respect of all. RETURN TO SCIENCE Perhaps no challenge to a scientist who has absented himself for some years is as great as that of returning to an active program of scientific investigation. The exponential advance
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of science necessarily implies that during a lapse of even two or three years from the laboratory, fundamental changes in understanding will have occurred to such an extent that the returned scientist's grasp of current knowledge and mastery of available techniques are outmoded. So it was with Painter, but his determination was indomitable. His colleagues testify that he spent more time in the library reading current periodicals and books than did any graduate student. He also asked to be reassigned to the teaching of cell biology to undergraduates and cytology to graduate students, and thus added to his burden all the reviewing and relearning required for teaching. As the Memorial Resolution prepared by his fellow faculty members records, he was successful: He developed a good knowledge of modern cellular molecular biology. Often he noticed that a researcher's data could be used to answer in part some classical biological problem, although the author had not mentioned that possibility. The interpretations were too narrow in coverage. As a consequence, Dr. Painter decided to teach his students the recent, chemicallyoriented discoveries and to make certain that they had a broader basic training in biology so that they could understand the biological implications of the discoveries. To Dr. Painter, a narrow channel of research may find answers for one small field of interest, but it will not serve the purpose of biology unless it has some major impact upon a basic biological problem.
One can verify his concern with the broader implications by glancing at eleven scientific papers written by Painter between 1955 and 1969. They seem to follow naturally from the earlier work on the salivary chromosomes of dipterans and the endomitosis in the nurse cells of the ovary. But they all probe the greater question of how it is that the hereditary materials passed down from one generation to another in the course of reproduction are converted into a multiplicity of end products in different tissues. Working with J. J. Biesele again—and with the advantage of electron microscopy—Painter was able to show how the
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precursors needed for the secretion of royal jelly (the only food consumed by the queen bee) are produced in the honey bee in special gland cells of young worker bees. Producing as many as 1000 eggs a day, the queen bee requires a considerable supply of both proteins and DNA, which is supplied by the royal jelly. When workers feed heavily on bee bread, their gland cells develop and produce the royal jelly. According to George E. Palade, Keith Porter, and others, royal jelly gland cells in the young worker bees produce the proteins by means of an extensively developed endoplasmic reticulum. Painter and Biesele searched for the origin of this cellular structure of endoplasmic tubules that apparently derive from outpockets of the nuclear membrane of the cell as the gland cell undergoes endomitosis. As this process enters a stage comparable to the prophase of ordinary mitosis, the numerous nuclei in the gland cell fragment and a myriad of ribosome-like bodies pass out through nuclear pores to become the polyribosomes attached to the walls of the endoplasmic tubules. This process clearly shows how an ovum becomes enriched with protein and nucleotides. In his final paper, Painter advised young researchers from his own experience: ''I get the impression that young people [today] master some sophisticated technique such as labeling cellular structures with radioactive isotopes followed by autoradiography, DNA and RNA hybridization, ultracentrifugation in gradients and all the rest and then look around to see how they can use their acquired skills! From my experience I think you should first select and define some broad biological problems, select a suitable material upon which to work and use any available techniques for the solution of your problem. The most important thing is for you to have a biological and not a test tube approach." (1971,1)
How well his own research exemplified that ability to identify the problem, find the right material, and develop the necessary techniques!
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Although research always stood foremost in his heart, Painter found time and energy for many other activities. He served on the University of Texas Premedical, Predental, and Library committees. He frequently attended the meetings of scientific societies and, in addition to serving on other committees of the American Philosophical Society, was a member of its Council from 1965 to 1967. He served for six years on the Council of the National Academy of Sciences and six more on its Finance Committee. He was a member of the American Society of Zoologists, the Genetics Society of America, the Association of American Anatomists, the American Society of Naturalists, and the Società Italiana di Biologia Sperimentale. He was a member of the Boy Scouts of America Committee (1935-40), an advisor to the Dental Research Council (1949-52), and advisor on research to the American Cancer Society. He served on the Commission on Colleges and Universities of the Southern Association and was its chairman for three years; the Southern Regional Education Board; the National Committee on Accreditation; and the Board of the Institute of Nuclear Studies at Oak Ridge. He was a National Lecturer for Sigma Xi in 1936-37. Locally, he was a member of the Rotary Club, Town and Gown, and the English Speaking Union. He was elected to the Hall of Fame for Famous Americans, served as president of the American Society of Zoologists in 1940, and received the first M. D. Anderson Award for Scientific Creativity and Teaching from the M. D. Anderson Hospital and Tumor Institute in 1969. Perhaps what he regarded most highly among his honors was his elevation to the rank of distinguished professor of the University of Texas in 1939. It was characteristic of him that he died as he had lived—suddenly, on his return home to Fort Stockton, Texas, from
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a hunting trip, in his eighty-first year and as active as ever. Two papers—"The Origin of the Nucleic Acid Bases Found in the Royal Jelly of the Honey bee" (1969,1) and "Chromosomes and Genes Viewed from a Perspective of Fifty Years" (1971,1)—appeared posthumously. The author of this memoir is deeply indebted to the University of Texas Faculty Committee that prepared the Memorial Minute on T. S. Painter that is quoted above. Members of this Committee were C. P. Oliver, chairman; J. J. Biesele; and R. P. Wagner. I would also like to acknowledge with deep gratitude the receipt of various documents, both published and unpublished, from Mrs. T. S. Painter. Without access to them there would have been serious gaps in the account, especially in respect to T. S. Painter's administrative career.
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Selected Bibliography
1913 On the dimorphism of the males of Maevia vittata, a jumping spider. Zool. Jahrb. Abt. Syst. Oekol. Geogr. Tiere, 35:625-35. 1914 Spermatogenesis in spiders. I. Zool. Jahrb. Abt. Anat. Ontog. Tiere, 38:509-76. The effect of carbon dioxide on the eggs of Ascaris. Proc. Soc. Exp. Biol. Med. , 11:62-64. 1915 An experimental study in cleavage. J. Exp. Zool., 18:299-323. The effects of carbon dioxide on the eggs of Ascaris. J. Exp. Zool., 19:355-85. 1916 Some phases of cell mechanics. Anat. Rec., 10:232-33. Contributions to the study of cell mechanics. I. Spiral asters. J. Exp. Zool., 20:509-27. 1917 A wing mutation in Piophila casei. Am. Nat., 51:306-8. 1918 Contributions to the study of cell mechanics. II. Monaster eggs and narcotized eggs. J. Exp. Zool., 24:445-97. 1919 The spermatogenesis of Anolis carolinensis. Anat. Rec., 17:328-29. 1921 Studies in reptilian spermatogenesis. I. The spermatogenesis of lizards. J. Exp. Zool., 34:281-327. The Y-chromosome in mammals. Science, 503-4.
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1922 Studies in mammalian spermatogenesis. I. The spermatogenesis of the opossum (Didelphys virginiana) . J. Exp. Zool., 35:13-38. The sex chromosomes of the monkey. Science, 56:286-87. 1923 Studies in mammalian spermatogenesis. II. The spermatogenesis of man. J. Exp. Zool., 37:291-336. Further observations on the sex chromosomes of mammals. Science, 58:247-48. 1924 A technique for the study of mammalian chromosomes. Anat. Rec., 27:77-86. Studies in mammalian spermatogenesis. III. The fate of the chromatin-nucleolus in the opossum. J. Exp. Zool., 39:197-227. Studies in mammalian spermatogenesis. IV. The sex chromosomes of monkeys. J. Exp. Zool., 39:433-62. Studies in mammalian spermatogenesis. V. The chromosomes of the horse. J. Exp. Zool., 39:229-47. The sex chromosomes of man. Am. Nat., 58:506-24. 1925 Chromosome numbers in mammals. Science, 61:423-24. A comparative study of the chromosomes of mammals. Am. Nat., 59:385-409. The chromosomes of the rabbit. Anat. Rec., 31:304. A comparative study of the chromosomes of the largest and the smallest races of rabbits. Anat. Rec., 31:304. A comparative study of mammalian chromosomes. Anat. Rec., 31:305. 1926 The chromosomes of rodents. Science, 64:336. Studies in mammalian spermatogenesis. VI. The chromosomes of the rabbit. J. Morphol. Physiol., 43:1-43.
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1927 The chromosome constitution of Gates' "non-disjunction" (v-o) mice. Genetics, 12:379-92. 1928 A comparison of the chromosomes of the rat and mouse with reference to the question of chromosome homology in mammals. Genetics, 13:180-89. The chromosome constitution of the Little and Bagg abnormaleyed mice. Am. Nat., 62:284-86. Cell size and body size in rabbits. J. Exp. Zool., 50:441-53. 1929 With H. J. Muller. Parallel cytology and genetics of induced translocations and deletions in Drosophila. J. Hered., 20:287-98. With H. J. Muller. The cytological expression of changes in gene alignment produced by X-rays in Drosophila. Am. Nat., 63:193-200. 1930 Recent work on human chromosomes. J. Hered., 21:61-64. Translocations, deletions, and breakage in Drosophila melanogaster . Anat. Rec., 47:392. 1931 With J. T. Patterson. A mottled-eyed Drosophila. Science, 73:530-31. A cytological map of the X-chromosome of Drosophila melanogaster. Science, 73:647-48. (Also in: Anat. Rec., 51:111.) 1932 With H.J. Muller. A cytological map of the X-chromosome of Drosophila. Proc. 6th Int. Congr. Genetics (Ithaca), 2:147-48. With H. J. Muller. The differentiation of the sex chromosomes of Drosophila into genetically active and inert regions. Z. Indukt. Abstamm. Vererbungsl., 62:316-65.
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1933 A method for the qualitative analysis of the chromosomes of Drosophila melanogaster. Anat. Rec., 57 (Suppl.):90. A new method for the study of chromosome rearrangements and the plotting of chromosome maps. Science, 78:585-86. 1934 A new method for the study of chromosome aberrations and the plotting of chromosome maps in Drosophila melanogaster. Genetics, 19:175-88. The morphology of the X-chromosome in salivary glands of Drosophila melanogaster and a new type of chromosome map for this element. Genetics, 19:448-69. A new type of cytological map of the X-chromosome in Drosophila melanogaster. Am. Nat., 68:75-76. (Also in: Genetics Soc. Am., 2:45-46.) Salivary chromosomes and the attack on the gene. J. Hered., 25:464-76. 1935 Salivary gland chromosomes in Drosophila melanogaster. Am. Nat., 69:74. The morphology of the third chromosome in the salivary gland of Drosophila melanogaster and a new cytological map of this element. Genetics, 20:301-26. With Wilson S. Stone. Chromosome fusion and speciation in Drosophile. Genet., 20:327-41. With J. T. Patterson. Localization of gene loci in the third chromosome of Drosophila melanogaster. Rec. Genetics Soc. Am., 4:76. (Also in: Am. Nat., 70:59.) Some recent advances in our knowledge of chromosomes. Advances in Modern Biology, 2:216-23. Moscow: State Biological and Medical Press. 1937 Eds. T. S. Painter and J. B. Gatenby. Bolles Lee's The Microtomist's Vade-Mecum, 10th ed. Philadelphia: P. Blakiston's Son & Co. With Allen B. Griffen. The origin and structure of the salivary
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gland chromosomes of Simulium virgatum. Genet., 22:202-3. (Also in: Rec. Genetics Soc. Am., 5:202-3.) With Allen B. Griffen. The structure and the development of the salivary gland chromosome of Simulium. Genetics, 22:612-33. 1939 The structure of salivary gland chromosomes. Am. Nat., 73:315-30. An acetocarmine method for bird and mammalian chromosomes. Science, 90:307-8. With Elizabeth C. Reindorp. Endomitosis in nurse cells of the ovary of Drosophila melanogaster. Chromosoma, 1:276-83. Chromosomes in relation to heredity. In: Science in Progress. Ed. George Baitsell. New Haven: Yale University Press, 1:210-32. 1940 The chromosomes of the chimpanzee. Science, 91:74-75. On the synthesis of cleavage chromosomes. Proc. Natl. Acad. Sci. USA, 26:95-100. A review of some recent studies of animal chromosomes. J. R. Microsc. Soc., 60:161-76. With A. N. Taylor. Nuclear changes associated with the growth of the oocytes in the toad. Anat. Rec., 78(Suppl.):84. 1941 The effects of an alkaline solution (pH 13) on salivary gland chromosomes. Genet., 26:163-64. (Also in: Rec. Genetics Soc. Am., 9:163-64.) An experimental study of salivary chromosomes. Cold Spring Harbor Symp. Quant. Biol., 9:47-53. 1942 With A. N. Taylor. Nucleic acid storage in the toad's egg. Proc. Natl. Acad. Sci. USA, 28:311-17. With J. J. Biesele and H. Poyner. Nuclear phenomena in mouse cancer. Univ. Tex. Publ., 4243:1-68. 1943 With L. J. Cole. The genetic sex of Pigeon-Ring Dove hybrids as determined by their sex chromosomes. J. Morphol., 72:411-39.
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Cell growth and nucleic acids in the pollen of Rhoeo discolor. Bot. Gaz., 105:58-68. The effects of alkali and urea on different types of chromosomes. Anat. Rec., 87:462. 1944 A cytologist looks forward. Tex. Rep. Biol. Med., 2:206-22. The effects of urea and alkali on chromosomes and the interpretative value of the dissolution images produced. J. Exp. Zool., 96:53-76. 1945 Chromatin diminution. Trans. Conn. Acad. Arts Sci., 36:443-48. Nuclear phenomena associated with secretion in certain gland cells with special reference to the origin of the cytoplasmic nucleic acid. J. Exp. Zool., 100:523-47. 1953 Some cytological aspects of the nucleic acid problem. Tex. Rep. Biol. Med., 11:709-14. 1954 Regional cooperation in education. Proc. Am. Philos. Soc., 93:266-69. 1955 Do nuclei of living cells contain more DNA than is revealed by the Feulgen stain? Tex. Rep. Biol. Med., 13:659-66. 1958 The selection and recruitment of graduate students—"First, you must catch your hare." Grad. J., 1:41-50. 1959 The elimination of DNA from soma cells. Proc. Natl. Acad. Sci. USA, 45:897-902. Some values of endomitosis. In: Biological Contributions. Ed. Marshall Wheeler. University of Texas, 5914:235-40.
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1964 Fundamental chromosome structure. Proc. Natl. Acad. Sci. USA, 51:1282-85. With J. J. Biesele and R. W. Riess. Fine structure of the honey bee's royal jelly gland. Tex. J. Sci., 16:478. 1965 John Thomas Patterson (November 3, 1878-December 4, 1960). In: Biographical Memoirs, vol. 38, pp. 223-62. New York: Columbia University for the National Academy of Sciences. 1966 With J. J. Biesele. The fine structure of the hypopharyngeal gland cell of the honey bee during development and secretion. Proc. Natl. Acad. Sci. USA, 55:1414-19. With J. J. Biesele. A study of the royal jelly gland cells of the honey bee as revealed by electron microscopy. In: Studies in Genetics . III. Morgan Centennial Issue, ed. Marshall R. Wheeler. Univ. Tex. Publ., 6615:475-99. The role of the E-chromosomes in Cecidomyiidae. Proc. Natl. Acad. Sci. USA, 56:853-55. With J. J. Biesele. Endomitosis and polyribosome formation. Proc. Natl. Acad. Sci. USA, 56:1920-25. (Also in: Science, 154:426.) 1969 The origin of the nucleic acid bases found in the royal jelly of the honey bee. Proc. Natl. Acad. Sci. USA, 64:64-66. 1971 Chromosomes and genes viewed from a perspective of fifty years of research. L. J. Stadler Memorial Symp., 1:33-42.
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George Pólya December 13, 1887-September 7, 1985
By R. P. Boas George (György) Pólya made many significant contributions to mathematics and at the same time—rather unusually for a distinguished research mathematician—was an effective advocate of improved methods for teaching mathematics. His research publications extend from 1912 to 1976; his publications about teaching began in 1919 and continued throughout his life. For several decades, he was steadily initiating new topics and making decisive contributions to more established ones. Although his main mathematical interest was in analysis, at the peak of his career he was contributing not only to real and complex analysis, but also to probability, combinatorics, occasionally to algebra and number theory, and to the theory of proportional representation and voting. His work typically combined great power and great lucidity of exposition. Although much of his work was so technical that it can be fully appreciated only by specialists, a substantial number of his theorems can be stated simply enough to be appreciated by anyone who has a moderate knowledge of mathematics. As a whole, Pólya's work is notable for its fruitfulness. All his major contributions have been elaborated on by other mathematicians and have become the foundations of important branches of mathematics. In addition to his more sub
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stantial contributions, Pólya made many brief communications, ranging from the many problems that he proposed to brief remarks—a considerable number of which became the germs of substantial theories in the hands of other mathematicians. A student who needs a topic for research could do worse than look through Pólya's short papers. Pólya's papers were published in four volumes: the first two devoted to complex analysis, the third to other branches of analysis including mathematical physics, the fourth to probability, combinatorics, and teaching and learning in mathematics.1 ORIGINS AND CAREER Pólya was born in Budapest on December 13, 1887, and died in Palo Alto, California, September 7, 1985. In 1918 he married Stella Vera Weber, who survived him; they had no children. He received his doctorate in mathematics— first having studied law, language, and literature—from the University of Budapest in 1912. After two years at Göttingen and a short period in Paris, he accepted a position as Privatdocent at the Eidgenössische Technische Hochschule (Swiss Federal Institute of Technology) in Zürich in 1914 and rose to full professor there in 1928. In 1924 he was the first International Rockefeller Fellow and spent the year in England. In 1933 he was again a Rockefeller Fellow at Princeton. He emigrated to the United States in 1940, held a position at Brown for two years, spent a short time at Smith College, and in 1952 became a professor at Stanford. He retired in 1954 but continued to teach until 1978. He was elected to the National Academy of Sciences in 1976.
1 George Pólya, Collected Papers, 4 vols. (Cambridge, Massachusetts, and London, England: MIT Press, 1974-1984).
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PROBABILITY Pólya's first paper was in this field, and during his career he contributed perhaps thirty papers to various problems in probability theory. These papers contain many results that have now become textbook material, or even exercises, so that every student of probability encounters Pólya's work. One of Pólya's best known results is typical of his style, being unexpected but simple enough to prove once it was thought of. The Fourier transform
of a one-dimensional probability measure is known as the characteristic function. Pólya discovered (1918,3; 1923,2) that a sufficient condition for a realvalued function to be a characteristic function is that f(0) = 1, f(∞) = 0, f(t) = f(t), and f is convex, t > 0. This is the only useful general test for characteristic functions, even though the most famous characteristic function, exp(-t2), is not covered by it. In 1921, Pólya initiated the study of random walks (which he named), proving the striking and completely unintuitive theorem that a randomly moving point returns to its initial position with probability 1 in one or two dimensions, but not in three or more dimensions (1921,4). His other contributions to the subject are less easily explained informally but, like those just mentioned, have served as starting points for extensive theories. These include limit laws (Pólya also named the central limit theorem), the continuity theorem for moments, stable distributions, the theory of contagion and exchangeable sequences of random variables, and the roots of random polynomials.
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COMPLEX ANALYSIS Complex analysis is the study of analytic functions in two dimensions— the field to which Pólya made his most numerous contributions. As every student of the subject learns at an early stage, a function f that is analytic at a point, say 0 (for the sake of simplicity), is represented by a convergent power series
and conversely such a series, if convergent, represents an analytic function. In principle the sequence {an} of coefficients contains all the properties of the function. The problem is to make the sequence surrender the desired information. The most attractive results are those that connect a simple property of the coefficients with a simple property of the function. Pólya made many contributions to this subject. He proved that the circle of convergence of a power series is ''usually" a natural boundary for the function— that is, a curve past which the sum of the series cannot be continued analytically (1916,1; 1929,1). It is, in fact, always possible to change the signs of the coefficients in such a way that the new series cannot be contained outside the original circle of convergence. According to Fabry's famous gap theorem, the circle of convergence of a power series is a natural boundary if the density of zero coefficients is 1. Pólya proved that no weaker condition will suffice for the same conclusion. He also extended this theorem in several ways and found analogs of Fabry's theorem for Dirichlet series, which have a more complex theory.
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In 1929, Pólya systematized his methods for dealing with problems about power series (1929,1). This very influential paper deals with densities of sequences of numbers, with convex sets and with entire functions of exponential type—that is, with functions analytic in the whole complex plane whose absolute values grow no faster than a constant multiple of some exponential function eA|z|. Functions of this kind have proved widely applicable in physics, communication theory, and in other branches of mathematics. The central theorem is Pólya's representation of a function f as a contour integral that resembles a Laplace transform,
a representation important in contexts far beyond those that Pólya originally envisioned. Another topic that interested Pólya was how the general character of a function is revealed by the behavior of the function on a set of isolated points. The whole subject originated with Pólya's discovery (1915, 2) that 2z is the "smallest" entire function, not a polynomial, that has integral values at the positive integers. There are many generalizations, on which research continued at least into the 1970s, and the theory is still far from complete. Pólya also contributed to many other topics in complex analysis, including the theory of conformal mapping and its extensions to three dimensions. One of Pólya's favorite topics was the connections between properties of an entire function and the set of zeros of polynomials that approximate that function. He and I. Schur introduced two classes (now known as Pólya-Schur or Laguerre-Pólya functions) that are limits of polynomials that have either only real zeros or only real positive zeros. There are now many more applications, both in pure and applied
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mathematics, than Pólya himself envisaged, including, for example, the inversion theory of convolution transforms and the theory of interpolation by spline functions. Another series of papers starting in 1927 was devoted to zeros of trigonometric integrals,
Pólya was much interested in the Riemann hypothesis about the zeros of the zeta function. His work on trigonometric integrals was inspired by the fact that a sufficiently strong theorem about their zeros would establish the hypothesis. That this hope has, so far, proved illusory, does not diminish the importance of Pólya's results in both mathematics and physics. Pólya devoted a great deal of attention to the question of how the behavior in the large of an analytic or meromorphic function affects the distribution of the zeros of the derivatives of the function. One of the simplest results (simplest to state, that is) is that when a function is meromorphic in the whole plane (has no singular points except for poles), the zeros of its successive derivatives become concentrated near the polygon whose points are equidistant from the two nearest poles. The situation for entire functions is much more complex, and Pólya conjectured a number of theorems that are only now becoming possible to prove. REAL ANALYSIS, APPROXIMATION THEORY, NUMERICAL ANALYSIS Pólya's most important contributions to this area are contained in the book on inequalities he wrote in collaboration with Hardy and Littlewood (1934,2). This was the first systematic study of the inequalities used by all working analysts in their research and has never been fully superseded by any of the more recent books on the subject.
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Peano's space-filling curve passes through every point of a plane area but passes through some points four times. In 1913, Pólya produced a construction for a similar curve that has, at most, triple points, the smallest possible number. In keeping with Pólya's principle of drawing pictures whenever possible, the construction is quite geometrical (1913,1). Pólya devoted two papers more than fifty years apart to Graeffe's method for numerical solution of polynomial equations (1914,2; 1968,1). Although this method is useful for functions other than polynomials as well, it was not highly regarded originally because of the large amount of computation it requires. With the availability of modern high-speed computers, however, the method is becoming more useful. His pioneering investigation of the theory of numerical integration (1933,1) is still important today in numerical analysis. COMBINATORICS Combinatorics addresses questions about the number of ways there are to do something that is too complicated to be analyzed intuitively. Pólya's chief discovery was the enumeration of the isomers of a chemical compound, that is, the chemical compounds with different properties but the same numbers of each of their constituent elements. The problem had baffled chemists. Pólya treated it abstractly as a problem in group theory and was able to obtain formulas that made the solution of specific problems relatively routine. With the abstract theory in hand, Pólya could solve many concrete problems in chemistry, logic, and graph theory. His ideas and methods have been still further developed by his successors. A related problem is the study of the symmetry of geometric figures, for example, tilings of the plane by tiles of particular shapes. Pólya's paper (1937,2) came to the attention of the artist M. C. Escher, who used it in constructing his famous pictures of interlocked figures.
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The theory of symmetries also plays an important role in Pólya's work in mathematical physics. MATHEMATICAL PHYSICS Physical problems in two or three dimensions usually depend in essential ways on the shape of the domain in which the problems are considered. For example, the shape of a drumhead affects the sound of the drum; the electrostatic capacitance of an object depends on its shape. Except for very simple shapes, such as circles or spheres, the mathematical equations that describe the properties are too difficult to solve exactly; the solutions must be approximated in some way. Pólya's contributions to mathematical physics consisted of developing methods for such approximations. These methods, like his work in other fields, were subsequently developed further by others. Pólya was interested in estimating quantities of physical interest connected with particular domains, as, for example, electrostatic capacitance, torsional rigidity, and the lowest vibration frequency. Usually one wants an estimate for some property of a domain in terms of another. The simplest problem of this kind (and the oldest—it goes back to antiquity) is the isoperimetric problem, in which the area inside a curve is compared with the perimeter, or the volume of a solid is compared with its surface area. Problems of this kind, consequently, go by the generic name of isoperimetric problems. One method to which Pólya devoted a great deal of work, including the production of a widely read book (1951,1), is to replace a given domain by a more symmetric one with one property (say, the area inside a curve) the same, and for which the other property is more easily discussed. If we know that symmetrization increases or decreases the quantity in which we are interested, the result is an inequality for the other property.
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One of the earliest successes of this technique was a simple proof of Rayleigh's conjecture that a circular membrane has the lowest vibration frequency (that is, the smallest eigenvalue of the corresponding differential equation) among all membranes of a specified area. For different problems, different kinds of symmetrization are needed. Many physical quantities that are determined as the solutions of extremal problems can be estimated by making appropriate changes of variable, a technique known as transplantation. Pólya exploited this technique in a long paper in collaboration with M. Schiffer (1954,1). He also contributed several refinements to the standard technique of approximating solutions of partial differential equations by solving difference equations (1952,1; 1954,2). TEACHING AND LEARNING MATHEMATICS Pólya believed that one should learn mathematics by solving problems. This led him to write, with G. Szego, Problems and Theorems in Analysis (1925,1 [2 vols., in German]; 1972,1 [vol. 1], and 1976,1 [vol. 2] revised and enlarged English translation) in which topics are developed through series of problems. Besides their use for systematic instruction, these volumes are a convenient reference for special topics and methods. Pólya thought a great deal about how people solve problems and how they can learn to do so more effectively. His first book on this subject (1945,2) was very popular and has been translated into many languages. He wrote two additional books (1954,3; 1962,1) and many articles on the same general theme. Pólya also stressed the importance of heuristics (essentially, intelligent guessing) in teaching mathematics and in mathematical research. In the preface to Problems and Theorems, for example, he and Szego remarked that—since a straight line is determined by a point and a parallel—geom
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etry suggests, by analogy, ways of approaching problems that have nothing to do with geometry. One can hope both to generate new problems and to guess methods for solving them by generalizing a well-understood problem, by interpolating between two problems or by thinking of a parallel situation. One can see these principles at work in some of Pólya's research, and many other mathematicians have found them helpful. Whether heuristics can really be successful on a large scale as a teaching technique has not yet been established. Some researchers in artificial intelligence have not found it effective for teaching mathematics. It is not clear, however, whether these results reflect more unfavorably on Pólya or artificial intelligence. It does seem clear that putting Pólya's ideas into practice on a large scale would entail major changes both in the mathematics curriculum and in the training of teachers of mathematics. Pólya also stressed geometric visualization of mathematics wherever possible, and "Draw a figure!" was one of his favorite adages. In preparing this memoir I have drawn on the introductions and notes in the Collected Papers and, to a large extent, on the more detailed memoir prepared by G. L. Alexanderson and L. H. Lange for the Bulletin of the London Mathematical Society, which I had the opportunity of seeing in manuscript and to which I also contributed.
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Selected Bibliography
1913 Über eine Peanosche Kurve. Bull. Acad. Sci. Cracovie, A, 305-13. Sur un algorithme toujours convergent pour obtenir les polynomes de meilleure approximation de Tchebychef pour une fonction continue quelconque. C. R. Acad. Sci. (Paris), 1957:840-43. Über Annäherung durch Polynome mit lauter reelen Wurzeln. Rend. Circ. Mat. Palermo, 36:279-95. Über Annäherung durch Polynome deren sämtliche Wurzeln in einen Winkelraum fallen. Nachr. Ges. Wiss. Göttingen, 1913:326-30. 1914 With G. Lindwart. Über einen Zusammenhang zwischen der Konvergenz von Polynomfolgen und der Verteilung ihrer Wurzeln. Rend. Circ. Mat. Palermo, 37:297-304. Über das Graeffesche Verfahren. Z. Mat. Phys., 63:275-90. With I. Schur. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 144:89-113. Sur une question concernant les fonctions entières. C. R. Acad. Sci. (Paris), 158:330-33. 1915 Algebraische Untersuchungen über ganze Functionen vom Geschlechte Null und Eins. J. Reine Angew. Math., 145:224-49. Über ganzwertige ganze Funktionen. Rend. Circ. Mat. Palermo, 40:1-16. 1916 With A. Hurwitz. Zwei Beweise eines von Herrn Fatou vermuteten Satzes. Acta Math., 40:179-83. Über den Zusammenhang zwischen dem Maximalbetrage einer analytischen Funktion und dem grössten Gliede der zugehörigen Taylorschen Reihe. Acta Math., 40:311-19. Über Potenzreihen mit ganzzahligen Koeffizienten. Math. Ann., 77:497-513.
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1917 Uber geometrische Wahrscheinlichkeiten. S.-B. Akad. Wiss. Denksch. Philos. Hist. Kl., 126:319-28. Über die Potenzreihen, deren Konvergenzkreis natürliche Grenze ist. Acta Math., 41:99-118. 1918 Uber Potenzreihen mit endlich vielen verscheidenen Koeffizienten. Math. Ann., 78:286-93. Über die Verteilung der quadratischen Reste und Nichtreste. Nachr. Ges. Wiss. Göttingen, 1918:21-29. Über die Nullstellen gewisser ganzer Funktionen. Math. Z., 2:352-83. 1919 Über das Gauss'sche Fehlergesetz. Astronom. Nachr., 208:186-91; 209:111. Proportionalwahl und Wahrscheinlichkeitsrechnung. Z. Gesamte Staatswiss., 74:297-322. 1920 Arithmetische Eigenschaften der Reihenentwicklungen rationaler Funktionen. J. Reine Angew. Math., 151:1-31. Über den zentralen Grenzwertsatz der Wahrscheinlichkeitsrechnung und das Momentenproblem. Math. Z., 8:171-81. Uber ganze ganzwertige Funktionen. Nachr. Ges. Wiss. Göttingen, 1920:1-10. 1921 Bestimmung einer ganzen Funktion endlichen Geschlechts durch viererlei Stellen. Mat. Tidsskr. B.: 16-21. Ein Mittelwertsatz für Funktionen mehrerer Veränderlichen. Tôhoku Math. J., 19:1-3. Über eine Aufgabe der Wahrscheinlichkeitsrechnung betreffend die Irrfahrt im Strassennetz. Math. Ann., 84:149-60. 1922 Über die Nullstellen sukzessiver Derivierten. Math. Z., 12:36-60.
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1923 Sur les séries entières à coefficients entiers. Proc. London Math. Soc., 21:22-38. Herleitung des Gauss'schen Fehlergesetzes aus einer Funktionalgleichung. Math. Z., 18:96-108. Bemerkungen über unendliche Folgen und ganze Funktionen. Math. Ann., 88:169-83. Über die Existenz unendlich vieler singulärer Punkte auf der Konvergenzgeraden gewisser Dirichletscher Reihen. S.-B. Preuss. Akad. Wiss. Göttingen Math. Phys. Kl. Abh. Folge 3., 1923:45-50. With F. Eggenberger. Über die Statistik verketteter Vorgänge. Z. Angew. Math. Mech., 3:279-89. On the zeros of an integral function represented by Fourier's integral. Mess. Math., 52:185-88. 1924 Über die Analogie der Krystallsymmetrie in der Ebene. Z. Kristall., 60:278-82. On the mean-value theorem corresponding to a given linear homogeneous differential equation. Trans. Am. Math. Soc., 24:312-24. 1925 With G. Szego. Aufgaben und Lehrsätze aus der Analysis. 2 vols. Berlin: Springer-Verlag. 1926 On an integral function of an integral function. J. London Math. Soc., 1:12-15. On the minimum modulus of integral functions of order less than unity. J. London Math. Soc., 1:78-86. 1927 With G. H. Hardy and A. E. Ingham. Theorems concerning mean values of analytic functions. Proc. R. Soc. A., 113:542-69. Über trigonometrische Integrale mit nur reelen Nullstellen. J. Reine Angew. Math., 158:6-18. Über die algebraisch-funktionentheoretischen Untersuchungen
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von J. L. W. V. Jensen. Kgl. Danske Vidensk. Selsk. Math. Fys. Medd., 7(17). Eine Verallgemeinerung des Fabryschen Lückensatzes. Nachr. Ges. Wiss. Göttingen, 1927:187-95. 1928 Über gewisse notwendige Determinantenkriterien für die Fortsetzbarkeit einer Potenzreihe. Math. Ann., 99:687-706. Beitrag zur Verallgemeinerung des Verzerrungssatzes auf mehrfach zusammenhängende Gebiete. S.B. Preuss. Akad. Wiss. Göttingen Math. Phys. Kl. Abh. Folge 3., 1928:228-32, 280-82; 1929:55-62. 1929 Untersuchungen über Lücken und Singularitäten von Potenzreihen. Math. Z., 29:549-640. 1931 With G. Szego. Über den transfiniten Durchmesser (Kapazitätskonstante) von ebenen und räumlichen Punktmengen. J. Reine Angew. Math., 165:4-49. 1932 With A. Bloch. On the roots of certain algebraic equations. Proc. London Math. Soc., 33:102-14. 1933 Über die Konvergenz von Quadraturverfahren. Math. Z., 37:264-86. Qualitatives über Wärmeausgleich. Z. Angew. Math. Mech., 13:125-28. Untersuchungen über Lücken und Singularitäten von Potenzreihen. II. Ann. of Math. (2), 34:731-77. 1934 Über die Potenzreihenentwicklung gewisser mehrdeutiger Funktionen. Comment. Math. Helv., 7:201-21. With G. H. Hardy and J. E. Littlewood. Inequalities. Cambridge: Cambridge University Press.
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1936 Algebraische Berechnung der Anzahl der Isomeren einiger organischer Verbindungen. Z. Kristall. (A), 93:315-43. 1937 With M. Plancherel. Fonctions entières et intégrales de Fourier multiples. Comment. Math. Helv., 9:224-48; 10:110-63. Kombinatorische Anzahlbestimmungen für Gruppen, Graphen und chemische Verbindungen. Acta Math., 68:145-254. Über die Realität der Nullstellen fast aller Ableitungen gewisser ganzer Funktionen. Math. Ann., 114:622-34. 1938 Sur la promenade au hasard dans un réseau de rues. Actual. Sci. Ind., 734:25-44. 1942 On converse gap theorems. Trans. Am. Math. Soc., 52:65-71. With R. P. Boas. Influence of the signs of the derivatives of a function on its analytic character. Duke Math. J., 9:406-24. 1943 On the zeros of the derivatives of a function and its analytic character. Bull. Am. Math. Soc., 49:178-91. 1945 With G. Szego. Inequalities for the capacity of a condenser. Am. J. Math., 67:1-32. How To Solve It. A New Aspect of Mathematical Method. Princeton: Princeton University Press. 1948 Torsional rigidity, principal frequency, electrostatic capacity and symmetrization. Q. Appl. Math., 6:276-77. 1949 With H. Davenport. On the product of two power series. Can. J. Math., 1:1-5.
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1950 With A. Weinstein. On the torsional rigidity of multiply connected cross sections. Ann. Math., 52:154-63. 1951 With G. Szego. Isoperimetric Inequalities in Mathematical Physics. Princeton: Princeton University Press. 1952 Sur une interprétation de la méthode des différences finies qui peut fournir des bornes supérieures ou inférieures. C. R. Acad. Sci. (Paris), 235:1079-81. 1954 With M. Schiffer. Convexity of functionals by transplantation. J. Analyse Math., 3:245-345. Estimates for eigenvalues. In: Studies in Mathematics and Mechanics Presented to Richard von Mises, New York: Academic Press, pp. 200-7. Mathematics and Plausible Reasoning. Vol. 1, Induction and Analogy in Mathematics. Vol. 2, Patterns of Plausible Inference. Princeton: Princeton University Press. 1956 With L. E. Payne and H. F. Weinberger. On the ratio of consecutive eigenvalues. J. Math. Phys., 35:289-98. 1958 With I. J. Schoenberg. Remarks on de la Vallée-Poussin means and convex conformal maps of the circle. Pacific J. Math., 8:295-334. 1959 With M. Schiffer. Sur la representation conforme de l'extérieur d'une courbe fermee convexe. C.R. Acad. Sci. (Paris), 248:2837-39.
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1961 On the eigenvalues of vibrating membranes, In memoriam Hermann Weyl. Proc. London Math. Soc., 11:419-33. 1962 Mathematical Discovery: On Understanding, Learning, and Teaching Problem Solving. 2 vols. New York: John Wiley & Sons. 1968 Graeffe's method for eigenvalues. Numer. Math., 11:315-19. 1972 With G. Szego. Problems and Theorems in Analysis, Vol. 1. New York, Heidelberg, Berlin: Springer-Verlag. Revised and enlarged English language version of 1925,1. (See 1976,1, for Vol. 2). 1974 Collected Papers. Vol. 1, Singularities of Analytic Functions. Vol. 2, Location of Zeros, ed. R. P. Boas. Cambridge: MIT Press. 1976 With G. Szego. Problems and Theorems in Analysis, vol. 2. New York, Heidelberg, Berlin: Springer-Verlag. Revised and enlarged English language version of 1925,1. (See 1972,1, for Vol. 1). 1984 Collected Papers, Vol. 3, Analysis, eds. J. Hersch and G. C. Rota. Vol. 4, Probability, Combinatorics, Teaching and Learning Mathematics , ed. G. C. Rota. Cambridge: MIT Press.
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Edward Lawrie Tatum December 14, 1909-November 7, 1975
By Joshua Lederberg In the history of biology Edward Lawrie Tatum's name is linked with that of George Wells Beadle for their pioneering studies of biochemical mutations in Neurospora.1 First published in 1941, these studies have endured as the prototype of the investigation of gene action to the present day. A still more enduring legacy is their development of experimental techniques for the mutation analysis of biochemical pathways used daily by modern biologists. Though this sketch is written as a biography of Edward Tatum, these singular scientific accomplishments were—in practice and attribution— intimately shared with Beadle. Tatum brought to the work a background in microbiology and a passion for the concept of comparative biochemistry; Beadle, great sophistication in ''classical genetics" and the leadership and drive to replace the underbrush of vitalistic thinking with a clear-cut, mechanistic view of the gene and the processes of life. Little more than the bare outlines of Edward Tatum's personal history can be documented, because of his own aversion to accumulating paper and the fact that most of his corre
1 George W. Beadle died on June 9, 1989, when this essay was in press. His memoir, by Norman H. Horowitz, is also included in this volume.
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spondence was discarded during his various moves. His scientific achievements, however, were largely and appropriately recognized. In 1952 he was elected to the National Academy of Sciences and in 1958, with George Beadle and Joshua Lederberg, won the Nobel Prize in Physiology or Medicine. Tatum was also known for his commitment to nurturing younger scientists, with whom he zestfully enjoyed every aspect of laboratory work. A still more enduring legacy of their work has been the everyday use of experimental mutation analysis of biochemical pathways in modern biology since then. EDUCATION AND EARLY LIFE Edward Lawrie Tatum was born in Boulder, Colorado, on December 14, 1909, the first surviving son of Arthur L. (1884-1955) and Mabel Webb Tatum. A twin, Elwood, died shortly after birth. At the time of Edward's birth his father was an instructor in chemistry at the University of Colorado at Boulder, where Mabel Webb's father had been Superintendent of Schools. Arthur's own father, Lawrie Tatum, a Quaker who had settled in the Iowa Territory, had been an Indian agent after the Civil War and written a book, Our Red Brothers. In rapid succession the Tatum family moved to Madison, Wisconsin; Chicago, Illinois; Philadelphia, Pennsylvania; Vermillion, South Dakota; and, back—in 1918—to Chicago. During this period the elder Tatum held a succession of teaching positions while earning a Ph.D. in physiology and pharmacology from The University of Chicago and an M.D. from Rush Medical College. By 1925 he was settled at the University of Wisconsin at Madison as professor of pharmacology in a department that was a major center for the training of professors of pharmacology. Among his research accomplishments were the introduction of picrotoxin as an
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antidote for barbiturate poisoning and the validation of arsenoxide (mapharsen) for the chemotherapy of syphilis,2 the most effective drug for this purpose until the introduction of penicillin. Edward, having the double advantage of this remarkable family background and the Laboratory School at The University of Chicago, continued his education at Wisconsin, earning a bachelor's degree in 1931. At Wisconsin he came upon the tradition of research in agricultural microbiology and chemistry that was then flourishing under the leadership of E. B. Fred (later president of the University) and W. H. Peterson.3 Tatum's first research was a bachelor's thesis (published 1932) on the effect of associated growth of bacterial species Lactobacillus and Clostridium septicum giving rise to racemic lactic acid. (In 1936 he demonstrated that the C. septicum racemized the d-lactic acid produced by the lactic acid bacteria.) He continued his graduate work at Wisconsin with financial support from the Wisconsin Alumni Research Foundation—the beneficiary of royalties from Steenbock's patents on vitamin D milk. His Ph.D. dissertation (1935) concerned the stimulation of C. septicum by a factor isolated from potato, identified as a derivative of aspartic acid and later shown to be asparagine. This was followed by collaborations with H. G. Wood and Esmond E. Snell in a series of pioneering studies
2 John Patrick Swann, "Arthur Tatum, Parke-Davis, and the Discovery of Mapharsen as an Antisyphilitic Agent," Journal of the History of Medicine and Allied Sciences, 40 (1985):167-87. F. E. Shideman, "A. L. Tatum, Practical Pharmacologist," Science, 123 (1956):449. Anonymous, "Profile of a Research Scientist,'' Bulletin of Medical Research, National Society for Medical Research, 8(1954):7-8. 3 The roots of their work can be traced to Koch, Tollens, and Kossel in Germany. See I. L. Baldwin, "Edwin Broun Fred, March 22, 1887-January 16, 1981," Biographical Memoirs of the National Academy of Sciences , Vol. 55, pp. 247-290; and Conrad A. Elvehjem, "Edwin Bret Hart, 1874-1953," Biographical Memoirs, Vol. 28, pp. 117-161. See also E. H. Beardsley, Harry L. Russell and Agricultural Science in Wisconsin (Madison, Wisconsin: University of Wisconsin Press, 1969).
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on the role of vitamins in bacterial nutrition. In 1936 they studied the growth factor requirements of propionic acid bacteria, fractionating one factor from an acetone extract of milk powder. Its physical properties suggested that the factor might be thiamine, and indeed crystalline thiamine was fully active as an essential growth factor. Vitamins had long been recognized to share a role in the nutrition of animals, man, and yeast. Tatum's work with Snell, Peterson, and Wood initiated a genre of studies showing that many bacterial species had diverse requirements for these identical substances. This was outstanding confirmation of the basic tenet of comparative biochemistry—the evolutionary conservation of biochemical processes—that produced common processes in morphologically diversified species. Tatum's education and doctoral research coincided with the culmination of understanding that all of the basic building blocks of life— amino acids, sugars, lipids, growth factors (and later nucleic acids)—existed in fundamentally similar chemical structures among all forms of life. Hence the most fruitful way to study a problem in animal metabolism might be to begin with a microbe, which might well prove more convenient for experimental manipulation and bioassay and—as the future would show—genetic analysis and alteration. Tatum then won a General Education Board postdoctoral fellowship that took him, his wife (the former June Alton, a fellow student at Wisconsin), and their infant daughter, Margaret, to Fritz Kögl's laboratory at Utrecht, The Netherlands, for a year. Kögl had just purified and crystallized biotin as a growth factor for yeast, and this enabled and inspired further studies on its nutritional role for other microorganisms. (Not until 1940 was the nutritional significance of biotin for animals recognized.) By Tatum's own account, his brief time at Utrecht, spent in efforts to isolate further growth factors for staphylococci,
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never achieved a sharp research focus. More importantly, he befriended Nils Fries, another research fellow from Uppsala, Sweden, who was using the newly available biotin to define the specific nutritional requirements of an ever wider range of fungi. Fries and Kögl were able to demonstrate striking examples of nutritional symbiosis—the compensation for complementary deficits in mixed cultures of various fungi. Tatum's report to the General Education Board records his gratification at having been able to meet, as well, A. J. Kluyver at Delft, and B. C. J. G. Knight and P. Fildes in England—then already well known as leading investigators of bacterial chemistry and nutrition from a comparative perspective. (J. H. Mueller at Harvard and A. Lwoff in Paris had also stressed how microbial nutrition reflected evolutionary losses of biochemical synthetic competence—a concept that can be traced to Twort and Ingram in 19114—though they had not as yet adopted the language or conceptual framework of genetics that would eventually describe such variations as gene mutations affecting biosynthetic enzymes.) THE STANFORD YEARS (1937-1945) That same year, 1937, Beadle was on the point of moving from Harvard to Stanford. His research program in physiological genetics was to continue the work on the genetics of Drosophila eye pigments that he had initiated in collaboration with Boris Ephrussi, first at Caltech, then in Paris. The Rockefeller Foundation's support of this enterprise was one of Warren Weaver's most foresighted initiatives in the gestation of molecular biology.5 Looking out for a possible position for Tatum, his profes
4 F. W. Twort and G. L. Y. Ingram, "A Method for Isolating and Cultivating the Mycobacterium enteritidis chronicae pseudotuberculosae Johne," and "Some Experiments on the Preparation of a Diagnostic Vaccine for Pseudo-tuberculous Enteritis of Bovines," Proceedings, Royal Society, London, Series B, 84(1911-12):517-42. 5 See also Mina Rees, "Warren Weaver, July 17, 1894-November 24, 1978," Biographical Memoirs, Vol. 57, pp. 493-530.
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sors at Wisconsin forwarded Beadle's solicitation for a research associate "biochemist to work on hormone-like substances that are concerned with eye pigments in Drosophila." But, practical-minded, they recommended that the young man undertake research on the chemical microbiology of butter, writing him that "this field is certainly getting hot." With jobs scarce, economic realities weighed as heavily as intellectual appeal in the choice between insect eyes and dairy microbiology. Arthur Tatum, Edward's father, was much concerned that, if his son undertook a hybrid role, he would find himself an academic orphan, disowned by each of the disciplines of biochemistry, microbiology, and genetics. In the event, however, Tatum accepted Beadle's offered position, and the multiple challenges of comparative biochemistry that went with it. Though the economic importance of butter research was far more obvious at the time, it is certain that Edward Tatum could not have chosen better than Drosophila as a means for contributing to the field of biotechnology. Joining Beadle at Stanford, Tatum was engaged between 1937 and 1941 with the arduous task of extracting pigment-precursors from Drosophila larvae. Ephrussi and Beadle's earlier transplantation experiments had demonstrated that a diffusible substance or hormone produced by wild-type flies was critically lacking in the mutant strain. Yet Tatum and Beadle's own experience differed significantly from the report published by Ephrussi and Chevais. According to this report, normal eye color could be restored in cultures supplemented with tryptophane. Tatum, however, could confirm this only with cultures carrying a bacterial contaminant. Far from discarding such a contaminant as an interfering variable, Tatum cultured the organism (a Bacillus species) to prove that it was a source of the elusive hormone. The interchangeability of growth factors for bacteria and animals and the knowledge that many microbes synthesized vitamins required by other species undoubtedly bolstered this theory.
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A. J. Haagen-Smit, whom Beadle had known at Harvard, was now at the California Institute of Technology, and Tatum visited him to learn microchemical techniques, then set out to isolate the "V + hormone" from the bacterial culture. He succeeded in doing this in 1941, only to be anticipated by Butenandt et al. in the identification of V + as kynurenine. (Butenandt, astutely noting—from a Japanese publication—that kynurenine was a metabolite of tryptophane in dog urine, had tested the substance for eye color hormone activity.) The jarring experience of having their painstaking work overtaken in so facile a way impelled Beadle and Tatum to seek another organism more tractable than Drosophila for biochemical studies of gene action. Neurospora and the One Gene-One Enzyme Theory In winter quarter 1941, Tatum (although a research associate without teaching responsibilities) volunteered to develop and teach a then unprecedented comparative biochemistry course for both biology and chemistry graduate students. In the course of his lectures he described the nutrition of yeasts and fungi, some of which exhibited well-defined blocks in vitamin biosynthesis. Attending these lectures, Beadle recalled B. O. Dodge's elegant work on the segregation of morphological mutant factors in Neurospora that he had heard in a seminar at Cornell in 1932,6 work that was followed up by C. C. Lindegren at Caltech. Neurospora, with its immediate manifestation of segregating genes in the string of ascospores, has an ideal life-cycle for genetic analysis. Fries's work suggested that Neurospora might also be cultured readily on a well defined medium. It was soon established that Neurospora required only biotin as
6 See also W.J. Robbins, "Bernard Ogilvie Dodge, April 18, 1872-August 9, 1960," Biographical Memoirs, Vol. 36, pp. 85-124.
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a supplement to an inorganic salt-sucrose medium and did indeed prove an ideal organism in which to seek mutations with biochemical effects demonstrated by nutritional requirements. By February 1941,7 the team was X-raying Neurospora and seeking these mutants. Harvesting nutritional mutants in microorganisms in those days was painstaking hand labor; it meant examining single-spore cultures isolated from irradiated parents for their nutritional properties—one by one. No one could have predicted how many thousands of cultures would have to be tested to discover one that would have a biochemical defect marked by a nutritional deficiency. Isolate #299 proved to be the first recognizable mutant, requiring as it did pyridoxine. The trait, furthermore, segregated in crosses according to simple Mendelian principles, which foretold that it could in due course be mapped onto a specific chromosome of the fungus. Therewith, Neurospora moved to center stage as an object of genetic experimentation. By May of the same year, Beadle and Tatum were ready to submit their first report of their revolutionary methods to the Proceedings of the National Academy of Sciences. In that report they noted "there must exist orders of directness of gene control ranging from one-to-one relations to relations of great complexity." The characteristics of mutations affecting metabolic steps suggested a direct and simple role for genes in the control of enzymes. The authors
7 G. W. Beadle, "Recollections," Annual Revue of Biochemistry, 43 (1974):1-13. In his chapter, "Biochemical Genetics, Some Recollections," in Phage and the Origins of Molecular Biology, eds. J. Cairns, G. S. Stent, and J. D. Watson (Cold Spring Harbor, New York: C. S. H. Biol. Labs, 1966), Beadle confused the 1940-41 meeting of the Society of American Naturalists in Philadelphia, which made no reference to Neurospora, with that of the Genetics Society in Dallas in December 1941. The net effect is to date the Neurospora experiments to 1940 rather than to 1941. H. F. Judson repeated the error in The Eighth Day of Creation (New York: Simon & Schuster, 1979), and it is bound to plague future historians.
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hypothesized, therefore, that enzymes were primary products of genes. Indeed, in some cases, genes themselves might be enzymes. This was what came to be labelled the one geneone enzyme theory, the precursor of today's genetic dogma. We shall return to it later. In that same year Tatum was recruited as an assistant professor to the regular faculty of Stanford's Biology Department, where he developed an increasingly independent research program exploiting the use of Neurospora mutants for the exploration of biochemical pathways. Despite the exigencies of the war effort, an increasing number of talented graduate students and postdoctoral fellows flocked to Stanford to learn the new discipline. Their participation rapidly engendered a library of mutants blocked in almost any anabolite that could be replaced in the external nutrients. Today, that catalog embraces over 500 distinct genetic loci and well over a thousand publications from laboratories the world over.8 Anticipating the One Gene-One Enzyme Theory Would that contemporaries could anticipate what future historians will ask or what errors they will promulgate! How many simple questions we neglect to ask, or fail to record the answers, that might have settled continuing controversies. Among these is the place of Archibald E. Garrod's work and thought in anticipation of the one gene-one enzyme hypothesis. The following discussion is offered in some detail in order to correct some prevalent misconstructions of that history. In 1908, Garrod published his study of what was then called "inborn errors of metabolism," including alcaptonuria
8 D. D. Perkins, A. Radford, D. Newmeyer, and M. Bjorkman, "Chromosomal loci of Neurospora crassa," Microbiological Reviews, 46 (1982):426-570.
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in man.9 This work is sometimes portrayed as a forgotten precursor of Beadle and Tatum's investigation of gene action. Indeed, many geneticists who specialized in maize or Drosophila, including Beadle himself, lamented not knowing of this pioneering work earlier—it having received remarkably little comment from geneticists until after Neurospora was launched in 1941.10 Yet Garrod's basic findings on alcaptonuria, which parallel the metabolic blocks in Neurospora mutants, were widely quoted in medical texts. J. B. S. Haldane cited them in a well-read essay in 1937. Tatum likewise referred to them in his course in comparative biochemistry before beginning his own experiments on Neurospora. Beadle, in his Nobel Prize lecture in 1958, was careful to acknowledge these antecedents, though widely quoted reminiscences have blurred the details of just when Beadle and Tatum became aware of Garrod's work.11 Haldane, in his 1937 article, cited the difficulty of experimentation on rare human anomalies as an important reason to seek other research paradigms— which Neurospora would eventually provide.12 But Garrod himself never quite made
9 "The Croonian Lectures of the Royal College of Physicians," Lancet 2(1908): 17, 73-79, 142-148, 214-220. 10 H. Harris, ed., Garrod's Inborn Errors of Metabolism (Oxford: Oxford University Press, 1963); and B. Childs and C. R. Scriver. eds., Inborn Factors in Disease by A. E. Garrod (Oxford: Oxford University Press, 1989), include extensive discussion and bibliography on the history of his ideas. On the neglect of Garrod's work, see also R. Olby, The Path to the Double Helix (London, Macmillan Press, 1974). 11 Though G. W. Beadle implies in PATOOMB (Phage and the Origins of Molecular Biology, see footnote 7 above), that he and Tatum were unaware of Garrod until perhaps 1945, they referred to Garrod in a paper on their Drosophila-pigment work delivered January 1, 1941 (see American Naturalist, 75:107-16). Garrod's findings were also prominent in Tatum's winter 1941 course on comparative biochemistry at Stanford. I first read about Garrod in Meyer Bodansky's Introduction to Physiological Chemistry (New York: Wiley & Sons, 1934), and the late Sewall Wright advised me that he had taught that material in Chicago since 1925. 12 J. B. S. Haldane, "The Biochemistry of the Individual," in Perspectives in Biochemistry, J. Needham and D. E. Green, eds. (Cambridge: Cambridge University
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the leap from the anomaly provoked by the mutant gene to the positive functioning of its normal allele. Nor did he recognize enzymes as the direct products of genes in their normal function, but rather referred to mutational anomalies as freaks or aberrations to be compared with the effects of infection or intoxication. Theoretical biology in Garrod's time believed in "protoplasm" as an almost mystical, living colloid. When altered, genes might influence the workings of that protoplasm but were not yet thought to be the exclusive, or nearly exclusive, seat of hereditary information (to use an anachronistically modern expression).13 In their 1941 paper, Beadle and Tatum cited the (now quaint) "rapidly disappearing belief that genes are concerned only with the control of 'superficial' characters." It would appear, then, that while Garrod understood how genetic anomalies could assist in the unravelling of metabolic pathways and that biochemical individuality was a hallmark of human nature, he had no comprehensive theory of gene action. Any geneticist, however, would wish to give alcaptonuria—a textbook example of a biochemical genetic defect—full credit as a paradigm on par with the pigment mutation in flowers or in insect eyes. Before 1941, simple metabolic effects on gene mutation could be inferred in a handful of cases like these, but the vast majority of mutants studied in, say, Drosophila, were complex morphogenetic traits that defied (and still very nearly defy) simple analysis. The experimental material available made it impossible to arrive at any simple theory of gene action. Even more exasperatingly, it offered almost no avenue
Press, 1937). Haldane remarked that "Garrod's pioneer work on congenital human metabolic abnormalities such as alcaptonuria and cystinuria had a very considerable influence both on biochemistry and genetics. But alcaptonuric men are not available by the dozen for research work. . . ." 13 See J. Sapp, Beyond the Gene: Cytoplasmic Inheritance and the Struggle for Authority in Genetics (Oxford: Oxford University Press, 1987).
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for continued investigation. How frustrated Tatum and Beadle were between 1937 and 1941 in their efforts with Drosophila pigments! It was the conceptual and experimental methodology they developed using nutritional mutants that provided the breakthrough. Today, four decades later, analyzing developmental and physiological pathways by systematically cataloguing mutants that block them is standard procedure and Beadle and Tatum's papers are rarely cited. Taken for granted, this methodology is yet central to sophisticated studies in physiology, development, and gene action and is of incalculable consequence to biotechnology. Tryptophane and E. coli K-12 The biosynthesis of tryptophane, possibly harking back to Drosophila eye color, remained one of Tatum's central interests. At one point, Tatum and Bonner inquired whether the dismutation of tryptophane into indole + serine was a simple reversal of the synthetic reaction. Though this analogy has been complicated by further knowledge, we now know that there are indeed interesting similarities between the tryptophane-cleaving enzyme and one subunit of the synthetase. In order to perform studies on tryptophanases, Tatum retrieved a stock strain of Escherichia coli from the Stanford Bacteriology Department's longstanding routine strain collection. By this accident, E. coli K-12 came to be the object of further genetic experimentation. Its name will reappear shortly in our story. With Beadle's encouragement, Tatum used his familiarity with bacteria to recruit Acetobacter and E. coli as experimental objects for biochemical analysis to parallel Neurospora. Despite the lack of any theoretical or experimental basis for expecting bacteria to have a genetic organization similar to that of higher organisms, Tatum intuitively favored a com
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monality of biological structure to match what comparative biochemistry had revealed in the realm of nutrition. Tatum's prompt demonstration that biochemical mutants like those in Neurospora could also be induced in E. coli was, in itself, strong provocation to apply some form of gene theory to bacteria. As their part in the wartime mobilization during 1944 and 1945, Tatum's laboratory was asked to use its expertise in fungal genetics in an OSRDsponsored, multi-laboratory search for better penicillin-yielding strains of Penicillium. Though Stanford made significant improvements in yield, their efforts were outstripped by developments elsewhere. Tatum and Lederberg—Genetic Recombination in Bacteria The team of Beadle and Tatum by this time had become world famous. But at Stanford, under President Tressider's troubled leadership, the exigencies of finance added to the academic politicking in the Biology Department and left little promise for innovative scientific development. The role of a chemist in a department of biology as then understood was particularly controversial, and C. B. van Niel's unequivocal support for Tatum was of no avail. Despite Tatum's success, his father's foreboding premonition had materialized, and, foreseeing a bleak academic future at Stanford, he sought a post where he could continue to work at the hybrid frontiers of microbiology, genetics, and biochemistry. In 1945, after a trial semester at Washington University in St. Louis, where Carl Lindegren hoped to find a niche for him, Tatum accepted a position at Yale University. A year later Beadle and his formidable team left Stanford en bloc to reshape the biology program at Caltech. At Yale Tatum held a tenured chair and was charged with developing a biochemically-oriented microbiology program with the Department of Botany. His arrival proved a seren
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dipitous break for this author, Joshua Lederberg, then a Columbia medical student studying Neurospora genetics with Francis J. Ryan—an apprenticeship begun at Columbia College in 1942. In 1941 Ryan had gone to Stanford for a year's postdoctoral fellowship, where he became one of the first disciples of Neurospora biochemical genetics. When he returned to Columbia, he brought back with him his enthusiasm for the new field. At Stanford, Ryan had established a warm friendship with Tatum, and—hearing that he was moving to Yalesent him Lederberg's proposals for studying genetic recombination in bacteria. On the strength of Ryan's commendation Tatum invited Lederberg to join his laboratory at New Haven starting March 1946, where he was supported financially by the Jane Coffin Childs Fund. What was to have been a few months' diversion from medical school exceeded Lederberg's wildest expectations. At the Cold Spring Harbor Symposium in July 1946, Tatum's laboratory could report a newly discovered genetic recombination in E. coli K-12, vindicating Tatum's gamble that, indeed, E. coli had genes!14 Our use of E. coli strain K-12 for these studies derived from Tatum's prior development of single, then double, mutants blocked at different nutritionalbiochemical steps. The use of such multiply-marked stocks averted a number of technical artifacts in recombination experiments. Only later did we learn that K-12 itself was a remarkably lucky choice of experimental material: Only about one in twenty randomly chosen strains would have given positive results in experiments designed according to our protocols. In particular, strain B—which had become the standard material for work on bacteriophage—would have been stubbornly unfruitful.
14 J. Lederberg, ''Genetic Recombination in Bacteria: A Discovery Account," Annual Review of Genetics, 21 (1987):23-46.
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Subsequently K-12 also proved to be a remarkably rich source of the plasmids F and lambda, which have become the objects of major experimental programs in their own right. The serendipity that so often marked Tatum's career cannot be attributed to any personal skill or insight on his part. But his receptivity to "far out" proposals from a medical student visiting his laboratory was typical of the man's unique combination of generosity of spirit and scientific vision. RETURN TO STANFORD (1948-1956) During his period at Yale, Tatum also recruited David Bonner to continue joint research on the biosynthesis of tryptophane and bolster the academic program in microbiology. But he was once again disappointed in the University's level of commitment to biochemically-oriented research in a department still heavily dominated by morphological-systematic tradition. In 1948, when Douglas Whitaker took over the leadership of biological research at Stanford, Tatum was persuaded to accept a full professorship in the department that had passed him over just three years before. From this time forward Tatum, with his particular brand of biochemical insights, pursued and supervised research projects that reconciled a variety of interests introduced by his students and colleagues. In early anticipation of the now famous Ames Screening Test, he became increasingly interested in the analogy between mutagenesis and carcinogenesis. If the induction of nutritionally dependent mutants in Neurospora was a rather laborious way to demonstrate mutagenicity of a chemical compound, it at least had the advantage of adding to the library of useful strains for biochemical pathway analysis. Many of us felt that E. coli was technically superior to Neurospora, both for biochemical and genetic studies (at least in the ease with which vast numbers of mu
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tants could be obtained and propagated; Tatum generally left the exploitation of this material to the students)—and while it was plain that Neurospora was Tatum's first love throughout his career, he leaned over backwards to give his intellectual heirs the utmost leeway for their own development. During the decade 1948 to 1958, Stanford made a bid to become a major center of scholarship, while California grew in economic, technological, demographic, and political influence. Stanford's then new president, the late J. E. Wallace Sterling, though himself a historian, warmly nurtured scientific and technical development. He supported an ambitious program to reconstruct the School of Medicine on the Stanford campus, transforming a hospital-based school in San Francisco with nominal connection to the University into a major center for medical and biological research. Under the leadership of Fred Terman, similar institution-building was taking place in Stanford's School of Engineering, nourished by vigorous federal support for science and technology in the wake of World War II. In short order the San Francisco Bay area was transformed into a center for high technology in the electronics and pharmaceuticals industries—a transformation that owed much to Sterling's and Terman's encouragement of University interaction with industry. With regard to academic policy at Stanford, Tatum proved an energetic spokesman for the rapidly emerging discipline of biochemistry. As a member of the National Science Board he was an influential exponent of predoctoral and postdoctoral fellowship support for creative talent in the new field. In this he no doubt recalled that critical stage in his own career: his postdoctoral experience at Utrecht, that foreshadowed his work with Beadle. He was also a strong advocate of international cooperation among scientists and played an important role in setting up a joint program with Japan.
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At Stanford he gave strong encouragement to the development of a new, science-oriented curriculum in medical education and to the whole enterprise— fraught with fiscal and managerial risks—of rebuilding the Medical School. In 1956 he was appointed to head a new Department of Biochemistry, an appointment that would take full effect in 1959 with the completion of the new medical center. Conflicts in his personal life, however, overshadowed his other plans and he left Stanford, separating from his wife and two daughters. THE ROCKEFELLER INSTITUTE (1957-1975) In 1953 Detlev Bronk, president of the National Academy of Sciences, left Johns Hopkins to assume the presidency of The Rockefeller Institute in New York, marking the expansion of the Institute into a graduate university. In 1955, Whitaker was recruited from Stanford as vice-president for administration. Between 1953 and 1957, Frank Brink, Keffer Hartline, Paul Weiss, and Fritz Lipmann joined the Institute faculty—not to mention the elevation to full membership of Theodore Shedlovsky, George Palade, and Keith Porter. Tatum was induced to join this illustrious group in 1957, and he remained there until his death in 1975. In New York, Tatum married Viola Kantor, a staff employee at the National Foundation/March of Dimes where he donated a great deal of time as scientific adviser. This rebuilding of his personal life was, however, to be scarred by Viola's illness and untimely death from cancer in 1974. As a professor at Rockefeller, Tatum concerned himself with institutional affairs just as he had at Stanford. He was also involved with science policy on a national scale and served on the National Science Board. His special aim was to strengthen fellowship programs and other measures that would bolster support for young people entering scientific work. He was also chairman of the board of the Cold Spring
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Harbor Biological Laboratory during a period of fiscal crisis and interpersonal turbulence that, according to one of his associates, was the most grievous episode of his professional life. THE NOBEL PRIZE (1958) The Nobel Prize came to Tatum in 1958, a year after his move to the Rockefeller Institute. In his Prize lecture, Tatum reviewed the history of biochemical genetics in his and Beadle's hands. Comparing microbial cultures to populations of tissue cells, he saw cancer as a genetic change subject to natural selection. From this vantage he looked forward to "the complete conquering of many of man's ills, including hereditary defects in metabolism and the momentarily more obscure conditions such as cancer and the degenerative diseases . . . . Perhaps within the lifetime of some of us here, the code of life processes tied up in the molecular structure of proteins and nucleic acids will be broken. This may permit the improvement of all living organisms by processes that we might call biological engineering." Tatum's prophecy erred mainly in its diffidence; the breaking of the genetic code was well under way by 1961, with the reports of M. W. Nirenberg and J. H. Matthaei that matched specific triplets of RNA with individual amino acids in the assembly of polypeptides. These rules of correspondence were the realization in explicit chemical structural terms of the expectations of the one gene-one enzyme theory. In his own laboratory, Tatum was especially notable for nurturing independent-minded fellows in the pursuit of their own ideas. He was prouder of having cultivated them as gifted investigators than of his own contributions to their research. He strongly encouraged young faculty members at the Rockefeller, like Norton Zinder, and they have acknowledged the debt.
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His personal research interests during this phase centered on the use of Neurospora as a model for the genetic control of development. The effects of inositol deprivation or the addition of substances like sorbose on the morphology of the fungus never failed to intrigue him. Features like mycelial branching, subsurface versus aerial hyphae, and the formation of peritheciae and micro-and macro-conidia were thought to be models for the more complex developmental patterns in animal embryogenesis. Such studies are only just now coming into their own. There is no doubt that mutational alteration of developmental patterns can throw a great deal of light on the interactions between genes and environment that lead to morphological elaboration. This type of material has yet to give us, however, those quasi-stable, epigenetic states—expressed in higher plant and animal cells propagated in tissue culture—whose biochemical genetic analysis would be extraordinarily helpful. IN CONCLUSION The ability to balance critical scientific objectivity, personal ambition, and interdependence on others—which some scientists take a lifetime to learn—was ingrained in Ed Tatum from the beginning. Despite misfortune in his personal life, he yet enjoyed the rare and well-earned pleasure of having so many of his fellow scientists look to him warmly as to a father or brother. At the time of Viola Tatum's death, Ed Tatum's health was already failing, and his friends could only watch with anguish the multiplying pains that attended a life to which he clung with the same doggedness that made him a committed cigarette smoker. He died on November 7, 1975, from heart failure complicated by progressive, chronic emphysema. Edward Lawrie Tatum was survived by two daughters
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from his first marriage: Margaret (Mrs. John Easter) and Barbara. His brother Howard worked for many years with the Population Council doing research on contraception. His late sister, Besse, was married to A. Frederick Rasmussen, professor of microbiology at UCLA. This memoir was completed more than a decade after Tatum's death— forty-seven years after the climactic initiation of microbial genetics in 1941. Half a century may be almost enough time to see that work in historical perspective and yet allow for some brief overlap to call testimony from contemporaries. My own familiarity with Neurospora, dating to 1942 when Ryan returned from Stanford to Columbia, qualifies me only barely.15 The one gene-one enzyme theory that a gene acts by controlling the formation of a specific enzyme in some fairly simple manner was implicit in earlier research on pigment biosynthesis. Before 1941 J. B. S. Haldane's speculative discussion came close but never jelled into a concrete theory that would lead to such effective lines of enquiry. Though the Neurospora work suggested that all biochemical traits could be studied in like fashion, it was Beadle and Tatum who extrapolated—from diverse examples—that all such traits would have an equally direct relationship to the corresponding genes. This fundamental observation is now stated as the DNA sequence providing the information for protein structure (though the numerics are sometimes more complex). Many genes, and sometimes families of enzymes, can be involved in the quantitative regulation and environmental responsiveness of enzyme synthesis. Enzymes are sometimes
15 Tatum's departure from Stanford in 1957 denied me the chance to be his colleague when I arrived there in 1959. His death in 1975 likewise predated my arrival at The Rockefeller in 1978. In sum, our academic careers ran in curiously parallel but dissynchronous tracks at Wisconsin, Stanford, and Rockefeller. Our sole congruence was at Yale for a year-and-a-half in 1946-47.
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complex multi-chain ensembles and can contain nonprotein cofactors requiring the participation of many genes. Understanding the role of RNA as a message intermediary between DNA and protein, the complexities of intervening sequences in RNA, RNA-processing, and post-translational processing came later and required more sophisticated biochemical analysis—but all derived from the concepts and the tools of the Neurospora studies. Beadle and Tatum's contribution, then, comprised the following: 1) A methodology for the investigation of gene-enzyme relationships that exploited experimentally-acquired genetic mutations affecting specific biosynthetic steps. 2) A conceptual framework—the one gene-one enzyme theory—from which to search for and characterize these mutants. This framework was derived from the model that chromosomal genes contain (substantially) all of the blueprints for development and that enzymes (and other proteins) are the mediators of gene action. 3) The dethronement of Drosophila as the prime experimental material for physiological genetic research in favor of the fungus Neurospora. This further helped open the way to use of bacteria and viruses in genetic research and the culture of tissue cells as if they were microbes. These methods and concepts have been the central paradigm for experimental biology since 1941. Beadle and Tatum shared many awards in addition to the 1958 Nobel Prize in recognition of these innovations. In 1952, Tatum was individually honored by election to the National Academy of Sciences. In 1953 he received the Remsen Award of the American Chemical Society and was elected to the American Philosophical Society. He was president of the Harvey Society (1964-65) and the recipient of at least seven honorary degrees. He served on the NAS Carty Fund Committee from 1956 to 1961. For the NRC, he took part in a number of panels
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and committees having to do with genetics and biology and was a member of the Advisory Committee on the Biological Effects of Ionizing Radiations from 1970 to 1973. He also did yeoman service on advisory committees for the National Institutes of Health, American Cancer Society, the National Foundation (March of Dimes), and other bodies concerned with the award of fellowships and grants. He was chairman of the Scientists' Institute for Public Information and an advisor to the City of Hope Medical Center, Rutgers University Institute of Microbiology, and Sloan-Kettering Institute for Cancer Research, and a consultant in microbiology for Merck and Co. He worked actively on many scientific publications, including Annual Reviews, Science, Biochemica et Biophysica Acta, Genetics, and the Journal of Biological Chemistry. Testifying to a Congressional committee on behalf of the National Science Foundation in 1959, Tatum said: "The general philosophy [of the NSF] is concentration on excellence . . . making it possible for [the scientist] to use his capacities, both for research and for training the next generation . . . whether it is a particular research program in a given area, whether it may or may not be immediately practicable in its application . . . freedom to develop the intellectual curiosity and abilities of the individual . . . ."
At this time Beadle and Tatum's legacy is embodied in published work that has influenced biological research through several scientific generations. The original papers are "classics" and taken for granted. Personal recollections of Tatum are fading, and this report can hardly do justice to his humor, his hobbies (including the French horn), his zest for experiments, his love of microbes, his attachment to students, friends, and family —the trauma of divorce notwithstanding—the tragedy of his final year of bereavement and of an illness that left him gasping for breath. He touched the lives of many young scientists.
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The enduring appreciation of his role in their development is the memorial he would have cherished most. The tantalizingly few personal papers of Edward Tatum now extant are on deposit at the Rockefeller University Archive Center. I am particularly indebted to Professor Carlton Schwerdt for having preserved and made available his lecture notes on Tatum's 1941 course on comparative biochemistry, to June Alton Tatum for making available to me materials regarding Tatum's life before 1946, and to the staff of the Rockefeller University Archive Center. I am also indebted to the following important studies for information that appears in this account: R. M. Burian, Jean Gayon, and Doris Zallen, "The Singular Fate of Genetics in the History of French Biology," Journal of the History of Biology, 21(1988):357-402, on the Beadle-Ephrussi collaboration that led directly to Beadle and Tatum's work on Drosophila eye color "hormones" and discusses the use of that terminology for what would later be termed "precursors." Lily E. Kay, "Selling Pure Science in Wartime: The Biochemical Genetics of G. W. Beadle," Journal of the History of Biology, 22 (1989):73-101, reviews the Beadle-Tatum work on penicillin improvement during World War II.
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Selected Bibliography16
1932 With W. H. Peterson and E. B. Fred. Effect of associated growth on forms of lactic acid produced by certain bacteria. Biochem. J., 26:846-52. 1934 Studies in the biochemistry of microorganisms. Ph.D. Dissertation, University of Wisconsin, Madison. 1936 With H. G. Wood and W. H. Peterson. Essential growth factors for propionic acid bacteria. II. Nature of the Neuberg precipitate fraction of potato: Replacement by ammonium sulphate or by certain amino acids. J. Bacteriol., 32:167-74. With H. G. Wood and W. H. Peterson. Growth factors for bacteria. V. Vitamin B, a growth stimulant for propionic acid bacteria. Biochem. J., 30:1898-1904. 1937 With E. E. Snell and W. H. Peterson. Growth factors for bacteria. III. Some nutritive requirements of Lactobacillus delbrückii. J. Bacteriol., 33:207-25. With W. H. Peterson and E. B. Fred. Enzymatic racemization of optically active lactic acid. Biochem. J., 30:1892-97. 1938 With G. W. Beadle. Development of eye colors in Drosophila: Some properties of the hormones concerned. J. Gen. Physiol., 22:239-53. 1939 Development of eye colors in Drosophila: Bacterial synthesis of v+ hormone. Proc. Natl. Acad. Sci. USA, 25:486-90. Nutritional requirements of Drosophila melanogaster. Proc. Natl. Acad. Sci. USA, 25:490-97.
16
A complete bibliography can be found in the Archives of the National Academy of Sciences and in the Rockefeller University Archive Center.
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1940 With G. W. Beadle. Crystalline Drosophila eye color hormone. Science, 91:458. 1941 With G. W. Beadle. Experimental control of development and differentiation. Am. Nat., 75:107-16. Vitamin B requirements of Drosophila melanogaster. Proc. Natl. Acad. Sci. USA, 27:193-97. With A. J. Haagen-Smit. Identification of Drosophila v+ hormone of bacterial origin. J. Biol. Chem., 140:575-80. With G. W. Beadle. Genetic control of biochemical reactions in Neurospora. Proc. Natl. Acad. Sci. USA, 27:499-506. 1942 With G. W. Beadle. Genetic control of biochemical reactions in Neurospora: An ''aminobenzoicless" mutant. Proc. Natl. Acad. Sci. USA, 28:234-43. 1943 With L. Garnjobst and C. V. Taylor. Further studies on the nutritional requirements of Colpoda duodenaria. J. Cell. Comp. Physiol., 21:199-212. With F. J. Ryan and G. W. Beadle. The tube method of measuring the growth rate of Neurospora. Am. J. Bot., 30:784-99. With D. Bonner and G. W. Beadle. The genetic control of biochemical reactions in Neurospora: A mutant strain requiring isoleucine and valine. Arch. Biochem., 3:71-91. With D. M. Bonner. Synthesis of tryptophan from indole and serine by Neurospora. J. Biol. Chem., 151:349. 1944 With D. Bonner. Indole and serine in the biosynthesis and breakdown of tryptophan. Proc. Natl. Acad. Sci. USA, 30:30-37. Biochemistry of fungi. Annu. Rev. Biochem., 13:667-704. With C. H. Gray. X-ray induced growth factor requirements in bacteria. Proc. Natl. Acad. Sci. USA, 30:404-10.
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1945 With N. H. Horowitz, D. Bonner, H. K. Mitchell, and G. W. Beadle. Genic control of biochemical reactions in Neurospora. Ann. Nat., 79:304-17. With G. W. Beadle. Biochemical genetics of Neurospora. Ann. Mo. Bot. Garden, 32:125-29. X-ray induced mutant strains of E. coli. Proc. Natl. Acad. Sci. USA, 31:215-19. With G. W. Beadle. Neurospora II. Methods of producing and detecting mutations concerned with nutritional requirements. Am. J. Bot., 32:678-86. 1946 With T. T. Bell. Neurospora III. Biosynthesis of thiamin. Am. J. Bot., 33:15-20. With J. Lederberg. Novel genotypes in mixed cultures of biochemical mutants of bacteria. Cold Spring Harbor Symp. Quant. Biol., 11:113-14. Induced biochemical mutations in bacteria. Cold Spring Harbor Symp. Quant. Biol., 11:278-84. 1947 Chemically induced mutations and their bearing on carcinogenesis. Ann. N.Y. Acad. Sci., 49:87-97. With J. Lederberg. Gene recombination in the bacterium Escherichia coli. J. Bacteriol., 53:673-84. 1950 With R. W. Barratt, N. Fries, and D. Bonner. Biochemical mutant strains of Neurospora produced by physical and chemical treatment. Am. J. Bot., 37:38-46. With R. C. Ottke and S. Simmonds. Deuteroacetate in the biosynthesis of ergosterol by Neurospora. J. Biol. Chem., 186:581-89. With D. D. Perkins. Genetics of microorganisms. Annu. Rev. Microbiol., 4:129-50. With E. A. Adelberg. Characterization of a valine analog accumulated by a mutant strain of Neurospora crassa. Arch. Biochem., 29:235-36.
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1951 With E. A. Adelberg and D. M. Bonner. A precursor of isoleucine obtained from a mutant strain of Neurospora crassa. J. Biol. Chem., 190:837-41. With E. A. Adelberg. Origin of the carbon skeletons of isoleucine and valine. J. Biol. Chem., 190:843-52. 1954 With S. R. Gross, G. Ehrensvard, and L. Garnjobst. Synthesis of aromatic compounds by Neurospora. Proc. Natl. Acad. Sci. USA, 40:271-76. With D. Shemin. Mechanism of tryptophan synthesis in Neurospora. J. Biol. Chem., 209:671-675. 1956 With S. R. Gross and R. D. Gafford. The metabolism of protocatechuic acid in Neurospora. J. Biol. Chem., 219:781-96. With S. R. Gross. Physiological aspects of genetics. Ann. Rev. Physiology, 18:53-68. With R. A. Eversole. Chemical alteration of crossing-over frequency in Chlamydomonas. Proc. Nat. Acad. Sci. USA, 42:68-73. With L. Garnjobst. A temperature independent riboflavin requiring mutant of Neurospora crassa. Am. J. Bot., 43:149-57. With R. C. Fuller. Inositol-phospholipid in Neurospora and its relationship to morphology. Am. J. Bot., 43:361-65. 1958 With R. W. Barratt. Carcinogenic mutagens. Ann. N.Y. Acad. Sci., 71:1072-84. Molecular basis of the cause and expression of somatic cell variation. J. Cell Comp. Physiol., 52:313-36. 1959 A case history in biological research. Science, 129:1711-15. Also in: Les prix Nobel en 1958, Stockholm, pp. 160-9. With A. J. Shatkin. Electron microscopy of Neurospora crassa mycelia. J. Biophys. Biochem. Cytol., 6:423-26.
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1961 With James F. Wilson and Laura Garnjobst. Heterocaryon incompatibility in Neurospora crassa —Micro-injection studies. Am. J. Bot., 48:299-305. With Noel de Terra. Colonial growth of Neurospora. Science, 134:1066-68. With E. Reich, R. M. Franklin, and A. J. Shatkin. Effect of actinomycin D on cellular nucleic acid synthesis and virus production. Science, 134:556-57. 1962 Biochemical genetics and evolution. Comp. Biochem. Physiol., 4:241-48. With A. J. Shatkin, E. Reich, and R. M. Franklin. Effect of mitomycin C on mammalian cells in culture. Biochem. Biophys. Acta, 55:277-89. With E. Reich, R. M. Franklin, and A. J. Shatkin. Action of actinomycin D on animal cells and viruses. Proc. Nat. Acad. Sci. USA, 48:1238-45. 1963 With Noel de Terra. A relationship between cell wall structure and colonial growth in Neurospora crassa. Am. J. Bot., 50:669-77. With B. Mach and E. Reich. Separation of the biosynthesis of the antibiotic polypeptide tyrocidine from protein biosynthesis. Proc. Nat. Acad. Sci. USA, 50:175-81. 1965 Perspectives from physiological genetics. In: The Control of Human Heredity and Evolution, ed. E. Sonneborn, New York: Macmillan, pp. 20-34. With E. G. Diacumakos and L. Garnjobst. A cytoplasmic character in Neurospora crassa. The role of nuclei and mitochondria. J. Cell Biol., 26:427-43. With C. W. Slayman. Potassium transport in Neurospora. III. Isolation of a transport mutant. Biochem. Biophys. Acta, 109: 184-93.
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1966 With Z. K. Borowska. Biosynthesis of edeine by Bacillus brevis Vm4: In vivo and in vitro. Biochem. Biophys. Acta, 114:206-9. The possibility of manipulating genetic change. In: Genetics and the Future of Man, First Nobel Conference, Gustavus Adolphus College. Ed., J. D. Roslansky, New York: AppletonCentury-Crofts, pp. 51-61. With B. Mach. The biosynthesis of antibiotic polypeptides. In: Ninth International Congress for Microbiology, Moscow, London: Pergamon Press, pp. 57-63. With S. Brody. The primary biochemical effect of a morphological mutation in Neurospora crassa. Proc. Nat. Acad. Sci. USA, 56:1290-7. Molecular biology, nucleic acids, and the future of medicine. Perspec. Biol. Med., 10:19-32. 1967 With B. Crocken. Sorbose transport in Neurospora crassa. Biochem. Biophys. Acta, 135:100-5. With E. Pina. Inositol biosynthesis in Neurospora crassa. Biochem. Biophys. Acta, 136:265-71. With S. Brody. Phosphoglucomutase mutants and morphological changes in Neurospora crassa. Proc. Nat. Acad. Sci. USA, 68:923-30. With L. Garnjobst. A survey of new morphological mutants in Neurospora crassa. Genet., 57:579-604. With M. P. Morgan and L. Garnjobst. Linkage relations of new morphological mutants in linkage group V of Neurospora crassa. Genet., 57:605-12. With P. R. Mahadevan. Localization of structural polymers in the cell wall of Neurospora crassa. J. Cell Biol., 35:295-302. 1970 With N. C. Mishra. Phosphoglucomutase mutants of Neurospora sitophila and their relation to morphology. Proc. Nat. Acad. Sci. USA, 66:638-45. With L. Garnjobst. New crisp genes and crisp modifiers in Neurospora crassa. Genetics, 66:281-90.
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1971 With W. A. Scott. Purification and partial characterization of glucose-6-phosphate dehydrogenase from Neurospora crassa. J. Biol. Chem., 246:6347-52. 1972 With E. G. Diacumakos. Fusion of mammalian somatic cells by microsurgery. Proc. Nat. Acad. Sci. USA, 69:2959-62. 1973 With N. C. Mishra. Non-Mendelian inheritance of DNA-induced inositol independence in Neurospora. Proc. Nat. Acad. Sci. USA, 70:3875-79. 1974 With C. R. Wrathall. Hyphal wall peptides and colonial morphology in Neurospora crassa. Biochem. Genet., 12:59-68.
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Cornelis Bernardus Van Niel November 4, 1897-March 10, 1985
By H. A. Barker and Robert E. Hungate Cornelis Bernardus Van Niel—Kees to his friends and students—is best known for his discovery of multiple types of bacterial photosynthesis, his deduction that all types of photosynthesis involve the same photochemical mechanism, and his extraordinary ability to transmit his enthusiasm for the study of microorganisms to his students. His interest in purple and green bacteria developed in his first year as a graduate student. After thoughtful analysis of the confusing literature dealing with these bacteria, he carried out a few simple experiments on their growth requirements. Interpreting the results in accordance with the theories of his professor, A. J. Kluyver, on the role of hydrogen transfer in metabolism, he developed a revolutionary concept of the chemistry of photosynthesis that was to influence research on the topic for many years. As a teacher he was unsurpassed. Although he taught in a small, somewhat remote institution with modest facilities, the force of his personality, his eloquence and scholarship made the Hopkins Marine Station a mecca for students of general microbiology throughout the western world. EDUCATION AND EARLY LIFE Van Niel was born in Haarlem, The Netherlands, into a family steeped in a highly conservative Calvinist tradition.
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His father and several uncles were businessmen and did not have a professional education. His father died when he was seven years old, and thereafter his mother largely depended on his uncles for advice in educating her young son. Since family tradition decreed that a son should succeed to his father's business, Kees was sent to a secondary school with a curriculum designed to prepare students for a commercial career. At the end of his third year in high school when he was fifteen years old, an event occurred that changed the course of his education. The family was spending their summer vacation as guests of a friend on a large estate in northern Holland devoted to various agricultural activities. A part was set aside for testing the effectiveness of various soil treatments on crop production, and van Niel has described how his host introduced him to the methods of agricultural research and how impressed he was to learn that "one could raise a question and obtain a more or less definitive answer to it as a result of an experiment ... particularly because I had grown up in a milieu where any kind of question was invariably answered by the stereotyped reply: 'Because somebody (usually a member of the family) said so'" (1967,1, p. 2). Van Niel's interest and enthusiasm for these activities led his family to reevaluate his education, and he was finally allowed to transfer to a college preparatory high school. Under the influence of one of his teachers in the new school he developed a strong interest in chemistry. He liked analytical chemistry so much that he set up a small laboratory at home and analyzed samples of fertilizer in his spare time. His academic record in high school was sufficient to obtain admission to the Chemistry Division of the Technical University in Delft on graduation without taking the usual entrance examination. He entered the University in autumn 1916 but, after only
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three months, was inducted into the Dutch army, in which he served until the end of December 1918. Life in the army was both a traumatic and a highly educational experience. Removed from the protective environment of his family for the first time, he was exposed to the rough and impersonal life of military training. He later wrote that up to this time he had been "utterly unaware of the many problems to which man is exposed and with which he must learn to cope." Fortunately, he received practical and intellectual support from a former high school classmate inducted at the same time, Jacques de Kadt. After a few days in a primitive military camp on the outskirts of Amersfoort, Jacques proposed that they rent a room in the city where they could spend their free time in greater comfort. They were soon joined by a friend of Jacques, and the three comrades spent their leisure hours discussing many subjects. Jacques was an intellectual with a cosmopolitan background. He introduced van Niel to new worlds of literature, art, and philosophy. Under his influence, van Niel read many of the works of Zola, Anatole France, Ibsen, Strindberg, Shaw, and Nietzsche. Their ideas frequently conflicted with van Niel's Calvinist background and led to what he later described as the rebellious phase of his life. On returning to the University after army service, van Niel was undecided whether he should continue the study of chemistry or take up the study of literature. But, discussing the alternatives with an aunt whose judgment he trusted "at least in part because of her unconventional attitudes and behavior," he was finally persuaded to continue on in chemistry. Still, his mental turmoil was such that he could not immediately switch back into the normal academic routine. He spent the first six months reading French, English, Scandinavian, and Russian 19th century literature and was not prepared to take the first year chemistry examination in June 1919.
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In the autumn, however, he finally settled down to serious study and by intensive effort was able in June 1920 to pass both the first-and second-year chemistry examinations. During the following year, van Niel took several courses in biology in addition to the prescribed chemistry program, including G. van Iterson's courses in genetics and plant anatomy and chemistry and M. W. Beijerinck's courses in general and applied microbiology. By November 1921, van Niel had completed all the requirements for the chemical engineering degree except a year of work in a specialized area of his own choosing. Already strongly attracted to microbiology from his exposure to Beijerinck's courses, he decided to specialize in it after hearing the inaugural lecture of A. J. Kluyver, who succeeded Beijerinck that year. Kluyver suggested that van Niel investigate the longevity of yeast in a medium containing sugar but little or no nitrogen. This problem provided some experience with microbiological and analytical methods and met the requirements for the degree, though the results were unimpressive. As a side project, van Niel checked a published report that a nonmotile Sarcina could develop flagella and motility by repeated transfer in a special medium. His first publication (van Niel 1923) showed that the previous author had confused Brownian movement with true motility and that his so-called flagella were artifacts of the staining method. DELFT: WORKING WITH KLUYVER After receiving his Chem. E. degree van Niel accepted a position as assistant to Kluyver. His duties consisted of caring for a large, pure culture collection of bacteria, yeasts and fungi; assisting undergraduates; and preparing demonstrations for Kluyver's two lecture courses. One of the courses dealt with the microbiology of water and sewage in which
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iron and sulfur bacteria play a role. Since Kluyver was unfamiliar with these organisms, he assigned van Niel the task of learning to culture them so that he could provide material for class demonstrations. To fulfill this assignment, van Niel read the publications of Winogradsky, Engelmann, Molisch, and Bavendamm on the colorless and purple sulfur bacteria and concluded that fundamental disagreements concerning the metabolism of these organisms needed clarification. Finding the purple bacteria ''aesthetically pleasing," he continued studying them after the lecture demonstrations were completed. During the next two years, while continuing as Kluyver's assistant, and later as conservator of the Institute, van Niel demonstrated that purple sulfur bacteria could grow in glass-stoppered bottles completely filled with a mineral medium containing sulfide and bicarbonate that were exposed to daylight. (No growth occurred in the dark.) He also isolated pure cultures of a Chromatium species and Thiosarcina rosea and showed that the yield of cells was proportional to the amount of sulfide provided and much greater than that of colorless aerobic sulfur bacteria in a similar medium. These observations and the earlier demonstration that O2 is not produced by purple bacteria were interpreted (in accordance with Kluyver's theory that most metabolic reactions are transfers of hydrogen between donor and acceptor molecules) to mean that purple sulfur bacteria carry out a novel type of photosynthesis in which carbon dioxide is reduced by hydrogen derived from hydrogen sulfide with the aid of energy from light. Mentioned briefly in Kluyver and Donker's treatise, "The Unity in Biochemistry,"1 without supporting evidence, this interpretation was probably based on van Niel's
1 A. J. Kluyver and H. J. L. Donker, "Die Einheit in der Biochemie," Chemie der Zelle und Gewebe, 13(1926): 134-90.
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work. Kluyver was not a coauthor of any of van Niel's early papers on photosynthetic bacteria. During this period Kluyver and van Niel published two papers: one dealing with a new type of yeast, Sporobolomyces (thought on the basis of its mode of spore formation to be a primitive basidiomycete), and another providing an explanation for the unusual morphology of a spore-forming bacterium that grew in liquid media as a tightly twisted, multistranded rope. While van Niel expected to continue his study of purple bacteria for his Ph.D. dissertation, he also developed, as a side project, an effective method for isolating propionic acid bacteria from Swiss cheese. When Kluyver pointed out that a study of this group would provide a faster path to the doctorate than a completion of his investigations of the slow-growing purple bacteria, van Niel reluctantly agreed. He spent the next two years, therefore, studying the biochemistry and taxonomy of the propionic acid bacteria. These biochemical studies were the first to provide a quantitative picture of the products derived from the fermentations of lactate, glycerol, glucose, and starch. His taxonomic studies provided a sound basis for recognition of the species of Propionibacterium. Van Niel's dissertation, written in English, was published in 1928. An unexpected byproduct of the study of the propionic acid bacteria was the identification of diacetyl as the compound responsible for the characteristic aroma of high quality butter. Van Niel noticed that cultures of one of his propionic acid bacteria grown on a glucose medium smelled like butter, then correlated this odor with the distinctive ability of the organism to produce acetylmethylcarbinol, an odorless compound that is readily oxidized to diacetyl, the actual source of the aroma. Van Niel spent almost seven years in the Delft laboratory,
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a stimulating period during which Kluyver was developing his ideas about the importance of hydrogen-transfer reactions in metabolism and the similarity of basic biochemical reactions in different organisms (the "unity in biochemistry" theory). Van Niel considered these ideas to be among the most fundamental and fruitful of that era. Revering Kluyver (whom he always referred to as "the Master"), as one of the great scientists of the age, he was yet able at a later time to point out some of Kluyver's errors in the analysis of specific phenomena and his occasional excessive reliance on generalizations lacking adequate experimental support (1959,1). PACIFIC GROVE: HOPKINS MARINE STATION In late 1927, L. G. M. Baas-Becking of Stanford University came to Delft looking for a microbiologist to fill a position at the new Jacques Loeb Laboratory at the Hopkins Marine Station on the Monterey Peninsula. Greatly impressed by van Niel's research accomplishments and his capacity for lucid communication, he offered him an appointment as associate professor. Put off by the reputed materialism of American society, van Niel was yet attracted by Becking's enthusiasm for the new laboratory and—encouraged by Kluyver— decided to strike out on his own. He arrived in California at the end of December 1928 and was immediately impressed by the charm of Carmel, the beautiful site of the Jacques Loeb Laboratory, and the freedom from outside pressures that the Marine Station provided. In later years he could never be persuaded to leave, even to succeed Kluyver at the Delft laboratory. PHOTOSYNTHESIS STUDIES At the Hopkins Marine Station van Niel continued his studies of purple and green bacteria with emphasis on the quantitative relations among substrates consumed and prod
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ucts formed. Progress was accelerated by the finding that the bacteria grew more rapidly under continuous artificial illumination. He demonstrated that the green bacteria oxidized hydrogen sulfide only as far as sulfur, whereas the purple sulfur bacteria further oxidized the sulfur to sulfate. Both coupled these oxidations with an essentially stoichiometric conversion of carbon dioxide to cellular materials in light-dependent reactions. The nonsulfur bacteria (Athiorhodaceace, which Molisch had grown aerobically on various organic compounds) were shown to develop anaerobically, but only in the presence of carbon dioxide and light. These and other observations led van Niel to conclude that photosynthesis is essentially a light-dependent reaction in which hydrogen from a suitable oxidizable compound reduces carbon dioxide to cellular materials having the approximate composition of carbohydrate. This was expressed by the generalized equation:
According to this formulation, H2O is the hydrogen donor in green plant photosynthesis and is oxidized to O2, whereas H2S or another oxidizable sulfur compound is the hydrogen donor for purple and green sulfur bacteria, and the oxidation product is sulfur or sulfate, depending on the organism. The nonsulfur purple bacteria that require suitable organic compounds in addition to carbon dioxide for anaerobic growth in light were presumed to use these compounds as hydrogen donors and to oxidize them—either partially or completely. Later, the purple sulfur bacteria were also shown to use some organic compounds in place of H2S in their photometabolism. These observations and interpretations, the results of some six years of investigation, were first presented at a small meeting of the Western Society of Naturalists in Pacific Grove
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at the end of 1929. Two years later van Niel published a detailed account of the culture, morphology and physiology of purple and green sulphur bacteria (1931,1), bringing his interpretation of their metabolism and its implications for green-plant photosynthesis to the attention of a wider audience. All of the purple sulfur bacteria he isolated were relatively small organisms, belonging to what he called Chromatium, Thiocystis, and Pseudomonas types. In material collected in nature (and in some enrichment cultures) he observed a number of larger forms but, despite numerous attempts, was unsuccessful in isolating them. The cultivation of these organisms was not accomplished until many years later, when N. Pfennig and H. G. Schlegel, both onetime associates of van Niel, discovered that nutritional and environmental requirements are more complex than had been previously recognized.2 Van Niel published a large monograph covering many years of work on the culture, general physiology, morphology and classification of the nonsulfur purple and brown bacteria in 1944 (1944,2). He classified over 150 strains isolated from natural sources into six species in two genera— Rhodospeudomonas and Rhodospirillum. He described the morphology of the organisms, their pigments, nutritional requirements, and metabolism in the presence and absence of light. As in all his publications, van Niel also reviewed the historical background and current literature of the subject critically and thoroughly. Following the recognition of several types of photosynthesis using different hydrogen donors, van Niel began to
2 H. G. Schlegel and N. Pfennig, "Die Anreicherungskultur einiger Schwefelpurpurbakterien. "Archiv für Mikrobiologie, 38(1961):1-39, and N. Pfennig and K. D. Lippert, "Über das Vitamin B12 Bedurfnisphototropher Schwefelbakterien." Archiv für Mikrobiologie, 55(1966):245-56.
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consider how radiant energy participates in these reactions. There appeared to be two possibilities, both considered by earlier investigators: radiant energy could be used to activate either carbon dioxide or the hydrogen donor. Initially, van Niel and Muller (1931,2) were inclined to believe that light is used primarily to activate carbon dioxide, a relatively stable compound and the common reactant in all photosynthetic systems. But they did not exclude the second possibility, that light also activated the hydrogen donor. In this connection they noted a correlation between the presence of nonchlorophyll yellow and red pigments and the nature of the hydrogen donor used by different organisms. These pigments, lacking in the green sulfur bacteria that utilize the easily oxidizable hydrogen sulfide, occur exclusively in organisms utilizing water or sulfur, then thought to require a greater activation. This led van Niel to undertake a series of studies of the pigments of the purple and green bacteria. Van Niel and Arnold (1938,1) developed a convenient spectrophotometric method for determining the amount of bacteriochlorophyll in photosynthetic purple and brown bacteria under conditions avoiding interference by the red carotinoid pigments. They also reported that van Niel and E. Wiedemann, working in A. Stoll's laboratory, had examined the green pigments of six different strains of purple and brown bacteria and concluded that they were identical with the chlorophyll of the purple sulfur bacterium, Thiocystis, previously studied by H. Fischer. Van Niel and Smith (1935,2) began a study of the chemistry of the major red pigment of the nonsulfur purple bacterium, Rhodospirillum rubrum. By solvent extraction and repeated crystallization, they isolated about 100 milligrams of an apparently homogeneous carotinoid they called "spirilloxanthin." Its empirical composition was found to be
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C48H66O3, and it contained fifteen double bonds, no more than one hydroxyl group, and no free carboxyl group—making it the most highly unsaturated carotinoid then known. Rhodoviolascin, a red pigment that had almost the same absorption spectrum and melting point as spirilloxanthin, was later isolated by Karrer and Solmssen from a nonsulfur purple bacterium identified as Rhodovibrio.3 This compound contained two methoxyl groups and had the empirical formula C40H54 (OCH3)2. Polgár, van Niel, and Zechmeister (1944,1) redetermined the molecular weight and composition of spirilloxanthin using material purified by column chromatography and concluded that the formula established by Karrer and Solmssen was correct and that rhodoviolascin and spirilloxanthin are identical. They also found that spirilloxanthin is unstable and reversibly converts, under relatively mild conditions, to two compounds designated neospirilloxanthin-A and B, which can be separated from spirilloxanthin chromatographically. A study of the absorption spectra of these compounds under various conditions led to the conclusion that spirilloxanthin is an all-trans compound, whereas neospirilloxanthin-A probably contains two cis double bonds, one of which is centrally located. In a broader review of the known properties of red pigments derived from various nonsulfur purple bacteria, van Niel (1944,2) concluded that, in addition to spirilloxanthin, at least two other pigments occur in these organisms, distinguishable by their melting points and absorption spectra. When anaerobic cultures are exposed to oxygen, some strains of nonsulfur purple bacteria undergo a dramatic color change from yellow-brown to deep red. Van Niel (1947,1) investigated this phenomenon in L. Zechmeister's
3 P. Karrer and U. Solmssen, "Die Carotinoide der Purpurbakterien I," Helvetica Chimica Acta 18(1935):1306-15. Parts II and III of this article appear in Helvetica Chimica Acta 19(1936):3 and 1019.
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laboratory. Using cells of Rhodopseudomonas spheroides grown under semianaerobic conditions in continuous light, he isolated the two most abundant red and yellow carotinoid pigments as crystalline products. Both pigments were shown to have all-trans configurations and, as previously shown for spirilloxanthin, were easily converted to the cis-isomers. In order to follow the pigment changes associated with exposure of anaerobically-grown cells to oxygen, a spectrophotometric method was developed to determine the amounts of red and yellow pigments in a mixture obtained by extracting a cell suspension. Using this method the yellow pigment was shown to be partially and irreversibly converted to the red pigment when anaerobically-grown cells were exposed to oxygen. As previously noted by C. S. French, the conversion of the yellow carotinoid to red occurred only in the presence of actively metabolizing bacteria. The nature of the chemical transformation responsible for the color change was not determined. Studies in several laboratories of the role of various pigments in photosynthesis and phototaxis by Rhodospirillum rubrum had resulted in conflicting conclusions as to whether spirilloxanthin with absorption maxima at 550, 510 and 480 nm, or another pigment with maxima at 530, 490 and 460 nm was the photoactive compound. A possible explanation for this discrepancy was provided by L. N. M. Duysen,4 who observed that the absorption spectrum, and therefore presumably the pigment composition, of R. rubrum changed with the age of the culture. Young cultures showed a minor 530 nm absorbance peak, gradually replaced by the 550 nm peak of spirilloxanthin as the culture aged. This observation was confirmed by van Niel and Airth (unpublished work, 1954) with two strains of R. rubrum. Van Niel, Goodwin, and Sissins
4
L. N. M. Duysens, unpublished doctoral thesis for the University of Utrecht, 1952.
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(1956,1) subsequently identified the carotinoids in young cultures and showed that these indeed decreased with time, while spirilloxanthin increased from about twenty percent of the total carotinoids in a one-day-old culture to about ninety percent in a five-day culture. These studies provided information concerning the identity and properties of the pigments of photosynthetic bacteria but did little to clarify the role of the pigments in photosynthesis. By 1936 van Niel's interpretation of the role of the photochemical system in photosynthesis had changed radically (1936,3). He had abandoned the earlier theory that radiant energy participated directly in carbon dioxide activation when he recognized that various nonphotosynthetic bacteria, including several chemoautotrophic species, methanogenic bacteria and propionic acid bacteria, readily utilized carbon dioxide in the dark. Furthermore, the idea that each of the many inorganic and organic compounds used as substrates by the photosynthetic bacteria were directly involved in a photochemical reaction appeared unlikely, particularly since van Niel had shown that certain organic compounds used by the nonsulfur purple bacteria are oxidized both in the dark with O2 or in the light in the absence of O2. He later demonstrated that even the rates of organic substrate oxidation are the same in the dark and in the photosynthetic reaction, provided the light intensity is sufficiently high (1941,2; 1949,2). Van Niel finally concluded that both plant and bacterial photosynthetic reactions have a common photochemical reaction: the photolysis of water to form a strong reducing agent and a strong oxidizing agent. He postulated that the reducing agent was used, through a series of enzymatic reactions, to convert carbon dioxide to cellular constituents; whereas the oxidizing agent was used either to generate O2
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in green plant photosynthesis or to oxidize the hydrogen donor in bacterial photosynthesis. Van Niel's unified interpretation of the photochemical event in photosynthesis is similar in principle to the current interpretation of this process, although a special type of chlorophyll (rather than water) is now considered to be the source of the light-generated oxidizing and reducing species. In collaboration with H. Larsen and C. S. Yocum, van Niel investigated the energetics of photosynthesis in green sulfur bacteria supplied with different reducing agents with the object of determining whether the energy released by oxidation of the reducing agents was used to reduce carbon dioxide (1952,3). They determined the number of light quanta used to convert one molecule of CO2 into cell material when either H2, thiosulfate, or tetrathionate was used as the reducing agent. Photosynthesis with H2 was expected to require about 28,000 calories less than with the other substrates because of the large energy change associated with H2 oxidation, but—finding that the number of light quanta required to reduce one molecule of CO2 was approximately the same with all three substrates—they concluded that the energy obtained by the oxidation of the electron donor is not used for CO2 assimilation. Several other postdoctoral fellows who studied with van Niel made significant contributions to understanding the biology and physiology of photosynthetic bacteria. Providing background and inspiration for these investigations, van Niel gave encouragement and advice during the experimental work, evaluated results critically, and aided in preparing the manuscripts—but was seldom willing to become a coauthor of the final publications. Many of his own scientific contributions, consequently, are embedded in the publications of his associates, as in F. M. Muller's 1933 publications on the utilization of organic compounds by purple sulfur bacteria;
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J. W. Foster's 1944 paper on the coupling of CO2 reduction to the oxidation of isopropanol to acetone by nonsulfur purple bacteria; H. Larsen's works in 1952 and 1953 on the culture and physiology of green sulfur bacteria; and R. K. Clayton's 1955 report on the relation between photosynthesis and respiration in Rhodospirillum rubrum. Van Niel's influence can also be seen in Pfennig's work on the nutrition and ecology of photosynthetic bacteria. METHANE PRODUCTION AND CARBON DIOXIDE UTILIZATION Van Niel's studies of photosynthetic bacteria led him to consider other processes in which carbon dioxide utilization might occur. In the early 1930s he had postulated that methane formation from organic compounds by anaerobic bacteria was the result of carbon dioxide reduction. This idea was based upon the investigations of N. L. Söhngen, a student of Beijerinck who had studied the decomposition of lower fatty acids by methanogenic enrichment cultures under anaerobic conditions and found that formate and lower fatty acids with an even number of carbon atoms are converted quantitatively to carbon dioxide and methane. The identity of the products, therefore, was independent of the chainlength of the substrate. Söhngen's cultures, furthermore, could convert hydrogen and carbon dioxide to methane according to the equation:
Since carbon dioxide is clearly reduced to methane in this reaction, van Niel concluded that this also occurs in the fermentation of fatty acids. Carbon dioxide, in other words, was postulated to serve as hydrogen acceptor for the oxidation of fatty acids to carbon dioxide and water. This could explain why methane is the only reduced compound formed in the
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methane fermentation of organic compounds—a theory that received support from the 1939-1940 demonstration by H. A. Barker that a purified culture of a methanogen apparently coupled the oxidation of ethanol to acetic acid with the reduction of carbon dioxide to methane. In 1967, however, M. P. Bryant et al. found that the culture contained two kinds of bacteria—one which oxidizes ethanol to acetate and H 2, and the methanogen that converts H2 and carbon dioxide to methane. The formation of methane from all but a few organic compounds now appears to require a similar participation of a non-methanogenic bacterium. Van Niel's carbon dioxide reduction theory of methane formation from organic compounds, consequently, is valid only for the syntrophic association of two species. Following the early studies of S. Ruben and M. D. Kamen at the University of California, Berkeley, on biological carbon dioxide fixation by use of the short-lived carbon isotopes 11C, van Niel and some of his students collaborated in similar studies with propionic acid bacteria,5 fungi,6 and protozoa (1942,3). The experimenters sought to confirm and extend the unexpected discovery of H. G. Wood and C. H. Werkman that succinic acid is formed in part from carbon dioxide. The ciliate Tetrahymena geleii was also shown to incorporate carbon dioxide into succinate, whereas the fungi Rhizopus nigricans and Aspergillus niger incorporated carbon dioxide into the carboxyl groups of fumarate and citrate, respectively. Van Niel's special contribution to these investigations was his attempt to understand the general requirement of non
5 S. F. Carson, J. W. Foster, S. Ruben, and H.A. Barker, ''Radioactive carbon as an indicator of carbon dioxide utilization. V. Studies on the propionic acid bacteria." PNAS 27(1941):229-35. 6 J. W. Foster, S. F. Carson, S. Ruben, and M. D. Kamen, "Radioactive carbon as an indicator of carbon dioxide utilization. VII. The assimilation of carbon dioxide by molds. PNAS 27(1941):590-96.
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photosynthetic microorganisms for carbon dioxide and the mechanism of its fixation (1942,2). He concluded that carbon dioxide fixation generally occurs by carboxylation reactions and that carbon dioxide is probably required to counteract the decarboxylation of oxaloacetate, which "constitutes a 'leak' through which certain essential cell constituents are drained off." In 1935 H. A. Barker, at van Niel's suggestion, undertook a study of the respiratory activity of the colorless algae Prototheca zopfii . His original objective was to use Otto Warburg's manometric method to identify the organic compounds that the organism could oxidize and to determine the quantities of O2 consumed and CO2 produced from a known quantity of each substrate. The data showed that the amounts of O2 and CO2 were far below those required for complete oxidation, the gas exchange accounting for only seventeen to fifty percent of that required for complete oxidation depending on the particular substrate. The rest of the substrate was apparently converted into storage or cellular materials with the approximate empirical composition of carbohydrate. This unexpectedly high conversion of respiratory substrates to cell materials became known as oxidative assimilation. In Kluyver's laboratory, G. Giesberger and C. E. Clifton subsequently obtained similar results with several bacteria. Because of its apparent relation to the synthesis of cell materials in photosynthesis and the general problem of the utilization of the products and energy of respiration for assimilatory purposes, van Niel maintained a continuing interest in this phenomenon. He and his students studied assimilation reactions of both yeast and bacteria, the most interesting result being the demonstration that yeast but not lactic acid bacteria assimilate about thirty percent of the glucose decomposed under anaerobic conditions—a process they called "fermentative assimilation."
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Bacterial Taxonomy One of van Niel's most enduring scientific interests outside of photosynthesis and photosynthetic bacteria was bacterial taxonomy. In his doctoral dissertation he had reviewed what he called "the main features of bacterial taxonomy" and proposed a possible sequence for the evolution of various morphological types of bacteria. Starting from a presumably primitive, nonmotile, spherical cell, it progressed along three postulated evolutionary lines to polarly flagellated spirilla, peritrichously flagellate sporulating rods, and permanently immotile rods forming conidia. With small modifications, this concept of morphological evolution formed the basis of the taxonomic system proposed by Kluyver and van Niel (1936,4). Four morphological families defined by cell shape, type of flagellation, and sporulation were subdivided by morphology into eleven tribes. The organisms in the morphological tribes were further assigned to sixty-three genera on the basis of types of energy metabolism, substrate utilization and—among chemoheterotrophic anaerobes—products of metabolism. Although recognizing that this taxonomic system was an oversimplification, the authors believed that it was more rational and "natural," i. e., phylogenetic, than previous systems. In 1941, van Niel and R. Y. Stanier undertook an analysis of the problems of classification of the larger taxonomic units among microorganisms (1941,3). After pointing out glaring deficiencies in the definitions of major microbial groups in Bergey's Manual,7 they concluded that for larger taxa, morphological characteristics should be given priority over physiological characteristics. On this basis they decided that the blue-green algae (Myxophyta) and the bacteria (Schizomy
7 Bergey's Manual of Determinative Bacteriology, ed. D. H. Bergey, Baltimore: Williams and Wilkins Co., 5th edition, 1936.
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cetae) should be combined in the kingdom, Monera, which comprises organisms without true nuclei, plastids, and sexual reproduction. The Schizomycetae were then separated into four classes: Eubacteriae, Myxobacteriae, Spirochaetae, and a heterogeneous group of organisms not falling into the other classes. The Eubacteriae were further separated into three orders (Rhodobacteriales, Eubacteriales, Actinomycetales) on the basis either of type of metabolism (photosynthetic, nonphotosynthetic) or cell organization (unicellular, mycelial). Each of these groups was defined as precisely as possible with the information available, the authors emphasizing that the proposed system was a first draft and subject to revision as new information accumulated. By 1946, van Niel no longer believed that a taxonomic system based on phylogenetic considerations was possible in view of the relatively few morphological properties of bacteria, the general absence of developmental processes, and the probability of the occurrence of both convergent and divergent evolution in the development of existing groups (1946,1). He pointed out that attempts to classify bacteria in a single system by the use of morphological, physiological, nutritional, and ecological properties was only partially successful. Since different properties often overlapped, a single organism could be assigned to more than one taxonomic group or could not be readily assigned to any. He concluded that attempts to accommodate all known bacteria in a single taxonomic system should be abandoned until more information on phylogenetic relationships was available. He went so far as to suggest that the use of binomial nomenclature should be discontinued until phylogenetic relationships could be firmly established and proposed, in the meantime, bacteria could be identified more readily by multiple keys based upon any of several conspicuous and readily determinable properties. By 1955, van Niel had become skeptical of the possibility
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of separating bacteria and blue-green algae from other organisms on the basis that they lacked nuclei, plastids, and sexual reproduction. New developments had weakened or destroyed these negative criteria as differential characters. He noted that some bacteria contained "discrete structures that might be considered, on the basis of their behavior and chemical nature, as nuclei"; that photosynthetic pigments of some purple bacteria and blue-green algae were located in uniform spherical particles rather than being distributed evenly throughout the cells; and that, in E. coli, an exchange of genetic characters between cells had been clearly demonstrated (1955,1). On the basis of new information developed since van Niel's 1955 paper, Stanier and van Niel (1962,2) again examined the criteria used to distinguish bacteria and blue-green algae from viruses and other protists. In agreement with Lwoff,8 they noted that the structures and modes of reproduction of viruses differ from those of bacteria and that no ambiguity existed as to the taxonomic position of rickettsia, pleuro-pneumonia-like organisms, and other obligately parasitic bacteria. The bacteria and blue-green algae were separated from all other protists by the procaryotic nature of their cells. They distinguished the procaryotic from the eucaryotic cell by the absence of internal membranes separating nuclear material and—when present—respiratory and photosynthetic apparatuses from each other and from the cytoplasm. In addition, the nuclei of procaryotes divide by fission rather than by mitosis, their cell walls contain mucopeptides as a strengthening element, and the structure of the flagella, when present, is unique. The authors concluded that there was no adequate basis for separating bacteria from blue-green algae.
8 A. Lwoff, "The concept of virus." Journal of General Microbiology , 17 (1957):239-53.
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DENITRIFICATION Van Niel published two papers dealing with aspects of the chemistry of denitrification. Allen and van Niel (1952,1) investigated the pathway of nitrite reduction by Pseudomonas stutzeri. Initially they tested the possibility that the conversion of nitrite to N2 may involve a reaction between nitrite and an amine, but no supporting evidence could be obtained. They then tested possible intermediates in nitrite reduction by the technique of simultaneous adaptation and the use of various inhibitors and found that neither N2O nor hyponitrite could fulfill this role. Nitramide, H2N. NO2, however, was found to be reduced readily to N2 at about the same rate as nitrite and the utilization of both compounds was inhibited by cyanide to the same extent. Nitramide, consequently, was considered to be a possible intermediate in denitrification. In 1920 Warburg and Negelein reported that algae exposed to light in a nitrate solution produce O2 in the absence of added carbon dioxide. They postulated that the algae used nitrate to oxidize cellular organic compounds to carbon dioxide, which was then used for O2 production by photosynthesis.9 Van Niel, Allen, and Wright proposed the alternative interpretation that nitrate replaces carbon dioxide as the electron acceptor in photosynthesis (1953,1). They showed that when nitrate-adapted Chlorella is exposed to high light-intensity in a medium containing excess carbon dioxide, the rate of O2 production increased with the addition of nitrate. This increased rate could not have been caused by an increase in carbon dioxide production, since the reaction was already saturated with this compound. The higher rate, then, could
9 O. Warburg and E. Negelein, "Über die Reduktion der Saltpetersäure in grünen Zellen." Biochemische Zeitschrift, 110(1920):66-115.
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only result from the utilization of nitrate as an additional electron acceptor. VAN NIEL THE GENERALIST As his reputation as a scientist and teacher spread, van Niel responded to many invitations to lecture and write reviews. In the early part of his career these mostly dealt with bacterial photosynthesis and its relation to plant photosynthesis. Later he often dealt with broader topics such as "The Delft School and the Rise of General Microbiology" (1949,4), "The Microbe as a Whole" (1955,4), ''Natural Selection in the Microbial World" (1955,3), "Evolution as Viewed by the Microbiologist" (1956,2c), and "Microbiology and Molecular Biology" (1966,1). He always displayed an impressive command of historical background and current literature and a notably clear, analytical, and elegant style of presentation. "On radicalism and conservatism in science" (1955,2), his presidential address to the Society of American Bacteriologists in 1954, was a clear statement of van Niel's personal philosophy—a strong preference for the heretical and unconventional over established and accepted dogma, despite his recognition of the weaknesses and strengths of both. For him the essence of science was the development of an attitude of mind that "accepts experience as the guiding principle by which it is possible to test the relative merits of opposing viewpoints by means of carefully conducted, controlled experiments," and "recognizes equally keenly that our knowledge and capacities are exceedingly limited, not merely if considered from the standpoint of the individual, but even with reference to the combined experience of the human race." He concluded that the most desirable mental characteristics of a scientist are objectivity and tolerance, and that his greatest satisfaction should derive from "having enriched the experience of his fellow men."
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Van Niel's chapter on "Evolution as Viewed by the Microbiologist" (1956,2c) provided a stimulating synthesis of ideas concerning the origin of life and the relation of living to nonliving systems. By the application of both logic and intuition to the available scientific information and theory, he developed the hypothesis that life is a special property of matter that inevitably appears when chemical systems reach a state of sufficient complexity under suitable conditions. His generalized concept of evolution comprised physical, chemical, biochemical, and biological phases of which only the last corresponds with evolution in the Darwinian sense. TEACHER AND COLLEAGUE In addition to being an outstanding investigator, van Niel was a superlative teacher, and his greatest contribution to science may well have been his teaching of general microbiology and comparative biochemistry. Soon after coming to the Hopkins Marine Station he began offering a tenweek laboratory course in microbiology. Initially, the content of the course was similar to that given at Delft and consisted of an introduction to methods of isolating and identifying microorganisms in commercial yeast, milk, water, and soil. But van Niel soon realized that neither Beijerinck's elective culture methods —based on the principle of natural selection—nor Kluyver's ideas about comparative biochemistry were appreciated in this country. He therefore undertook to develop a course emphasizing these approaches to microbiology and biochemistry. Van Niel's students learned how numerous morphological and physiological types of bacteria, when their nutritional and environmental requirements were known, could be enriched and isolated from natural sources. He discussed the metabolism of each group, emphasizing the most recent findings regarding intermediary metabolism, similarities and differences in deg
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radative pathways, and the chemical and energetic relations between degradative metabolism and the synthesis of cellular components. He examined the structure of bacterial cells, aspects of bacterial genetics, variation and adaptation, bacterial and yeast taxonomy, and the philosophy of science. The course was organized as a series of relatively simple experiments for which van Niel provided the background, rationale, and interpretation of results. He was always in the laboratory guiding the work and commenting on each student's observations and results and often used the Socratic method, stimulating students to make judgments about the meaning of their observations and sometimes intentionally leading them to some plausible but incorrect conclusion so that a later experiment, already planned, would reveal the error. After a topic or phenomenon had been introduced in a laboratory experiment, he would launch into a presentation of its historical background, usually starting with the most primitive ideas and progressing to the latest developments. He always placed great emphasis on possible alternative interpretations of the available information at each phase of scientific development and on the frequently slow and difficult process of moving from clearly erroneous to more nearly correct—but never immutable—conclusions. His lectures often lasted for several hours and were presented with such clarity and histrionic skill as to capture the complete attention and stimulate the enthusiasm of his students. As the course developed over the years along with the literature of microbiology, lectures took up a larger proportion of the available time. The course expanded from three afternoons to three days a week, with class hours often extending from eight in the morning to well into the evening, with time out only for lunch and tea and coffee breaks. The course was very strenuous for van Niel, who was never particularly robust, and in his later years he was so exhausted by its end he needed some weeks to recuperate.
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During the early years, only a few students attended, but as van Niel's reputation as a teacher spread, the class had to be limited, initially to eight, and later to fourteen students—the number that could be accommodated in the small Marine Station laboratory. The students were initially undergraduate or graduate students from Stanford, but later a large proportion came from other institutions. In 1950, for example, only one of the thirteen students was from Stanford. The others were from Washington University, Wisconsin, Michigan, Missouri, California Institute of Technology, Connecticut, Illinois, Cambridge, and the University of California at Los Angeles. In addition there were eleven auditors of the discussions and lectures who did not do the experiments— mostly postdoctoral fellows or established scientists who wished to extend their background in general microbiology. The lists of students and auditors who attended van Niel's course between 1938 and 1962 reads like a Who's Who of biological scientists in the United States, with several, as well, from other countries. Both directly, and indirectly through his students, van Niel exerted a powerful influence on teaching and research in general microbiology for a generation. Although his own research was concerned mainly with photosynthetic bacteria, van Niel was interested in the biology and metabolism of many other groups of microorganisms. He did not believe in directing the research of his younger associates but rather encouraged them to follow their own interests, some of which had been stimulated by his lectures and personal discussions. As a consequence, the range of phenomena investigated in his laboratory was exceedingly wide and included the culture and physiology of blue-green algae and diatoms, nutritional and taxonomic studies of plant-pathogenic bacteria, biological methane formation, pteridine and carbohydrate metabolism of protozoa, germination of mold spores, biology of caulobacteria, cultivation of free-living spirochetes, induction of fruiting bodies
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in myxobacteria, decomposition of cellulose, the role of microorganisms in the food cycle of aquatic environments, adaptation of bacteria to high salt concentrations, cultivation of spirilla and colorless sulfur bacteria, bacterial fermentations, thermophylic bacteria, denitrification, pyrimidine metabolism, and the thermodynamics of living systems. To all students van Niel gave freely of his time, advice and enthusiasm, drawing on his own extraordinary knowledge of the literature. RETIREMENT Following his retirement from the Marine Station in 1962, van Niel held a visiting professorship at the University of California at Santa Cruz from 1964 to 1968, teaching part of a freshman-level biology course in collaboration with K. V. Thimann and L. Blinks. After 1972, van Niel gave up teaching and research entirely and disposed of his scientific library and large collection of reprints. Thereafter he lived quietly with his wife, Mimi, in Carmel and spent his leisure reading classical and modern literature and listening to classical music, which he greatly enjoyed. He was often visited by former students who continued to be impressed by the warm hospitality of his home, the charm of his personality, the breadth of his understanding, and the comprehensiveness of his memory.
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HONORS AND DISTINCTIONS Degrees And Honorary Degrees 1923 Chemical Engineering, Technical University, Delft 1928 D. Sci., Technical University, Delft 1946 D. Sci. (Honorary), Princeton University 1954 D. Sci. (Honorary), Rutgers University 1968 LL. D., University of California, Davis Fellowships And Professional Appointments 1925-1928 Conservator, Laboratorium voor Microbiologie, Delft 1928-1935 Associate Professor of Microbiology, Stanford University, Hopkins Marine Station 1935-1936 Rockefeller Foundation Fellow 1935-1946 Professor of Microbiology, Stanford University 1945 John Simon Guggenheim Fellow 1946-1963 Herstein Professor of Biology, Stanford University 1955-1956 John Simon Guggenheim Fellow 1963-1985 Herstein Professor, Emeritus, Stanford University 1964-1968 Visiting Professor, University of California, Santa Cruz Awards And Honors 1942 Stephen Hales Prize, American Society of Plant Physiology 1964 Emil Christian Hansen Medalist, Carlsberg Foundation of Copenhagen 1964 National Medal of Science 1966 Charles F. Kettering Award, American Society of Plant Physiology 1967 Rumford Medal, American Society of Arts and Sciences 1967 Honorary Volume, Archiv für Mikrobiologie 1970 Antonie van Leeuwenhoek Medal, Royal Netherlands Academy of Sciences Learned Societies 1945 National Academy of Sciences 1948 American Philosophical Society 1950 American Academy of Arts and Sciences 1952 Charles Reid Barnes Life Membership, American Society of Plant Physiology
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1954 President, American Society for Microbiology 1954 Corresponding Member, Academy of Sciences, Göttingen, Germany 1958 American Academy of Microbiology 1963 Honorary Member, Société Française de Microbiologie 1967 Honorary Member, Society of General Microbiology 1968 Honorary Member, Royal Danish Academy of Sciences and Letters
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Selected Bibliography
1923 Über die Beweglichkeit und das Vorkommen von Geisseln bei einigen Sarcina Arten. Zentralbl. Bakteriol. Parasitenkd. Infektionskr. Hyg., Abt. II., 60:289-98. 1924 With A. J. Kluyver. Über Spiegelbilder erzeugende Hefearten und die neue Hefegattung Sporobolomyces. Zentralbl. Bakteriol. Parasitenkd. Infektionskr. Hyg., Abt. II., 63:1-20. 1925 With F. Visser't Hooft. Die fehlerhafte Anwendung biologischer Agenzien in der organischen Chemie. Eine Warnung. Ber. Dtsch. Chem. Ges., 58:1606-10. 1926 With A. J. Kluyver. Über Bacillus funicularis n. sp. nebst einigen Bemerkungen über Gallionella ferruginea Ehrenberg. Planta, 2:507-26. 1927 With A. J. Kluyver. Sporoboloymces-ein Basidiomyzet? Ann. Mycol. Notitiam Sci. Mycol. Univ., 25:389-94. Notiz über die quantitativ Bestimmung von Diacetyl und Acetylmethylcarbinol. Biochem. Z., 187:472-78. 1928 The Propionic Acid Bacteria. (Doctoral Dissertation.) Haarlem, The Netherlands: Uitgeverszaak J. W. Boissevain & Co. 1929 With A. J. Kluyver and H. G. Derx. De bacteriën der roomverzuring en het boteraroma. Verslag gewone Vergader. Afd. Naturrkd. Nederl. Akad. Wetensch., 38:61-2. With A. J. Kluyver and H. G. Derx. Über das Butteraroma. Biochem. Z., 210:234-51.
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1930 Photosynthesis of bacteria. In: Contributions to Marine Biology, Stanford: Stanford University Press, pp. 161-69. 1931 On the morphology and physiology of the purple and green sulfur bacteria. Arch. Mikrobiol., 3:1-112. With F. M. Muller. On the purple bacteria and their significance for the study of photosynthesis. Rec. Trav. Bot. Neer., 28:245-74. 1935 Photosynthesis of bacteria. Cold Spring Harbor Symp. Quant. Biol., 3:138-50. With J. A. C. Smith. Studies on the pigments of the purple bacteria. I. On spirilloxanthin, a component of the pigment complex of Spirillum rubrum. Arch. Mikrobiol., 6:219-29. A note on the apparent absence of Azotobacter in soils. Arch. Mikrobiol., 6:215-18. 1936 On the metabolism of the Thiorhodaceae. Arch. Mikrobiol., 7:323-58. With D. Spence. Bacterial decomposition of the rubber in Hevea latex. Ind. Eng. Chem., 28:847-50. Les photosynthèses bactériennes. Bull. Assoc. Diplomes Microbiol. Fac. Pharm. Nancy, 13:3-18. With A. J. Kluyver. Prospects for a natural system of classification of bacteria. Zentralbl. Bakteriol. Parasitenkd. Infektionskr. Hyg. Abt. II, 94:369-403. 1937 The biochemistry of bacteria. Ann. Rev. Biochem., 6:595-615. 1938 With W. Arnold. The quantitative estimation of bacteriochlorophyll. Enzymologia, 5:244-50.
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1939 A. J. Kluyver. Als mikrobiolog en als biochemikus. Chem. Weekbl., 36:1-109. 1940 The biochemistry of microorganisms: An approach to general and comparative biochemistry. Am. Assoc. Adv. Sci. Publ., 14:106-19. 1941 With E. H. Anderson. On the occurrence of fermentative assimilation. J. Cell. Comp. Physiol., 17:49-56. The bacterial photosyntheses and their importance for the general problem of photosynthesis. Adv. Enzymol., 1:263-328. With R. Y. Stanier. The main outlines of bacterial classification. J. Bacteriol., 42:437-66. 1942 With A. L. Cohen. On the metabolism of Candida albicans. J. Cell. Comp. Physiol., 20:95-112. With S. Ruben, S. F. Carson, M. D. Kamen, and J. W. Foster. Radioactive carbon as an indicator of carbon dioxide utilization. VIII. The role of carbon dioxide in cellular metabolism. Proc. Natl. Acad. Sci. USA, 28:8-15. With J. O. Thomas, S. Rubin, and M. D. Kamen. Radioactive carbon as an indicator of carbon dioxide utilization. IX. The assimilation of carbon dioxide by protozoa. Proc. Natl. Acad. Sci. USA, 28:157-61. 1943 Biochemistry of microorganisms. Ann. Rev. Biochem., 12:551-86. Biochemical problems of the chemoautotrophic bacteria. Physiol. Rev., 23:338-54. 1944 With A. Polgár and L. Zechmeister. Studies on the pigments of the purple bacteria. II. A spectroscopic and stereochemical investigation of Spirilloxanthin. Arch. Biochem., 5:243-64. The culture, general physiology, morphology, and classification of
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the nonsulfur purple and brown bacteria. Bacteriol. Rev., 8:11-18. Recent advances in our knowledge of the physiology of microorganisms. Bacteriol. Rev., 8:225-34. 1946 The classification and natural relationships of bacteria. Cold Spring Harbor Symp. Quant. Biol., 11:285-301. 1947 Studies on the pigments of the purple bacteria. III. The yellow and red pigments of Rhodopseudomonas spheroides. Antonie van Leeuwenhoek J. Microbiol., 12:156-66. 1948 Propionibacterium, pp. 372-79; Rhodobacterineae, pp. 838-74; Beggiatoaceae, pp. 988-96; Achromatiaceae, pp. 997-1001. In: Bergey's Manual of Determinative Bacteriology, 6th ed., eds. R. S. Breed, E. G. D. Murray, and A. P. Hitchens, Baltimore: Williams and Wilkins Co. 1949 The kinetics of growth of microorganisms. In: The Chemistry and Physiology of Growth, ed. A. K. Parpart, Princeton: Princeton University Press, pp. 91-105. The comparative biochemistry of photosynthesis. In: Photosynthesis in Plants, eds. J. Franck and W. E. Loomis, Ames: Iowa State College Press, pp. 437-95. Comparative biochemistry of photosynthesis. Am. Sci., 37:371-83. The ''Delft school" and the rise of general microbiology. Bacteriol. Rev., 13:161-74. 1952 With M. B. Allen. Experiments on bacterial denitrification. J. Bacteriol., 64:397-412. Bacterial photosynthesis. In: The Enzymes, vol. 2, part 2, eds. J. B. Sumner and K. Myrback, New York: Academic Press, pp. 1074-88. With H. Larsen and C. S. Yocum. On the energetics of the photosyntheses in green sulfur bacteria. J. Gen. Physiol., 36:161-71.
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With M. B. Allen. A note on Pseudomonas stutzeri. J. Bacteriol., 64:413-22. 1953 With M. B. Allen and B. E. Wright. On the photochemical reduction of nitrate by algae. Biochim. Biophys. Acta, 12:67-74. Introductory remarks on the comparative biochemistry of microorganisms. J. Cell. Comp. Physiol., 41(Suppl. 1): 11-38. 1954 The chemoautotrophic and photosynthetic bacteria. Annu. Rev. Microbiol., 8:105-32. 1955 Classification and taxonomy of the bacteria and blue-green algae. In: A Century of Progress in the Natural Sciences 1853-1953, ed. E. L. Kessel, San Francisco: California Academy of Sciences, pp. 89-114. On radicalism and conservatism in science. Bacteriol. Rev., 19: 1-5. Natural selection in the microbial world. J. Gen. Microbiol., 13:201-17. The microbe as a whole. In: Perspectives and Horizons in Microbiology , ed. S. A. Waksman, New Brunswick, N. J.: Rutgers University Press, pp. 3-12. 1956 With T. W. Goodwin and M. E. Sissins. Studies in carotenogenesis. 21. The nature of the changes in carotinoid synthesis in Rhodospirillum rubrum during growth. Biochem. J., 63:408-12. Phototrophic bacteria: Key to the understanding of green plant photosynthesis, pp. 73-92; Trial and error in living organisms: Microbial mutations, pp. 130-54; Evolution as viewed by the microbiologist, pp. 155-76. In: The Microbes Contribution to Biology. A. J. Kluyver and C. B. van Niel. Cambridge: Harvard University Press. With G. Milhaud and J. P. Aubert. Études de la glycolyse de Zymosarcina ventriculi, Ann. Inst. Pasteur, 91:363-68. In memoriam: Professor Dr. Ir. A. J. Kluyver. Antonie van Leeuwenhoek J. Microbiol. Sérol., 22:209-17.
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1957 Rhodobacteriineae, pp. 35-67; Propionibacterium, pp. 569-76; Achromatiaceae, pp. 851-53. In: Bergey's Manual of Determinative Bacteriology, 7th ed., eds. R. S. Breed, E. G. D. Murray, and N. R. Smith. Baltimore: Williams and Wilkins. Albert Jan Kluyver, 1888-1956. J. Gen. Microbiol., 16:499-521. 1959 Kluyver's contributions to microbiology and biochemistry. In: Albert Jan Kluyver, His Life and Work, eds. A. F. Kamp. J. W. M. La Rivière, and W. Verhoeven, Amsterdam: NorthHolland Publishing Co. and New York: Interscience Publishers, pp. 68-155. With R. Y. Stanier. Bacteria. In: Freshwater Biology, ed. W. T. Edmondson, New York: John Wiley & Sons, pp. 16-46. 1962 The present status of the comparative study of photosynthesis. Annu. Rev. Plant Physiol., 13:1-26. With R. Y. Stanier. The concept of a bacterium. Arch. Mikrobiol., 42:17-35. 1963 With L. R. Blinks. The absence of enhancement (Emerson effect) in the photosynthesis of Rhodospirillum rubrum. In: Studies on Microalgae and Photosynthetic Bacteria, ed. Japanese Society for Plant Physiology, Tokyo: University of Tokyo Press, pp. 297-307. A brief survey of the photosynthetic bacteria. In: Bacterial Photosynthesis , eds. H. Gest, A. San Pietro, and L. P. Vernon, Yellow Springs, Ohio: Antioch Press, pp. 459-67. Ed. C. B. van Niel, Selected Papers of Ernest Georg Pringsheim. New Brunswick, N. J.: Institute of Microbiology, Rutgers University. 1965 On aquatic microbiology today. Science, 148:353. 1966 Microbiology and molecular biology. Q. Rev. Biol., 41:105-12. Lipmann's concept of the metabolic generation and utilization of
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phosphate bond energy: A historical appreciation. In: Current Aspects of Biochemical Energetics, eds. N. O. Kaplan and E. P. Kennedy, New York: Academic Press, pp. 9-25. 1967 The education of a microbiologist: Some reflections. Annu. Rev. Microbiol., 21:1-30. 1971 Techniques for the enrichment, isolation, and maintenance of the photosynthetic bacteria. In: Methods in Enzymology, eds. S. P. Colowick and N. O. Kaplan, New York: Academic Press, vol. 23(A), pp. 3-28. 1972 With G. E. Garner and A. L. Cohen. On the mechanism of ballistospore discharge. Arch. Mikrobiol., 84:129-40.
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Robert H. Whittaker December 27, 1920-October 20, 1980
By Walter E. Westman, Robert K. Peet, And Gene E. Likens Robert Harding Whittaker was one of the preeminent community ecologists of the twentieth century. By studying the interactions of plant populations at the biogeochemical, species, and community levels, he made contributions to basic knowledge in several subdisciplines of biology. He developed new approaches for the analysis of plant communities and provided exemplary insight into the patterns of composition, productivity, and diversity of land plants. He brought clarity to such disparate fields as the classification and ordination of plant communities, plant succession, allelochemistry, evolution and measurement of species diversity, niche theory, and the systematics of kingdoms of organisms. In several influential monographs he detailed the vegetational patterns of various montane regions of the United States, and-during the last six years of his life extended his research to Mediterranean- and arid-climate regions of the United States, Israel, Australia, and South Africa. Whittaker's most cited work is his undergraduate textbook, Communities and Ecosystems (1970,3; second edition, 1975,3), which not only introduced thousands of students throughout the world to ecology but also provided a succinct summary of a highly diverse literature and new insights useful to professional ecologists.
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EDUCATION AND EARLY LIFE Robert H. Whittaker, the youngest of three children, was born on December 27, 1920, in Wichita, Kansas, to Clive Charles and Adeline Harding Whittaker. His mother encouraged Whittaker's abiding interest in languages, while his father stimulated an early interest in natural history. Whittaker entered Washburn Municipal College (now University) in Topeka, Kansas, in 1938. He received a Bachelor of Arts degree in biology and languages in 1942 but postponed his plans to pursue graduate work in ecology to enlist in the Army. He was stationed in the United States and in England until 1946 as an Army-Air Force weather observer and forecaster. Upon his return to civilian life in 1946, he entered graduate school at the University of Illinois, where he completed his Ph.D. two-and-a-half years later. When Whittaker applied for graduate standing in the Department of Botany at Illinois, his application was denied because of insufficient background in botany, but he was admitted to the Zoology Department and awarded a fellowship. In February 1946, he began his graduate studies under the direction of Victor Shelford, who retired from active teaching that summer. Charles Kendeigh replaced Shelford as Whittaker's adviser in September, and though Whittaker worked with him and acknowledged his debt, Whittaker was also heavily influenced by the University of Illinois botanist Arthur G. Vestal, whom Whittaker called his "second adviser." SCIENTIFIC WORK The Continuum of Plant Species Distribution Whittaker was particularly taken by classroom lectures in which Vestal questioned rigid Clementsian notions of plant association and discussed Gleason's opposing idea of individ
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ualistic species distributions. From later conversations it was apparent that Whittaker keenly appreciated Vestal's influence in shaping his own theoretical approach and, in his later years at Cornell, was pleased to play a similar role for the graduate students of others. Whittaker's doctoral dissertation (1948,1) examined patterns of plant species change along an altitudinal gradient in the Great Smoky Mountains of Tennessee. In seeking to understand underlying patterns of species change, he plotted plant species' distributions along axes of environmental change. He then was able to show that the ecological importance of plant species (as measured by density or cover) rose and fell in a Gaussian fashion along key environmental gradients, with each species showing an individualistic distribution. Though Whittaker had hypothesized the occurrence of groups of coadapted species with parallel distributions, what emerged from his work was a validation of Gleason's hypothesis and rejection of his own: Most species were distributed independently along environmental gradients. The significance of this work was obvious. It supported the "continuum" concept of species distribution and extended the statistical basis for gradient analysis in general. W. H. Camp wrote Whittaker that his manuscript was "probably the most important ecological paper of the present century" and that his method would revolutionize the field.1 Plant and Insect Population Patterns, and Element Cycling In 1948, Whittaker was appointed instructor in the Department of Zoology at Washington State College (now University) in Pullman, Washington. While at Washington State
1 Despite this assessment, Whittaker's doctoral dissertation was not published until eight years later (1956,1). By that time—along with J. T. Curtis and the Wisconsin school —he had developed a series of detailed gradient analyses, but it took another ten to fifteen years before his thesis was widely accepted.
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he began field work on the vegetation of the Klamath region and Siskiyou Mountains of Oregon and California, including a comparative study of vegetation on serpentine and quartzdiorite soils. Returning to the original focus of his dissertation work, Whittaker completed a manuscript on foliage insects in the Great Smokies, building on his vegetation analysis there. At the same time he conducted a uniquely thorough study of copepod communities of small ponds in the Columbia basin. Whittaker left Washington State in 1951 to become a senior scientist in the Hanford Laboratories Aquatic Biology Unit, Department of Radiological Sciences, in Richland, Washington. Quick to see the value of radioactive tracers for unraveling complex ecological problems, he studied in detail the movement of radioactive phosphorus in aquarium microcosms. His results were important to understanding the fate of radionuclides in the environment and for evaluating the movement and storage of nutrients in ecosystems. At Hanford (and later at Brookhaven National Laboratory with George Woodwell), he also contributed to the first large-scale study of the effect of chronic gamma radiation on the structure and function of forest ecosystems. While at the Hanford Laboratories, Bob met Clara Caroline Buehl, and the two were married on New Year's Day, 1953. Although Clara had an M.S. in biology, her role in the marriage soon became that of wife and mother rather than scientific collaborator. The Whittakers raised three sons: John Charles, Paul Louis, and Carl Robert. Dimension Analysis and the Classification of the Kingdoms In 1954, Whittaker was hired as an instructor in the Department of Biology of Brooklyn College, the City University of New York, where he would remain for ten years. During the summers he returned to the Great Smoky Mountains,
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where he initiated a major effort to obtain measurements of the biomass and productivity of the forest communities along an elevational gradient. Because he was interested in the entire production of plants above ground, he began to develop methods for measuring productivity of shrubs and herbs and other parts of trees in addition to trunks. He used a volumetric measurement based on growth rings and succeeded, through laborious calculations, in obtaining productivity estimates for the major plant communities in the mountain range. His efforts provided a basis for the subsequent development of the dimension analysis methodology still widely in use. Throughout his career—in addition to conducting model studies of a variety of ecological systems—he also maintained an interest in the problem of classification and speciation. In 1957 he proposed a new classification for the kingdoms of organisms based on the evolution of trophic structures and nutritional energy sources (1957,1). Later updated (1969,4), this system of classification eventually was accepted widely and used in biology textbooks. Desert and Forest: Structure and Function From 1963 to 1965, Whittaker and W. A. Niering published a series of studies of the Arizona Saguaro cactus desert—among the first studies of a desert community to emphasize functional rather than structural attributes. For this work the authors received the Ecological Society of America's 1966 Mercer Award for the best paper published in the preceding two years by a young ecologist. In 1964 another colleague and future collaborator, George M. Woodwell, persuaded Whittaker to take a year's leave from Brooklyn College to work with him at Brookhaven National Laboratory in New York State. The two developed a profound respect and fondness for each other, and
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throughout the 1960s the team of Whittaker and Woodwell was one of the most productive and influential in plant ecology. Together they produced eight papers on the surface area, biomass, production and nutrient flow, and effects of gamma radiation on structure and diversity of forested ecosystems in the Brookhaven oak-pine forest and surrounding vegetation. Just before leaving Brookhaven in 1966, Whittaker had initiated studies— with Gene E. Likens and F. Herbert Bormann—on the biomass, productivity, and nutrient content of the Hubbard Brook Experimental Forest in New Hampshire. These subsequently led to two major monographs about this northern hardwood forest ecosystem (1970,1; 1974,3). With Likens, Whittaker also compiled the widely cited summary tables of plant production, biomass, and associated characteristics for ecosystems of the world. Species Diversity, Ordination Methods In 1966 Whittaker decided to accept the offer of a professorship at the new Irvine campus of the University of California. He took up this new post with great enthusiasm and anticipation but was dismayed by the rapid pace of urbanization around Irvine. In September 1968 he accepted an invitation to move to Cornell University as professor of biology in the Section of Ecology and Systematics, where his last years were marked by a significant expansion and solidification of his reputation. Once again pursuing his early interest in species diversity, Whittaker was stimulated in part by the attention G. E. Hutchinson, R. H. MacArthur, and their students had given to the topic. His concise paper in Science (1965,3) remains a classic review of the field. When general theories to explain patterns of plant species diversity did not emerge, Whittaker emphasized factors influencing local patterns, based on pe
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culiarities of site history and environment. In association with Hugh Gauch, Jr., and others he also explored techniques for ordinating species data—techniques that helped computerize earlier gradient analyses he had developed along with J. T. Curtis and the Wisconsin school. TEACHER, DIPLOMAT, HONORED RESEARCHER At Irvine and Cornell, Whittaker had the opportunity to supervise graduate students for the Ph.D. for the first time. Of the twelve he trained, eight went on to complete their dissertations under his supervision. Through personal diplomacy, furthermore, he built bridges between American and European ecologists, calming the waters he himself had troubled with his challenges to phytosociological theories and methods of classification. His reviews of classification and ordination studies and his global studies of diversity and productivity helped inspire North American ecologists to increase contacts and collaboration with ecologists beyond their borders. In his later years Robert Whittaker reaped the rewards of a prolific intellectual career. He enjoyed a solid reputation among his peers, who elected him vice president of the Ecological Society of America in 1971. He was elected to the National Academy of Sciences in 1974 and named Cornell's Charles A. Alexander Professor of Biological Sciences in 1976. Elected to the American Academy of Arts and Sciences in 1979, he also held honorary memberships in the British Ecological Society and the Swedish Phytogeographical Society. At the time of his death he was president of the American Society of Naturalists. HEALTH PROBLEMS In 1974 Whittaker's wife contracted cancer. Clara's struggle with the disease lasted three years, and at Christmas
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time in 1977, she finally succumbed. Though her prolonged illness upset Whittaker greatly, he remained stoically silent, and many of his students and colleagues were not aware of the events that were troubling him. Turning to his traditional values for support, he increased the intensity with which he pursued his work. Following Clara's death, Whittaker developed a close friendship with his doctoral student, Linda Olsvig. In October 1979 the two were married, and Linda, taking an active interest in his research, accompanied Whittaker into the field on visits to Israel and South Africa. There were no children from this marriage. Four months after his second marriage Whittaker complained of hip pain. X-rays revealed cancer in hip and lungs, but he set himself to complete as much of his work as possible. His health failed in September and he died on October 20, 1980. Shortly before his death, the Ecological Society of America honored him with its highest award, that of Eminent Ecologist. IN CONCLUSION Difficult as it to assess which of Whittaker's many contributions to the science of ecology will prove most profound or long lasting, one hallmark stands out. Demonstrating the continuity of species' response to environmental gradients, he challenged the classificatory approach to vegetation structure. Though Whittaker credited Ramensky, Gleason, Curtis, and McIntosh with much, it was his own theory, method, and empirical evidence that solidified gradient analysis into a scientifically accepted approach. In the preparation of this memoir, the authors often referred to a short biography by W. E. Westman and R. K. Peet published shortly after Whittaker's death, ''Robert H. Whittaker
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(1920-1980)—The Man and His Work," Vegetatio 48(1982):97-122. A memorial volume has been published by Whittaker's students and colleagues: R. K. Peet, ed., Plant Community Ecology. Papers in Honor of Robert H. Whittaker (Dordrecht: Junk, 1985). In 1975 Robert Whittaker supplied the National Academy with an autobiographical note, which remains on file with the Archives of the National Academy of Sciences in Washington, D.C.
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Selected Bibliography
1944 With D. B. Stallings. Notes on seasonal variation in Lepidoptera. Entomol. News, 53 (3):67-71; (4):87-92. 1948 A vegetation analysis of the Great Smoky Mountains (doctoral dissertation). University of Illinois, Department of Zoology. 1951 A criticism of the plant association and climatic climax concepts. Northwest Sci., 26:17-31. 1952 A study of summer foliage insect communities in the Great Smoky Mountains. Ecol. Monogr., 22:1-44. 1953 A consideration of climax theory: the climax as a population and pattern. Ecol. Monogr., 23:41-78. 1954 Plant populations and the basis of plant indication. (German summary.) Angew. Pflanzensoziol. (Wien), 1:183-206. The ecology of serpentine soils. I. Introduction. Ecology, 35:258-59. The ecology of serpentine soils. IV. The vegetational response to serpentine soils. Ecology, 35:275-88. 1956 Vegetation of the Great Smoky Mountains. Ecol. Monogr., 26:1-80. In honor of Edwin Aichinger. Review of Festscrhift für Edwin Aichinger zum 60 Geburtstag. 1954. Ecology, 37:296-97. A new Indian Ecological Journal. Review of Bulletin of the Indian Council of Ecological Research, vol. 1. Ecology, 37:628.
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1957 Recent evolution of ecological concepts in relation to the eastern forests of North America. Am. J. Bot., 44:197-206. Also in: Fifty Years of Botany: Golden Jubilee Volume of the Botanical Society of America, ed., W. C. Steere, New York: McGraw-Hill, pp. 340-58. The kingdoms of the living world. Ecology, 38:536-38. Review of H. Ellenberg. 1950-1954. Gradient analysis in agricultural ecology. Landwirtsch. Pflansensoziol. Ecology, 38:363-64. Two ecological glossaries and a proposal on nomenclature. Ecology, 38:371. 1958 With C. W. Fairbanks. A study of plankton copepod communities in the Columbia Basin, southeastern Washington. Ecology, 39:46-65. Also in: Readings in Population and Community Ecology, ed., W. E. Hazen, Philadelphia: W. B. Saunders, pp. 369-88. A manual of phytosociology. Review of F. R. Bharucha and W. C. de Leeuw, A Practical Guide to Plant Sociology for Foresters and Agriculturalists (1957). Ecology, 38:182. The Pergamon Institute and Russian journals of ecology. Ecology, 39:182-83. 1959 On the broad classification of organisms. Q. Rev. Biol., 34:210-26. 1960 Ecosystem. In: McGraw-Hill Encyclopedia of Science and Technology, New York: McGrawHill, pp. 404-8. Vegetation of the Siskiyou Mountains, Oregon and California. Ecol. Monogr., 30:279-338. A vegetation bibliography for the northeastern states. Review of F. E. Egler, A Cartographic Guide to Selected Regional Vegetation Literature-Where Plant Communities Have Been Described (1959). Ecology, 41:245-46. 1961 Estimation of net primary production of forest and shrub communities. Ecology, 42:177-80.
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Experiments with radio-phosphorus tracer in aquarium microcosm. Ecol. Monogr., 31:157-88. Vegetation history of the pacific coast states and the "central" significance of the Klamath Region. Madroño, 16:5-23. New serials. Ecology, 42:616. The chemostat as a model system for ecological studies. In: Modern Methods in the Study of Microbial Ecology, ed. T. Rosswell, Uppsala: Swedish National Sciences Research Council, pp. 347-56. 1962 Classification of natural communities. Bot. Rev., 28:1-239. Reprinted: New York: Arno Press (1977). Net production relations of shrubs in the Great Smoky Mountains. Ecology, 43:357-77. With V Garfine. Leaf characteristics and chlorophyll in relation to exposure and production in Rhododendron maximum. Ecology, 43:190-25. The pine-oak woodland community. Review of J. T. Marshall, Birds of Pine-Oak Woodland in Southern Arizona and Adjacent Mexico (1957). Ecology, 43:180-81. 1963 With W. A. Niering and C. H. Lowe. The saguaro: A population in relation to environment. Science, 142:15-23. Essays on enchanted islands. Review of G. E. Hutchinson, The Enchanted Voyage and Other Studies (1962). Ecology, 44:425. Net production of heath balds and forest heaths in the Great Smoky Mountains. Ecology, 44:176-82. With N. Cohen and J. S. Olson. Net production relations of three tree species at Oak Ridge, Tennessee. Ecology, 44:806-10. With W. A. Niering. Vegetation of the Santa Catalina Mountains. Prog. Agric. Ariz., 15:4-6. 1964 With W. A. Niering. Vegetation of the Santa Catalina Mountains, Arizona. I. Ecological classification and distribution of species. J. Ariz. Acad. Sci., 3:9-34.
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1965 With W. A. Niering. The saguaro problem and grazing in southwestern national monuments. Natl. Parks Mag., 39:4-9. Branch dimensions and estimation of branch production. Ecology, 46:365-70. Dominance and diversity in land plant communities. Science, 147:250-60. With W. A. Niering. Vegetation of the Santa Catalina Mountains, Arizona. II. A gradient analysis of the south slope. Ecology, 46:429-52. With W. A. Niering. The saguaro problem and grazing in southwestern national monuments. Nat. Parks Mag., 39:4-9. 1966 Forest dimensions and production in the Great Smoky Mountains. Ecology, 47:103-21. With G. M. Woodwell and W. M. Malcolm. A-bombs, bugbombs, and us. NAS-NRC Symposium on "The Scientific Aspects of Pest Control," the Brookhaven National Laboratory, U.S. Atomic Energy Commission. Washington, D.C.: Atomic Energy Commission. 1967 Ecological implications of weather modification. In: Ground Level Climatology, ed. R. H. Shaw, Washington, D.C.: AAAS, pp. 367-84. Gradient analysis of vegetation. Biol. Rev., 42:207-64. With G. M. Woodwell. Surface area relations of woody plants and forest communities. Am. J. Bot., 54:931-39. With G. M. Woodwell. Primary production and the cation budget of the Brookhaven forest. In: Symposium on Primary Productivity and Mineral Cycling in Natural Ecosystems, ed. H. E. Young, Orono: University of Maine Press, pp. 151-66. 1968 With I. Frydman. Forest associations of southeast Lublin Province, Poland. (German summary.) Ecology, 49:896-908. With S. W. Buol, W. A. Niering, and Y. H. Havens. A soil and
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vegetation pattern in the Santa Catalina Mountains, Arizona. Soil Sci., 105:440-50. With W. A. Niering. Vegetation of the Santa Catalina Mountains, Arizona. III. Species distribution and floristic relations on the north slope. J. Ariz. Acad. Sci., 5:3-21. With W. A. Niering. Vegetation of the Santa Catalina Mountains, Arizona. IV. Limestone and acid soils. J. Ecol., 56:523-44. With G. M. Woodwell. Effects of chronic gamma irradiation on plant communities. Q. Rev. Biol., 43:42-55. With G. M. Woodwell. Primary production in terrestrial ecosystems. Am. Zool., 8:19-30. 1969 A view toward a National Institute of Ecology. Ecology, 50:169-70. Een nieuwe indeling van de organismen. Nat. Tech., 37:124-32. Evolution of diversity in plant communities. In: Diversity and Stability in Ecological Systems, Brookhaven Symposia in Biology, Upton, New York: Brookhaven Natl. Lab. Publ. 50175 (C-56), No. 22, pp. 178-95. New concepts of kingdoms of organisms. Science, 163:150-60. With G. M. Woodwell. Structure, production, and diversity of the oak-pine forest at Brookhaven, New York. J. Ecol., 57:155-74. 1970 With F. H. Bormann, T. G. Siccama, and G. E. Likens. The Hubbard Brook ecosystem study: Composition and dynamics of the tree stratum. Ecol. Monogr., 40:373-88. With W. L. Brown and T. Eisner. Allomones and kairomones: Transspecific chemical messengers. BioScience, 20:21-22. Communities and Ecosystems. New York: Macmillan. (Japanese edition, Tokyo, 1974.) Neue Einteilung der Organismenreiche. Umschau, 16:514-15. Taxonomy. In: McGraw-Hill Yearbook of Science and Technology 1970, New York: McGraw-Hill, pp. 365-69. The biochemical ecology of higher plants. In: Chemical Ecology, eds. E. Sondheimer and J. B. Simeone, New York: Academic Press, pp. 43-70. The population structure of vegetation. In: Gesellschaftsmorphologie (Strukturforschung) (German summary), ed. R. Tuxen, Ber.
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Symp. Int. Ver. Vegetationskunde, Rinteln, 1966. The Hague: Junk., pp. 39-62. With G. M. Woodwell. Ionizing radiation and the structure and functions of forests. In: Gesellschaftsmorphologie (Strukturforschung) (German summary), ed. R. Tuxen, Ber. Symp. Int. Ver. Vegetationskunde, Rinteln, 1966. The Hague: Junk, pp. 334-39. 1971 With P. P. Feeny. Allelochemics: Chemical interactions between species. Science, 171:757-70. With G. M. Woodwell. Evolution of natural communities. In: Ecosystem Structure and Function, Proc. 31st Ann. Biol. Colloq., ed. J. A. Wiens, Corvallis: Oregon State University Press. pp. 137-56. With G. M. Woodwell. Measurement of net primary production of forests. In: Productivity of Forest Ecosystems (French summary), ed. P. Duvigneaud, Proc. Brussels Symp., 1969, Paris: UNESCO, pp. 159-75. With P. F. Brussard, A. Levin, and L. N. Miller. Redwoods: A population model debunked. Science, 175:435-36. Dry weight, surface area, and other data for individuals of three tree species at Oak Ridge, Tennessee. In: Dry weight and Other Data for Trees and Woody Shrubs of the Southeastern United States, (Publ. ORNL-IBP-71-6), eds. P. Sollins and R. M. Anderson, Oak Ridge, Tennessee: Oak Ridge National Laboratory, pp. 37-38. The chemistry of communities. In: Biochemical Interactions Among Plants. Washington, D.C.: National Academy of Sciences, pp. 10-18. 1972 With H. G. Gauch, Jr. Coenocline simulation. Ecology, 53:446-51. With H. G. Gauch, Jr. Comparison of ordination techniques. Ecology, 53:868-75. Convergences of ordination and classification. In: Basic Problems and Methods in Phytosociology (German summary), ed. R. Tuxen, Ber. Symp. Int. Ver. Vegetationskunde, Rinteln, 1970. The Hague: Junk, pp. 39-55.
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Evolution and measurement of species diversity. Taxon, 21:213-51. A hypothesis rejected: The natural distribution of vegetation. In: Botany: An Ecological Approach, eds. W. A. Jensen and F. B. Salisbury, Belmont, Calif.: Wadsworth, Inc., pp. 689-91. Also in: Biology, ed. W. A. Jensen et al., Belmont, Calif.: Wadsworth, Inc., 1979, pp. 474-76. 1973 With G. Cottam and F. G. Goff. Wisconsin comparative ordination. In: Ordination and Classification of Communities, ed. R. H. Whittaker, The Hague: Junk, pp. 193-221. Approaches to classifying vegetation. In: Handbook of Vegetation Science, V Ordination and Classification of Vegetation, ed. R. H. Whittaker, The Hague: Junk, pp. 325-54. Community, biological. In: Encyclopaedia Britannica, 15th ed., pp. 1027-35. Direct gradient analysis: Results. In: Handbook of Vegetation Science, V Ordination and Classification of Vegetation, ed. R. H. Whittaker, The Hague: Junk, pp. 9-31. Dominance-types. In: Handbook of Vegetation Science, V Ordination and Classification of Vegetation, ed. R. H. Whittaker, The Hague: Junk, pp. 389-92. (Editor). Handbook of Vegetation Science, V Ordination and Classification of Vegetation. The Hague: Junk. Introduction. In: Handbook of Vegetation Science, V Ordination and Classification of Vegetation, ed. R. H. Whittaker, The Hague: Junk, pp. 1-6. With H. G. Gauch, Jr. Evaluation of ordination techniques. In: Handbook of Vegetation Science, V Ordination and Classification of Vegetation , ed. R. H. Whittaker, The Hague: Junk, pp. 289-321. With S. A. Levin and R. B. Root. Niche, habitat, and ecotope. Am. Nat., 107:321-38. With G. E. Likens. Carbon in the biota. In: Carbon and the Biosphere , Symp. 2d. Cong. Am. Inst. of Bio. Sci., Miami, Florida, 1971. Human Ecol., 1:301-2. With G. E. Likens. Primary production: The biosphere and man. In: Handbook of Vegetation Science, ed. R. H. Whittaker, The Hague: Junk, pp. 55-73.
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1974 With H. G. Gauch, Jr., and G. B. Chase. Ordination of vegetation samples by Gaussian species distributions. Ecology, 55:1382-90. Climax concepts and recognition. In: Handbook of Vegetation Science, VIII. Vegetation Dynamics, ed. R. Knapp, pp. 139-54. With F. H. Bormann, G. E. Likens, and T. G. Siccama. The Hubbard Brook ecosystem study: Forest biomass and production. Ecol. Monogr., 44:233-54. 1975 With H. Leith (eds.). The Primary Productivity of the Biosphere. New York: Springer-Verlag. With W. E. Westman. The pygmy forest region of northern California: Studies on biomass and primary productivity. J. Ecol., 62:493-520. Communities and Ecosystems, 2d ed. New York: Macmillan. (Japanese ed., Tokyo, 1978.) Functional aspects of succession in deciduous forests. In: Sukzessionsforschung (German summary), ed. W. Schmidt., Ber. Symp. Int. Ver. Vegetationskunde, Rinteln, 1973, pp. 377-405. The design and stability of plant communities. In: Unifying Concepts in Ecology (Rep. Plenary Sessions, 1st Int. Cong. Ecology, The Hague, 1974), eds. W. H. van Dobben and R. H. Lowe McConnell, The Hague: Junk, and Wageningen: Pudoc, pp. 169-81. Vegetation and parent material in the western United States. In: Vegetation and Substrate (German summary), ed. H. Dierschke, Ber. Symp. Int. Ver. Vegetationskunde, Rinteln, 1969, pp. 443-65. With S. A. Levin (eds.). Niche: Theory and Application, Benchmark Papers in Ecology. Stroudsburg: Dowden, Hutchinson, and Ross. With S. A. Levin and R. B. Root. On the reasons for distinguishing "niche, habitat, and ecotope." Am. Nat., 109:479-82. With G. E. Likens. The biosphere and man. In: Primary Productivity of the Biosphere, eds. H. Leith and R. H. Whittaker, New York: Springer-Verlag, pp. 305-23. With P. L. Marks. Methods of assessing terrestrial productivity. In:
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Primary Productivity of the Biosphere, eds. H. Leith and R. H. Whittaker, New York: Springer-Verlag, pp. 55-118. With W. A. Niering. Vegetation of the Santa Catalina Mountains, Arizona. V. Biomass, production, and diversity along the elevation gradient. Ecology, 56:771-90. With G. M. Woodwell and R. A. Houghton. Nutrient concentrations in plants in the Brookhaven oak-pine forest. Ecology, 56:318-32. 1976 With H. G. Gauch, Jr. Simulation of community patterns. Vegetatio, 33:13-16. With R. B. Hanawalt. Altitudinally coordinated patterns of soils and vegetation in the San Jacinto Mountains, California. Soil Sci., 121:114-24. With S. R. Kessell. Comparisons of three ordination techniques. Vegetatio, 32:21-29. 1977 With H. G. Gauch, Jr., and T. R. Wentworth. A comparative study of reciprocal averaging and other ordination techniques. J. Ecol., 65:157-74. With R. B. Hanawalt. Altitudinal patterns of Na, K, Ca and Mg in soils and plants in the San Jacinto Mountains, California. Soil Sci., 123:25-36. With R. B. Hanawalt. Altitudinal gradients of nutrient supply to plant roots in mountain soils. Soil Sci., 123:85-96. With I. Noy-Meir. Continuous multivariate methods in community analysis: Some problems and developments. Vegetatio, 33:79-98. Animal effects on plant species diversity. In: Vegetation and Fauna , ed. R. Tuxen, Ber. Symp. Int. Ver. Vegetationskunde, Rinteln, 1976. Vaduz, The Netherlands: Cramer, pp. 409-25. Broad classification: The kingdoms and the protozoans. In: Parasitic Protozoa, ed. J. Krier, New York: Academic Press, vol. 1, pp. 1-34. Evolution of species diversity in land communities. In: Evolutionary Biology , eds. M. K. Hecht, W C. Steere, and B. Wallace, New York: Plenum Publishing Corp., vol. 10, pp. 1-67.
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With S. A. Levin. The role of mosaic phenomena in natural communities. Theor. Pop. Biol., 12:117-39. 1978 With I. Noy-Meir. Recent developments in continuous multivariate techniques. In: Ordination of Plant Communities, ed. R. H. Whittaker, The Hague: Junk, pp. 337-78. (Editor). Classification of Plant Communities. The Hague: Junk. With H. G. Gauch, Jr. Evaluation of ordination techniques. In: Ordination of Plant Communities, ed. R. H. Whittaker, The Hague: Junk, pp. 277-336. With L. Margulis. Protist classification and the kingdoms of organisms. BioSystems, 10:3-18. With G. M. Woodwell, W. A. Reiners, G. E. Likens, C. C. Delwiche, and D. B. Botkin. The biota and the world carbon budget. Science, 199:141-46. Review of Terrestrial Vegetation of California, eds. M. G. Barbour and J. Major, Vegetatio, 38:124-25. 1979 With Z. Naveh. Measurements and relationships of plant species diversity in Mediterranean shrublands and woodlands. In: Ecological Diversity in Theory and Practice, eds. F. Grassle, G. P. Patil, W. Smith, and C. Taillie, Fairland, Md.: Int. Co-op. Pub. House, pp. 219-39. With L. S. Olsvig and J. F. Cryan. Vegetational gradients of the pine plains and barrens of Long Island. In: Pine Barrens: Ecosystem and Landscape, ed. R. T. T. Forman, New York: Academic Press, pp. 265-82. With S. R. Sabo. Bird niches in a subalpine forest: An indirect ordination. PNAS, 76:1338-42. With A. Shmida. Convergent evolution of deserts in the old and new world. In: Werden and Vergehen von Pflanzengesellschaften, eds., O. Wilmanns and R. Tuxen, Ber. Symp. Int. Ver. Vegetationskunde, Rinteln, 1978, Vaduz, The Netherlands: Cramer, pp. 437-50. Vegetational relationships of the pine barrens. In: Pine Barrens: Ecosystem and Landscape, ed., R. T. T. Forman, New York: Academic Press, pp. 315-31.
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With L. E. Gilbert and J. H. Connell. Analysis of two-phase pattern in a mesquite grassland, Texas. J. Ecol., 67:935-52. With D. Goodman. Classifying species according to their demographic strategy. I. Population fluctuations and environmental heterogeneity. Am. Nat., 113:185-200. With G. E. Likens, F. H. Bormann, J. S. Eaton, and T. G. Siccama. The Hubbard Brook ecosystem study: Forest nutrient cycling and element behavior. Ecology, 60:203-20. With Z. Naveh. Analysis of two-phase patterns. In: Contemporary Quantitative Ecology and Related Econometrics, eds. G. P. Patil and M. Rosenzweig, Fairland, Md.: Int. Co-op. Pub. House, pp. 157-65. With W. A. Niering and M. D. Crisp. Structure, pattern, and diversity of a mallee community in New South Wales. Vegetatio, 39:65-76. 1980 With Z. Naveh. Structural and floristic diversity of shrublands and woodlands in northern Israel and other Mediterranean areas. Vegetatio, 41: 171-90. 1981 With H. G. Gauch, Jr., and S. B. Singer. A comparative study of nonmetric ordinations. J. Ecol., 69:135-52. With A. Shmida. Pattern and biological microsite effects in two shrub communities, southern California. Ecology, 62:234-51. With K. D. Woods. Canopy-understory interaction and the internal dynamics of mature hardwood and hemlock-hardwood forests. In: Forest Succession: Concepts and Application, eds. D. West, H. H. Shugart and D. B. Botkin, New York: Springer-Verlag, pp. 305-23. With H. G. Gauch, Jr. Hierarchical classification of community data. J. Ecol., 69:537-57. 1984 With J. Morris and D. Goodman. Pattern analysis in savanna woodlands at Nylsvley, South Africa. Mem. Bot. Surv. S. Africa, 49.
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Maxwell Myer Wintrobe October 27, 1901-December 9, 1986
By William N. Valentine When Maxwell Myer Wintrobe died in Salt Lake City on December 9, 1986, his distinguished career in medicine and in his subspecialty of hematology had spanned some sixty years—from the conquest of pernicious anemia to the present. His scientific achievements are recorded in more than 400 publications. His Clinical Hematology, first published in 1942 and currently in its eighth edition, remains a prototype of excellence and for many years stood alone as the premier text in his chosen field. In 1943 Max Wintrobe became the founding chairman of the Department of Medicine at the University of Utah—a post he filled with great energy and ability until 1967. From that time until his death he continued an active and productive career at Utah as Distinguished Professor. By all accounts, Max was a world leader in hematology, a role attested to by a legion of honors, visiting professorships, memberships, and activities in national and international scientific societies, consultantships, editorial responsibilities, and—perhaps most importantly—by the large cadre of students who had flocked to be under his tutelage and who themselves went on to be leaders in their medical communities and in academia.
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EDUCATION AND EARLY LIFE Max Wintrobe was born October 27, 1901, in Halifax, Nova Scotia. His parents (both of whom had emigrated from Austria) adjusted rapidly to the new community, adding the English language to their repertoire of German, Polish, and Yiddish. Their educational background was limited and their lifestyle frugal, as dictated by modest means. His mother's family, the Zwerlings, was large and had been in Canada for many years. Max, an only child, responded to his mother's deep interest in education and her urgings to study, work hard, and achieve. In 1912, the family moved to Winnipeg, Manitoba, where, however, there were few friends and no family. A better-than-average student, Max entered the University of Manitoba at age fifteen. Having already determined on a medical career, he also made the decision to spend four undergraduate years before entering medical school, though only one year was required at the time. At the University he showed his facility for language, favoring English, Latin, and French and winning gold medals in the latter and in political economy. He also discovered his love of history and the well-turned phrase—so important to his later career. Entering medical school at twenty, Max developed a special interest in the Johns Hopkins Medical Center through the writings of William Osler, but limited circumstances prevented any thought of transferring. Throughout his undergraduate and medical school years he worked at a variety of odd jobs to further his education and to help the family finances. Of his teachers at Manitoba he remembered William Boyd, professor of pathology—a flowery and exciting lecturer with a rich Scottish brogue—as the most stimulating. But as graduation neared, Max, who had achieved an outstanding record, became increasingly aware of his lack of
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desire to go into private practice, though other opportunities were few and resources limited. After his internship and receipt of the M.D. degree in 1926, the dilemma was resolved by the offer of the first Gordon Bell Fellowship, named in honor of the dean of the University who had just retired. Wintrobe was first assigned the task of determining the relative prevalence of achlorhydria in certain western Canadian communities where the incidence of pernicious anemia—a subject of especially great interest in 1926—was believed to vary widely. A second assignment, pursued energetically but fruitlessly, was to produce achylia gastrica in dogs. Thus was launched a distinguished, lifelong academic career in the field of hematology. THE TULANE YEARS (1927-1930): ''ANEMIA OF THE SOUTH," NORMAL BLOOD VALUES, THE WINTROBE HEMATOCRIT, AND CORPUSCULAR CONSTANTS In September 1927, Max arrived in New Orleans, having accepted the offer of an appointment as assistant in medicine at Tulane University from Dean C. Bass. Assured of an annual stipend of $1,800 and a small laboratory next to Roy Turner—a Hopkins graduate and the consummate erudite clinician—it was possible to get married. Max returned to Winnipeg and shortly thereafter, on January 1, 1928, brought his bride, nee Becky Zamphir, from the -50°F of Winnipeg to bright, sunny New Orleans. Max's New Orleans years were both pleasant and productive. Charity Hospital offered a wealth of clinical material, including nutritional and other anemias of all types, tropical disease, tuberculosis, and every variety of neoplasia. John H. Musser, the distinguished chief of medicine, suggested that Wintrobe find out if the widely believed "anemia of the South" myth actually existed. Though Max could not identify any such entity, the study allowed him to collect data and
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develop techniques that became an integral part of the clinical evaluation of all patients, not only those with blood and marrow disorders. He first worked to document statistically normal values for hematologic parameters in normal adults and children. Accepted round numbers of normality at that time were derived from only a few counts and from observations some seventy-five years old. A "normal" hemoglobin value in men was expressed as 100%. Wintrobe's careful observations made on Tulane medical students and women from Sophie Newcomb College—together with observations by Russell Haden in Cleveland, Edwin Osgood in Portland, and a few made in Europe— served as basic data for establishing normality in terms of quantitatively accurate observations. Max's second important contribution was the invention of the Wintrobe hematocrit, which universally replaced the leaky, awkwardly calibrated and poorly conceived devices of the 1920s. Wintrobe's calibrated, straight-sided tube held about a milliliter of blood. Most importantly, any venous blood sample being measured in the tube was anticoagulated with a combination of potassium and ammonium oxalate that did not cause cells to shrink or swell. Although many millions of the Wintrobe hematocrits have been sold, neither Wintrobe nor Tulane profited. Since the instrument was intended for the public good, Wintrobe refused all royalties and applied for no patent. Another important innovation came to Wintrobe in the middle of the night while puzzling over the inadequacies of the various indices then in vogue. These included color, volume, and saturation indices derived indirectly from ratios based on "percent of normal" for red cell numbers, hemoglobin content, etc. Wintrobe's method permitted direct calculation of the average cell size, MCV (mean corpuscular volume in cubic microns), MCH (mean hemoglobin content in
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picograms), and MCHC (mean corpuscular hemoglobin concentration in percent) —quantifications that are standard procedure in research and clinical laboratories today. J. H. Musser's invitation to assist him in rewriting the section on diseases of the blood for the ten-volume looseleaf set of the Tice Practice of Medicine (1931,3) marked a new step in Wintrobe's career. The new section was documented with great care and had a lengthy bibliography, not a common practice at the time. This desire for full bibliographical documentation later resulted in one of the most valuable features of Wintrobe's textbook Clinical Hematology (1942,5). During his three years in New Orleans, Wintrobe worked toward his Ph.D. degree. His thesis, The Erythrocyte in Man (1930,3), represented a review of world literature and of his own studies in that field. In his efforts to apply appropriate statistical methods to his own data, Wintrobe had contacted Raymond Pearl at Johns Hopkins, author of the helpful Introduction to Medical Biometry and Statistics. With the assistance of Dean Bass, Wintrobe was able to journey to Hopkins, see Pearl, and meet Alan Chesney, dean of the Medical School. When searching for a suitable publication for his thesis sometime later, Wintrobe hit upon the review journal Medicine; serendipitously, Chesney was its editor. Chance again favored Max, his thesis was published, and his long-cherished wish to study and work at Hopkins became a reality. He was offered an appointment as instructor in the Division of Clinical Microscopy. JOHNS HOPKINS (1930-1943) The Wintrobes found some aspects of life in Baltimore less than pleasing, but medically and scientifically Hopkins was all they had hoped for. Max directed the second-and third-year courses in clinical microscopy, stimulating his students by integrating laboratory findings with clinical prob
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lems and diagnoses. In addition he worked in the Outpatient Department and gave consultations as requested—a practice that burgeoned as his reputation spread. The student caliber was good, the faculty talented and in the forefront of medicine. The times were busy but the Great Depression had brought austerity to all. Max had no secretarial assistance and there were no funds to train technicians. He trained his own assistants (including Becky) but could pay them nothing. Instead, he bartered training for their services. To assist in studies in comparative hematology, Becky first mastered the art of venipuncture on fish, and she subsequently became chief technician at the diagnostic clinic. Max carried out studies of comparative hematology on animals in the Washington, D.C., Zoo, and—during one enjoyable summer—at Mountain Desert Island in Maine, where Homer Smith, Jim Shannon, and other distinguished scientists were also working. The Wintrobes spent other summers pleasantly working at Woods Hole in Massachusetts. Baltimore was the site of much intellectual exchange in medicine, and Wintrobe enjoyed and benefited from discussions with his many colleagues, including George Minot, Bill Castle, and others of the Boston group. Max's career-long interest in pernicious anemia, for instance, was furthered by his admiration of Castle's classic experiments, and Castle appropriately authored the foreword to his last book, Hematology, the Blossoming of a Science (1985,1). It was also fitting that Irving Sherman, a Hopkins student working with Wintrobe, incidentally noted birefringence of sickled red cells in the course of his studies on the role of deoxygenation in producing sickling. Bill Castle later brought this finding to the attention of Linus Pauling in a chance conversation, giving birth to studies that would define the molecular lesion of hemoglobin responsible for sickle cell anemia and usher in the era of molecular biology.
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In 1933 Becky and Max, backed by a six-month leave and a half-year's pay, embarked on the first of their many trips to Europe. During these months they visited a large number of institutions and met many of the current and future leaders of hematology in England and on the continent. Among many others were Otto Naegli of Zurich, acknowledged as the outstanding hematologists in Europe, Isidore Snapper, whose clinic was in Holland, Paul Morawitz of Leipzig, and Janet Vaughn of England. Although Max's first paper was published in 1928 in the New Orleans Medical and Surgical Journal, by 1933 he had already achieved a considerable reputation in the field of hematology. At Hopkins he sought to expand his data on normal blood values and on the uses of the hematocrit. He demonstrated that the hematocrit effectively measured erythrocyte sedimentation rate and that, when proper centrifugation was employed, the volume of packed red cells could be ascertained accurately and the mass of leukocytes and platelets roughly approximated. The supernatant plasma was also a convenient medium for determining icterus. With the hematocrit, Wintrobe was also able to demonstrate a cryoglobulin in blood and to diagnose a previously unsuspected case of multiple myeloma. As he and Buell reported in the Bulletin of the Johns Hopkins Hospital (1933,2), the temperature dependent, reversible turbidity evident in supernatant plasma in a hematocrit temporarily placed in a refrigerator, had led the researchers to this diagnosis. After returning from Europe, Max resumed a busy schedule of writing and research. In 1940 he published a study of forty members of three Italian families, some of whom suffered from splenomegaly, mild icterus, and blood changes recognized as a mild form of thalassemia. In a footnote he pointed out that the same condition had been observed in the parents of a patient with Cooley's anemia, also cited in a
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later table in his first edition of Clinical Hematology. This observation coincided with independent observations by Dameshek and Strauss in America and Silvestroni and Bianco, somewhat later, in Italy, to establish the recessive transmission of thalassemia. In 1938, another study by Wintrobe and Robert H. Williams (later to head the Department of Medicine at the University of Washington in Seattle) demonstrated that nonautolyzed yeast in sufficient amounts could induce a hemopoietic response in patients with pernicious anemia. As a house officer, Williams was able to sequester suitable subjects from the eye of Professor Longcope, who was unenthusiastic about the study. The hemopoietic response presumably arose from large amounts of folic acid in the yeast supplement. Other studies conducted with H. B. Schumacker, who later became chief of surgery at Indiana University, centered on the significance of macrocytosis and its association with liver disease. Struck by the fact that macrocytosis occurred in both human and animal fetal development, Max, his students, and coworkers began studying fetal blood in experimental animals. The opossum proved unaccommodating and was abandoned, but the domestic pig proved more tractable. Wintrobe's early work with this animal model provided a basis for his later studies in nutritional anemia, vitamin deficiency, and trace metal metabolism carried out at Utah. Though attempts to produce pernicious anemia in animals proved fruitless, other studies brought about diverse scientific contributions in many areas: the role of splenectomy in thrombocytopenic purpura, the etiology and management of the anemias, and the diverse manifestations of the leukemias. Quantitatively determined corpuscular constants became universally accepted as a basis for classifying red cell disorders. All of these investigations, both clinical and in the laboratory, followed Max's modus operandi. Experiments were
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done meticulously, records were fully documented and maintained, all available literature was explored thoroughly, and compendious bibliographies were compiled. Max consistently involved both students and house officers in his research activities, and his association with fine investigators (such as pathologist Arnold Rich) stimulated the flow of ideas while building valuable contacts. Many of these students and house officers later achieved fame, including George Eastman Cartwright, who worked with Max as a second-year student, followed him to Salt Lake City, and in 1967 succeeded him as Utah's chairman of medicine. On Pearl Harbor Day, December 7, 1941, Max was working to complete the index of the first edition of Clinical Hematology (1942,5). Since the authorities insisted he remain in Baltimore he began studying chemical warfare agents with Professor Longcope and Val Jaeger, then a house officer. At Utah, he and Jaeger later continued the work begun at Baltimore's U.S. Army Edgewood Arsenal (in Baltimore). In 1943, Max was called to be the chairman of Medicine at the newly established University of Utah Medical School—the first four-year medical school between Denver and the Pacific Coast from Canada to Mexico. THE UTAH YEARS (1943-1986) Max was now an established leader in hematology in charge of the Clinic for Nutritional, Gastrointestinal and Hemopoietic Disorders and an associate physician at Hopkins. Clinical Hematology, published in 1942, had filled a major void in the field and was well on its way to becoming the leading hematological reference work. But when the Wintrobes and their young daughter, Susan Hope (born in Baltimore in 1937), considered moving to Utah in 1943, they did so with considerable trepidation. As Canadians, they knew little about Utah, but two Hopkins
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men—Phillip Price and A. Louis Dippel—were going there as, respectively, chief of Surgery and chief of Obstetrics-Gynecology at the new school. In addition, Alan Gregg, vice-president of the Rockefeller Foundation, and Isaiah Bowman, president of Johns Hopkins, both urged Max to accept, stressing the importance of this opportunity to open a new frontier. But if Utah offered "opportunity," it offered, in Max's own word, "absolutely nothing more." The hospital's clinical facilities and plant were run down and poorly administered. The medical school was housed in a dormitory constructed for World War I cavalry officers. The promised new medical center materialized only after twenty-two years, to be dedicated two years before Max's retirement as chief of medicine. In 1943, as far as he was concerned, faculty in all departments had to be recruited, medical care improved, student scholastic standards raised, goals reoriented, research projects and facilities established, and supporting funds obtained. Despite these hard facts, all the departments continued to grow steadily, and their chairmen functioned well together. By 1950, the Department of Medicine faculty numbered ten and included high-caliber, enthusiastic recruits dedicated to the goal of establishing a first-rate medical school. The Hematology Division enjoyed worldwide fame, attracting young physicians from North America and elsewhere in large numbers. Max instituted a program (later widely emulated) whereby students, house officers, and fellows initially examined all patients, whether private or nonpaying, as subjects for undergraduate and graduate teaching. Between 1947 and 1984, 170 graduate students were trained in hematology and participated in research activities at Utah. Well over half remained in academic medicine, a number as leaders, and
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several later shared authorship with Max in the seventh and eighth editions of Clinical Hematology. The National Institutes of Health first research grant went to the Utah study of muscular dystrophy and other hereditary and metabolic disorders. Encouraged to seek federal support by Senator Elbert Thomas of Utah, chairman of the Senate Committee on Health, Max had applied. Senator Thomas and U.S. Surgeon General Parran wanted to continue peacetime support of medical research, and the Utah senator was also an enthusiastic supporter of his state's new four-year school. The initial bill provided $100,000 a year, which was subsequently renewed for twenty-three years, providing the new school monies for faculty recruitment in many fields other than medicine. The grant supported work that would bring recognition and renown to the school and its staff. Muscular dystrophy of a hereditary type affected a considerable number of Utah families, and the Mormon reservoir of genealogical data was a substantial aid to research. Max served as director of the Laboratory of Hereditary and Metabolic Disorders from 1945 to 1973 and was succeeded by Frank Tyler, who had, from its inception, been head of its Clinical Division. Among Utah's more distinguished recruits was Emil Smith, who began his important studies in biochemistry in shacks, all the research facilities then available. During the years when Max served as Utah's founding chairman of medicine he also became an international leader in his chosen field, well beyond the University confines. He served as a visiting professor throughout the world and received honors and filled high positions too abundant to mention. His twenty-five years of participation in the work of the Research Grants Division of NIH began in 1949 and included four years on the Council of the Institute of Arthritis
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and Metabolic Diseases, four years on the Allergy and Infectious Disease Council, and service on the Study Sections of Biochemistry and Hematology (including chairmanship of the latter) and on a variety of NIH committees with special charges. His many other responsibilities included consultantships to the Army, the Atomic Energy Commission, and the World Health Organization; chairmanship of the Advisory Committee of the Leukemia Society; and nine years in various capacities with the American Medical Association's Council on Drugs. From 1964 to 1974, Max served as member and chairman of the Scientific Advisory Committee, Scripps Clinic and Research Foundation, La Jolla. He was president of a large number of prestigious learned societies including the Western Association of Physicians, the Association of Professors of Medicine, the Association of American Physicians, and the American and International Societies of Hematology. He became a master of the American College of Physicians in 1973 and the same year received the Robert H. Williams Award of the Association of Professors of Medicine. In 1974, Cecil Watson presented him with the coveted Kober Medal of the Association of American Physicians. Elected to the National Academy of Sciences in 1973, he became the first chairman of the Section on Human Genetics, Hematology, and Oncology and, for three years, secretary of the Class on Medical Sciences. The Utah Group and the Wintrobe Legacy Despite this plethora of commitments, Max's hematology research program at Utah flourished and expanded. As his own involvement in national and international activities increased, G. E. Cartwright, then head of Hematology, assumed direction of the Research and Training Programs.
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Yet if Max was less active in the laboratory, he continued to be involved with the University, particularly in the area of training. He also wrote more than two dozen papers on the pathogenesis of the anemia of infections—including studies of erythrocyte life span, marrow response, and the impaired return of iron from the macrophage to plasma. Extending Wintrobe's original Baltimore experiments with pigs (recorded in some seventeen papers), the Utah group established the pig as a model experimental animal. They defined deficiencies of the vitamin B complex and neurologic lesions but were unable to produce pernicious anemia in the pig. They documented megaloblastic anemias responsive to folic acid and B12 when folic acid antagonists and a nonabsorbable sulfonamide were added to a base diet lacking folate and B12. Cartwright et al. reported in detail the striking changes involving blood, marrow, the central nervous system, and the liver that responded fully and specifically to the addition of pyridoxine to a vitamin B6deficient diet. Pigs also served as subjects for important studies of iron, copper, and porphyrin metabolism—studies later extended to man. Cartwright's investigations of hepatolenticular degeneration (Wilson's disease) and hereditary hemochromatosis were particularly noteworthy, while G. Richard Lee made important observations on the involvement of copper in iron metabolism, the role of the copper transport protein ceruloplasmin, and sideroblastic anemia. The Utah group (particularly Jack Athens, G. E. Cartwright, A. M. Mauer, and Dane Boggs) also made highly significant investigations of leukocyte physiology and kinetics. Athens succeeded Cartwright as head of hematology in 1967. Boggs later transferred his studies of host defense mechanisms, leukocyte kinetics, and the hematopoietic stem cell to the University of Pittsburgh. There were many others in the Utah group—students, residents, fellows, and faculty— who
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contributed to clinical and bench investigations of the leukemias, aplastic and sideroblastic anemias, the spleen, the hemoglobinopathies, coagulation disorders, and other aspects of the spectrum of hematologic disease. At Utah, Jaeger and Wintrobe continued studies on chemical warfare agents they had begun in Baltimore during World War II. The effects of nitrogen mustard on hematopoiesis they observed led them to investigate its therapeutic usefulness in human neoplasia reported by Goodman et al. in 1946. Independently initiated therapeutic trials were reported about the same time by Jacobsen et al. in Chicago. During the Utah years, the Wintrobes exploited Becky's talent as a hostess to initiate an annual garden party for newcomers, faculty, fellows, house staff, and town friends. The list of those attending this summer function eventually grew to more than 400 guests. They also enjoyed departmental picnics and bonfires at dusk in the canyons. Within an hour's drive lay the beautiful Wasatch Mountains, the snows of Alta, and some of the world's finest skiing. It became a tradition that, on Wednesday afternoons, the Department of Medicine at Utah was to be found skiing in the mountains. Max, along with George Cartwright, grew to love this recreation, and Cecil Watson described the Wednesday afternoon jaunts as his ''Maxiavellian" plan to promote morale and friendship within the Department and among the disciplines. Watson speculated that Max and George's love for, and skill at, skiing were aided by their physical constitutions and centers of gravity. Max had studied the violin in high school and carried over a love for chamber music. Though absorption in his profession caused him to abandon music for many years, at Utah he again took up his violin, studying with the concertmaster of the Utah Symphony. He enjoyed playing chamber music with friends. On receiving the prestigious Ferrata Prize in
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Rome, he used some of the associated monies to purchase an Enrico Politi violin. From 1963 to 1965 he served as a member of the Utah Symphony Board, becoming a member of its National Advisory Board after 1976. But there was also tragedy and adversity in Utah. In 1952, while in a car driven by friends, a collision on a slippery road resulted in the deaths of the Wintrobe's son Paul, born in 1944, and of their friends' child. Max, Becky, and their daughter were also injured in the accident. Clinical Hematology; Principles Of Internal Medicine; Blood, Pure And Eloquent It would be difficult to overestimate the impact of Clinical Hematology on students, house officers, and hematologists since its initial publication in 1942. Authoritatively written, compendious, heavily and meticulously referenced and indexed, there is no doubt that it was the premier textbook in hematology of its time. Nor can we appreciate how narrow was the scope and restricted the outlook of the field even as recently as the 1920s. The tenth edition of Osler's Principles and Practice of Medicine, published about the middle of that decade, devotes appreciably less space (thirty pages) to all the disorders of blood combined than to the discussion of typhoid fever (forty-two pages). The eighth and most recent edition of Clinical Hematology (1981) ran to more than 2,000 pages. Max had written and edited the first six editions by himself, though always depending on the unreserved, critical peer review and proofreading of his talented colleagues at Utah, with Becky, as he said, his severest and most helpful critic. The seventh and eighth editions were coauthored with several former fellows and associates. The eighth edition appeared in 1981, and Max was at work on the ninth at the time of his death in 1986. While recent years have seen other equally authorita
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tive and compendious hematology texts, Clinical Hematology was the prototype and remains a model of excellence in the field. A second publishing endeavor highly valued by Max was the Principles of Internal Medicine (1950,1; 1954,3; 1974,1), with Tinsley R. Harrison as editorin-chief. In 1950 when Principles was first published, Cecil and Loeb's excellent text enjoyed a near monopoly in its field. Harrison's Principles, with its emphasis on the pathophysiology and biochemistry of disease, opened the way for a new approach. Principles recommended diagnosis and treatment based not only on the signs and symptoms that brought the patient to the physician, but also on this pathophysiology. The original authors included Harrison, Resnick, Dock, Keefer, and Wintrobe, who were later joined by Paul Beeson, George Thorn, and others. Max was coeditor of this highly successful text through five editions, and the book was translated into Portuguese, Italian, Polish, and Greek. For the sixth and seventh editions, he served as editor-in-chief. Max's final literary efforts sprang from a long-standing interest in medical history. Blood, Pure and Eloquent (1980,1), edited and partly authored by Max, was (like Clinical Hematology) dedicated "To Becky." It includes his own chapters on classic early discoveries in hematology, followed by chapters written by contemporary hematologists who themselves had made significant contributions to the subject areas of which they wrote. Most recently, his Hematology, the Blossoming of a Science: A Story of Inspiration and Effort (1985,1) tells the human history of many contributors to the field through more than 500 biographical sketches. Writing this book as part memoir, part history, Wintrobe yet realized that he could never cover the lives of all who had contributed to "the Golden Age of hematology."
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RETIREMENT FROM THE CHAIR OF MEDICINE In 1967 Max was succeeded at Utah as head of the Department of Medicine and physician-in-chief at the University Hospital by George Cartwright; it can hardly be said he retired. As Distinguished Professor of Internal Medicine he continued to see patients and, of course, write. He continued old activities, initiated new ones, and received a cascade of honors and awards after becoming emeritus. His curriculum vitae shows more than twenty visiting professorships at major universities in the United States and abroad after 1967. In 1977 the Wintrobes purchased a condominium in Palm Desert and thereafter spent the winter months in the more gentle climate of southern California. This meant an end to skiing but the opportunity to golf, write, edit, and relax. Many agencies—private and governmental—continued their demand for Max's participation. As a senior statesman and ambassador his style underwent little change. He spoke in deep, carefully measured tones, and when he was in charge, he ran a tight ship. He never dispensed the fruits of experience and wisdom with the benignity of a Bernard Baruch, from a park bench. Fair and decisive, he held strong opinions, and he did not hesitate to express them and would scrap for a cause he believed in. Reminiscing in 1984, he stated that he was unequivocally happy to have accepted the challenge and come to Utah in 1943. As he looked back over the forty years since leaving Hopkins, a time that had been full of opportunities and crowned with achievement, he and Becky could only conclude they were glad they had ventured. When Max received his M.D. in 1926, the death sentence of a diagnosis of pernicious anemia had just been commuted and the discipline of hematology (essentially based on morphology) would never be the same. That same year Cooley
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was to identify the anemia that bears his name, but the thalassemia syndromes— their genetics, expression in heterozygotes, and molecular basis—remained unknown. The first hospital-operated blood bank would not appear for more than another decade. The Rh-antigen system was undiscovered. The Coombs' test and autoimmune disease were unknown and the erythroenzymopathies unsuspected. The genetic code, the hemoglobinopathies and their molecular basis were not the subject of any text. Nobody knew of erythropoietin or discussed "B" and "T" lymphocytes, "colony stimulating factor," lymphokines, or granulocyte metabolism and kinetics. There were no chemotherapeutic agents for malignant blood dyscrasias except the arsenical Fowler's solution employed in treating chronic granulocytic leukemia. No one had thought of marrow transplants, genetic engineering, or the role of oncogenes. These fragments of the explosion of information uncovered between 1926 and Max's death in 1986 give some small idea of what he liked to call the Golden Age of Hematology. It was indeed a golden era—and Max Wintrobe was one of its chief architects and ambassadors to the world. Max is survived by his wife, Becky; his daughter, Susan; and his four grandsons, Andrew, Stephen, Timothy, and David Brown.
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Selected Bibliography
1929 With M. W. Miller. Normal blood determinations in the South. Arch. Intern. Med., 43:96. Hemoglobin standards in normal men. Proc. Soc. Exp. Biol. Med., 26:868. The volume and hemoglobin content of the red blood corpuscles. Am. J. Med. Sci., 177:513. A simple and accurate hematocrit. J. Lab. Clin. Med., 15:287. 1930 Blood of normal young women residing in a subtropical climate. Arch. Intern. Med., 45:287. Classification of the anemias on the basis of differences in the size and hemoglobin content of the red corpuscles. Proc. Soc. Exp. Biol. Med., 27:1071. The erythrocyte in man. Medicine, 9:195. 1931 Hemoglobin content, volume and thickness of the red blood corpuscle in pernicious anemia and sprue and the changes associated with liver therapy. Am. J. Med. Sci., 181:217. The direct calculation of the volume and hemoglobin content of the erythrocyte. Am. J. Clin. Path., 1:147. With J. H. Musser. Diseases of the blood. In: Tice Practice of Medicine . Hagerstown: W. Prior. Vol. 6, p. 739. 1932 The size and hemoglobin content of the erythrocyte. J. Lab. Clin. Med., 17:899. With L. J. Soffer. The metabolism of leukocytes from normal and leukemic blood. J. Clin. Invest., 11:661. 1933 Macroscopic examination of the blood. Am. J. Med. Sci., 185:58. With M. V. Buell. Hyperproteinemia associated with multiple myeloma. Bull. Johns Hopkins Hosp., 52:156. With R. T. Beebe. Idiopathic hypochromic anemia. Medicine, 12:187.
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With H. S. Shumacker, Jr. The occurrence of macrocytic anemia in association with disorder of the liver. Bull. Johns Hopkins Hosp. 52:387. Variations in the size and hemoglobin content of erythrocytes in the blood of various vertebrates. Fol. Haematol., 51:32. With J. W. Landsberg. Blood of normal men and women. Bull. Johns Hopkins Hosp., 53:118. 1935 With J. W. Landsberg. A standardized technique for the blood sedimentation test. Am. J. Med. Sci., 189:102. With H. B. Shumacker, Jr. Comparison of hematopoiesis in the fetus and during recovery from pernicious anemia. J. Clin. Invest., 14:837. 1936 With H. B. Shumacker, Jr. Erythrocyte studies in the mammalian fetus and newborn. Am. J. Anat., 58:313. 1937 The application and interpretation of the blood sedimentation test in clinical medicine. Med. Clin. North Am., 21:1537. 1939 The antianemic effect of yeast in pernicious anemia. Am. J. Med. Sci., 197:286. Diagnostic significance of changes in leukocytes. Bull. N.Y. Acad. Med., 15:223. With M. Samter and H. Lisco. Morphologic changes in the blood of pigs associated with deficiency of water-soluble vitamins and other substances contained in yeast. Bull. Johns Hopkins, 64:399. Nutritive requirements of young pigs. Am. J. Physiol., 126:375. 1940 With E. Matthews, R. Pollack, and B. M. Dobyns. A familial hemopoietic disorder in Italian adolescents and adults. J. Am. Med. Assoc., 114:1530. With J. L. Miller and H. Lisco. The relation of diet to the occur
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rence of ataxia and degeneration in the nervous system of pigs. Bull. Johns Hopkins Hosp., 67:377. 1941 Attempts to produce pernicious anemia experimentally. Bull. N. Engl. Med. Cent., 3:13. 1942 With C. Mushatt, J. L. Miller, Jr., L. C. Kolb, H. J. Stein, and H. Lisco. The prevention of sensory neuron degeneration in the pig with special reference to the role of various liver fractions. J. Clin. Invest., 21:71. With H. J. Stein, M. H. Miller, R. H. Follis, Jr., V. Najjar, and S. Humphreys. A study of thiamine deficiency in swine. Bull. Johns Hopkins Hosp., 71:141. With M. H. Miller, R. H. Follis, Jr., H. J. Stein, C. Mushatt, and S. R. Humphreys. Sensory neuron degeneration in pigs. IV. Protection afforded by calcium pantothenate and pyridoxine. J. Nutr., 24:345. With M. H. Miller, R. H. Follis, Jr., and H. J. Stein. What is the antineuritic vitamin? Trans. Assoc. Am. Physicians, 57:55. Clinical Hematology. Philadelphia: Lea & Febiger. 1943 With R. H. Follis, Jr., M. H. Miller, H. J. Stein, R. Alcayaga, S. Humphreys, A. Suksta, and G. E. Cartwright. Pyridoxine deficiency in swine, with particular reference to anemia, epileptiform convulsions and fatty liver. Bull. Johns Hopkins Hosp., 72:1. 1944 With G. E. Cartwright and S. Humphreys. Studies on anemia in swine due to pyridoxine deficiency, together with data on phenylhydrazine anemia. J. Biol. Chem., 153:171. With G. E. Cartwright, P. Jones, M. Lauritsen, and S. Humphreys. Tryptophane derivatives in the urine of pyridoxine-deficient swine. Bull. Johns Hopkins Hosp., 75:35. With W. Buschke, R. H. Follis, Jr., and S. Humphreys. Riboflavin deficiency in swine, with special reference to the occurrence of cataracts. Bull. Johns Hopkins Hosp., 75:102.
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With R. H. Follis, Jr., S. Humphreys, H. Stein, and M. Lauritsen. Absence of nerve degeneration in chronic thiamine deficiency in pigs. J. Nutr., 28:283. 1945 Relation of nutritional deficiency to cardiac dysfunction. Arch. Intern. Med., 76:341. With H. J. Stein, R. H. Follis, Jr., and S. Humphreys. Nicotinic acid and the level of protein intake in the nutrition of the pig. J. Nutr., 30:395. 1946 With G. E. Cartwright, M. A. Lauritsen, S. Humphreys, P. J. Jones, and I. M. Merrill. The anemia associated with chronic infection. Science, 103:72. With G. E. Cartwright, M. A. Lauritsen, P. J. Jones, and P. J. Merrill. The anemia of infection. I. Hypoferremia, hypercupremia, and the alterations in porphyrin metabolism in patients. J. Clin. Invest., 25:65. With G. E. Cartwright, M. A. Lauritsen, S. Humphreys, P. J. Jones, and I. M. Merrill. The anemia of infection. II. The experimental production of hypoferremia and anemia in dogs. J. Clin. Invest., 25:81. With L. S. Goodman, W. Dameshek, M. J. Goodman, A. Gilman, and M. T. McLennan. Nitrogen mustard therapy. Use of methyl-bis (beta-chloroethyl) amine hydrochloride and tris (betachloroethyl) amine hydrochloride for Hodgkin's disease, lymphosarcoma, leukemia and certain allied and miscellaneous disorders. J. Am. Med. Assoc., 132:126. With G. R. Greenberg and G. E. Cartwright. The pathogenesis of the anemia of infection. Trans. Assoc. Am. Physicians, 59:110. 1947 With G. R. Greenberg, S. R. Humphreys, H. Ashenbrucker, W. Worth, and R. Kramer. The anemia of infection. III. The uptake of radioactive iron in iron-deficient and in pyridoxinedeficient pigs before and after acute inflammation. J. Clin. Invest., 26:103. The mammalian red corpuscle. Blood, 2:299. With M. Grinstein, J. J. Dubash, S. R. Humphreys, H. Ashen
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brucker, and W. Worth. The anemia of infection. VI. The influence of cobalt on the anemia associated with inflammation. Blood, 2:323. With C. M. Huguley, Jr., M. T. McLennan, and L. P. C. Lima. Nitrogen mustard as a therapeutic agent for Hodgkin's disease, lymphosarcoma and leukemia. Ann. Intern. Med., 27:529. With M. T. McLennan and C. M. Huguley, Jr. Clinical experiences with nitrogen mustard therapy. In: Approaches to Tumor Chemotherapy, p. 347. 1948 With G. E. Cartwright. Studies on free erythrocyte protoporphyrin, plasma copper, and plasma iron in normal and in pyridoxine-deficient swine. J. Biol. Chem., 172:557. Nitrogen mustard therapy. Am. J. Med., 4:313. With M. Grinstein and J. A. Silva. The anemia of infection. VII. The significance of free erythrocyte protoporphyrin, together with some observations on the meaning of the "easily split-off" iron. J. Clin. Invest., 27:245. With G. E. Cartwright, J. Fay, and B. Tatting. Pteroylglutamic acid deficiency in swine: effects of treatment with pteroylglutamic acid, liver extract and protein. J. Lab. Clin. Med., 33:397. With C. M. Huguley, Jr. Nitrogen-mustard therapy for Hodgkin's disease, lymphosarcoma, the leukemias, and other disorders. Cancer, 1:357. With G. E. Cartwright. Studies on free erythrocyte protoporphyrin, plasma copper, and plasma iron in protein-deficient and iron-deficient swine. J. Biol. Chem., 176:571. 1949 With G. E. Cartwright, B. Tatting, and H. Ashenbrucker. Experimental production of a nutritional macrocytic anemia in swine. Blood, 4:301. With G. E. Cartwright. Further studies on nutritional macrocytic anemia in swine. Trans. Assoc. Am. Physicians, 62:294. 1950 Eds., M. M. Wintrobe, T. R. Harrison, P. B. Beeson, W. H. Resnik, G. W. Thorn. Principles of Internal Medicine. Philadelphia: The Blakiston Company.
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1951 With G. E. Cartwright, M. E. Lahey, and C. J. Gubler. The role of copper in hemopoiesis. Trans. Assoc. Am. Physicians, 64:310. Clinical Hematology, 3d ed. Philadelphia: Lea & Febiger. 1952 Factors and mechanisms in the production of red corpuscles. In: Harvey Lectures, 45. Springfield: C. C. Thomas. With R. K. Smiley and G. E. Cartwright. The anemia of infection. XVII. A review. Ad. Intern. Med., 5:165. 1953 With G. E. Cartwright and C. J. Gubler. Studies on the function and metabolism of copper. J. Nutr., 50:395. Shotgun antianemic therapy. Am. J. Med., 15:141. 1954 With G. E. Cartwright, P. Fessas, A. Haut, and S. J. Altman. Chemotherapy of leukemia, Hodgkin's disease and related disorders. Ann. Intern. Med., 41:447. With G. E. Cartwright, R. E. Hodges, C. J. Gubler, J. P. Mahoney, K. Daum, and W. B. Bean. Studies on copper metabolism. XIII. Hepatolenticular degeneration. J. Clin. Invest., 33:1487. Eds. M. M. Wintrobe, T. R. Harrison, R. D. Adams, P. B. Beeson, W. H. Resnik, and G. W. Thorn. Principles of Internal Medicine. New York: McGraw-Hill. 1955 With P. Fessas and G. E. Cartwright. Angiokeratoma corporis diffusum universale (Fabry). Arch. Intern. Med., 68:42 and 95:469. 1956 Clinical Hematology. 4th ed. Philadelphia: Lea & Febiger. With G. E. Cartwright. Blood disorders caused by drug sensitivity. Arch. Intern. Med., 96:559.
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MAXWELL MYER WINTROBE
471
1957 The search for an experimental counterpart of pernicious anemia (The George Minot Lecture). Arch. Intern. Med., 100:862. 1961 Clinical Hematology, 5th ed. Philadelphia: Lea & Febiger. 1964 The therapeutic millennium and its price. Adverse reactions to drugs. In: Drugs in Our Society. Baltimore: Johns Hopkins Press. With G. E. Cartwright. Copper metabolism in normal subjects. Am. J. Clin. Nutr., 14:224. With G. E. Cartwright and J. W. Athens. The kinetics of granulopoiesis in normal man. Blood, 24:780. 1965 The problems of drug toxicity in man—a view from the hematopoietic system. Ann. N.Y. Acad. Sci., 123:316. The virtue of doubt and the spirit of inquiry (Presidential Address, Assoc. Am. Phys., Atlantic City, May 1965). Trans. Assoc. Am. Physicians, 78:1. 1966 The problem of adverse drug reactions. Am. Med. Assoc., 196:404. 1967 A hematological odyssey, 1926-66. Johns Hopkins Med. J., 120:287. 1969 The therapeutic millennium and its price: A view from the hematopoietic system. J. R. Coll. Phys. (London), 3:99. Anemia, serendipity, and science. J. Am. Med. Assoc., 210:318. 1974 Ed. M. M. Wintrobe. Harrison's Principles of Internal Medicine, 7th ed. New York: McGrawHill. With others. Clinical Hematology, Philadelphia: Lea & Febiger, 7th ed.
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1980 With others. Blood, Pure and Eloquent. A Story of Discovery of People, and of Ideas. New York: McGraw-Hill. 1981 With others. Clinical Hematology. Philadelphia: Lea & Febiger, 8th ed. 1982 Medical education in Utah (Medical schools of the west). West. J. Med., 136:357. 1985 Hematology, the Blossoming of a Science: A Story of Inspiration and Effort. Philadelphia: Lea & Febiger.
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CUMULATIVE INDEX
473
Cumulative Index
Volumes 1 Through 59
A Abbe, Cleveland 8:469-508 Abbot, Henry Larcom 13:1-101 Abel, John Jacob 24:231-57 Adams, Comfort Avery 38:1-16 Adams, Leason Heberling 52:3-33 Adams, Roger 53:3-47 Adams, Walter Sydney 31:1-31 Adkins, Homer Burton 27:293-317 Agassiz, Alexander 7:289-305 Agassiz, Louis 2:39-73 Aitken, Robert Grant 32:1-30 Albert, Abraham Adrian 51:2-22 Albright, Fuller 48:3-22 Alexander, John H. 1:213-26 Alexander, Stephen 2:249-59 Allee, Warder Clyde 30:3-40 Allen, Charles Elmer 29:3-15 Allen, Eugene Thomas 40:1-17 Allen, Joel Asaph 21(1):1-20 Ames, Joseph Sweetman 23:181-201 Anderson, Edgar 49:3-23 Anderson, John August 36:1-18 Anderson, Rudolph John 36:19-50 Angell, James Rowland 26:191-208 Armsby, Henry Prentiss 19:271-84 Astwood, Edwin Bennett 55:3-42 Atkinson, George Francis 29:17-44 Avery, Oswald Theodore 32:31-49 B Babcock, Ernest Brown 32:50-66
Babcock, Harold 45:1-19 Bache, Alexander Dallas 1:181-212d Bachmann, Werner Emmanuel 34:1-30 Badger, Richard McLean 56:3-20 Baekeland, Leo Hendrik 24:281-302 Bailey, Irving Widmer 45:21-56 Bailey, Percival 58:3-46 Bailey, Solon Irving 15:193-203 Bain, Edgar Collins 49:25-47 Baird, Spencer Fullerton 3:141-60 Ball, Eric Glendinning 58:49-73 Balls, Arnold Kent 41:1-22 Barbour, Thomas 27:13-45 Barnard, Edward Emerson 21(14):1-23 Barnard, Frederick Augustus Porter 20:259-72 Barnard, John Gross 5:219-29 Barrell, Joseph 12:3-40
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CUMULATIVE INDEX
Bartelmez, George William 43:1-26 Bartlett, William H. C. 7:171-93 Bartter, Frederic C. 59:2-24 Barus, Carl 22:171-213 Bateman, Harry 25:241-56 Beadle, George Wells 59:26-52 Beams, Jesse Wakefield 54:3-49 Becker, George Ferdinand 21(2):1-19 Beecher, Charles Emerson 6:57-88 Bell, Alexander Graham 23:1-29 Benedict, Francis Gano 23:67-99 Benedict, Stanley Rossiter 27:155-77 Benioff, Victor Hugo 43:27-40 Berkey, Charles Peter 30:41-56 Berry, Edward Wilber 45:57-95 Berson, Solomon A. 59:54-70 Bigelow, Henry Bryant 48:51-80 Billings, John Shaw 8:375-416 Birge, Raymond Thayer 59:72-84 Bishop, George Holman 55:45-66 Blackwelder, Eliot 48:83-103 Blake, Francis Gilman 28:1-29 Blakeslee, Albert Francis 33:1-38 Blalock, Alfred 53:49-81 Blichfeldt, Hans Frederik 26:181-89 Bliss, Gilbert Ames 31:32-53 Boas, Franz 24:303-22 Bogert, Marston Taylor 45:97-126 Bolton, Elmer Keiser 54:51-72 Boltwood, Bertram Borden 14:69-96 Bonner, Tom Wilkerson 38:17-32 Boring, Edwin Garrigues 43:41-76 Borthwick, Harry Alfred 48:105-22 Boss, Lewis 9:239-60 Bowditch, Henry Pickering 17: 183-96 Bowen, Ira Sprague 53:83-119 Bowen, Norman Levi 52:35-79 Bowie, William 26:61-98 Bowman, Isaiah 33:39-64 Bradley, Wilmot Hyde 54:75-88 Bramlette, Milton Nunn 52:81-92 Branner, John Casper 21(3):1-20 Bray, William Crowell 26:13-24 Breasted, James Henry 18:95-121 Brewer, William Henry 12:289-323 Bridges, Calvin Blackman 22:31-48 Bridgman, Percy Williams 41:23-67 Brillouin, Leon Nicolas 55:69-92 Britton, Nathaniel Lord 19:147-202 Bronk, Detlev Wulf 50:3-87 Brooks, William Keith 7:23-70 Brouwer, Dirk 41:69-87 Brown, Ernest William 21:243-73 Brown-Sequard, Charles Edouard 4:93-97
474
Brush, George Jarvis 17:107-12 Bucher, Walter Hermann 40:19-34 Buckley, Oliver Ellsworth 37:1-32 Buddington, Arthur Francis 57:3-24 Bueche, Arthur M. 56:23-40 Bumstead, Henry Andrews 13:105-24 Burgess, George Kimball 30:57-72 Burkholder, Paul Rufus 47:3-25 Bush, Vannevar 50:89-117 Byerly, Perry 55:95-105 C Campbell, Angus 56:43-58 Campbell, Douglas Houghton 29:45-63 Campbell, William Wallace 25:35-74 Cannan, Robert Keith 55:107-33 Carlson, Anton Julius 35:1-32 Carmichael, Leonard 51:25-47 Carothers, Wallace Hume 20:293-309 Carty, John Joseph 18:69-91
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CUMULATIVE INDEX
Casey, Thomas Lincoln 4:125-34 Castle, William Ernest 38:33-80 Caswell, Alexis 6:363-72 Cattell, James McKeen 25:1-16 Chamberlin, Rollin Thomas 41:89-110 Chamberlin, Thomas Chrowder 15:307-407 Chandler, Charles Frederick 14:127-81 Chandler, William Henry 59:86-115 Chaney, Ralph Works 55:135-61 Chapman, Frank Michler 25:111-45 Chauvenet, William 1:227-44 Child, Charles Manning 30:73-103 Chittenden, Russell Henry 24:59-104 Clark, Henry James 1:317-28 Clark, William Bullock 9:1-18 Clark, William Mansfield 39:1-36 Clarke, Frank Wigglesworth 15:139-65 Clarke, Hans Thacher 46:3-20 Clarke, John Mason 12:183-244 Clausen, Jens Christian 58:75-107 Clausen, Roy Elwood 39:37-54 Cleland, Ralph Erskine 53:121-39 Cleveland, Lemuel Roscoe 51:49-60 Clinton, George Perkins 20:183-96 Cloos, Ernst 52:95-119 Coblentz, William Weber 39:55-102 Cochran, William Gemmell 56:61-89 Cochrane, Edward Lull 35:33-46 Coffin, James Henry 1:257-64 Coffin, John Huntington Crane 8:1-7 Coghill, George Ellett 22:251-73 Cohn, Edwin Joseph 35:47-84 Cole, Rufus 50:119-39 Compton, Arthur Holly 38:81-110 Comstock, Cyrus Ballou 7:195-201 Comstock, George Cary 20:161-82 Conant, James Bryant 54:91-124 Condon, Edward Uhler 48:125-51 Conklin, Edwin Grant 31:54-91 Cook, George Hammell 4:135-44 Cooke, Josiah Parsons 4:175-83 Coolidge, William David 53:141-57 Coon, Carleton Stevens 58:109-30 Cope, Edward Drinker 13:127-317 Cottrell, Frederick Gardner 27:1-11 Coues, Elliott 6:395-446 Coulter, John Merle 14:99-123 Councilman, William Thomas 18:157-74 Cox, Gertrude Mary 59:116-132 Crafts, James Mason 9:159-77 Craig, Lyman Creighton 49:49-77 Crew, Henry 37:33-54 Cross, Charles Whitman 32: 100-112
475
Curme, George Oliver, Jr. 52:121-37 Curtis, Heber Doust 22:275-94 Cushing, Harvey 22:49-70 D Dall, William Healey 31:92-113 Dalton, John Call 3:177-85 Daly, Reginald Aldworth 34:31-64 Dana, Edward Salisbury 18:349-65 Dana, James Dwight 9:41-92 Danforth, Charles Haskell 44:1-56 Davenport, Charles Benedict 25:75-110 Davidson, George 18:189-217 Davis, Bergen 34:65-82 Davis, Charles Henry 4:23-55 Davis, William Morris 23:263-303 Davisson, Clinton Joseph 36:51-84 Day, Arthur Louis 47:27-47
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CUMULATIVE INDEX
Debye, Peter Joseph Wilhelm 46:23-68 DeGolyer, Everette Lee 33:65-86 Demerec, Milislav 42:1-27 Dempster, Arthur Jeffrey 27:319-33 Dennison, David Mathias 52:139-59 Detwiler, Samuel Randall 35:85-111 Dewey, John 30:105-24 Dobzhansky, Theodosius 55:163-213 Dochez, Alphonse, Raymond 42:29-46 Dodge, Bernard Ogilvie 36:85-124 Dodge, Raymond 29:65-122 Donaldson, Henry Herbert 20:229-43 Dragstredt, Lester Reynold 51:63-95 Draper, Henry 3:81-139 Draper, John William 2:349-88 Dryden, Hugh Latimer 40:35-68 Duane, William 18:23-41 DuBois, Eugene Floyd 36:125-45 Dubos, Rene Jules 58:133-61 Duggar, Benjamin Minge 32:113-31 DuMond, Jesse W. 52:161-201 Dunn, Gano Sillick 28:31-44 Dunn, Leslie Clarence 49:79-104 Dunning, John Ray 58:163-86 Durand, William Frederick 48:153-93 Dutton, Clarence Edward 32:132-45 E Eads, James Buchanan 3:59-79 East, Edward Murray 23:217-42 Echart, Carl Henry 48:195-219 Edison, Thomas Alva 15:287-304 Eigenmann, Carl H. 18:305-36 Einstein, Albert 51:97-117 Eisenhart, Luther Pfahler 40:69-90 Elkin, William Lewis 18:175-88 Elvehjem, Conrad Arnold 59:134-167 Emerson, Alfred Edward 53:159-75 Emerson, Ralph 55:231-45 Emerson, Robert 35:112-31 Emerson, Rollins Adams 25:313-23 Emmert, William Le Roy 22:233-50 Emmons, Samuel Franklin 7:307-34 Engelmann, George 4:1-21 Erlanger, Joseph 41:111-39 Evans, Griffith Conrad 54:127-55 Evans, Herbert McLean 45:153-92 Ewing, James 26:45-60 Ewing, William Maurice 51:119-93 F
476
Farlow, William Gilson 21(4):1-22 Fenn, Wallace Osgood 50:141-73 Fermi, Enrico 30:125-55 Fernald, Merritt Lyndon 28:45-98 Ferrel, William 3:265-309 Fewkes, Jesse Walter 15:261-83 Fischer, Hermann Otto Laurenz 40:91-112 Fisk, James Brown 56:91-116 Fleming, John Adam 39:103-40 Folin, Otto (Knut Olaf) 27:47-82 Foote, Paul Darwin 50:175-94 Forbes, Alexander 40:113-41 Forbes, Stephen Alfred 15:3-54 Fraenkel, Gottfried Samuel 59:168-195 Francis, Thomas, Jr. 44:57-110 Frazer, John Fries 1:245-56 Fred, Edwin Broun 55:247-90
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CUMULATIVE INDEX
Freeman, John Ripley 17:171-87 Frost, Edwin Brant 19:25-51 G Gabb, William More 6:345-61 Gamble, James Lawder 36:146-60 Gay, Frederick Parker 28:99-116 Genth, Frederick Augustus 4:201-31 Gerald, Ralph Waldo 53:179-210 Gesell, Arnold Lucius 37:55-96 Gherardi, Bancroft 30:157-77 Gibbon, John Hersham, Jr. 53:213-47 Gibbs, Josiah Willard 6:373-93 Gibbs, William Francis 42:47-64 Gibbs, Wolcott 7:1-22 Gilbert, Grove Karl 21(5):1-303 Gill, Theodore Nicholas 8:313-43 Gilliland, Edwin Richard 49:107-27 Gilliss, James Melville 1:135-79 Gilluly, James 56:119-32 Godel, Kurt 56:135-78 Goldmark, Peter Carl 55:293-303 Goldschmidt, Richard Benedict 39:141-92 Gomberg, Moses 41:141-73 Gooch, Frank Austin 15:105-35 Goodale, George Lincoln 21(6):1-19 Goode, George Brown 4:145-74 Goodpasture, Ernest William 38:111-44 Gorini, Luigi 52:203-21 Gortner, Ross Aitken 23:149-80 Gould, Augustus Addison 5:91-113 Gould, Benjamin Apthorp 17:155-80 Graham, Clarence Henry 46:71-89 Graham, Evarts Ambrose 48:221-50 Gray, Asa 3:151-75 Gregory, William 46:91-133 Guyot, Arnold 2:309-47 H Haagen-Smit, Arie Jan 58:189-217 Hadley, James 5:247-54 Hague, Arnold 9:21-38 Haldeman, Samuel Stedman 2:139-72 Hale, George Ellery 21:181-241 Hall, Asaph 6:241-309 Hall, Edwin Herbert 21:73-94 Hall, Granville Stanley 12:135-80 Hallowell, Alfred 51:195-213 Halsted, William Stewart 17:151-70 Handler, Philip 55:305-53 Hanson, William Webster 27:121-37
477
Harkins, William Draper 47:49-81 Harlow, Harry Frederick 58:219-57 Harned, Herbert Spencer 51:215-44 Harper, Robert Almer 25:229-40 Harrar, J. George 57:27-56 Harrison, Ross Granville 35:132-62 Hart, Edwin Bret 28:117-61 Hartline, Haldan Keffer 59:196-213 Harvey, Edmund Newton 39:193-266 Hassid, William Zev 50:197-230 Hastings, Charles Sheldon 20:273-91 Haworth, Leland John 55:355-82 Hayden, Ferdinand Vandiveer 3:395-413 Hayford, John Fillmore 167:157-292 Heidelberger, Charles 58:259-302 Hektoen, Ludvig 28: 163-97
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CUMULATIVE INDEX
Henderson, Lawrence Joseph 23:31-58 Hendricks, Sterling Brown 56:181-212 Henry, Joseph 5:1-45 Herget, Paul 57:59-86 Herrick, Charles Hudson 43:77-108 Herskovits, Melville Jean 42:65-93 Herty, Charles Holmes, Jr. 31:114-26 Hess, Harry Hammond 43:109-28 Hewett, Donnel Foster 44:111-26 Hibbert, Harold 32:146-80 Hilgard, Eugene Woldemar 9:95-155 Hilgard, Julius Erasmus 3:327-38 Hill, George William 8:275-309 Hill, Henry Barker 5:255-66 Hillebrand, William Francis 12:43-70 Hitchcock, Edward 1:113-34 Hoagland, Dennis Robert 29:123-43 Holbrook, John Edward's 5:47-77 Holdren, Edward Singleton 8:347-72 Holmes, William Henry 17:223-52 Hoover, Herbert Clark 39:267-91 Horsfall, Frank Lappin, Jr. 50:233-67 Houston, William Vermillion 44:127-37 Hovgaard, William 36:161-91 Howard, Leland Ossian 33:87-124 Howe, Henry Marion 21(7):1-11 Howe, Marshall Avery 19:243-69 Howell, William Henry 26:153-80 Hrdlicka, Ades 23:305-38 Hubbard, Joseph Stillman 1:1-34 Hubble, Edwin Powell 41:175-214 Hubbs, Carl Leavitt 56:215-49 Hudson, Claude Silbert 32:181-220 Hulett, George Augustus 34:83-105 Hull, Albert Wallace 41:215-33 Hull, Clark Leonard 33:125-41 Humphreys, Andrew Atkinson 2:201-15 Hunt, Edward B. 3:29-41 Hunt, Reid 26:25-41 Hunt, Thomas Sterry 15:207-38 Hunter, Walter Samuel 31:127-55 Huntington, George Summer 18:245-84 Hyatt, Alpheus 6:311-25 I Ipatieff, Vladimir Nikolevich 47:83-140 Isaacs, John Dove III 57:89-122 Ives, Herbert Eugene 29:145-89 J Jackson, Charles Loring 37:97-128
478
Jackson, Dunham 33:142-79 Jacobs, Walter Abraham 51:247-78 Jennings, Herbert Spencer 47:143-223 Jewett, Frank Baldwin 27:239-64 Johnson, Douglas Wilson 24:197-230 Johnson, Samuel William 7:203-22 Johnson, Treat Baldwin 27:83-119 Jones, Donald Forsha 46:135-56 Jones, Lewis Ralph 31:156-79 Jones, Walter (Jennings) 20:79-139 Jordan, Edwin Oakes 20:197-228 Joy, Alfred Harrison 47:225-47 Julian, Percy Lavon 52:223-66 K Kac, Mark 59:214-235 Kasner, Edward 31:180-209 Keeler, James Edward 5:231-46 Keith, Arthur 29:191-200 Kelley, Walter Pearson 40:143-75
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CUMULATIVE INDEX
Kellogg, Remington 46:159-89 Kellogg, Vernon Lyman 20:245-57 Kelly, Mervin Joe 46:191-219 Kelser, Raymond Alexander 28:199-221 Kemp, James Furman 16:1-18 Kendall, Edward C. 47:249-90 Kennelly, Arthur Edwin 22:83-119 Kent, Robert Harrington 42:95-117 Kettering, Charles Franklin 34:106-22 Kharasch, Morris Selig 34:123-52 Kidder, Alfred Vincent 39:293-322 Kimball, George Elbert 43:129-46 King, Clarence 6:25-55 Kirtland, Jared Potter 2:127-38 Kluckhohn, Clyde Kay Maben 37:129-59 Knopf, Adolf 41:235-49 Kofoid, Charles Atwood 26:121-51 Kohler, Elmer Peter 27:265-91 Kok, Bessel 57:125-48 Kompfner, Rudolf 54:157-80 Kraus, Charles August 42:119-59 Krayer, Otto 57:151-225 Kroeber, Alfred Louis 36:192-253 Kunitz, Moses 58:305-17 Kunkel, Louis Otto 38:145-60 L Lamb, Arthur Becket 29:201-34 Lambert, Walter Davis 43:147-62 La Mer, Victor Kuhn 45:193-214 Lancefield, Rebecca Craighill 57:227-46 Landsteiner, Karl 40:177-210 Lane, Jonathan Homer 3:253-64 Langley, Samuel Pierpont 7:245-68 Langmuir, Irving 45:215-47 LaPorte, Otto 50:269-85 Larsen, Esper Signius, Jr. 37:161-84 Lashley, Karl Spencer 35:163-204 Lasswell, Harold Dwight 57:249-74 Latimer, Wendell Mitchell 32:221-37 Laufer, Berthold 18:43-68 Lauritsen, Charles Christian 46:221-39 Lauritsen, Thomas 55:385-96 Lawrence, Ernest Orlando 41:251-94 Lawson, Andrew Cowper 37:185-204 Lazarsfeld, Paul F. 56:251-82 Lea, Matthew Carey 5:155-208 Le Conte, John 3:369-93 Le Conte, John Lawrence 2:261-93 Le Conte, Joseph 6:147-218 Leidy, Joseph 7:335-96 Leith, Charles Kenneth 33:180-204 Leopold, Aldo Starker 59:236-255
479
Lesley, J. Peter 8:155-240 Lesquereux, Leo 3:187-212 Levene, Phoebus Aaron Theodor 23:75-126 Leverett, Frank 23:203-15 Lewis, George William 25:297-312 Lewis, Gilbert Newton 31:210-35 Lewis, Howard Bishop 44:139-73 Lewis, Warren Harmon 39:323-58 Lillie, Frank Rattray 30:179-236 Lim, Robert Kho-Seng 51:281-306 Linton, Ralph 31:236-53 Little, Clarence Cook 46:241-63 Loeb, Jacques 13:318-401 Loeb, Leo 35:205-51 Loeb, Robert Frederick 49:149-83 Long, Cyril Norman Hugh 46:265-309 Long, Esmond R. 56:285-310 Longcope, Warfield Theobald 33:205-25 Longstreth, Miers Fisher 8:137-40
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CUMULATIVE INDEX
Longwell, Chester Ray 53:249-62 Loomis, Alfred Lee 51:309-41 Loomis, Elias 3:213-52 Lothrop, Samuel Kirkland 48:253-72 Lovering, Joseph 6:327-44 Lucas, Howard Johnson 43:165-76 Lueschner, Armin Otto 49:129-47 Lush, Jay Laurence 57:277-305 Lusk, Graham 21:95-142 Lyman, Theodore 5:141-53 Lyman, Theodore 30:237-56 M MacArthur, Robert Helmer 58:319-27 MacCallum, William George 23:339-64 Macelwane, James B., S. J. 31:254-81 MacInnes, Duncan Arthur 41:295-317 Mackin, Joseph Hoover 45:249-62 MacLeod, Colin Munro 54:183-219 MacNider, William deBerneire 32:238-72 Mahan, Dennis Hart 2:29-37 Mall, Franklin Paine 16:65-122 Mann, Frank Charles 38:161-204 Marsh, George Perkins 6:71-80 Marsh, Othniel Charles 20:1-78 Marshall, Eli Kennerly, Jr. 56:313-52 Mason, Max 37:205-36 Maxcy, Kenneth Fuller 42:161-73 Mayer, Alfred Marshall 8:243-72 Mayer, Manfred Martin 59:256-280 Mayer, Maria Gappert 50:311-28 Mayor, Alfred Goldsborough 21(8):1-14 Mayo-Smith, Richmond 17:73-77 McCollum, Elmer Verner 45:263-335 McDermott, Walsh 59:282-307 McElvain, Samuel Marion 54:221-48 McLean, William B. 55:399-409 McMaster, Philip Dursee 50:287-308 McMath, Robert Raynolds 49:185-202 Mead, Margaret 58:329-54 Mead, Warren Judson 35:252-71 Meek, Fielding Bradford 4:75-91 Meek, Walter Joseph 54:251-68 Mees, Charles Edward Kenneth 42:175-99 Meggers, William Frederick 41:319-40 Meigs, Montgomery Cunningham 3:311-26 Meltzer, Samuel James 21(9):1-23 Mendel, Lafayette Benedict 18:123-55 Mendenhall, Charles Elwood 18:1-22 Mendenhall, Thomas Corwin 16:331-51 Mendenhall, Walter Curran 46:311-28 Merica, Paul Dyer 33:226-40
480
Merriam, Clinton Hart 24:1-57 Merriam, John Campbell 26:209-32 Merrill, Elmer Drew 32:273-333 Merrill, George Perkins 17:33-53 Merrill, Paul Willard 37:237-66 Meyer, Karl Friedrich 52:269-332 Meyerhoff, Otto 34:153-82 Michael, Arthur 46:331-66 Michaelis, Leonor 31:282-321 Michelson, Albert Abraham 19:121-46 Midgley, Thomas, Jr. 24:361-80 Miles, Walter Richard 55:411-32 Miller, Alden Holmes 43:177-214 Miller, Dayton Clarence 23:61-74 Miller, George Abram 30:257-312 Millikan, Clark Blanchard 40:211-25
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CUMULATIVE INDEX
Millikan, Robert Andrews 33:241-82 Minkowski, Rudolf Leo Bernhard 54:271-98 Minot, Charles Sedgwick 9:263-85 Minot, George Richards 45:337-83 Mitchell, Henry 20:141-50 Mitchell, Samuel Alfred 36:254-76 Mitchell, Silas Weir 32:334-53 Modjeski, Ralph 23:243-61 Moore, Carl Richard 45:385-412 Moore, Eliakim Hastings 17:83-102 Moore, Joseph Haines 29:235-51 Moore, Stanford 56:355-85 Morgan, Lewis Henry 6:219-39 Morgan, Thomas Hunt 33:283-325 Morley, Edward Williams 21(10):1-8 Morse, Edward Sylvester 17:3-29 Morse, Harmon Northrop 21(11): 1-14 Morton, Henry 8:143-51 Moulton, Forest Ray 41:341-55 Mueller, John Howard 57:307-21 Murphree, Eger Vaughan 40:227-38 Murphy, James Bumgardner 34:183-203 N Nachmansohn, David 58:357-404 Nef, John Ulric 34:204-27 Newberry, John Strong 6:1-24 Newcomb, Simon 17: 1-69 Newton, Hubert Anson 4:99-124 Newton, John 4:233-40 Nicholas, John Spangler 40:239-89 Nichols, Edward Leamington 21:343-66 Nichols, Ernest Fox 12:99-131 Nicholson, Seth Barnes 42:201-27 Niemann, Carl 40:291-319 Nissen, Henry Wieghorst 38:205-22 Norris, James Flack 45:413-26 Norton, William A. 2:189-99 Novy, Frederick George 33:326-50 Noyes, Arthur Amos 31:322-46 Noyes, William Albert 27:179-208 O Oliver James Edward 4:57-74 Olson Harry F. 58:407-23 Opie, Eugene Lindsay 47:293-320 Osborn, Henry Fairfield 19:53-119 Osborne, Thomas Burr 14:261-304 Osterhout, Winthrop John Vanleven 44:213-49
481
P Packard, Alpheus Spring 9:181-236 Painter, Theophilus Shickel 59:308-337 Palache, Charles 30:313-28 Parker, George Howard 39:359-90 Patterson, Bryan 55:435-50 Patterson, John Thomas 38:223-62 Paul, John Rodman 47:323-68 Pearl, Raymond 22:295-347 Pecora, William Thomas 47:371-90 Peirce, Benjamin Osgood 8:437-66 Penfield, Samuel Lewis 6:119-46 Peters, John Punnett 31:347-75 Pickering, Edward Charles 15:169-89 Pierce, George Washington 33:351-80
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CUMULATIVE INDEX
Pillsbury, Walter Bowers 37:267-91 Pincus, Gregory Goodwin 42:229-70 Pirsson, Louis Valentine 34:228-48 Pitts, Robert Franklin 57:323-44 Pólya, George 59:338-355 Pourtales, Louis Francois de 5:79-89 Powell, John Wesley 8:11-83 Prudden, Theophil Mitchell 12:73-98 Pumpelly, Raphael 16:23-62 Pupin, Michael Idvorsky 19:307-23 Putnam, Frederic Ward 16:125-52 R Ransome, Frederic Leslie 22:155-70 Ranson, Stephen Walker 23:365-97 Raper, John Robert 57:347-70 Reeside, John Bernard, Jr. 35:272-91 Reid, Harry Fielding 26:1-12 Remsem, Ira 14:207-57 Rice, Oscar Knefler 58:425-56 Rich, Arnold Rice 50:331-50 Richards, Alfred Newton 42:271-318 Richards, Dickinson Woodruff 58:459-87 Richards, Theodore William 44:251-86 Richtmyer, Floyd Karker 22:71-81 Riddle, Oscar 45:427-65 Ridgway, Robert 15:57-101 Ritt, Joseph Fels 29:253-64 Rivers, Thomas Milton 38:263-94 Robertson, Howard Percy 51:343-64 Robertson, Oswald Hope 42:319-38 Robinson, Benjamin Lincoln 17:305-30 Rodebush, Worth Huff 36:277-88 Rodgers, John 6:81-92 Rogers, Fairman 6:93-107 Rogers, Robert Empie 5:291-309 Rogers, William Augustus Part I, 4:185-99 Part II, 6:109-17 Rogers, William Barton 3:1-13 Romer, Alfred Sherwood 53:265-94 Rood, Ogden Nicholas 6:447-72 Rosa, Edward Bennett 16:355-68 Ross, Frank Elmore 39:391-402 Rossby, Carl-Gustaf Arvid 34:249-70 Rous, Francis Peyton 48:275-306 Rowland, Henry Augustus 5:115-40 Royce, Josiah 33:381-96 Rubey, William Walden 49:205-23 Ruedemann, Rudolf 44:287-302 Russell, Henry Norris 32:354-78 Russell, Richard Joel 46:369-94 Rutherford, Lewis Morris 3:415-41
482
Ryan, Harris Joseph 19:285-306 S Sabin, Florence Rena 34:271-319 Sabine, Wallace Clement Ware 21 (13):1-19 St. John, Charles Edward 18:285-304 Sargent, Charles Sprague 12:247-70 Saunders, Frederick Albert 29:403-16 Sauveur, Albert 22:121-33 Savage, John Lucian 49:225-38 Sax, Karl 57:373-97 Saxton, Joseph 1:287-316 Scatchard, George 52:335-77
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CUMULATIVE INDEX
Schiff, Leonard Isaac 54:301-23 Schlesinger, Frank 24:105-44 Schmidt, Gerhard 57:399-429 Scholander, Per Fredrik Thorkelsson 56:387-412 Schott, Charles Anthony 8:87-133 Schuchert, Charles 27:363-89 Schultz, Adolf Hans 54:325-49 Schultz, Jack 47:393-422 Scott, William Berryman 25:175-203 Scudder, Samuel Hubbard 17:81-104 Seares, Frederick Hanley 39:417-44 Seashore, Carl Emil 29:265-316 Setchell, William Albert 23:127-47 Shaffer, Philip Anderson 40:321-36 Shane, Charles Donald 58:489-511 Shapley, Harlow 49:241-91 Shedlovsky, Theodore 52:379-408 Sherman, Henry Clapp 46:397-433 Shope, Richard Edwin 50:353-75 Silliman, Benjamin, Sr. 1:99-112 Silliman, Benjamin, Jr. 7:115-41 Sinnott, Edmund Ware 54:351-72 Slater, John Clarke 53:297-321 Slipher, Vesto Melvin 52:411-49 Small, Lyndon Frederick 33:397-413 Smith, Alexander 21(12):1-7 Smith, Edgar Fahs 17:103-49 Smith, Erwin Frink 21:1-71 Smith, Gilbert Morgan 36:289-313 Smith, Homer William 39:445-70 Smith, James Perrin 38:295-308 Smith, John Lawrence 2:217-48 Smith, Sidney Irving 14:5-16 Smith, Theobald 17:261-303 Sperry, Elmer Ambrose 28:223-60 Spier, Leslie 57:431-58 Squier, George Owen 20:151-59 Stadie, William Christopher 58:513-28 Stadler, Lewis John 30:329-47 Stebbins, Joel 49:293-316 Steenrod, Norman Earl 55:453-70 Stein, William H. 56:415-40 Steinhaus, Edward Arthur 44:303-27 Stejneger, Leonhard Hess 24:145-95 Stern, Curt 56:443-73 Stern, Otto 43:215-36 Stevens, Stanley Smith 47:425-59 Stewart, George W. 32:379-98 Stieglitz, Julius 21:275-314 Stillwell, Lewis Buckley 34:320-28 Stimpson, William 8:419-33 Stock, Chester 27:335-62 Stone, Wilson Stuart 52:451-68
483
Stratton, George Malcolm 35:292-306 Stratton, Samuel Wesley 17:253-60 Streeter, George Linius 28:261-87 Strong, Theodore 2:1-28 Sullivant, William Starling 1:277-85 Sumner, Francis Bertody 25:147-73 Sumner, James Batcheller 31:376-96 Sutherland, Earl W. 49:319-50 Swain, George Fillmore 17:331-50 Swanton, John Reed 34:329-49 Swasey, Ambrose 22:1-29 Szilard, Leo 40:337-47 T Taliaferro, William Hay 54:375-407 Tate, John Torrence 47:461-84 Tatum, Edward Lawrie 59: 356-386 Taylor, Charles Vincent 25:205-25 Taylor, David Watson 22:135-53
About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as the authoritative version for attribution.
CUMULATIVE INDEX
Tennent, David Hilt 26:99-119 Terman, Lewis Madison 33:414-61 Teuber, Hans-Lukas 57:461-90 Thaxter, Roland 17:55-68 Thom, Charles 38:309-44 Thompson, Thomas Gordon 43:237-60 Thomson, Elihu 21:143-79 Thorndike, Edward Lee 27:209-37 Thurstone, Louis Leon 30:349-82 Timoshenko, Stephen 53:323-49 Tolman, Edward Chace 37:293-324 Tolman, Richard Chace 27:139-53 Torrey, John 1:265-76 Totten, Joseph Gilbert 1:35-97 Tozzer, Alfred Marston 30:383-97 Trelease, William 35:307-32 Trowbridge, Augustus 18:219-44 Trowbridge, John 14:185-204 Trowbridge, William P. 3:363-67 Trumbull, James Hammond 7:143-69 Tuckerman, Edward 3:15-28 Turner, Richard Baldwin 53:351-65 Tyzzer, Ernest Edward 49:353-73 V Van Hise, Charles Richard 17:145-51 Van Niel, Cornelis Bernardus 59:388-423 Van Slyke, Donald Dexter 48:309-60 Van Vleck, Edward Burr 30:399-409 Van Vleck, John Hasbrouck 56:501-40 Vaughan, Thomas Wayland 32:399-437 Veblen, Oswald 37:325-41 Verrill, Addison Emery 14:19-66 Vestine, Ernest Harry 51:367-85 Vickery, Hubert Bradford 55:473-504 Vigneaud, Vincent du 56:543-95 von Bekesy, Georg 48:25-49 von Karman, Theodore 38:345-84 von Neumann, John 32:438-57 W Walcott, Charles Doolittle 39:471-540 Walker, Francis Amasa 5:209-18 Warren, Gouverneur Kemble 2:173-88 Washburn, Edward Wight 17:69-81 Washburn, Margaret Floy 25:275-95 Watson, James Craig 3:43-57 Watson, Sereno 5:267-90 Weaver, Warren 57:493-530 Webster, Arthur Gordon 18:337-47 Webster, David Locke II 53:367-400
484
Welch, William Henry 22:215-31 Wells, Harry Gideon 26:233-63 Wells, Horace Lemuel 12:273-85 Werkman, Chester Hamlin 44:329-70 Wetmore, Alexander 56:597-626 Wheeler, William Morton 19:203-41 White, Abraham 55:507-36 White, Charles Abiathar 7:223-43 White, David 17:189-221 White, Henry Seely 25:17-33
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CUMULATIVE INDEX
485
Whitehead, John Boswell 37:343-61 Whitman, Charles Otis 7:269-88 Whitmore, Frank Clifford 28:289-311 Whitney, Willis Rodney 34:350-67 Whittaker, Robert H. 59:424-444 Wiggers, Carl John 48:363-97 Wilczynski, Ernest Julius 16:295-327 Williams, John Harry 42:339-55 Willier, Benjamin Harrison 55:539-628 Willis, Bailey 35:333-50 Williston, Samuel Wendell .17: 115-41 Wilson, David Wright 43:261-84 Wilson, Edmund Beecher 21:315-42 Wilson, Edwin Bidwell 43:285-320 Wilson, Henry Van Peters 35:351-83 Wilson, Ralph Elmer 36:314-29 Wilson, Robert Erastus 54:409-34 Winlock, Joseph 1:329-43 Winstein, Saul 43:321-53 Wintrobe, Maxwell Mayer 59:446-472 Wolfrom, Melville Lawrence 47:487-549 Wood, Horatio C. 33:462-84 Wood, William Barry, Jr. 51:387-418 Woodruff, Lorande Loss 52:471-85 Woodward, Joseph Janvier 2:295-307 Woodward, Robert Simpson 19:1-24 Woodworth, Robert Sessions 39:541-72 Worthen, Amos Henry 3:339-62 Wright, Arthur Williams 15:241-57 Wright, Frederick Eugene 29:317-59 Wright, Orville 25:257-74 Wright, William Hammond 50:377-96 Wyman, Jeffries 2:75-126 Y Yerkes, Robert Mearns 38:385-425 Young, Charles Augustus 7:89-114 Z Zinsser, Hans 24:323-60
NOTE: An asterisk () indicates volumes 17 and 21 of the scientific Memoir series, which correspond to volumes 10 and 11, respectively, of the Biographical Memoirs.
Report "Biographical memoirs. 9780309041980, 0309041988"
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Set
The notion of a set is one of the fundamental notions in mathematics that is difficult to precisely define. Of course, we have plenty of synonyms for the word "set," like collection, ensemble, group, etc., but those names really do not define the meaning of the word set; all they can do is replace it in various sentences. So, instead of defining what sets are, one has to define what can be done with them or, in other words, what axioms the sets satisfy. These axioms are chosen to agree with our intuitive concept of a set, on one hand, and to allow various, sometimes quite sophisticated, mathematical constructions on the other hand. For the full collection of these axioms, see Zermelo-Fraenkel Axioms. In this article we shall present just a brief discussion of the most common properties of sets and operations related to them.
Contents
[hide]
1 Rough Definition
2 Relation of Belonging
3 Specifying a Set
3.1 Listing its Elements
3.2 Stating the Common Property of its Elements
3.3 Set Notation
4 Subsets
5 Power Sets
6 Operations
6.1 Union and Intersection
6.2 Cartesian Product
7 Empty Set
8 Infinite Sets
9 Cardinality
10 Problems
10.1 Introductory
10.2 Intermediate
10.3 Olympiad
11 See Also
12 External Links
Rough Definition
A set is a collection of objects. The objects can be anything: numbers, letters, libraries that have at least 20 male staff, or absolutely nothing. Order does not matter. What does matter is what is in the set. There might be a finite number of objects in the set, in which case it is called a finite set. Otherwise we call it an infinite set. The objects in a set are called the elements of the set. A common misconception is that a set can have multiple indistinct elements, such as the following: Such an entity is actually called a multiset.
Relation of Belonging
The most important property of sets is that, for every object and a set , we can say whether belongs to (written as ), or not (written as ). Two sets and are equal if they include the same objects, i.e., if for every object , we have if and only if .
Specifying a Set
There are many ways to specify a set, using different notation.
Listing its Elements
This means that in order to identify a particular set, it suffices to tell which objects belong to this set. If the set contains just several such objects, all you need to do is list them. So, you can specify the set consisting of the numbers , and , for example. (The standard notation for this set is . Note that the order in which the terms are listed is completely unimportant: we have to follow some order when writing things in one line, but you should actually imagine those numbers flowing freely inside those curly braces with no preference given to any of them. What matters is that these four numbers are in the set and everything else is out). But how do you specify sets that have very many (maybe infinitely many) elements? You cannot list them all even if you spend your entire life writing!
Stating the Common Property of its Elements
Another way to specify a set is to use some property to tell when an object belongs to this set. For instance, we may try to think (alas, only try!) of the set of all objects with green hair. In this case, we do not even try to list all such objects. We just decide that something belongs to this set if it has green hair and doesn't belong to it otherwise. This is a wonderful way to describe a set.
Unfortunately, this method has several potential pitfalls. It turns out, counter-intuitively, that not every collection of objects with a certain property is a set. The most famous example of this problem is Russell's Paradox: consider the property, "is a set and does not contain itself." (Remember that, given a set, we should be able to tell about every object whether it belongs to this set or not; in particular, we can ask this question about the set itself). The set specified by this property can neither belong, nor not belong to itself. There are a variety of ways to resolve this paradox, but the problem is clear: this way to describe sets should be used with extreme caution. One way to avoid this problem is as follows: given a property , choose a known set . Then the collection of elements of which have property will always be a set. (In particular, for our previous example to lead to a paradox, we would need to choose . However, it turns out that it can be proven that the set of all sets does not exist -- the collection of all sets is too big to be a set.)
Set Notation
There is a notation used just for sets:
That symbolizes the set of all reals not equal to 0. This is probably the fastest way of describing a large set.
Also, the empty set can be specified using set notation:
Since there are no reals such that the square of it is less than 0, that set is the empty set.
Subsets
We say that a set is a subset of a set if every object that belongs to also belongs to . This is denoted or . For example, the sets and are subsets of the set , but the set is not. Thus we can say that two sets are equal if and only if each is a subset of the other. A special kind of subset is the empty set.
Power Sets
Main article: power set
The power set of a set , denoted is defined as the set of all its subsets. For example, the power set of is . If a is a finite set of size then has size .
Operations
Union and Intersection
The union of two or more sets is the set of all objects that belong to one or more of the sets. The union of A and B is denoted . For example, the union of and is . Unions can also be represented just as sums and products can be. would be the union of all sets that satisfy the statement. So, for example, would be the set of all natural numbers .
The intersection of two or more sets is the set of all objects that belong to all of the sets. The intersection of A and B is denoted . For example, the intersection of and is . Just like unions, intersections can be represented as such: . For example, , or the empty set defined next.
Cartesian Product
The Cartesian Product of two sets and is defined as the set of Ordered Pairs such that and
Empty Set
Main article: empty set
An empty set, denoted is a set with no elements.
An empty set has some special properties:
It is a subset of every other set.
The union of any other set and an empty set is the original set.
The intersection of any other set and an empty set is an empty set.
Infinite Sets
An infinite set can be defined as a set that has the same cardinality as one of its proper subsets. Alternatively, infinite sets are those which cannot be put into correspondence with any set of the form .
Cardinality
The cardinality of a set , denoted , is (informally) the size of the set. For a finite set, the cardinality is simply the number of elements. The empty set has cardinality 0.
iff there is a bijective function meaning that there is a function that maps all elements of to all the elements of with one-to-one correspondence.
iff there exists an injective function meaning there is a function that maps all elements of to (not necessarily all) elements of . can be defined similarily by expressing it as .
iff there exists an injective function and there is no bijective function meaning but . is defined similarly.
Although showing that and implies that is easy to prove when using finite sets, it is more complicated when using infinite sets. This theorem is called the Cantor-Bernstein-Schröder theorem and was proven by Georg Cantor, Felix Bernstein, and Ernst Schröder.
Problems
Introductory
The regular 5-point star is drawn and in each vertex, there is a number. Each and are chosen such that all 5 of them came from set . Each letter is a different number (so one possible way is ). Let be the sum of the numbers on and , and so forth. If and form an arithmetic sequence (not necessarily in increasing order), find the value of .
(Source)
Intermediate
Let set be a 90-elementsubset of and let be the sum of the elements of Find the number of possible values of
(Source)
Olympiad
Let be a fixed positive integer, and let be an infinite family of sets, each of size , no two of which are disjoint. Prove that there exists a set of size that meets each set in .
(Source)
See Also
Set theory
Function
External Links
Naive Set Theory by Paul R. Halmos.
Set Theory and Logic by Robert Roth Stoll.
Retrieved from "
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Definition
Set theory
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5991 | https://math.stackexchange.com/questions/1928757/find-integers-a-b-c-d-e-and-f-such-that-px-ax2-bx-c2-%E2%88%925xex | elementary number theory - Find integers $a, b, c, d, e$ and $f$ such that $P(X) = (aX^2 +bX +c)^2 −5X(eX +f)^2$ - Mathematics Stack Exchange
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Find integers a,b,c,d,e a,b,c,d,e and f f such that P(X)=(a X 2+b X+c)2−5 X(e X+f)2 P(X)=(a X 2+b X+c)2−5 X(e X+f)2
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Given
P(X)=X 4+X 3+X 2+X+1 P(X)=X 4+X 3+X 2+X+1
Find integers a,b,c,d,e a,b,c,d,e and f f such that
P(X)=(a X 2+b X+c)2−5 X(e X+f)2 P(X)=(a X 2+b X+c)2−5 X(e X+f)2
I am absolutely horrible at math in general. My class just started and this is the first bit of homework, I could use some guidance to get started on this problem. Thank's for any tips, it is very much appreciated.
elementary-number-theory
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edited Sep 16, 2016 at 3:14
msm
7,299 2 2 gold badges 16 16 silver badges 31 31 bronze badges
asked Sep 16, 2016 at 2:52
knamesknames
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Just expand and compare coefficients. Obviously a=1 a=1 N.S.JOHN –N.S.JOHN 2016-09-16 02:58:25 +00:00 Commented Sep 16, 2016 at 2:58
@N.S.JOHN could you give me an example of what expand and comparing coefficients is? I don't know what that is. Thanks!knames –knames 2016-09-16 03:01:28 +00:00 Commented Sep 16, 2016 at 3:01
(a x+b)2=x 2+2 x+1(a x+b)2=x 2+2 x+1, then a=b=1 a=b=1 is Aam example N.S.JOHN –N.S.JOHN 2016-09-16 03:04:37 +00:00 Commented Sep 16, 2016 at 3:04
I don't understand... I took (a x+b)2(a x+b)2 and got a 2 b 2+2 a b x+b 2 a 2 b 2+2 a b x+b 2 how did you get your equation? Why did you set a and b to 1?knames –knames 2016-09-16 04:13:52 +00:00 Commented Sep 16, 2016 at 4:13
Also, after expanding the equation fully, I don't understand how you compare coefficients. The x 4 x 4 is the only one by itself, i end up with 2 a b x 3 2 a b x 3 from the (a x 2+b x+c)2(a x 2+b x+c)2 and then theres −5 e 2 x 3−5 e 2 x 3 from the −5 x(e x+f)2−5 x(e x+f)2 knames –knames 2016-09-16 04:21:54 +00:00 Commented Sep 16, 2016 at 4:21
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The biggest tip for this kind of thing is that you're trying to find ways such that it is true for any possible value of X X, therefore you can work out what must be true by checking what happens with specific values of X X.
For example, P(0)=0 4+0 3+0 2+0+1=1 P(0)=0 4+0 3+0 2+0+1=1, and P(0)=c 2 P(0)=c 2. So clearly c 2=1 c 2=1, meaning that c=±1 c=±1. Then P(1)=5 P(1)=5 and P(1)=(a+b+c)2−5(e+f)P(1)=(a+b+c)2−5(e+f) which you can use to relate all the values together.
After a while, you'll be able to get some specific values in, at which point you might want to see if expanding things out will help. For example, you can see that in the second expression the coefficient of X 4 X 4 is just a 2 a 2, so a=±1 a=±1 too.
In general, you need at least n n equations to find a solution for n n unknowns, so try at least 5 different values of X X to substitute in.
EDIT: Since you've asked about comparing coefficients, here's a quick explanation:
You're trying to find values a,b,c,d,e,f a,b,c,d,e,f to make the two polynomials equivalent. The most obvious way to do that is to try to arrange the two of them so that they look similar, then compare the two to relate the various things to each other.
As an example, here's a simpler problem: given T(x)=x 3+6 x 2+12 x+25 T(x)=x 3+6 x 2+12 x+25, find values a,b,c a,b,c such that T(x)=(a x+b)3+c T(x)=(a x+b)3+c. We expand the second one, giving us T(x)=a 3 x+3 a 2 b x 2+3 a b 2 x+b 3+c T(x)=a 3 x+3 a 2 b x 2+3 a b 2 x+b 3+c. Then, we compare the coefficients of x 3 x 3, x 2 x 2, x x and 1 1 (i.e. the constant coefficient).
First, by looking at the coefficients of x 3 x 3, we get a 3=1 a 3=1, so a=1 a=1. Then, looking at the coefficients of x 2 x 2, we have 3 a 2 b=6 3 a 2 b=6, so using our knowledge about a a we get b=2 b=2. In this case, the coefficient of x x gives us no additional knowledge except that we haven't messed anything up. Finally, the constant terms give us b 3+c=25 b 3+c=25, and since b 3=8 b 3=8 we have c=17 c=17.
Do you see how you could apply that to your problem? You just have to be careful expanding out the (a X 2+b X+c)2(a X 2+b X+c)2 term.
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edited Sep 16, 2016 at 3:11
answered Sep 16, 2016 at 3:00
ConManConMan
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9
Don't you think this is slightly overkill? Comparing coefficients would do N.S.JOHN –N.S.JOHN 2016-09-16 03:03:29 +00:00 Commented Sep 16, 2016 at 3:03
Well in this case, comparing coefficients requires expanding a three-term quadratic, which is something I mess up 80% of the time. Also, it provides a useful trick for more complicated versions of the same problem.ConMan –ConMan 2016-09-16 03:05:07 +00:00 Commented Sep 16, 2016 at 3:05
So I came up with T(x)=(1 x+2)3+17 T(x)=(1 x+2)3+17 . Just making sure I'm correct. I think I can extrapolate this to my original question, thank you very much for your help!knames –knames 2016-09-16 03:31:57 +00:00 Commented Sep 16, 2016 at 3:31
After following through your steps, with a,c=+/−1 a,c=+/−1, how do you decide whether to use the positive or negative 1 in the next equations? I don't understand how this helps with so many variables knames –knames 2016-09-16 04:18:31 +00:00 Commented Sep 16, 2016 at 4:18
So you're going to wind up with a bunch of relationships between the variables. That's when you have to try to eliminate the variables one at a time to get to expressions involving individual variables. However, I have a sneaking suspicion that this particular problem doesn't have any real solutions.ConMan –ConMan 2016-09-16 05:14:25 +00:00 Commented Sep 16, 2016 at 5:14
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5992 | https://www.vedantu.com/formula/equilateral-triangle-formula | Equilateral Triangle Formula - Explanation, Area, Perimeter, and Solved Example
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Formula
Equilateral Triangle Formula
Equilateral Triangle Formula
Reviewed by:
Rama Sharma
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What is an Equilateral Triangle?
As the name suggests, ‘equi’ means equal, an equilateral triangle is the one in which all sides are equal. The internalangles of any given equilateral triangle are of the same measure, that is, equal to 60 degrees.
Triangles are classified into three sorts based on the length of their sides:
Scalene triangle: The sides and the angles of the scalene triangle are not equal.
Isosceles triangle: An isosceles triangle has two equal sides and two equal angles.
Equilateral triangle: All sides and angles of the equilateral triangle are equal.
Area of Equilateral Triangle
The region enclosed by the three sides of an equilateral triangle is defined as the area of the equilateral triangle. It is expressed in square units. The common units used to express the area of an equilateral triangle are in 2, m 2, cm 2 and yd 2
Below the area of the equilateral triangle formula, the altitude of the equilateral triangle formula, the perimeter of the equilateral triangle formula, and the semi-perimeter of an equilateral triangle are discussed.
Area of the Equilateral Triangle Formula
The area of an equilateral triangle is the amount of space that it occupies in a 2-dimensional plane. To recall, an equilateral triangle can be defined as a triangle in which all the sides are equal and the measure of all the internal angles is 60°. So, an equilateral triangle’s area can be calculated if the length of any one side of the triangle is known.
The area occupied between the sides of an equilateral triangle in a plane is calculated using the equilateral triangle area formula.
The formula for calculating the area of a triangle with a known base and height is:
Area = 1/2 × base × height
The following formula can be used to compute the area of an equilateral triangle:
Area = √3/4 × (side)2 square units
Perimeter of the Equilateral Triangle Formula
The perimeter of a triangle is equal to the sum of the length of its three sides, whether they are equal or not.
An equilateral triangle's perimeter is the sum of its three sides.
P = 3a is the basic formula for calculating the perimeter of an equilateral triangle, where 'a' denotes one of the triangle's sides. The sum becomes a + a + a = 3a since all three sides of an equilateral triangle are equal.
Height = √3a/ 2
Semi perimeter = (a + a + a)/2 = 3a/2
Formulas and Calculations for an Equilateral Triangle:
Perimeter of Equilateral Triangle: P = 3a
Semiperimeter of Equilateral Triangle Formula: s = 3a/2
Area of Equilateral Triangle Formula: K = (1/4) √3 a 2
The altitude of Equilateral Triangle Formula: h = (1/2) √3 a
Angles of Equilateral Triangle: A = B = C = 60 degrees
Sides of Equilateral Triangle: a equals b equals c.
Given the side of the triangle, find the perimeter, semiperimeter, area, and altitude.
a is known here; find P, s, K, h.
P equals 3a
s = 3a/2
K = (1/4) √3 a 2
h = (1/2) √3 a
Given theperimeter of the triangle , find the side, semiperimeter, area, altitude.
Perimeter(P) is known; find a, s, K, and h.
a = P/3
s = 3a/2
K = (1/4) √3 a 2
h = (1/2) √3 a
Given the semi perimeter of a triangle, find the side, perimeter, area, and altitude.
Semiperimeter (s) is known; find a, P, K, and h.
a = 2s/3
P = 3a
K = (1/4) √3 a 2
h = (1/2) √3 a
Given the area of the triangle find the side, perimeter, semiperimeter, and altitude.
K is known; find a, P, s and h.
a = √
(4/√3)∗K
(4/√3)∗K equals 2 √
K/√3
K/√3
P = 3a
s = 3a / 2
h = (1/2) √3 a
Given the altitude/height find the side, perimeter, semiperimeter, and area
Altitude (h) is known; find a, P, s, and K.
a = (2/√3) h
P = 3a
s = 3a/2
K = (1/4) √3 a 2
Solved Examples
Apply the equilateral triangle area formula and find the area of an equilateral triangle whose each side is 12 in.
Solution:
Side = 12 in
Applying the equilateral triangle area formula,
Area = √3/4 × (Side)2
= √3/4 × (12)2
= 36√3 in 2
Answer: Area of an equilateral triangle area 36√3 in 2
Calculate the perimeter and semi perimeter of an equilateral triangle with a side measurement of 12 units.
Solution:
The perimeter = 3a
Semi-perimeter = 3a/ 2
Given, side a = 12 units
Now, the perimeter of an equilateral triangle is equal to:
3 × 12 = 36 units
And, Semi-perimeter of an equilateral triangle is equal to:
36/2 = 18 units.
3.Suppose you have an equilateral triangle with a side of 5 cm. What will be the perimeter of the given equilateral triangle?
Solution) We know that the formula of the perimeter of an equilateral triangle is 3a.
Here, a = 5 cm
Therefore, Perimeter = 3 5 cm = 15 cm.
FAQs on Equilateral Triangle Formula
1. How do you define Napoleon’s Theorem?
If equilateral triangles are formed on the sides of any triangle, either all outward or all inward, the lines joining the centers of those equilateral triangles produce an equilateral triangle, according to Napoleon's theorem.
The result of this technique is the inner or outer Napoleon triangle. The size of the original triangle is equal to the area difference between the outer and inner Napoleon triangles.
2. Is the Pythagorean theorem applicable to equilateral triangles?
Equilateral triangles have three equal sides and angles that are all 60 degrees. Two right triangles are formed when a perpendicular bisector line is drawn through the vertex of an equilateral. To find the missing side lengths of an equilateral triangle, utilize the Pythagorean theorem and the height of right triangles within the equilateral triangle. The area of an equilateral triangle can therefore be calculated using the formula A = √3/4 (a²). Knowing how to calculate the height and area of a triangle with equal sides makes learning other trigonometric formulas much easier.
3. Is it possible for an equilateral triangle to be obtuse?
An equilateral triangle can’t be obtuse, because an equilateral triangle has equal sides and angles, each angle is sharp and measures 60 degrees. As a result, an equilateral angle cannot be obtuse.
A triangle can’t be both right-angled and obtuse-angled at the same time. As a right-angled triangle has only one right angle, the other two angles are acute. To summarise, an obtuse-angled triangle can never form a right angle and vice versa. The side opposite the obtuse angle of a triangle is the longest.
4. An equilateral triangle is always congruent. Explain.
For this, Consider two triangles ABC and DEF with sides AB and DE equal.
AB=DE
As they are equilateral, the other two pairs of sides are also equal.
BC=EF
AC=DF
∴△ABC≅△DEF by SSS congruence criteria.
Hence, two equilateral triangles with equal sides are always congruent.
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5993 | https://courses.lumenlearning.com/calculus3/chapter/spherical-coordinates/ | Spherical Coordinates | Calculus III
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Calculus III
Module 2: Vectors in Space
Search for:
Spherical Coordinates
Learning Objectives
Convert from spherical to rectangular coordinates.
Convert from rectangular to spherical coordinates.
In the Cartesian coordinate system, the location of a point in space is described using an ordered triple in which each coordinate represents a distance. In the cylindrical coordinate system, location of a point in space is described using two distances (r r and z z) and an angle measure (θ θ). In the spherical coordinate system, we again use an ordered triple to describe the location of a point in space. In this case, the triple describes one distance and two angles. Spherical coordinates make it simple to describe a sphere, just as cylindrical coordinates make it easy to describe a cylinder. Grid lines for spherical coordinates are based on angle measures, like those for polar coordinates.
DEFINITION
In thespherical coordinate system, a point P P in space (Figure 1) is represented by the ordered triple (ρ,θ,φ)(ρ,θ,φ) where
ρ ρ (the Greek letter rho) is the distance between P P and the origin (ρ≠0)(ρ≠0);
θ θ is the same angle used to describe the location in cylindrical coordinates;
φ φ (the Greek letter phi) is the angle formed by the positive z z-axis and line segment ¯¯¯¯¯¯¯¯O P O P¯, where O O is the origin and 0≤φ≤π 0≤φ≤π.
Figure 1. The relationship among spherical, rectangular, and cylindrical coordinates.
By convention, the origin is represented as (0,0,0)(0,0,0) in spherical coordinates.
THEOREM: converting among Spherical, cylindrical, and rectangular coordinates
Rectangular coordinates (x,y,z)(x,y,z) and spherical coordinates (ρ,θ,φ)(ρ,θ,φ) of a point are related as follows:
x=ρ sin φ cos θ These equations are used to convert from spherical coordinates to rectangular coordinates.y=ρ sin φ sin θ z=ρ cos φ and ρ 2=x 2+y 2+z 2 These equations are used to convert from rectangular coordinates to spherical coordinates.tan θ=y x φ=arccos(z√x 2+y 2+z 2)x=ρ sinφ cosθ These equations are used to convert from spherical coordinates to rectangular coordinates.y=ρ sinφ sinθ z=ρ cosφ and ρ 2=x 2+y 2+z 2 These equations are used to convert from rectangular coordinates to spherical coordinates.tanθ=y x φ=arccos(z x 2+y 2+z 2)
If a point has cylindrical coordinates (r,θ,z)(r,θ,z), then these equations define the relationship between cylindrical and spherical coordinates.
r=ρ sin φ These equations are used to convert from spherical coordinates to cylindrical coordinates θ=θ z=ρ cos φ and ρ=√r 2+z 2 These equations are used to convert from cylindrical coordinates to spherical coordinates θ=θ φ=arccos(z√r 2+z 2)r=ρ sinφ These equations are used to convert from spherical coordinates to cylindrical coordinates θ=θ z=ρ cosφ and ρ=r 2+z 2 These equations are used to convert from cylindrical coordinates to spherical coordinates θ=θ φ=arccos(z r 2+z 2)
The formulas to convert from spherical coordinates to rectangular coordinates may seem complex, but they are straightforward applications of trigonometry. Looking at Figure 2, it is easy to see that r=ρ cos φ r=ρ cosφ. Then, looking at the triangle in the x y x y-plane with r r as its hypotenuse, we have x=r cos θ=ρ sin φ cos θ x=r cosθ=ρ sinφ cosθ. The derivation of the formula for y y is similar.Figure 9 in Cylindrical Coordinatesalso shows that ρ 2=r 2+z 2=x 2+y 2+z 2 ρ 2=r 2+z 2=x 2+y 2+z 2 and z=ρ cos φ z=ρ cosφ. Solving this last equation for φ φ and then substituting ρ=√r 2+z 2 ρ=r 2+z 2 (from the first equation) yields φ=arccos(z√r 2+z 2)φ=arccos(z r 2+z 2). Also, note that, as before, we must be careful when using the formula tan θ=y x tanθ=y x to choose the correct value of θ θ.
Figure 2. The equations that convert from one system to another are derived from right-triangle relationships.
As we did with cylindrical coordinates, let’s consider the surfaces that are generated when each of the coordinates is held constant. Let c c be a constant, and consider surfaces of the form ρ=c ρ=c. Points on these surfaces are at a fixed distance from the origin and form a sphere. The coordinate θ θ in the spherical coordinate system is the same as in the cylindrical coordinate system, so surfaces of the form θ=c θ=c are half-planes, as before. Last, consider surfaces of the form φ=c φ=c. The points on these surfaces are at a fixed angle from the z z-axis and form a half-cone (Figure 3).
Figure 3. In spherical coordinates, surfaces of the form ρ=c ρ=c are spheres of radius ρ ρ (a), surfaces of the form θ=c θ=c are half-planes at an angle θ θ from the x x-axis (b), and surfaces of the form φ=c φ=c are half-cones at an angle φ φ from the z z-axis (c).
Example: converting from spherical coordinates
Plot the point with spherical coordinates (8,π 3,π 6)(8,π 3,π 6) and express its location in both rectangular and cylindrical coordinates.
Show Solution
Use the equations in Theorem: Converting among Spherical, Cylindrical, and Rectangular Coordinates to translate between spherical and cylindrical coordinates (Figure 4):
x=ρ sin φ cos θ=8 sin(π 6)cos(π 3)=8(1 2)1 2=2 y=ρ sin φ sin θ=8 sin(π 6)sin(π 3)=8(1 2)√3 2=2√3 z=ρ cos φ=8 cos(π 6)=8(√3 2)=4√3.x=ρ sinφ cosθ=8 sin(π 6)cos(π 3)=8(1 2)1 2=2 y=ρ sinφ sinθ=8 sin(π 6)sin(π 3)=8(1 2)3 2=2 3 z=ρ cosφ=8 cos(π 6)=8(3 2)=4 3.
Figure 4. The projection of the point in the x y x y-plane is 4 4 units from the origin. The line from the origin to the point’s projection forms an angle of π 3 π 3 with the positive x x-axis. The point lies 4√3 4 3 units above the x y x y-plane.
The point with spherical coordinates (8,π 3,π 6)(8,π 3,π 6) has rectangular coordinates (2,2√3,4√3)(2,2 3,4 3).
Finding the values in cylindrical coordinates is equally straightforward:
r=ρ sin φ=8 sin π 6=4 θ=θ z=ρ cos φ=8 cos π 6=4√3 r=ρ sinφ=8 sinπ 6=4 θ=θ z=ρ cosφ=8 cosπ 6=4 3.
Thus, cylindrical coordinates for the point are (4,π 3,4√3)(4,π 3,4 3).
try it
Plot the point with spherical coordinates (2,−5 π 6,π 6)(2,−5 π 6,π 6) and describe its location in both rectangular and cylindrical coordinates.
Show Solution
Figure 5. The point with spherical coordinates (2,−5 π 6,π 6)(2,−5 π 6,π 6)
Cartesian: (−√3 2,−1 2,√3)(−3 2,−1 2,3), cylindrical: (1,−5 π 6,√3)(1,−5 π 6,3)
Watch the following video to see the worked solution to the above Try IT.
You can view the transcript for “CP 2.58” here (opens in new window).
Example: converting from rectangular coordinates
Convert the rectangular coordinates (−1,1,√6)(−1,1,6) to both spherical and cylindrical coordinates.
Show Solution
Start by converting from rectangular to spherical coordinates:
ρ 2=x 2+y 2+z 2=(−1)2+1 2+(√6)2=8 tan θ=1−1 ρ=2√2 θ=arctan(−1)=3 π 4 ρ 2=x 2+y 2+z 2=(−1)2+1 2+(6)2=8 tanθ=1−1 ρ=2 2 θ=arctan(−1)=3 π 4
Because (x,y)=(−1,1)(x,y)=(−1,1), then the correct choice for θ θ is 3 π 4 3 π 4.
There are actually two ways to identify φ φ. We can use the equation φ=arccos(z√x 2+y 2+z 2)φ=arccos(z x 2+y 2+z 2). A more simple approach, however, is to use equation z=ρ cos φ z=ρ cosφ. We know that z=√6 z=6 and ρ=2√2 ρ=2 2, so
√6=2√2 cos φ, so cos φ=√6 2√2=√3 2 6=2 2 cosφ, so cosφ=6 2 2=3 2
and therefore φ=π 6 φ=π 6. The spherical coordinates of the point are (2√2,3 π 4,π 6)(2 2,3 π 4,π 6).
To find the cylindrical coordinates for the point, we need only find r r:
r=ρ sin φ=2√2 sin(π 6)=√2 r=ρ sinφ=2 2 sin(π 6)=2.
The cylindrical coordinates for the point are (√2,3 π 4,√6)(2,3 π 4,6).
Example: identifying surfaces in the spherical coordinate system
Describe the surfaces with the given spherical equations.
θ=π 3 θ=π 3
φ=5 π 6 φ=5 π 6
ρ=6 ρ=6
ρ=sin θ sin φ ρ=sinθ sinφ
Show Solution
The variable θ θ represents the measure of the same angle in both the cylindrical and spherical coordinate systems. Points with coordinates (ρ,π 3,φ)(ρ,π 3,φ) lie on the plane that forms angle θ=π 3 θ=π 3 with the positive x x-axis. Because ρ>0 ρ>0, the surface described by equation θ=π 3 θ=π 3 is the half-plane shown in Figure 6.
Figure 6. The surface described by equation θ=π 3 θ=π 3 is a half-plane.
2. Equation φ=5 π 6 φ=5 π 6 describes all points in the spherical coordinate system that lie on a line from the origin forming an angle measuring 5 π 6 5 π 6 rad with the positive z z-axis. These points form a half-cone (Figure 7). Because there is only one value for φ φ that is measured from the positive z z-axis, we do not get the full cone (with two pieces).
Figure 7. The equation φ=5 π 6 φ=5 π 6 describes a cone.
To find the equation in rectangular coordinates, use equation φ=arccos(z√x 2+y 2+z 2)φ=arccos(z x 2+y 2+z 2).
5 π 6=arccos(z√x 2+y 2+z 2)cos 5 π 6=z√x 2+y 2+z 2−√3 2=z√x 2+y 2+z 2 3 4=z√x 2+y 2+z 2 3 x 2 4+3 y 2 4+3 z 2 4=z 2 3 x 2 4+3 y 2 4−z 2 4=0.5 π 6=arccos(z x 2+y 2+z 2)cos5 π 6=z x 2+y 2+z 2−3 2=z x 2+y 2+z 2 3 4=z x 2+y 2+z 2 3 x 2 4+3 y 2 4+3 z 2 4=z 2 3 x 2 4+3 y 2 4−z 2 4=0.
This is the equation of a cone centered on the z z-axis.
Equation ρ=6 ρ=6 describes the set of all points 6 6 units away from the origin—a sphere with radius 6 6 (Figure 8).
Figure 8. Equation ρ=6 ρ=6 describes a sphere with radius 6 6.
To identify this surface, convert the equation from spherical to rectangular coordinates, using equations y=ρ sin φ sin θ y=ρ sinφ sinθ and ρ 2=x 2+y 2+z 2 ρ 2=x 2+y 2+z 2:
ρ=sin θ sin φ ρ 2=ρ sin θ sin φ Multiply both sides of the equation by φ x 2+y 2+z 2=y Substitute rectangular variables using the equations above.x 2+y 2−y+z 2=0 Subtract y from both sides of the equation.x 2+y 2−y+1 4+z 2=1 4 Complete the square.x 2+(y−1 2)2+z 2=1 4 Rewrite the middle terms as a perfect square.ρ=sinθ sinφ ρ 2=ρ sinθ sinφ Multiply both sides of the equation by φ x 2+y 2+z 2=y Substitute rectangular variables using the equations above.x 2+y 2−y+z 2=0 Subtract y from both sides of the equation.x 2+y 2−y+1 4+z 2=1 4 Complete the square.x 2+(y−1 2)2+z 2=1 4 Rewrite the middle terms as a perfect square.
The equation describes a sphere centered at point (0,1 2,0)(0,1 2,0) with radius 1 2 1 2.
try it
Describe the surfaces defined by the following equations.
ρ=13 ρ=13
θ=2 π 3 θ=2 π 3
φ=π 4 φ=π 4
Show Solution
a. This is the set of all points 13 13 units from the origin. This set forms a sphere with radius 13 13.
b. This set of points forms a half plane. The angle between the half plane and the positive x x-axis is θ=2 π 3 θ=2 π 3.
c. Let P P be a point on this surface. The position vector of this point forms an angle of φ=π 4 φ=π 4 with the positive z z-axis, which means that points closer to the origin are closer to the axis. These points form a half-cone.
Spherical coordinates are useful in analyzing systems that have some degree of symmetry about a point, such as the volume of the space inside a domed stadium or wind speeds in a planet’s atmosphere. A sphere that has Cartesian equation x 2+y 2+z 2=c 2 x 2+y 2+z 2=c 2 has the simple equation ρ=c ρ=c in spherical coordinates.
In geography, latitude and longitude are used to describe locations on Earth’s surface, as shown in Figure 9. Although the shape of Earth is not a perfect sphere, we use spherical coordinates to communicate the locations of points on Earth. Let’s assume Earth has the shape of a sphere with radius 4000 mi. We express angle measures in degrees rather than radians because latitude and longitude are measured in degrees.
Figure 9. In the latitude–longitude system, angles describe the location of a point on Earth relative to the equator and the prime meridian.
Let the center of Earth be the center of the sphere, with the ray from the center through the North Pole representing the positive z z-axis. The prime meridian represents the trace of the surface as it intersects the x z x z-plane. The equator is the trace of the sphere intersecting the x y x y-plane.
Example: converting latitude and longitude to spherical coordinates
The latitude of Columbus, Ohio, is 40∘40∘ N and the longitude is 83∘83∘ W, which means that Columbus is 40∘40∘ north of the equator. Imagine a ray from the center of Earth through Columbus and a ray from the center of Earth through the equator directly south of Columbus. The measure of the angle formed by the rays is 40∘40∘. In the same way, measuring from the prime meridian, Columbus lies 83∘83∘ to the west. Express the location of Columbus in spherical coordinates.
Show Solution
The radius of Earth is 4,000 4,000 mi, so ρ=4,000 ρ=4,000. The intersection of the prime meridian and the equator lies on the positive x x-axis. Movement to the west is then described with negative angle measures, which shows that θ=−83∘θ=−83∘, Because Columbus lies 40∘40∘ north of the equator, it lies 50∘50∘ south of the North Pole, so φ−50∘φ−50∘. In spherical coordinates, Columbus lies at point (4,000,−83∘,50∘)(4,000,−83∘,50∘).
try it
Sydney, Australia is at 34∘34∘S and 151∘151∘E. Express Sydney’s location in spherical coordinates.
Show Solution
(4,000,151∘,124∘)(4,000,151∘,124∘)
Cylindrical and spherical coordinates give us the flexibility to select a coordinate system appropriate to the problem at hand. A thoughtful choice of coordinate system can make a problem much easier to solve, whereas a poor choice can lead to unnecessarily complex calculations. In the following example, we examine several different problems and discuss how to select the best coordinate system for each one.
Example: choosing the best coordinate system
In each of the following situations, we determine which coordinate system is most appropriate and describe how we would orient the coordinate axes. There could be more than one right answer for how the axes should be oriented, but we select an orientation that makes sense in the context of the problem.Note: There is not enough information to set up or solve these problems; we simply select the coordinate system (Figure 10).
Find the center of gravity of a bowling ball.
Determine the velocity of a submarine subjected to an ocean current.
Calculate the pressure in a conical water tank.
Find the volume of oil flowing through a pipeline.
Determine the amount of leather required to make a football.
Figure 10. (credit: (a) modification of work by scl hua, Wikimedia, (b) modification of work by DVIDSHUB, Flickr, (c) modification of work by Michael Malak, Wikimedia, (d) modification of work by Sean Mack, Wikimedia, (e) modification of work by Elvert Barnes, Flickr)
Show Solution
Clearly, a bowling ball is a sphere, so spherical coordinates would probably work best here. The origin should be located at the physical center of the ball. There is no obvious choice for how the x x-, y y– and z z-axes should be oriented. Bowling balls normally have a weight block in the center. One possible choice is to align the z z-axis with the axis of symmetry of the weight block.
A submarine generally moves in a straight line. There is no rotational or spherical symmetry that applies in this situation, so rectangular coordinates are a good choice. The z z-axis should probably point upward. The x x– and y y-axes could be aligned to point east and north, respectively. The origin should be some convenient physical location, such as the starting position of the submarine or the location of a particular port.
A cone has several kinds of symmetry. In cylindrical coordinates, a cone can be represented by equation z=k r z=k r, where k k is a constant. In spherical coordinates, we have seen that surfaces of the form φ=c φ=c are half-cones. Last, in rectangular coordinates, elliptic cones are quadric surfaces and can be represented by equations of the form z 2=x 2 a 2+y 2 b 2 z 2=x 2 a 2+y 2 b 2. In this case, we could choose any of the three. However, the equation for the surface is more complicated in rectangular coordinates than in the other two systems, so we might want to avoid that choice. In addition, we are talking about a water tank, and the depth of the water might come into play at some point in our calculations, so it might be nice to have a component that represents height and depth directly. Based on this reasoning, cylindrical coordinates might be the best choice. Choose the z z-axis to align with the axis of the cone. The orientation of the other two axes is arbitrary. The origin should be the bottom point of the cone.
A pipeline is a cylinder, so cylindrical coordinates would be best the best choice. In this case, however, we would likely choose to orient our z z-axis with the center axis of the pipeline. The x x-axis could be chosen to point straight downward or to some other logical direction. The origin should be chosen based on the problem statement. Note that this puts the z z-axis in a horizontal orientation, which is a little different from what we usually do. It may make sense to choose an unusual orientation for the axes if it makes sense for the problem.
A football has rotational symmetry about a central axis, so cylindrical coordinates would work best. The z z-axis should align with the axis of the ball. The origin could be the center of the ball or perhaps one of the ends. The position of the x x-axis is arbitrary.
Watch the following video to see the worked solution to Example: Choosing the Best Coordinate System.
You can view the transcript for “Ex 2.67” here (opens in new window).
try it
Which coordinate system is most appropriate for creating a star map, as viewed from Earth (see the following figure)?
Figure 11. Star map as viewed from Earth.
How should we orient the coordinate axes?
Show Solution
Spherical coordinates with the origin located at the center of the earth, the z z-axis aligned with the North Pole, and the x x-axis aligned with the prime meridian.
Candela Citations
CC licensed content, Original
CP 2.58. Authored by: Ryan Melton. License: CC BY: Attribution
CP 2.67. Authored by: Ryan Melton. License: CC BY: Attribution
CC licensed content, Shared previously
Calculus Volume 3. Authored by: Gilbert Strang, Edwin (Jed) Herman. Provided by: OpenStax. Located at: License: CC BY-NC-SA: Attribution-NonCommercial-ShareAlike. License Terms: Access for free at
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CC licensed content, Original
CP 2.58. Authored by: Ryan Melton. License: CC BY: Attribution
CP 2.67. Authored by: Ryan Melton. License: CC BY: Attribution
CC licensed content, Shared previously
Calculus Volume 3. Authored by: Gilbert Strang, Edwin (Jed) Herman. Provided by: OpenStax. Located at: License: CC BY-NC-SA: Attribution-NonCommercial-ShareAlike. License Terms: Access for free at
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compound Summary
(2H8)Styrene
PubChem CID
88025
Structure
Primary Hazards
Laboratory Chemical Safety Summary (LCSS) Datasheet
Molecular Formula
C 8 H 8
Synonyms
19361-62-7
(2H8)Styrene
Benzene-d5, ethenyl-d3-
EINECS 242-995-6
DTXSID601015444
View More...
Molecular Weight
112.20 g/mol
Computed by PubChem 2.2 (PubChem release 2025.09.15)
Dates
Create: 2005-03-27
Modify: 2025-09-20
1 Structures
1.1 2D Structure
Structure Search
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Chemical Structure Depiction
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1.2 3D Conformer
Structure Search
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PubChem
2 Names and Identifiers
2.1 Computed Descriptors
2.1.1 IUPAC Name
1,2,3,4,5-pentadeuterio-6-(1,2,2-trideuterioethenyl)benzene
Computed by Lexichem TK 2.9.3 (PubChem release 2025.09.15)
PubChem
2.1.2 InChI
InChI=1S/C8H8/c1-2-8-6-4-3-5-7-8/h2-7H,1H2/i1D2,2D,3D,4D,5D,6D,7D
Computed by InChI 1.07.4 (PubChem release 2025.09.15)
PubChem
2.1.3 InChIKey
PPBRXRYQALVLMV-GDALLCCDSA-N
Computed by InChI 1.07.4 (PubChem release 2025.09.15)
PubChem
2.1.4 SMILES
[2H]C1=C(C(=C(C(=C1[2H])[2H])C(=C([2H])[2H])[2H])[2H])[2H]
Computed by OEChem 4.2.0 (PubChem release 2025.09.15)
PubChem
2.2 Molecular Formula
C 8 H 8
Computed by PubChem 2.2 (PubChem release 2025.09.15)
PubChem
2.3 Other Identifiers
2.3.1 CAS
19361-62-7
CAS Common Chemistry; ChemIDplus; EPA Chemicals under the TSCA; EPA DSSTox; European Chemicals Agency (ECHA); New Zealand Environmental Protection Authority (EPA)
2.3.2 Related CAS
27732-42-9
Compound: Benzene-1,2,3,4,5-d5, 6-(ethenyl-1,2,2-d3)-, homopolymer
CAS Common Chemistry
2.3.3 European Community (EC) Number
242-995-6
European Chemicals Agency (ECHA)
2.3.4 DSSTox Substance ID
DTXSID601015444
EPA DSSTox
2.3.5 Nikkaji Number
J644.067C
Japan Chemical Substance Dictionary (Nikkaji)
2.4 Synonyms
2.4.1 Depositor-Supplied Synonyms
19361-62-7
(2H8)Styrene
Benzene-d5, ethenyl-d3-
EINECS 242-995-6
DTXSID601015444
Benzene-1,2,3,4,5-d5, 6-(ethenyl-1,2,2-d3)-
RefChem:396739
DTXCID501473728
242-995-6
Styrene-d8
Styrene D8
27732-42-9
1,2,3,4,5-pentadeuterio-6-(1,2,2-trideuterioethenyl)benzene
Styrene-d8 (Stabilized with Hydroquinone)
MFCD00044231
MFCD00145139
Styrene D8 100 microg/mL in Methanol
STYRENE (D8, 98%) + ~50 PPM BHT
1,2,3,4,5-pentadeuterio-6-(1,2,2-trideuteriovinyl)benzene
Styrene-d8, stab.
Benzene-d5-, ethenyl-d3-
orb2299672
SCHEMBL3480734
MSK9075D8
PPBRXRYQALVLMV-GDALLCCDSA-N
AKOS015889099
AS-86115
SY061549
Styrene-D8, polymerised. >98 Atom % D
1-(1,2,2-?H?)ethenylbenzene
D98578
A935667
BENZENE-1,2,3,4,5-D5,6-(ETHENYL-1,2,2-D3)-
Styrene-d8, >=98 atom % D, >=98% (CP), contains 4-t-butylcatechol as stabilizer
PubChem
3 Chemical and Physical Properties
3.1 Computed Properties
Property Name
Property Value
Reference
Property Name
Molecular Weight
Property Value
112.20 g/mol
Reference
Computed by PubChem 2.2 (PubChem release 2025.09.15)
Property Name
XLogP3
Property Value
2.9
Reference
Computed by XLogP3 3.0 (PubChem release 2025.09.15)
Property Name
Hydrogen Bond Donor Count
Property Value
0
Reference
Computed by Cactvs 3.4.8.24 (PubChem release 2025.09.15)
Property Name
Hydrogen Bond Acceptor Count
Property Value
0
Reference
Computed by Cactvs 3.4.8.24 (PubChem release 2025.09.15)
Property Name
Rotatable Bond Count
Property Value
1
Reference
Computed by Cactvs 3.4.8.24 (PubChem release 2025.09.15)
Property Name
Exact Mass
Property Value
112.112814223 Da
Reference
Computed by PubChem 2.2 (PubChem release 2025.09.15)
Property Name
Monoisotopic Mass
Property Value
112.112814223 Da
Reference
Computed by PubChem 2.2 (PubChem release 2025.09.15)
Property Name
Topological Polar Surface Area
Property Value
0 Ų
Reference
Computed by Cactvs 3.4.8.24 (PubChem release 2025.09.15)
Property Name
Heavy Atom Count
Property Value
8
Reference
Computed by PubChem
Property Name
Formal Charge
Property Value
0
Reference
Computed by PubChem
Property Name
Complexity
Property Value
68.1
Reference
Computed by Cactvs 3.4.8.24 (PubChem release 2025.09.15)
Property Name
Isotope Atom Count
Property Value
8
Reference
Computed by PubChem
Property Name
Defined Atom Stereocenter Count
Property Value
0
Reference
Computed by PubChem
Property Name
Undefined Atom Stereocenter Count
Property Value
0
Reference
Computed by PubChem
Property Name
Defined Bond Stereocenter Count
Property Value
0
Reference
Computed by PubChem
Property Name
Undefined Bond Stereocenter Count
Property Value
0
Reference
Computed by PubChem
Property Name
Covalently-Bonded Unit Count
Property Value
1
Reference
Computed by PubChem
Property Name
Compound Is Canonicalized
Property Value
Yes
Reference
Computed by PubChem (release 2025.09.15)
PubChem
4 Related Records
4.1 Related Compounds with Annotation
Follow these links to do a live 2D search or do a live 3D search for this compound, sorted by annotation score. This section is deprecated (see the neighbor discontinuation help page for details), but these live search links provide equivalent functionality to the table that was previously shown here.
PubChem
4.2 Related Compounds
Same Connectivity Count
28
Same Parent, Connectivity Count
546
Mixtures, Components, and Neutralized Forms Count
2
Similar Compounds (2D)
View in PubChem Search
Similar Conformers (3D)
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PubChem
4.3 Substances
4.3.1 PubChem Reference Collection SID
505392469
PubChem
4.3.2 Related Substances
All Count
85
Same Count
83
Mixture Count
2
PubChem
4.3.3 Substances by Category
PubChem
4.4 Entrez Crosslinks
PubMed Count
3
PubChem
4.5 NCBI LinkOut
NCBI
5 Chemical Vendors
PubChem
6 Use and Manufacturing
6.1 General Manufacturing Information
EPA TSCA Commercial Activity Status
Benzene-1,2,3,4,5-d5, 6-(ethenyl-1,2,2-d3)-: ACTIVE
EPA Chemicals under the TSCA
7 Safety and Hazards
7.1 Hazards Identification
7.1.1 GHS Classification
Pictogram(s)
Signal
Warning
GHS Hazard Statements
H226 (97.9%): Flammable liquid and vapor [Warning Flammable liquids]
H315 (95.7%): Causes skin irritation [Warning Skin corrosion/irritation]
H319 (95.7%): Causes serious eye irritation [Warning Serious eye damage/eye irritation]
H332 (97.9%): Harmful if inhaled [Warning Acute toxicity, inhalation]
Precautionary Statement Codes
P210, P233, P240, P241, P242, P243, P261, P264, P264+P265, P271, P280, P302+P352, P303+P361+P353, P304+P340, P305+P351+P338, P317, P321, P332+P317, P337+P317, P362+P364, P370+P378, P403+P235, and P501
ECHA C&L Notifications Summary
Aggregated GHS information provided per 47 reports by companies from 5 notifications to the ECHA C&L Inventory. Each notification may be associated with multiple companies.
Information may vary between notifications depending on impurities, additives, and other factors. The percentage value in parenthesis indicates the notified classification ratio from companies that provide hazard codes. Only hazard codes with percentage values above 10% are shown. For more detailed information, please visit ECHA C&L website.
European Chemicals Agency (ECHA)
7.1.2 Hazard Classes and Categories
Flam. Liq. 3 (97.9%)
Skin Irrit. 2 (95.7%)
Eye Irrit. 2 (95.7%)
Acute Tox. 4 (97.9%)
European Chemicals Agency (ECHA)
7.2 Regulatory Information
New Zealand EPA Inventory of Chemical Status
Benzene-d5, ethenyl-d3-: Does not have an individual approval but may be used under an appropriate group standard
New Zealand Environmental Protection Authority (EPA)
8 Literature
8.1 Consolidated References
PubChem
8.2 Springer Nature References
Springer Nature
8.3 Thieme References
Thieme Chemistry
8.4 Chemical Co-Occurrences in Literature
PubChem
8.5 Chemical-Gene Co-Occurrences in Literature
PubChem
8.6 Chemical-Disease Co-Occurrences in Literature
PubChem
8.7 Chemical-Organism Co-Occurrences in Literature
PubChem
9 Patents
9.1 Depositor-Supplied Patent Identifiers
PubChem
Link to all deposited patent identifiers
PubChem
9.2 WIPO PATENTSCOPE
Patents are available for this chemical structure:
PATENTSCOPE (WIPO)
9.3 Chemical Co-Occurrences in Patents
PubChem
9.4 Chemical-Organism Co-Occurrences in Patents
PubChem
10 Classification
10.1 ChemIDplus
ChemIDplus
10.2 UN GHS Classification
GHS Classification (UNECE)
10.3 EPA DSSTox Classification
EPA DSSTox
10.4 EPA TSCA and CDR Classification
EPA Chemicals under the TSCA
10.5 MolGenie Organic Chemistry Ontology
MolGenie
10.6 Chemicals in PubChem from Regulatory Sources
PubChem
11 Information Sources
Filter by Source
CAS Common ChemistryLICENSE The data from CAS Common Chemistry is provided under a CC-BY-NC 4.0 license, unless otherwise stated. Benzene-1,2,3,4,5-d5, 6-(ethenyl-1,2,2-d3)-, homopolymer 6-(Ethenyl-1,2,2-d3)benzene-1,2,3,4,5-d5
ChemIDplusLICENSE Benzene-1,2,3,4,5-d5, 6-(ethenyl-1,2,2-d3)- ChemIDplus Chemical Information Classification
EPA Chemicals under the TSCALICENSE Benzene-1,2,3,4,5-d5, 6-(ethenyl-1,2,2-d3)- EPA TSCA Classification
EPA DSSToxLICENSE (2H8)Styrene CompTox Chemicals Dashboard Chemical Lists
European Chemicals Agency (ECHA)LICENSE Use of the information, documents and data from the ECHA website is subject to the terms and conditions of this Legal Notice, and subject to other binding limitations provided for under applicable law, the information, documents and data made available on the ECHA website may be reproduced, distributed and/or used, totally or in part, for non-commercial purposes provided that ECHA is acknowledged as the source: "Source: European Chemicals Agency, Such acknowledgement must be included in each copy of the material. ECHA permits and encourages organisations and individuals to create links to the ECHA website under the following cumulative conditions: Links can only be made to webpages that provide a link to the Legal Notice page. (2H8)styrene (2H8)styrene (EC: 242-995-6)
New Zealand Environmental Protection Authority (EPA)LICENSE This work is licensed under the Creative Commons Attribution-ShareAlike 4.0 International licence. Benzene-d5, ethenyl-d3-
Japan Chemical Substance Dictionary (Nikkaji)
Springer Nature
Thieme ChemistryLICENSE The Thieme Chemistry contribution within PubChem is provided under a CC-BY-NC-ND 4.0 license, unless otherwise stated.
PubChem Chemicals in PubChem from Regulatory Sources
GHS Classification (UNECE)GHS Classification
MolGenieLICENSE CC-BY 4.0 MolGenie Organic Chemistry Ontology
PATENTSCOPE (WIPO)SID 391132537
NCBI
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CONTENTS
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ArticlesPhysiologyNervous systemOverview of the nervous systemOligodendrocytes
Oligodendrocytes
Author: Christos Bozidis • Reviewer: Assist. Prof. Yannis V. Simos Last reviewed: January 29, 2025 Reading time: 14 minutes
Oligodendrocyte
Oligodendrocytus
Synonyms: none
For the human body to survive and function properly, the nervous system continuously interprets external signals and generates appropriate responses. Efficient signal transmission between neurons is crucial, and this rapid and uninterrupted communication is facilitated by the myelination of nerve fibers.
This article focuses on the structure and physiology of oligodendrocytes, the glial cells responsible for myelination within the central nervous system (CNS), and explores their essential role in maintaining nervous system homeostasis.
Key facts about the oligodendrocytes
| Location | CNS, mainly in the white matter |
| Origin | Oligodendrocyte precursor cells derived from the neural tube |
| Function | Round nucleusSmall cell bodyMultiple processes that extend to form myelin sheaths on axons |
| Differences between CNS and PNS myelination | CNS1. Each oligodendrocyte myelinates multiple axons making up to 50 myelin sheaths.2. The oligodendrocyte cell body does not attach to the axon.3. Axon support, nutrition and moderation comes from the extracellular space and is controlled by glia.PNS1. Every Schwann cell produces one myelin sheath.2. The Schwann cell body is wrapped around the axon.3. Axon support, nutrition and moderation is provided by connective tissue and basal lamina. |
Contents
Definition and general information
Location and Origin
Structure
Function and Physiology
Myelination
Differences between CNS and PNS myelination
The significance of oligodendroglia in the CNS
Myelination-irrelevant functions
Clinical notes
Demyelinating diseases
Other clinical significance
Sources
Show all
Definition and general information
The oligodendrocytes form myelin sheaths around axons in the CNS enhancing and insulating signal transduction between nerve cells. The name oligodendrocyte is derived from the Greek words "oligo" (meaning small), "dendro" (meaning tree), and "cyte" (meaning cell), which together translate to "small tree-like cell" and reflect the cell's appearance.
The oligodendrocytes, also called oligodendroglia, are part of the neuroglia, the supporting cells of the nervous system. More accurately, this cell type belongs to the subcategory of the macroglia, along with astrocytes and ependymal cells, due to their common origin and similarity in structure and location. On the contrary, there is microglia that makes up the rest of the neuroglia.
Location and Origin
Oligodendrocytes can be found throughout the CNS but they are present in exponentially higher numbers in the white matter where they cover the majority of the axons, producing myelin that gives white matter its distinct pale color.
During development, oligodendroglia is the last glial cell type to appear in the nervous tissue. Like the rest of the macroglia, its origin traces to the neural tube. The precursor cells of the macroglia are called glioblasts. In the case of oligodendroglia the progenitor cells are the oligodendrocyte precursor cells (OPCs) that migrate and spread across gray and white matter, at the same time differentiating into oligodendrocytes. The process of migration is completed a few weeks after birth. Before completely differentiating into oligodendroglia, OPCs become immature oligodendrocytes that express all the corresponding factors but are yet to start myelinating.
Structure
Oligodendrocyte
Oligodendrocytus
1/2
Synonyms: none
The oligodendroglia is among the largest cell populations in the CNS, consisting of cells that are in general smaller in size than astrocytes and bigger than microglia. These cells have a round and dense nucleus, surrounded by a small volume of cytoplasm with multiple processes that do not branch, rather extend and wrap around adjacent axons. A single oligodendrocyte can form processes to support the creation of up to 50 myelin sheaths. Multiple mitochondria and microtubules can be seen using electron microscopy due to the extended cytoskeleton needed to support the wrapping processes. Moreover an extended smooth endoplasmic reticulum and Golgi apparatus can be also observed that reflect the increased production of the different lipids and proteins myelin consists of.
Function and Physiology
The primary role of oligodendrocytes is axon myelination. However, like other glial cells, their functions extend beyond this, including helping maintain a stable microenvironment for neurons and playing a role in tissue repair.
Myelination
Each axon in the human body achieves myelination through a series of consecutive myelin sheaths separated by thin gaps known as the nodes of Ranvier. Each myelin sheath consists of multiple layers of the oligodendrocyte’s cytoplasmic membrane, which wraps around the axolemma, or axonal cell membrane. In the CNS, oligodendrocytes are responsible for creating these myelin sheaths, while Schwann cells perform the same role in the peripheral nervous system (PNS).
To form a myelin sheath, an oligodendrocyte extends a cellular process toward an unmyelinated axon, which wraps around it, making contact with the axolemma and forming a loop known as the inner mesaxon. This wrapping continues in successive loops until multiple layers surround the axon. As this occurs, the cytoplasm within the process is gradually squeezed back toward the glial cell body, leading to the thinning and tightening of the layers around the axon. The resulting layers are tightly compressed, producing structures known as the major dense lines and intraperiod lines, visible under electron microscopy.
The stability of this sheath structure is reinforced by tight junctions—termed autotypic junctions when between glial layers and heterotypic junctions between the glial membrane and the axolemma. Additionally, transmembrane proteins, such as proteolipid protein in the CNS, provide further resilience and integrity to the sheath, ensuring effective insulation and signal transmission along the axon.
Differences between CNS and PNS myelination
While the basic principle of forming a myelin sheath remains similar in the central and peripheral nervous systems, notable distinctions exist between the roles and mechanisms of oligodendrocytes in the CNS and Schwann cells in the PNS. In the CNS, each oligodendrocyte extends multiple processes, enabling it to support up to 50 separate myelin sheaths, which may encase the same or various axons. Consequently, the oligodendrocyte cell body and nucleus remain distant from the sheaths it forms. In contrast, each Schwann cell in the PNS creates only a single myelin sheath, staying closely associated with it by attaching its cell body and nucleus directly to the axolemma, eliminating the need for extended processes.
Structural and nutritional support also differ significantly. In the PNS, connective tissue and a basal lamina provide necessary support to the myelin sheaths. However, in the CNS, support relies on the stable chemical environment maintained by astrocytes, which regulate ion and molecular concentrations in the extracellular fluid. Astrocytes also extend their specialized end-foot processes to the nodes of Ranvier, where they influence ion balance and support the highly concentrated ion channels essential for neural conduction.
Learn all about the types of glial cells and their functions with this study unit:
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The significance of oligodendroglia in the CNS
The fact that most of the neuronal pathways in the human brain and spinal cord comprise myelinated axons is the reason our body is able to respond in time to new stimuli. When assessing the effects of myelination in the CNS due to the presence of the oligodendroglia, two main benefits arise for signal transmission
Insulation. Myelin consists mainly of lipids (galactocerebroside, sphingomyelin and cholesterol), connected to special proteins that stabilize its form. This unique composition along with the increased thickness of the multiple-layer sheaths insulates the axon from the extracellular space. Consequently, the leakage of ions is significantly limited compared to unmyelinated axons. This insulation provides the basis for greater precision in signaling, with less energy needed, allowing for more complex neuronal circuits.
Increased conduction speed. The consecution of myelin sheaths (called internodes) separated by the nodes of Ravier forms an axon wrap with alternating insulated and uninsulated regions. This arrangement leads to a phenomenon called “saltatory conduction”. When an action potential is initiated in an unmyelinated axon, it triggers nearby voltage gated channels that in turn activate to propagate the action potential along the axon. It is clear that in an unmyelinated axon the process of action potential propagation includes every part of the axon. On the other hand, the action potential in myelinated axons is propagated only between the nodes of Ravier, “jumping” from node to node and “skipping” the myelinated regions. This is made possible due to the increased concentration of voltage-gated channels at the nodes as well as the insulation of myelin that ensures the continuation of the intracellular ion current. This saltatory conduction of the myelinated axons results in significantly greater speed of signal transmission compared to unmyelinated axons and has made the survival of large multicellular organisms possible, laying the foundation for superior brain functions.
Nutrition and support. Oligodendrocytes, through myelin, provide to the axons nutrients, regulate ions and molecular levels, and enhance cytoskeletal function, strengthening the axon’s structural integrity.
Myelination-irrelevant functions
A percentage of oligodendrocytes does not show active myelinating activity, though being completely differentiated. These cells are classified as satellite oligodendrocytes and they are not attached via myelin to axons. They are located in the gray matter and their properties are related to the regulation of nervous tissue microenvironment and the replacement of other dysfunctional oligodendrocytes.
Furthermore, oligodendroglia generally plays a major role in neuronal metabolism regulation, expressing growth factors like glial cell line-derived neurotrophic factor (GDNF), or brain-derived neurotrophic factor (BDNF), thus upregulating neuronal growth when needed. Other functions of the oligodendroglia related to damage control and neuronal plasticity are a subject of ongoing studies and are yet to be completely defined.
Test yourself on the oligodendrocytes with this quiz!
Clinical notes
In clinical contexts, oligodendrocytes play a critical role in understanding the pathology of various neurological diseases, particularly demyelinating diseases and neurodegenerative conditions.
Demyelinating diseases
Demyelinating diseases are characterized by the degradation of myelin sheaths, leading to compromised nerve signal transmission. This deterioration often results from the failure of oligodendrocytes to effectively replace destroyed myelin, impairing neural communication and resulting in various neurological symptoms.
Immune-Mediated Diseases (Autoimmune Disorders). In certain autoimmune diseases, the immune system mistakenly targets myelin sheaths of the CNS, resulting in their destruction. Multiple sclerosis (MS) is a common example, where immune cells such as T cells, B cells, and macrophages attack CNS tissue. They release cytokines and other proinflammatory molecules, leading to increased vascular permeability and sustained inflammation. This inflammation, in turn, triggers oligodendrocyte apoptosis, stripping axons of their myelin sheaths and disturbing the structure of the nervous tissue. Astrocytes often respond by becoming hyperactive, exacerbating tissue disruption. In MS, hallmark pathological findings include plaques or lesions with inflammatory cells, disrupted astrocytes, and areas devoid of myelin. These lesions commonly affect the brain, spinal cord, and optic nerve, leading to the symptoms typical of MS.
Metabolic Diseases.Certain metabolic conditions impair myelin maintenance due to nutritional deficiencies or imbalances. For example, a deficiency in vitamin B12 can lead to demyelination, as this vitamin is essential for myelin synthesis and maintenance.
Genetic Diseases. Genetic mutations affecting oligodendrocytes can lead to various forms of demyelination due to disrupted myelin formation or toxicity within the CNS, like adrenoleukodystrophy, stemming from mutations in the ABCD1 gene that leads to the accumulation of long-chain fatty acids, which exert toxic effects on myelin. Pelizaeus-Merzbacher Disease is another example, with mutations affecting the production or function of myelin proteins, leading to either a reduction in functional oligodendrocytes and failure to form stable myelin sheaths.
Other clinical significance
Beyond demyelinating disorders, oligodendrocytes are increasingly recognized as key players in the pathology of neurodegenerative diseases, like Alzheimer’s and Parkinson’s disease. Their roles can vary, sometimes providing neuroprotective support, while in other cases contributing to disease progression. Additionally, oligodendroglioma represents a notable malignancy within the CNS. This aggressive tumor arises from oligodendrocyte precursors, illustrating how these cells can play a role in oncogenesis.
Sources
All content published on Kenhub is reviewed by medical and anatomy experts. The information we provide is grounded on academic literature and peer-reviewed research. Kenhub does not provide medical advice. You can learn more about our content creation and review standards by reading our content quality guidelines.
Silverthorn, D. U. (2019). Human Physiology: An Integrated Approach (8th ed.). New York, NY: Pearson Education, Inc.
Kierszenbaum, A. L., & Tres, L. L. (2011). Histology and cell biology : an introduction to pathology (3rd ed.). New York NY: Elsevier Inc.
Hill, R. A., Nishiyama, A., & Hughes, E. G. (2024). Features, Fates, and Functions of Oligodendrocyte Precursor Cells. Cold Spring Harbor perspectives in biology, 16(3), a041425. Link
Yu, Q., Guan, T., Guo, Y., & Kong, J. (2023). The Initial Myelination in the Central Nervous System. ASN neuro, 15, 17590914231163039. Link
Chen, J. F., Wang, F., Huang, N. X., Xiao, L., & Mei, F. (2022). Oligodendrocytes and myelin: Active players in neurodegenerative brains?. Developmental neurobiology, 82(2), 160–174. Link
Franklin, R. J. M., Bodini, B., & Goldman, S. A. (2024). Remyelination in the Central Nervous System. Cold Spring Harbor perspectives in biology, 16(3), a041371. Link
Dolma, S., & Joshi, A. (2023). The Node of Ranvier as an Interface for Axo-Glial Interactions: Perturbation of Axo-Glial Interactions in Various Neurological Disorders. Journal of neuroimmune pharmacology : the official journal of the Society on NeuroImmune Pharmacology, 18(1-2), 215–234. Link
Luo, J. X. X., Cui, Q. L., Yaqubi, M., Hall, J. A., Dudley, R., Srour, M., Addour, N., Jamann, H., Larochelle, C., Blain, M., Healy, L. M., Stratton, J. A., Sonnen, J. A., Kennedy, T. E., & Antel, J. P. (2022). Human Oligodendrocyte Myelination Potential; Relation to Age and Differentiation. Annals of neurology, 91(2), 178–191. Link
Zou, P., Wu, C., Liu, T. C., Duan, R., & Yang, L. (2023). Oligodendrocyte progenitor cells in Alzheimer's disease: from physiology to pathology. Translational neurodegeneration, 12(1), 52. Link
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5996 | https://medium.com/@jha.ameet/modernist-literature-and-the-break-from-tradition-e662546c4237 | Sitemap
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Modernist Literature and the Break from Tradition
Amit Jha
4 min readAug 26, 2024
The early 20th century marked a period of significant upheaval in the arts, as writers, artists, and thinkers sought to break away from the conventions of the past and respond to the rapidly changing world around them. This era gave birth to modernist literature, a movement characterized by its radical departure from traditional narrative forms, themes, and styles. Modernist writers challenged established norms, experimenting with new ways of expressing the complexities of human experience in a world that seemed increasingly fragmented and disjointed. This article explores the key features of modernist literature and how it represented a break from tradition.
The Historical Context of Modernism
Modernist literature emerged in the wake of major historical events and cultural shifts, including the devastation of World War I, the rise of industrialization, and the rapid urbanization of society. These changes led to a sense of disillusionment and a questioning of previously held beliefs. The Victorian era, which preceded modernism, was marked by a belief in progress, stability, and the importance of social order. However, the horrors of war and the disintegration of old social structures left many feeling that these ideals were no longer tenable.
In response, modernist writers sought to capture the sense of alienation, confusion, and fragmentation that characterized the modern world. They rejected the idea that literature should offer clear moral lessons or neatly resolved plots, instead embracing ambiguity, complexity, and the exploration of the inner workings of the human mind.
Characteristics of Modernist Literature
1. Stream of Consciousness and Psychological Depth
One of the most distinctive features of modernist literature is the use of stream of consciousness as a narrative technique. This style seeks to depict the flow of thoughts, feelings, and memories within a character’s mind, often in a fragmented and nonlinear manner. Pioneered by writers like James Joyce and Virginia Woolf, stream of consciousness allows for a deeper exploration of psychological complexity, blurring the lines between reality and perception.
For example, James Joyce’s Ulysses is renowned for its use of stream of consciousness, capturing the intricacies of the protagonist’s thoughts over the course of a single day. Similarly, Virginia Woolf’s Mrs. Dalloway delves into the inner lives of its characters, offering a rich and nuanced portrayal of their mental states.
2. Fragmentation and Nonlinear Narratives
Modernist literature often eschews traditional linear narratives in favor of fragmented, disjointed structures. This reflects the modernist belief that reality is not a coherent, orderly progression but rather a series of chaotic and disjointed experiences. As a result, modernist works often feature multiple perspectives, shifting timelines, and abrupt transitions.
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T.S. Eliot’s The Waste Land exemplifies this fragmentation, combining a collage of voices, images, and references from different cultures and time periods. The poem’s fragmented structure mirrors the dislocation and despair of the post-war world, challenging readers to piece together meaning from the disparate elements.
3. Rejection of Realism and Embrace of Symbolism
While 19th-century literature was dominated by realism — a focus on detailed, accurate representations of everyday life — modernist writers often rejected this approach. Instead, they embraced symbolism and abstraction, using metaphor, allegory, and allusion to convey deeper truths about the human condition.
In Franz Kafka’s The Metamorphosis, the protagonist’s transformation into a giant insect is not a realistic event but a powerful symbol of alienation and existential anxiety. Similarly, the surreal landscapes and cryptic imagery in the works of authors like Samuel Beckett and Jorge Luis Borges challenge readers to interpret the underlying meanings rather than take the narrative at face value.
4. The Role of Language and Form
Modernist writers were deeply concerned with the limitations and possibilities of language. They questioned whether language could ever fully capture the complexities of human experience and often experimented with new forms and styles to push the boundaries of literary expression. This experimentation extended to the structure of their works, with modernist texts often breaking with conventional forms of poetry, drama, and prose.
Ezra Pound’s call to “make it new” encapsulates the modernist drive to innovate and renew literary form. His Cantos and the works of other modernist poets like Gertrude Stein and Wallace Stevens exemplify this experimentation, challenging traditional poetic forms and syntax to create something entirely new.
Modernism’s Legacy and Influence
Modernist literature’s break from tradition has had a lasting impact on the literary world. The movement paved the way for later experimental writing, including postmodernism, and continues to influence contemporary authors. The themes of alienation, fragmentation, and the questioning of reality that modernist writers explored remain relevant in today’s literature, as authors grapple with the complexities of the modern world.
Moreover, modernism’s emphasis on the interior life and the subjectivity of experience has shaped the way we think about narrative and character development in fiction. The innovations of modernist writers have expanded the possibilities of what literature can achieve, allowing for more diverse and complex explorations of human consciousness.
Conclusion
Modernist literature represents a profound break from the traditions of the past, reflecting the disorienting and rapidly changing world of the early 20th century. By challenging established norms and experimenting with new forms of expression, modernist writers created a body of work that continues to resonate with readers today. Their legacy endures in the continued exploration of the themes and techniques they pioneered, ensuring that modernist literature remains a vital and influential force in the world of letters.
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Teaching addition strategies to students with learning difficulties
Irene Polo-Blanco irene.polo@unican.es and Eva M González LópezView all authors and affiliations
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Abstract
Background & aims
In recent years, there has been an increased interest in analyzing the mathematical performance of students with learning difficulties in order to provide them with teaching methods adapted to their needs. In particular, the importance of studying the type of informal strategy that students use when solving problems has been highlighted. Observing how these strategies emerge and develop in children with learning difficulties is crucial, as it allows us to understand how they develop a subsequent understanding of arithmetic operations. In this paper we study the effect of explicit instruction in addition strategies, focusing on the minimum addend strategy, and analyze the difficulties that arise during this process.
Methods
An adapted multiple-probe design across students with a microgenetic approach was employed to assess the effectiveness of the teaching instruction and the acquisition of the minimum addend strategy while solving addition word problems. The participants were three primary-school children (two boys and one girl) with learning difficulties, one of them diagnosed with autism spectrum disorder. The instruction on the minimum addend strategy was sequenced into levels of abstraction based on the addends represented with and without manipulatives.
Results
The results show that the three participants were able to acquire the minimum addend strategy and transfer it to two-step problems. They all showed difficulties during the instructional process, with quantity comparison difficulties predominating. The instruction provided to address these and other difficulties is detailed for each participant.
Conclusions
The teaching of the minimum addend strategy has proven effective, and all three students acquired it throughout the instruction. The results concerning the student diagnosed with autism spectrum disorder are especially interesting given the lack of studies that focus on the strategies employed by students with this disorder to solve arithmetic problems. In this sense, the use of the microgenetic approach was especially useful to observe the type of spontaneous strategies used by this participant, and how they varied in response to the instruction.
Implications
Each study participant faced different difficulties and needed different periods of time to assimilate the new strategy. Conclusions are drawn for educators to help children with learning difficulties advance to more sophisticated strategies, so they can acquire these and subsequent mathematical concepts.
Research suggests that at least 12% of school-age students exhibit difficulties over the course of learning mathematics (Geary, 2013). It is important to work from an early age to overcome difficulties in learning mathematical concepts, as many of these difficulties can persist also in the face of new knowledge (Montague, 2007). Studies that have evaluated the strategies manifested by students with learning difficulties (LD) show that these are often less varied and effective than those of their typically-performing peers (Geary, 1990). This means that they could benefit from specific instruction to acquire these strategies (Siegler, 1988; Timmermans & Van Lieshout, 2003). Despite this, there is a lack of research on the effect of explicitly teaching advanced strategies to students with LD. For this reason, we propose a study with primary-school students with LD in order to help them develop effective addition strategies. Specifically, we first identify the participant's solution strategies when solving addition word problems independently (spontaneous strategies), and then provide instruction aimed at acquiring the minimum addend strategy.
Literature review
Various researchers have shown that, from a very early age, young children exhibit informal mathematical knowledge that will develop over time (Geary, 1994). For example, they are able to acquire the concepts of cardinality and/or ordinality (Wynn, 1990) and anticipate correct solutions to addition and subtraction. This informal mathematics is an intermediate step between intuitive and formal mathematics (Baroody & Tiilikainen, 2003). Some researchers have studied the relationship between some numerical abilities and informal mathematical knowledge. For example, Libertus et al. (2013) showed that children's approximation abilities correlated with and predicted informal, but not formal, mathematics abilities.
Accordingly, researchers have delved into the skills of children involving non-symbolic, approximate arithmetic, and how they relate to subsequent mathematical learning (Gilmore et al., 2007, 2010). For example, Gilmore et al. (2007) carried out a study involving children who had mastered verbal counting but had not yet been taught symbolic arithmetic. They showed that children were able to solve problems of symbolic approximate arithmetic without resorting to guessing or comparison strategies. Their findings suggest that children retrieve their non-symbolic number knowledge when they solve new approximate symbolic arithmetic problems.
Various studies in the literature have analyzed the type of informal strategy that typically-performing students manifest when solving basic arithmetic operations. Observing how these strategies emerge and develop in children is crucial, as it allows us to understand how they develop a subsequent understanding of arithmetic operations. Following this line of research, Cookson and Moser (1980) conducted a longitudinal study of strategies with children from Grade 1 to Grade 3 when solving addition and subtraction problems. At the beginning of the study, the students primarily exhibited the counting all strategy, and to a lesser extent the first addend or max strategy (counting up from the first addend). Over time, the use of these strategies decreased as the use of the minimum addend strategy increased (counting up from the larger addend), until they eventually acquired number facts (Cookson & Moser, 1980).
Teaching addition facts to children with learning difficulties
In the case of students with LD, the study of these informal strategies is especially relevant because it provides information that will allow for adequate interventions (Geary, 1990). Research in this area shows that students with LD often use fewer and less effective strategies than their typically-performing peers when solving different mathematical reasoning tasks, which may make it difficult to formally learn arithmetic operations and other mathematical concepts later (Geary, 1990; Geary et al., 2004; Siegler, 2007).
The most effective informal addition strategy is the minimum addend (hereinafter, MA) strategy: the student identifies the larger addend and starts counting up from that cardinal value the number of units of the smaller addend. This strategy is an essential predictor of success in mathematics (Siegler, 1988). Although most students learn this strategy over time, other students with learning difficulties may not master it, so it is necessary to provide them with appropriate instructions so they can learn how to execute this and other effective strategies (Montague, 2007; Powell et al., 2009).
In order to contribute to this effort, some studies have attempted to provide instructions for teaching MA to children with LD. Such is the case of the study by Tournaki (2003), who compared the drill-and-practice approach with that of teaching the MA strategy to students with LD and general education students. The study shows how only the students with LD who followed the MA strategy method improved significantly, whereas general education students showed improvement with both methods.
Microgenetic approach
One approach that has proved especially useful in explaining how strategies are developed is the microgenetic approach. This approach is frequently used to investigate how learning takes place in children in very different areas. It has been applied successfully in areas such as arithmetic, scientific reasoning, spoken and written language, motor activity, memory and reading (Blöte et al., 2004; Fletcher et al., 1998; Siegler, 2006). Some of the advantages of this approach are: the change can be observed while it is happening; several aspects of the change can be studied (the sequence of behaviors, how quickly they occur, the degree of generalization, individual differences and their causes); it makes it possible to detect variability in the behavior of different individuals under similar circumstances (or tasks); and finally, it offers flexibility because it can be used for highly diverse concepts and from different theoretical positions (Siegler, 2006).
In the particular case of arithmetic strategies, this approach can be used to identify the trial in which the child uses a certain strategy for the first time, as well as what leads him to discovery, and how it is generalized to other contexts (Blöte et al., 2004; Fletcher et al., 1998; Siegler & Stern, 1998). This is particularly useful for observing how strategies are developed in response to instruction, and, therefore, can be used to understand how the instruction exerts its effects (Siegler, 2006).
The main criteria that define the microgenetic approach are: (1) the complete period during which the change in behavior takes place must be observed; (2) the density of observations of the behavior in question must be high; and (3) the data from these observations are examined through a “trial by trial” analysis of the behavior, both qualitative and quantitative, to study the aspects that cause the change (Siegler, 2006). Specifically, the microgenetic approach suggests that cognitive change can be analyzed from five dimensions, which, in the case of developing arithmetic strategies, can be specified as follows (Siegler & Stern, 1998; Zhang et al., 2011): source of change (refers to the causes that make children adopt new strategies); path of change (refers to the sequence of different strategies that students manifest as they progress); rate of change (refers to the amount of time or experience that elapses between the first use of a strategy and its consistent use); breadth of change (refers to how a new strategy is generalized to other problems); and variability of change (considers the differences between children in the other dimensions, and the changing set of strategies used by each child).
Several studies have used the microgenetic approach to analyze how arithmetic operations strategies evolve in typically-developing children (Siegler, 1988), although few researchers have used this method to study how children with LD develop these strategies (Huffman et al., 2004). Of note in the area of developing strategies in response to instruction in children with LD are the works of Zhang et al. (2011, 2014), who propose methods for teaching advanced multiplication strategies to children with LD. The microgenetic approach allowed these authors to observe how the multiplication strategies evolved during the instruction. In particular, the results showed that the unitary counting was frequent during the baseline sessions, but it decreased during the teaching experiment to be replaced by the double counting strategy. In order to describe participants’ performance and strategic development, five dimensions (source, path, rate, breadth and variability) of change were analyzed according to the framework of microgenetic studies (Siegler, 2006). The authors note the need to conduct more studies with this approach in children with LD to establish a functional relation between the intervention and the changes in the students’ performance (Zhang et al., 2014).
Research questions
In view of what was gleaned from the literature, in this paper we propose studying the strategies exhibited by three students with LD in mathematics when solving addition problems, and how instruction focused on teaching the MA strategy affects the acquisition of new strategies. In particular, the following research questions are posed:
1.
What strategies do students with LD exhibit when solving arithmetic addition problems in the baseline sessions?
2.
How does the instruction favor these students’ acquisition of new strategies (in particular the MA strategy) and what difficulties arise during this process?
3.
Do they generalize the new strategies to problems with two operations?
4.
To what extent do they retain the acquired knowledge over time?
Method
Design and data collection
An adapted multiple-probe design across students with a microgenetic approach was employed to assess the effectiveness of the teaching instruction and the acquisition of new strategies while solving addition word problems by three primary school students with LD. In order to provide a detailed analysis of people's learning process during the intervention, microgenetic studies generally employ single-participant designs (Zhang et al., 2011) and often focus on a small number of individuals (Siegler, 2006).
During baseline, several probes were performed consisting of change and combine addition problems. After a stable baseline was observed, the first student was provided the instruction. Once the student modified the type of strategy used, the instruction began with the next student, and so on. The phases of the experiment included: baseline, instruction, generalization to two-step problems and maintenance.
As in other studies of strategies that follow a microgenetic approach, evaluations were performed during the intervention in order to observe the acquisition of new strategies (Blöte et al., 2004; Siegler, 2006; Zhang et al., 2011). This method requires trial by trial observation and coding, so all the sessions were videotaped. All phases of the study were carried out individually, in weekly sessions, with each student in a classroom in the school without distractions. Nine instructional sessions were held with each student, each lasting approximately 30 min. At the end of each instruction session, the authors watched the videos and planned the next instruction to give each student.
Participants and setting
Three children with LD and a teacher participated in the study. The teacher who conducted the experiment held a degree in teaching, specializing in special education, and had previous experience with children with LD. The three children, who have been given the pseudonyms Peter, Jane and Robert, had been identified by their tutors as struggling with math and were receiving learning support. They were in different grades of primary education in the same mainstream school. Table 1 shows the demographic information on the three participants.
Table 1. Student demographics.
| Variable | Student | | |
--- --- |
| | Peter | Jane | Robert |
| Gender | Male | Female | Male |
| Ethnicity | Caucasian | Caucasian | Caucasian |
| Age (years:months) | 7:9 | 7:8 | 10:1 |
| Disability category | GDD | GDD | ASD |
| Schooling | Mainstream | Mainstream | Mainstream |
| Grade | 1st | 2nd | 4th |
| Socioeconomic status | Low | Med | Low |
| Learning support | 8 h/week | 10 h/week | 8 h/week |
| IQ score (RPM) | 71 (low) | 45 (very low) | 58 (very low) |
| Literacy skills | Prolec-r | Prolec-r | Prolec-r |
| Oral comprehension | Very low | Very low | Very low |
| Reading comprehension | Very low | Low | Low |
| Mathematics Achievement | TEMA-3 | TEMA-3 | TEMA-3 |
| Total score | 34 | 21 | 18 |
| Equivalent age (y:m) | 6:0 | 5:1 | 4:9 |
| Number skills | 70% | 43% | 48% |
| Number-Comparison | 67% | 50% | 0% |
| Calculation skills | 24% | 18% | 12% |
| Concepts | 47% | 24% | 29% |
| Number facts | 0% | 0% | 0% |
Note. GDD: Global Developmental Delay, ASD: Autism Spectrum Disorder, RPM: Raven's Progressive Matrices test (Raven, 2015), Prolec-r: Reading Processes Assessment Battery, revised version (Cuetos et al., 2014), TEMA-3: Test of Early Mathematics Ability (Ginsburg & Baroody, 2007). Based on parents’ profession and level of education, as per Hollingshead's occupational scale (Hollingshead, 1975).
Peter was a 7-year-old boy who was repeating the first grade of primary school. At two years of age, the child was detected with a bilateral hearing loss with a hearing threshold of around 70 dB, and started using hearing implants. He showed a development delay in language, particularly in the acquisition of vocabulary (at 5: 9 years he showed age of 4:11 in vocabulary according to the Vavel test, Spanish Vocabulary Assessment Test, 6–9 years) and a global development delay. Jane was a 7-year-old girl enrolled in the second grade of primary school. In addition to pedagogical and language therapy, she received two hours of physical therapy per week. Robert was a 10-year-old boy diagnosed with autism spectrum disorder with severe symptoms. He was in fourth grade with an individual significant curriculum adaptation with contents and objectives corresponding to the first year of primary school. He showed a good attitude towards schoolwork, but had serious difficulties in directing or maintaining attention.
Measures and tasks
The problems designed for the study were change and combine addition problems with the unknown in the final position, as these are the easiest to solve (Carpenter & Moser, 1984). An example of a problem considered in the instruction is the following combine addition problem: “I bought 4 strawberry and 9 mint candies. How many candies did I buy in total?”. The number sets considered in the problems contained addends with no result higher than 15, and they did not contain repeated addends (4 + 4 = 8) or sums equal to 10 (6 + 4 = 10) in order to avoid easy numerical combinations (Carpenter & Moser, 1984). In order to distinguish between the first addend and minimum addend strategies, and in keeping with the guidelines of similar studies (Carpenter & Moser, 1984), the smaller addend always appeared first in the problem statement.
For the purpose of assessing generalization to more complex problems, two-step problems were introduced after the 4th session. These problems where of the type change-change and combine-combine with the unknown in the final position, and required two sums for their resolution (e.g., “On a farm there are 2 cows, 3 horses, and 7 sheep. How many animals are there in total on the farm?”).
Procedures
Baseline
The three students completed three pre-tests during the baseline that contained change and combine addition problems. The students were asked to solve the problems independently. They were given access to manipulatives and were encouraged to use them during the problem-solving process as needed. No help or feedback was provided during the baseline phase.
Experimental procedures
After completing the baseline sessions, the participants were given the instruction sequentially. An adapted Strategic Training Program from Zhang et al. (2014) was followed, which consisted of: (1) progress monitoring and appropriately selected task assignment, (2) encouraging students to use their own solution strategies before explicit instruction on MA, (3) providing feedback to the student and (4) explicit instruction with a focus on the MA strategy. This approach consisted on providing the students addition problems of two different types (change and combine). They were also given access to manipulatives and encouraged to use them as needed. Each problem was first solved by them independently and the solution strategy they used was recorded. After that, the instructor provided feedback to the students both when the resolution was correct and when it was not, which has often proved beneficial to the students’ learning (Zhang et al., 2014). Finally, the explicit instruction with a focus on the MA strategy was carried out.
To demonstrate the MA strategy, a sequence adapted from Tournaki (2003, Appendix A) was followed: (i) read the problem, (ii) choose largest amount, (iii) add the smallest to that, and (iv) what is the final result? In addition, in keeping with the sequencing of strategies identified by authors such as Baroody and Tiilikainen (2003) in terms of the representation of addends, the MA strategy was demonstrated using a three-stage process: level 1 (two addends represented with manipulatives), level 2 (one addend represented with manipulatives) and level 3 (no addends represented with manipulatives). In order to provide visual support for working with the MA strategy at each of these levels, instructional sequences were designed using simple messages and adapted enhanced language material (pictographic symbols). The sequence of instruction for the three levels is illustrated in Figure 1).
In order to evaluate the generalization of the instruction to more complex problems, from the 4th session on, two-step problems were introduced provided that the following criteria were met: manifest a successful MA strategy at least 60% of the time, and achieve at least 50% right answers in the previous two sessions. In these problems, the instruction adhered to guidelines similar to those used for one-step problems. First, the largest of the three numbers was identified. Then, depending on the level of abstraction being used to execute the MA strategy (see Figure 1), the student was guided to use the manipulatives to represent some of the numbers in the problem before eventually adding the three addends.
Four weeks after finishing the instruction, each student completed a post-test similar to those in the baseline phase, consisting of change and combine one-step problems (maintenance).
Below is an instructional scenario with Robert, in which he first exhibits level-1 knowledge of the MA strategy (two addends represented with manipulatives)
Coding
The students’ performance while solving each problem was recorded in terms of the type of solution strategy and the success of its implementation. Specifically, the following strategies were considered: incorrect strategies (guess, given number, incorrect operation), counting all, first addend, minimum addend. For each session, the frequency of each solution strategy spontaneously exhibited by the participants was calculated, as was the success rate of each. Behavioral aspects of the participants during each session were also recorded, which was used to adapt the instruction in subsequent sessions.
Reliability
Interobserver reliability data were collected during the baseline, instruction and maintenance phases. An experienced mathematics education teacher, who did not know the hypotheses of the study, recoded 30% of the students’ strategies and performance. Interobserver agreement was calculated by dividing the number of agreements by the number of agreements plus disagreements and multiplying by 100. It ranged from 91% to 100% during baseline, 84% to 91% during instruction, and 83% during maintenance. The mean interobserver reliability agreement for strategy categorization for each student across all phases was 86% for Peter, 89% for Jane and 86% for Robert. The mean interobserver reliability agreement for solution accuracy was 100% for Peter, 96% for Jane and 95% for Robert.
Procedural reliability measured the instructor's performance regarding the planned behaviors, which were: the instructor: (1) provides the agreed number of problems, with the agreed amounts; (2) provides the agreed manipulatives for the session; (3) lets the students solve problems independently; (4) [only for instructional sessions] after an unsuccessful attempt by the student, demonstrates how to solve it using the MA strategy at the corresponding level of abstraction (level 1, 2 or 3); (5) [only for instructional sessions] congratulates the student and/or encourages him/her once the problem is solved. Procedural agreement was calculated by dividing the number of observed teacher behaviors by the number of planned behaviors and multiplying by 100 for 30% of the sessions across all phases. Procedural reliability was 100% for Peter and Jane and 95% for Robert.
Results
Figure 2 shows the participants’ performance in terms of the strategies used and accuracy during the baseline, intervention and maintenance phases. In addition, Table 2 shows the strategies used by participants on trials immediately before and after each child's first use of the MA Strategy.
Table 2. Example of instruction on the MA strategy with robert.
| |
| Tutor: We will read the problem together. Remember: What did we have to do first? |
| Robert: (looking at the instruction sequence) Count out the larger number using gray cubes. |
| Tutor: Very good, and which one is larger? |
| Robert: 6! [takes 6 gray cubes] |
| Tutor: Great. What now? |
| Robert: Count out the smaller one using white cubes. |
| Tutor: Very good, and which one is smaller? |
| Robert: 3 [takes 3 white cubes] |
| Tutor: Very good. And what do we do next? |
| Robert: [Looking at the sequence] Now we cover the gray ones, right? [Covers the gray ones with his hand and counts the white ones:] 1, 2, 3. But I counted wrong, didn’t I? |
| Tutor: You forgot the gray ones! How many were under your hand? |
| Robert: Oh, that's right. There were 6. |
| Tutor: So now add the white ones. |
| Robert: [Touching his hand] 6, [touching a white cube] 7, [another] 8, and [another] 9. |
| Tutor: Very good! See how well you did? |
To delve into how this strategy was acquired by each participant, its manifestation and success at each level was analyzed (see Figures 3–5).
Baseline
The participants used few strategies during baseline, with different success rates between them. None of the three students ever exhibited the MA strategy during baseline. Peter employed incorrect strategies in the first baseline session, and the first addend strategy in the resolution of all problems of the following two sessions. He had a high success rate (78%) throughout baseline. The mistakes were due to the use of incorrect strategies such as guessing and, on isolated occasions, to counting errors when using the first addend strategy. Jane exhibited the counting all strategy when solving 88% of the problems, which resulted in the right answer 63% of the time. The main errors noted were the incorrect execution of the counting all strategy and the use of the improper strategy of guessing. Robert did not solve any of the problems correctly during baseline, and mostly exhibited incorrect strategies (guessing, or given number).
Instruction
A functional relation between the teaching experiment and the development of new strategies was observed: first the students showed a stable baseline in terms of the variety of strategies, and used new strategies only after the intervention began.
Peter
The student had a success rate of 85% to 100% during all the instructional sessions. Peter was instructed on the use of the MA strategy in the first instructional session. He immediately showed an understanding of the strategy and used it to solve almost every problem in this session (see Figure 2). This strategy was initially employed at level 1 (representing both addends with manipulatives, see Figure 3), although he quickly stopped using the manipulatives. He exhibited the MA strategy at level 3 (no addend with manipulatives) during the third session (see Figure 3). The student was highly motivated during the first four sessions and responsive to the instructions that were given on how to solve the problems.
In order to evaluate the generalization of the MA strategy, and after satisfying the pre-defined criteria (having exhibited this strategy at least 60% of the time, with a 50% success rate in the last two sessions), two-step problems were introduced in the fifth session. Initially, Peter did not transfer the MA strategy to these problems, but instead went back to the first addend strategy (see Figure 2), with a high success rate (85%). In the sixth session, Peter transitioned from the first addend strategy to the MA strategy, and eventually stopped using the first one altogether. A high success rate (in excess of 85%) was maintained throughout the experience. It should be noted that although he finally used the MA strategy in all the two-step problems, he again resorted to manipulatives to represent some of the addends in the one- and two-step problems, thus regressing to levels 1 and 2 in the execution of the strategy. The following graph shows the frequency of the MA strategy, differentiated by the levels used to represent the addends during the instructional and maintenance sessions.
The incorrect answers during the instructional sessions were due in most cases to counting errors when executing the MA strategy. The types of errors and the number of trials per instructional and maintenance sessions are listed in Table 3.
Table 3. Errors by Peter during instruction and maintenance sessions.
| Session | Error type | | | Total number of errors | Total number of trials |
--- --- --- |
| Counting error | Forgetting data | Comparison error |
| S1 | | | 1 | 1 | 12 |
| S2 | | | | 0 | 14 |
| S3 | 1 | | | 1 | 11 |
| S4 | 2 | | | 2 | 13 |
| S5 | | 1 | | 1 | 10 |
| S6 | | | | 0 | 10 |
| S7 | | | 1 | 1 | 9 |
| S8 | 1 | 1 | | 2 | 10 |
| S9 | 1 | | | 1 | 8 |
| M | | | 1 | 1 | 12 |
As Table 3 shows, on some occasions Peter also made comparison errors when executing the MA strategy. He made these errors when adding the same addend twice, which we interpret as a difficulty when identifying the larger and smaller amount. The errors related to forgetting data occurred when two-step problems were introduced and Peter forgot to add one of the three addends.
Jane
She had a success rate above 50% during most of the instructional sessions. During the first instructional session, Jane was instructed on the MA strategy; however, she resorted most of the time to the counting all strategy, representing both addends with manipulatives, and only spontaneously exhibited the MA strategy at level 1 in the last two problems of the first session. During this session, it was very important to insist on the proper representation of the addends with the cubes, since she occasionally made coordination errors when counting. During the second session, she relied more on the MA strategy, which she successfully used in half of the problems. As a result, in the third session, the teacher demonstrated the MA strategy at level 2 (only one of the addends represented with manipulatives). Figure 4 shows the frequency of the MA strategy used by Jane during the instructional and maintenance sessions, differentiating between the three levels in the representation of the addends.
During the third session, she made numerous errors when executing the MA at level 2 that we attribute to difficulties comparing the addends. Specifically, she circled the larger addend and represented the smaller addend but starting counting from the latter, obtaining as a result double this addend. The table below shows the number and types of errors per session. As we can see, the number of comparison errors decreased, although she continued to make numerous counting mistakes throughout the instruction and maintenance sessions.
Since the student maintained a success rate of over 60% during sessions 4 and 5, two-step problems were introduced in session 6. From the beginning, the student was able to transfer the MA strategy to the two-step problems, although the percentage of right answers decreased considerably (see Figure 2). This was due primarily to mental counting mistakes that had not manifested themselves with the manipulatives. Although the student was encouraged to resort again to the manipulatives when executing the MA strategy in two-step problems - as explained in the procedures-, the girl insisted on not using them, responding on more than one occasion: “I want to do it in my head, I don’t like it with the manipulatives.” In addition to the counting errors mentioned, the student forgot to add one of the addends in some of the two-step problems (see Table 4).
Table 4. Errors by Jane during instruction and maintenance sessions.
| Session | Error type | | | Total number of errors | Total number of trials |
--- --- --- |
| Counting error | Forgetting data | Comparison error |
| S1 | 1 | | | 1 | 10 |
| S2 | 1 | | | 1 | 11 |
| S3 | 1 | | 7 | 8 | 12 |
| S4 | 1 | | 1 | 2 | 11 |
| S5 | 4 | | | 4 | 11 |
| S6 | 9 | | | 9 | 11 |
| S7 | 3 | 2 | 1 | 6 | 11 |
| S8 | 3 | 2 | | 5 | 8 |
| S9 | 1 | | | 1 | 13 |
| M | 4 | | 1 | 5 | 12 |
Robert
From the beginning Robert showed a success rate above 50%, which he maintained during most instructional sessions. Due to the number of counting errors (both partitioning and coordination) exhibited during the baseline by this student when representing the addends, the instruction focused on working with the counting all strategy with manipulatives to help him overcome these errors and allow him to progress toward instruction in the MA strategy. The student was receptive and spontaneously exhibited the counting all strategy from the first session, using it to achieve a high success rate in the first two sessions (see Figure 2). In order to prepare him for the instruction in the MA strategy, he was instructed to represent the larger addend with the gray cubes, and the smaller one with the white cubes (see Step 1, Level 1 in Figure 1). Sometimes he was unable to distinguishing the smaller addend from the larger one, so a preliminary step was added in which he was asked to represent both quantities by matching the set of cubes, thus coordinating the elements of the smaller addend with those of the larger one.
From the third session, the instruction in the MA strategy began at level 1, although the subject did not exhibit it spontaneously until the fifth session (see Figures 2 and 5).
The student spontaneously exhibited the MA strategy for the first time, at level 1, when solving the third problem in the fifth session (see Table 1 for the description of this instruction scenario). From this moment on, the student exhibited a significant improvement in terms of interest and concentration, which he maintained until the end of the study. After this point, he almost completely abandoned the counting all strategy and used the MA strategy, which he adapted without difficulty to the two-step problems after session 7, with a high success rate. The student wanted to resort to the manipulatives to represent both addends (level 1) or one of them (level 2) in every session, as shown in Figure 5.
During the instructional sessions, Robert made several mistakes. The most frequent one (as in Jane's case) was, after representing both addends with the appropriate number of cubes, adding the larger addend to itself, thus doubling it (comparison error). He also made numerous counting errors in almost every session, including maintenance, and he forgot an addend in two of the two-step problems. The type and number of errors per session are given in Table 5.
Table 5. Errors by Robert during instruction and maintenance sessions.
| Session | Error type | | | | Total number of errors | Total number of trials |
--- --- ---
| Counting error | Guessing | Forgetting data | Comparison error |
| S1 | 2 | 2 | | | 4 | 7 |
| S2 | | | | | 0 | 4 |
| S3 | 1 | | | 1 | 2 | 5 |
| S4 | 2 | | | 2 | 4 | 6 |
| S5 | 1 | | | 4 | 5 | 9 |
| S6 | 1 | | | 1 | 2 | 9 |
| S7 | 1 | | | 3 | 4 | 9 |
| S8 | 1 | | 1 | 3 | 5 | 13 |
| S9 | 1 | | 1 | | 2 | 11 |
| M | 3 | | | | 3 | 12 |
Maintenance
During the post-test (four weeks after finishing the instruction), Peter exhibited the MA strategy in 83% of the problems, and resorted to the first addend strategy on all other occasions. He maintained a high percentage of correct answers (92%). Jane's approach was similar to Peter's in the use of strategies during the post-test (83% MA strategy, and first addend in the rest). Although the percentage of correct answers by Peter with respect to baseline increased, it did not exceed 60%. Robert exhibited the MA strategy in 75% of the post-test problems and resorted to the counting all and first addend strategies in the remaining problems. This resulted in a large increase in the number of right answers, from 0% in baseline to 75% in the post-test.
It is interesting to note that the three subjects used the MA strategy again in the trials immediately after their first use. Table 6 shows the strategy used in the four trials immediately before and after the initial use of the MA by the three subjects. Note, for example, how Peter used this strategy again in three of the four trials after his first use, and Jane only in the one immediate after it, before again returning to the counting all or first addend strategies. In Robert's case, he used it again in the three trials immediately after his initial use.
Table 6. Strategies used in trials immediately before and after each child's first use of the MA strategy (trial 0).
| Trial | −4 | −3 | −2 | −1 | 0 | 1 | 2 | 3 | 4 |
--- --- --- --- --- |
| Peter | CA | FA | FA | FA | MA | FA | MA | MA | MA |
| Jane | CA | CA | CA | CA | MA | MA | CA | FA | FA |
| Robert | IS | IS | IS | IS | MA | MA | MA | MA | CA |
Note. CA: Counting all, FA: First addend, MA: Minimum addend, IS: Incorrect Strategy.
Discussion and conclusions
This paper presents the effect of explicit instruction in advanced strategies, focusing on the MA strategy, in three children with LD (one of them with ASD). The instruction in the strategy was sequenced into levels of abstraction based on the addends represented with and without manipulatives, which helped the students better acquire the knowledge. We also observed the three students develop strategies in terms of: changes in the set of existing strategies, construction of new strategies and disuse of old ones (Blöte et al., 2004; Siegler, 2006).
In relation to the sequence of strategies that students showed as they gained proficiency, various strategies were observed. It is worth distinguishing here between the study subjects. In the case of Peter and Jane, the two students used a sequence of strategies that was similar to those of typically-developing students (Cookson & Moser, 1980): both exhibited the counting all strategy at the beginning, combined with the first addend strategy, before the MA strategy finally prevailed. It should also be noted that both subjects exhibited the MA strategy in the first instructional session where it was explicitly demonstrated. In the case of Robert (diagnosed with ASD), a smaller variety of strategies was observed: he went directly from using the counting all strategy exclusively, to gradually introducing the MA strategy, exhibiting practically no other effective strategies, like the first addend strategy. Moreover, the first time he exhibited the MA strategy was not during the first session when it was taught (session 3), but two sessions later. In this dimension we also consider the sequence of levels in the manifestation of the MA strategy. In this sense, the three subjects showed a progression from a concrete to an abstract level in terms of the representation of the two addends that coincided with previous studies on strategies (Baroody & Tiilikainen, 2003), although not all attained or manifested all three levels of abstraction. The use of incorrect strategies, such as guessing or given number, was significantly reduced in all three subjects during the course of the instruction.
In relation to how the strategy was generalized to other problems, we again observe different behaviors in the three subjects. Peter consistently acquired the MA strategy from the first session, but almost completely stopped using it with the introduction of two-step problems during two sessions, and did not generalize its use until the seventh session. As for Jane, it was not until the fourth session that she consistently used the MA strategy and generalized its use to solve two-step problems, although this did considerably reduce her success rate. Finally, although Robert took longer to exhibit the MA strategy spontaneously, it replaced the counting all strategy almost from the beginning, and he used the MA quite successfully for the most complicated problems until the end of the study.
In general, the students resorted to increasingly abstract counting strategies over the course of the instruction. They barely resorted to guessing strategies to solve the problems, consistent with the work of Gilmore et al., 2007, which showed that they are able to solve symbolic addition problems without resorting to this strategy. When executing the MA strategy, all three participants exhibited comparison errors. Comparison is regarded as one of the main skills in index numerical magnitude processing (Holloway and Ansari, 2009). Various studies have shown a link between numerical comparison and mathematical competency in both typically and atypically developing children (Gilmore et al., 2010; Rousselle & Noël, 2007). In our work, all the participants in the study exhibited difficulties comparing quantities, although it was more evident in the two students with the lowest mathematical skills. This is consistent with studies that have demonstrated that the basic processing of numerical magnitude is affected in children who present with mathematical difficulties (Holloway & Ansari, 2009; Rousselle & Noël, 2007).
The use of manipulatives to represent addends helped overcome the difficulties mentioned, by allowing the transition from symbolic to non-symbolic representations of addends. This confirms other studies that have shown that children with mathematics disabilities had difficulty comparing Arabic digits (i.e., symbolic number magnitude), but not comparing collections (i.e., non-symbolic number magnitude) (Rousselle & Noël, 2007). More generally, the benefits of using concrete manipulatives during instruction coincide with previous research on teaching arithmetic operations to students with learning difficulties (Baroody & Tiilikainen, 2003), and specifically with ASD (Bruno et al., 2021; Hart & Cleary, 2015; Polo-Blanco et al., 2019; 2021). The use of sequences based on augmentative language to teach strategies to students with this disorder also proved beneficial, as other studies show (Hart & Cleary, 2015; Mirenda, 2003).
Our results also coincide with prior research (Zhang et al., 2011) in the sense that the three subjects exhibited a wider variety of strategies during the instruction than during the baseline. As mentioned earlier, they also showed differences in how they manifested the MA strategy (some in the first session when explicitly demonstrated by the tutor, and others later) and the variety in their use of other strategies (less variety in the case of the subject with ASD).
The results concerning the student with ASD are especially interesting given the lack of studies that focus on the strategies employed by students with this disorder to solve arithmetic problems (Hart & Cleary, 2015; Polo-Blanco et al., 2019; in press). In this sense, the use of the microgenetic approach was especially useful to observe the type of spontaneous strategies used by the subject with ASD, and how they varied in response to the instruction. Although our study provides limited data on only one subject, the results show notable behaviors, such as the later but more consistent acquisition of this MA strategy, and a lower variability in the use of strategies, aspects that can be further studied in future research with students with this disorder.
In relation to the implications to classroom practices, teachers must weigh the considerable evidence showing that many students with learning difficulties, including autism, have problems learning the basic arithmetic operations used in everyday life, which makes it important for educators to know methods to teach them (Geary, 1990). Accordingly, it is very important that educators of children with LD, and in particular of those diagnosed with ASD, observe the strategies employed by children before the arithmetic operations are formalized, and provide tasks that help them acquire more advanced strategies (Zhang et al., 2011). Also, given the frequency of two forms of error (counting and comparison) observed, and given the fact that a high rate of error was observed at maintenance in one of the students, it could also be useful in further research to attempt intervention specifically designed to address these skills outside the context of arithmetic.
In this study, the students received instruction on the MA strategy, sequenced into levels of abstraction. As we have seen, adaptations were made in terms of the instruction given to each student that are interesting to consider and that are simple to implement. In the case of the student with ASD, the counting all strategy had to be drilled beforehand to overcome counting errors, and the matching strategy to compare magnitudes. It was also essential to carefully observe each student's preference for the level of abstraction of the MA strategy. While the student with ASD seemed comfortable with the more concrete levels of representation of the addends (levels 1 and 2), the other two students soon showed a desire to forego (at least partially) the manipulatives, so they were encouraged and helped to execute the strategy at higher levels of abstraction.
One aspect that emerges from this work is that each study participant needed different periods of time to assimilate the new strategies. Numerous studies involving students with learning difficulties have shown that they require more time to acquire new mathematical concepts, meaning that teachers must plan carefully and make the necessary temporary adaptations to each student's needs.
Declaration of conflicting interests
The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.
Funding
This work was supported by projects EDU2017-84276-R and PID2019-105677RB-I00 / AEI/10.13039/501100011033.
ORCID iD
Irene Polo-Blanco
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Autism & Developmental Language Impairments
Volume 6
Article first published online: September 28, 2021
Issue published: January-December 2021
Keywords
Mathematics
learning difficulties
microgenetic
problem solving
strategies.
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Irene Polo-Blanco
irene.polo@unican.es
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Eva M González López
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Universidad de Cantabria, Santander, Spain
Notes
Irene Polo-Blanco, Departamento Matemáticas Estadística y Computación, Facultad de Ciencias, Universidad de Cantabria, Avda de los Castros s/n, 39005, Santander, Spain. Email: irene.polo@unican.es
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Figure 1. Instructional steps in each of the three levels of the MA strategy: level 1 (two addends represented with manipulatives); level 2 (one addend represented with manipulatives); and level 3 (no addends represented with manipulatives). Pictographic symbols adapted from ARASAAC ( created by Sergio Palao and distributed under Creative Commons license (BY-NC-SA).
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Figure 2. Percentage of strategy types and accuracy during the baseline, intervention and maintenance sessions of the three participants.
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Figure 3. Percentage of each level in the use of the MA strategy by Peter during the instruction and maintenance sessions. The arrow indicates the session where two-step problems are introduced.
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Figure 5. Percentage of each level in the use of the MA strategy by Robert during the instruction and maintenance sessions. The arrow indicates the session where two-step problems are introduced.
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Figure 4. Percentage of each level in the use of the MA strategy by Jane during the instruction and maintenance sessions. The arrow indicates the session where two-step problems are introduced.
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Table 1. Student demographics.
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Table 2. Example of instruction on the MA strategy with robert.
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Table 3. Errors by Peter during instruction and maintenance sessions.
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Table 4. Errors by Jane during instruction and maintenance sessions.
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Table 5. Errors by Robert during instruction and maintenance sessions.
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Table 6. Strategies used in trials immediately before and after each child's first use of the MA strategy (trial 0).
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5998 | https://www.sciencedirect.com/science/article/abs/pii/S1388198119302392 | Carotenoids in human skin - ScienceDirect
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Cited by (73)
Biochimica et Biophysica Acta (BBA) - Molecular and Cell Biology of Lipids
Volume 1865, Issue 11, November 2020, 158588
Review
Carotenoids in human skin☆
Author links open overlay panel Sarah Zerres, Wilhelm Stahl
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Highlights
•Carotenoid levels in skin increase after dietary intake or supplementation.
•Skin carotenoids provide moderate endogenous protection against UV-irradiation.
•Endogenous photoprotection is related to antioxidant effects of carotenoids.
•Carotenoids modulate cosmetically relevant skin properties.
Abstract
The skin is shielding our organism from exogenous threats including solar radiation. Carotenoids which are ingested with the diet accumulate in the skin with the highest levels occurring in skin of the forehead and in the palms of the hands. Blood and skin levels of carotenoids increase during supplementation and due to their antioxidant properties and UV-absorbing effects carotenoids are used as photoprotective agents. Systemic photoprotection with carotenoids after supplementation or ingestion of a carotenoid rich diet has been demonstrated in several human intervention studies. Although protection is only moderate it may contribute to UV protection in combination with other measures. Beyond photoprotection, ingestion of carotenoids has been postulated to be of additional benefit for cutaneous tissue and influences moisture and texture or elasticity of the skin. However, only a limited number of studies is available yet to substantiate such a claim.
Introduction
The skin is the largest organ of our body and one of its primary tasks is shielding the organism from exogenous threats of physical, chemical and biological origin including radiation or infective microorganisms. Further functions in maintaining health are related to the regulation of skin water levels, temperature and also include the light-dependent synthesis of vitamin D. Sensation of external signals such as heat, cold, and pain via the skin is essential for us to react and adapt to changes in the environment. Appearance of the skin plays a role in attraction; function and appearance depend among other factors upon proper nutrition supplying the tissue with suitable and balanced amounts of macro- and micro-nutrients [1,2].
Human skin is composed of several layers with specific tasks with the epidermis as the outermost structure. From top to bottom the epidermis is subdivided into the stratum corneum, stratum granulosum, stratum spinosum and the stratum basale. The main cell type in the epidermis is the keratinocyte, being provided with melanin from the melanocytes located in the basal layer. Keratinocytes mature and are transported from the bottom to the top. During this process called cytomorphosis they accumulate keratin and lose their nucleus, leading to an accumulation of dead skin cells in the stratum corneum as the outermost protective layer. Skin's second major layer is the dermis, with embedded blood vessels, sensing nerves, sweat glands, and hair follicles. The main cell type here is the fibroblast surrounded by a complex extracellular matrix, consisting mainly of different types of collagens, elastin and proteoglycans providing optimal structure for physiological function of the tissue. Located underneath the dermis is a layer of adipose tissue important for thermal regulation, shock absorption and storage of energy. Lipophilic compounds like the carotenoids tend to accumulate in lipophilic compartments. Carotenoids measurably contribute to normal human skin color, in particular the appearance of “yellowness” [3,4].
Light is a major environmental stressor which penetrates the skin and interferes with different biomolecules including the DNA. UV radiation as part of the electromagnetic wave spectrum has been associated with the induction and promotion of skin cancer which is among the most common cancers with increasing incidence worldwide. Rising skin cancer rates are likely caused by an increased exposure to UV light due to increased outdoor activities, clothing style, artificial tanning and increased longevity .
Penetration efficacy of UV light depends on structural features of the skin and on its pigmentation. With increasing wavelength the depth of penetration increases with UVA and visible light passing epidermis and dermis whereas UVB hardly passes beyond the epidermal layer. UVB combines tissue targeting and damaging properties in a way that most of the severe consequences of UV exposure are attributed to this wavelength range . Excessive exposure of the skin to ultraviolet radiation results in sunburn, an inflammatory response to light-induced lesions. In particular UVB irradiation, provokes an inflammation associated with local pain, reddening, and hyperthermia . UVA radiation is mainly involved in processes of photoaging and photocarcinogenesis, playing a major role in the pathogenesis of photodermatoses. There is also evidence from in vitro and in vivo studies that infrared radiation may play a role in photoaging. Upon light exposure, a cascade of photo-induced chemical and biological reactions takes place in the target tissue . Reactive oxygen species (ROS) are generated in photooxidative processes and damage molecules and cellular structures. The chemical reaction cascade leads to cellular biochemical responses including modified gene expression, impact on kinase-dependent regulatory pathways, immune and inflammatory events, or induction of apoptosis . Thus, sun light is a major environmental factor responsible for skin changes and avoidance of UV exposure is an important strategy to protect the skin. An adequate supply of the skin with nutrients is required for proper function of the skin and supplementation with selected dietary components including antioxidants may be suitable to support photoprotection and contribute to repair and healing processes in damaged tissue. But little is known how key nutrients influence principle phases in the wound-healing cascade . Much attention has been attributed to an adequate supply of the skin with minerals such as Se, Mn, Cu and Zn, which are key constituents of antioxidant enzymes, and the antioxidant vitamins C and E. Further important dietary constituents with antioxidant activity and skin protecting properties include the group of carotenoids.
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Carotenoids in the skin
The general structure of carotenoids consists of a tetraterpene backbone optionally flanked by terminating rings. These rings are either oxygenated or non-oxygenated and distinguish carotenoids between xanthophylls and carotenes, respectively. Carotenoids are pigments used in photosynthesis and coloration and they exist in a high variety. Depending on their structure, their conjugated electron systems absorb different wavelengths resulting in a broad color range being produced by the wide
Photoprotective properties of carotenoids
On average indoor-working Europeans experience an irradiation dose of about 10,000–20,000 J/m 2 per year. The doses described for Americans and Australians are considerably higher with 20,000–30,000 J/m 2 per year and 20,000–50,000 J/m 2 per year, respectively. With a standard vacation in sunny places, the yearly dose increases by about 30% or more . With increasing altitude and decreasing latitude UV doses that people are exposed increase too. Considering detrimental health effects, including
Systemic photoprotection with carotenoids - human intervention studies
For the evaluation of biological effects related to photoprotection in humans several dietary intervention studies were performed. Depending on the design, most of the studies showed that an increased intake of carotenoids is associated with a decrease of UV-induced erythema reaction (sunburn). An ameliorated sunburn reaction was determined after dietary intervention with carotenoid-rich food. After a 10 week ingestion of tomato paste providing about 16 mg of lycopene/day, serum levels of
Erythropoietic protoporphyria
The first clinical approaches to use carotenoids for photoprotection were in the treatment of erythropoietic protoporphyria (EPP) to ameliorate secondary side effects of light exposure . EPP is a genetic disorder with a disturbed porphyrin synthesis. Ferrochelatase deficiency leads to an accumulation of the heme precursor protoporphyrin IX which acts as a strong photosensitizing agent . Upon light exposure, excited triplet state molecules and singlet oxygen are generated triggering
Cosmetic properties of carotenoids
Beyond photoprotection, carotenoids and other micronutrients are of additional benefit for cutaneous tissue and influence moisture and texture as well as elasticity and structure of the skin . Claims on cosmetic effects of dietary constituents have been common, and several natural compounds are used in cosmetic products. The field of cosmeceuticals, however, is developing and endogenous delivery of nutrients to optimize skin appearance and application for skin care are developing as new
Conclusion
Dietary carotenoids are useful in the protection of skin against excess light. Human intervention studies demonstrated that β-carotene and lycopene rich products are suitable for endogenous photoprotection. It should be noted, however, that the protection in terms of sun protection factor is low and requires additional means especially upon high UV exposure. Claims on cosmetic effects of carotenoids have been supported by human intervention studies but the field is still in its developing
Declaration of Competing Interest
Research of WS has been funded by the Deutsche Forschungsgemeinschaft Project STA 699/3-1
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5999 | https://dl.nirasystem.com/nira-book/Field%20and%20Wave%20Electromagnetics%20Cheng.pdf | DAVID K. CHENG SYRACUSE UNIVERSITY A vv ADDISON-WESLEY PUBLISHING COMPANY Reading, Massachusetts Menlo Park, California London Amsterdam Don Mills, Ontario Sydney DAVID K. CHENG SYRACUSE UNIVERSITY A , vv ADDISON-WESLEY PUBLISHING COMPANY I 1 Reading, Massachusetts Menlo Park, California m on don' Amsterdam Don Mills, Ontario' Sydney I This book is in the API)ISON7WESLEY SERIES IN ELERRICAL ENGINEERING , . SPONSORING EDIT& Tom Robbins PRODUCTION EDITOR: Marilee Sorotskin TEXT DESIGNER: Mednda Grbsser ILLUSTRATOR: Dick Morton , - COVER DESIGNER AND ILLUSTRATOR: Richard H a n y s ART COORDINATOR: Dick Morton i.: PRODUCTION MANAGER: Herbert Nolan The text of this book was composed in Times Roman by Syntax International. Library of Congress Cataloging in Publication Data Cheng, David K. avid-~(euni dqte- Field and wave e ~ e c t r o ~ ~ e t i ~ s . Bibliography: p. I 1. Electromagnetism. 2. Fiqld th :ory (Physics) I. Title. I - QC760. C48 < 530.1'41 . 81-12749 ISBN 0-201-01239-1 AACR2 . 3 I I I COPY%^^ O 1983 by Addison-W+ey Publishing Company, Inc. rights reserved. NO part of @is publication may be reproduqd, btored in a retrieval system, or tnns- m,ltted. in any, form or by any means. electmnic, mechanica~. photocopying, recordmg, or othenu~se, without the prior written permission df the publisher. Prmted m the Uolted States of Amenca. Published simultaneously in Canada. , ISBN 0-201-01239-1 ABCDEFGHIJ-AL-89876543 I 1-1 Introduction 1-2 The electromagnetic model 1-3 SI units and universal constants Review questions Introduction Vector addition and subtraction Products of vectors 2-3.1 Scalar or dot product 2-3.2 Vector or cross product 2-3.3 Product of three vectors Orthogonal coordinate systems 2-4.1 Cartesian coordinates 2-4.2 Cylindrical coordinates 2-4.3 Spherical coordinates Gradient of a scalar field Divergence of a vector field Divergence theorein Curl of a vector field Stokes's theorem 2-10 Two null identities 2-10.1 Identity I 2-10.1 Identity I1 2-1 1 Helmholtz's theorem Review questions . Problems 3-1 Introduction 3-2 Fundamental postulates of electrostatics in free space - I J-2 Coulonb's law 3 ?.: Electric fitla ciue LO a system of discrete charges 3-3.2 Electric field due to a contipuous distribution of charge 3-4 Gauss's law and applications 3-5 Electric potential I 3-5.1 Electric potential due to a charge distribution 3-6 Condugtors in static electric field 3-7 Dielectrics in static electric field 3-7.1 Equivalent charge distributions of polarized dielectrics 3-8 Electric flux density and dielectric constant 3-8.1 ~ielectric'strength 3-9 Boundary conditions for electrostatic fields 3-10 Capacitance and ~apacitors 3 - 10.1 Series aqd parallel connections of capacitors 3-1 1 Electrostatic enerpy and forces . , 3- 11.1 ~lectrost'atic energy in terms of field quantities 3- 11.2 Electrostatic forces Review. questions Problems 6 , I - : 2 . Solution of Electrostati~ Problems . . 4-1 Introduction i 4-2 Poisson's and La~lace's equations 4-3 Uniqueness of eleptrgstatic solutions i 4-4 Method of imageg' : : I 4-4.1 Point charge and conducting planes 4-4.2 Line charge and parallel . I conducting cylindpr . 4-4.3 Point charge aqd conducting sbhera, 1 4-5 Boundary-value problems in Cartesian coordinates , 4-6 Boundary-value problems in cylindrical coordinates 4-7 Boundary-value problems in spherical coordinates Review questions ~roblems . Steady Electric Currents 5-1 Introduction 5-2 Current density and Ohm's law 5-3 Electromotive force and Kirchhoff's voltage law 5-4 Equation of continuity and Kirchhoff's current law 5-5 Power dissipation and Joulc's law 5-6 Boundary conditions for current density, 5-7 Resistance calculations Review questions Problems 1 S t a t i c Magnetic Fields , 6-1 Introduction 6-2 Fundamental postulates of magnetostatics in free space 6-3 Vector magnetic potential 6-4 Biot-Savart's law and applications 6-5 The magnetic dipole 6-5.1 Scalar magnetic potential 6-6 - Magnetization ;~ntl cquivalcnt currcnt dcnsiiics 6-7 Magnetic field intensity and relative permeability I 6-8 Magnetic circuits 6-9 Behavior of magnetic materials I 6-10 Boundary conditions for magnetostatic fields 1 6-1 1 Inductances and Inductors 1 . 6-12 Magnetic energy . 6-12.1 Magnetic energy in terms of field quantities / I CONTENTS xiii 1 1 . I 8-2.1 Transverse electromagnetic waves 312 8-2.2 Polarization of plane waves I 314 Plane waves in conducting media 1 317 8-3.1 Low-loss dielectirc I 318 8-3.2 Good conductor 8-3.3 Group velocity Flow of electromagnetic power and the Poynting vector 8-4.1 Instantaneous ana average power densities Normal incidence at a plane conducting boundary Oblique incidence at a plane conducting boundary 8-6.1 Perpendicular polarization 8-6.2 Parallel polarization Normal incidence at a plane dielectric boundary Normal incidence at multiple dielectric interfaces 8-8.1 Wave impedance of total field 8-8.2 Impedance transformation wish- multiple dielectrics Oblique incidence at a plane dielectric boundary 8-9.1 Total reflection 8-9.2 Perpendicular polarization 8-9.3 Parallel polarization Review questions Problems 9 Theay and Applications of Mnsmission Lines Introduction Transverse electromagnetic wave along a parallel-plate transmission line 9-2.1 Lossy parallel-plate transmission lines General transmission-line equations 9-3.1 Wave characteristics on an infinite transmission line 9-3.2 Tr:lnsrnission-li11c uurumctcl.~ 9-3.3 Attenuation constant from power relations Wave characteristics on finite transmission'lines 9-4.1 Transmission lines as circuit elements 9-4.2 Lines with resistive termination . I 1 I '! I I ? 1 I ' , I xiv CONTENTS 1 . . , ;. ' I < . ' . t + . - i . . i I i ( I 1 ' I ' J 1 '" 9-4.3 Lines with Gbjtiaq terminafian . , 404 9-4.4 ~ransmissioa-link circuits . 407 11 An : 9-5 The Smith chart 41 1 11 1 9-5.1, Smith-chart ca1culations for losiy lines I 420 11 9 -6 Transmission-line im&d&ce matching 1 - . , . 422 9 4 . 1 Impedance rnatc$ng by quarter: wave transformer , 423 ! 1 9-6.2 Single-stub matching 426 11 9-6.3 Double-stub matching 43 1 Review questions 1 . 435 11 Problems , , 437 F ( 1 0 Waveguides and Cavity R8so"ators , Introduction General wave behaviors along uniform ' guiding structures , - 10-2.1 Transverse electromagnetic waves 10-2.2 Transverse magnetic waves 10-2.3 Transverse e1ecp;ic waves , Parallel-plate waveg$de t 10-3.1 TM waves petween yarahlel plates 10-3.2 TE waves between parallel plates 10-3.3 Attenuatioq in.~~rallel-plate ' waveguides . t Rectangular waveguides 10-4.1 TM waves in rectangular wqvebides 10-4.2 TE waves in recrangular waveguides 10-4.3 ~ttenuiiti$n)rktan~ular waveguides Dielectric waveguide6 : ! 10-5.1 TM wayes lo^& a dielectric slab 10-5.2 TE waves ;)ong ii dielectric slab ' > , Cavity resonators \ 10-6. ! TM,,,, moqes I I , I 10-6.2 TE,,, modts 10-6.3 Quality factor &(cavity resonatbr Review questions Problems ... , I . . . , . . , . . , , . . , . . , . ' . . . , . . , , , CONTENTS xv .' ; - , $ .. ' , ' - 3 . . '. , 1.. . . . . , . . . . . , . < , . , . . . . , , . . . : h - , . ~ z l , ~ . . , , . , ~ . ., -. , J - , . ,, " , - ... ;. --.. .?;,; ... i ; ..: ;>"'.:.',' ' : . ., .. . , ', . .
, .. . ., . I , . ; , ; , ,:v ,,,, :;. f +.< . ' , i . . .
.
. : . , , ' , !!., ., 11 '.. , .., , . i , ~ ' " . , Antekms and Radiating Systems. " , . , ,. . . . , . I S Introduction ~ ' ~ 1 . , { 500 Radiation fields of elemental dipoles 502 11 -2.1 The elemental electric dipole 11 -2.2 The elemental magnetic dipole Antenna patiefh5 and at. tenna parameters . Thi;. linear antennas 11 -4.1 The halt-wave aipde Antenna arrays 11 -5.1 Two-element arrays 1 1-5.2 General uniform linear arrays Receiving antennas 11 -6.1 Internal impedance and directional pattern 11 -6.2 Effective area Some other antenna types 11 -7.1 Traveling-wave antenna '-\ 11-7.2 Yagi-Uda antenna 11 -7.3 Broadband antennas Apeiture Radiators References Review questions ProbIems Amendix A Symbols and Units A-1 Fundamental SI (rationalized MKSA) units A-2 Derived quantities A-3 Multiples and s~bmultiples of units Appendix B Some Useful Material Constants B-1 Constants of free space B-2 Physical constants.of electron and proton B-3 Relative permittivities (dielectric constants) \ ..I I I . ,. B-Q ~onductivities , I !i t .., . . B-5 Relative permeabilities : , Back Endpapers . Left: . , Gradient, divergence. cprl, and Laplacian opetations Right: Cylindrical coordinates Spherical coordinates . The many books on introductory electromagnetics can be roughly divided into two main groups. The first group takes the traditional development: starting with the experimental laws, generalizing them in steps, and finally synthesizing them in the form of Maxwell's equations. This is an inductive approach. The second group takes the axiomatic development: starting with Maxwell's equations, identifying each with the appropriate experimental law, and specializing the general equations to static and time-varying situations for analysis. This is a deductive approach. A few books begin with a treatment of the special theory ofrelativity and develop all of electro- magnetic theory from Coulomb's law of force; but this approach requires the dis- cussion and understanding of the special theory of relativity first and is perhaps best suited for a course at an advanced level. Proponents of the traditional development argue that it is the way electromag- netic theory was unraveled historically (from special experimental laws to Maxwell's equations), and that it is easier for the students to follow than the other methods. I feel, however, that the way a body of knowledge was unraveled is not necessarily the best way to teach the spbject to students. The topics tend to be fragmented and cannot take full advantage of the conciseness of vector calculus. Students are puzzled at, and often form a mental block to, the subsequent introduction of gradient, di- vergence. and curl operations. As ;I proccss for formu1;lting :in clcctroni:~gnctic model, this approach lacks co,hesivoness and elegance. The axiomatic development usually begins with the set of four Maxwell's equa- tions, either in differential or in integral form, as fundamental postulates. These are equations of considerable complexity and are difficult to master. They are likely to cause consternation and resistance in students who are hit with all of them at the beginning of a book. Alert students will wonder about the meaning of the field vectors and about the necessity and sufficiency of these general equations. At the initial stage students tend to be confused about the concepts of the electromagnetic model, and they are not yet comfortable with the associated mathematical manip- ulations. In any case, the general Maxwell's equations are soon simplified to apply to static fields, which allaw the consideration of electrostatic fields and magneto- static fields separately. Why then should the entire set of four Maxwell's equations be introduced at the outset? ' I vi PREFAC' It may be argued tfiat'Coulomb's law, i h o u ~ h based on experimental evidence, is in fact also a~postulate.' Cbnsider the tw6 stipulations of Coulomb's law: that the charged'bodies are very spa11 comparec/ with thcir distance of separation, and that the force between the qhafged bodies is isv'crscly proportional to t11c sclu;~rc of their distance. he question a~ises regarding the first stipulation: How small must the charged bodies be in ardqr to be considered "very small" compared with their dis- tance? I,, practice the,charged bodies cannot .be of vanrs' ag sizes (ideal poht charges), and there is dificplty in determinirig the?'true" distance between two bodies of finite dimensions. Fgr given body sizes the relative accuracy in distance measure- ments is better when {he separation is largdr. However, practical considerations, (weakness of force, existence of extraneous charged bodies, etc.) restrict the usable distance of separation in the laboratory, and experimental inaccuracies cannot be entirely avoided. This {ends to a more importa~if question concerning the invcrse- square relation of thc second stipulnlion. Even if thc clwrgcd bodies wcrc of vanishing . sizes, experiment?! measurements could not be of an infinite accuracy no matter how skillful and careful an experimentor was. ko+ then was it possible for Coulomb to know that the force,wns exactly inversely pr'pportional to thc syuure (not the 2.000001th or the 1.995j999th power) of the distance of separation? This question cannot be answered from an experimental1 viewQoint because it is not likely that during Coulomb'$ time experiments could fiave been accurate to the seventh place. We must therefore conclud'e.that Coulombis lawsis itself a postulate and that it is a law of nature discovered and assumed on the basis of his experiments of a limited accuracy (see Section 3-2). ' This book builds tpe dcctromagnetic inode] using ap axiomatic approach in steps: first for static elecrric fields (Chapter 3), then. for siatic magnetic fields (Chapter 6), and finally for timemrying .fields leading to ~axkefl's equations (Chapter 7). The mathematical basis:for each step is Halmhohr's theorem, which states that a vector field is determined to within an additive cbnst$bt if both its divergence and its curl are speciff?;d evfrywhere. Thus, for the::'development of the electrostatic model in free space.. it is 'pnly necessary to define n singb vector (namely. the electric liclcl iulcuaily E) by spccil'~iug its clivcrg~.rde and its cur! as postulates. All other relations in electrostati~s for free space, ihchding Coul~mb's law and Gauss's law, can be derivb- !hq two rather simple postulates. Relations in material media can be devclopedihydugh the concept of eguival&nt charge distributions of polarized dielectrics. r , f Similarly, for fhe magphstatic model .in .fr& space it is necessary to define only a single magnetic fluyidensity vector B by .specifying its divergence and its curl as postulates all other formulas can be derived frpm these two postulates. E, Relations in rnater!al medip ban be developed t$ough.fhe concept of equivalent current densities. Of coyrsp' the validity of the postulates lies in their ability to yield results that dpnfoqb pith experimentil evidince. For time-varying fields, 'the electric andYmagliftic field intensities are coupled. The curl E postula& for)hy electrostatic Godel rmst be modified to conform with Faraday's law. In addition, the curl B postdate the magnetostatic model must also be modified in'prder: to be consistent with of continuity. We have, = _ > L 2 ' 8 : I \ . as-. PREFACE vii bnental evidence, bb's law: that the hration, and that je square of their v small must the : d with their dis- izes (ideal point !ween two bodies ;istancc measurc- 11 considerations :strict the usable racies cannot be ling the inverse- vere of vanishing m c y no matter ble for Coulomb syuure (not the ' ! T . n p e s t i o n ; not . ~ ~ e l y that 2 > . '\ placu. ie and that it is :nts of a'lirnited ~ t c approach in ' fields (Chapter ns (Chapter 7). : h states that a divergence and le electrostatic rly, the electric ates. All other v and Gauss's )us in material lihbutions of isary to define rgenwd its wo t .tulates. o i -dent hei?,dity to s are coupled. conform with ;c model must lity. We have, then, the. four Maxwell's equations that constitute the electromagnetic model. , I believe that this gradual development of the electromagnetic model based on Helmholtz's theorem is novel, systematic, and more easily accepted by students. In the presentation of the material, I strive for lucidity and unity, and for smooth and logical flow of ideas. Many worked-out examples (a total of 135 in the book) are included to emphasize fundamental concepts and to illustra~e methods for solving typical problems. Review questions appear at the end of each chapter to test the students' retention 2nd ur.dex;anding of the esse~tial material in the chapter. Thc problcms in c;~ch chap~cr arc tlcsigncd rcinforcc >,;,!cnts1 comprchcnsion of the interrelationships b-tween the difTerent quantities in the formulas, and to extend their abilitywf appljling the formulas to solve practical problems. I do not believe in simple-minded drill-type problems that accomplish little more than sn exercise on a calculator. The subjects covered, besides the fundamentals of electromagnetic fields, include theory and applications of transmission lines, waveguides and resonators, and antennas and radiating systems. The fundamental concepts and the governing theory of electromagnetism do not change with the introduction of new eiectromaz- netic devices. Ample reasons and incentives for learning the fundamental principles of electromagnetics are given in Section 1-1.. I hope that the contents of this book strcngthened by the novel approach, will pr'oiide students with a secure and suf- ficient background for understanding and analyzing basic electromagneiic phe- nomena as well as prepare them for more advanced subjects in electromagnetic theory. There is enough material in this book for a two-semester sequence of courses. Chapters 1 through 7 contain the material on fields, and Chapters 8 through 11 on waves and applications. In schools where there is only a one-semester course on electromagnetics, Chapters 1 through 7, plus the first four sections of Chapter 8 would provide a good foundation on fields and an introduction to waves in un- bounded media. The remaining material could serve as a useful reference book on applications or as a textbook for a follow-up elective course. If one is pressed for time, some material, such as Example 2-2 in Section 2-2, Subsection 3-11.2 on electrostatic forces, Subsection 6-5.1 on scalar magnetic potential, Section 6-5 on magnctic circuits, and Subscctions 6-13.1 and 6-13.2 on magnetic forces and torqucs, may be omittcd. Schools on a quarter system could adjust the material to be covered in accordance with the total number of hours assigned to the subject of electromagnetics. ' The book in its manuscript form was class-tested several times in my classes on electromr\gnetics at Syracuse University. I would like to thank all of the students in those classes who gave me feedback on the covered material. I would also like to thank all the reviewers of the manuscript who offered encouragement and valuable suggestions. Special thanks are due Mr. Chang-hong Liang and Mr. Bai-lin Ma for their help in providing solutions to some of the problems. Syracuse, New York January 1983 \ ! 1-1 INTRODUCTION Stated in a simple fashion, electromugnetics is the study of the effects of electric charges at rest and i q motion. From elementary physics we know there are two kinds of' charges: posithe ahd negative. Both positive and negative charges are sources of an electric field Mb~in~charges produce a current, which gives rise to a magnetlc field. Here we tentatively speak of electric field and magnetic ficld in a general way; more definitive meanings+will bc attached to these terms later. Ajeld is a spatial distribution of a q u a d y , which may or may not b;e, s function, of time. A time-varying electric field is accompanied by a magnetic field, and vice versa. In other words, time-varying electric and magn4iic fields are coupled, r$suiting i i rin electromagnetlc field. Under certain conditibns, time-dependent electromagnetic fields produce waves that radiate from the sotlfce. The concept of fields and waves is essential in the explanation of action at a distance. In this book, Field mii Wave Electromugnetics, we study the principles a d applications of the laws of electromagnetism that govern electro- magnetic dhenomepa. Eleciromagnetks is of fundamental importance to physicists and electr~cal engineers. ElEctromagnetic theory is indispensable in the understanding of the principle of atdm smashers, cathode-ray oscillosco~s, radar, satellite communication, television reception, remote sensing, radio astronomy, microwave devices. optical fiber communication, instrument-landing systems, electromechanical energy con- version, and s? on. Circuit concepts represent a restricted version, a special case, of electromagni?tic caflcepts. As we shall see in chap& 7, when the source frequency IS very low so that th& dimensions of a conducting ntfwork are much smaller than the wavelenglh, 3 e have a quasi-static situation, which simplifies an electromagnetic problem circuit problem. However, we hasten to add that circuit theory is itself a highly deveroped, sophisticated discipline. It appHls to a different class of electrical engineering pr~blebs, rtnd it is certainly important in its own right. Two sitbations:illustrate the inadequacy of circuit-theory concepts and the need - - . of electroma~netic-field concepts. Figure !
-I depicts a monopole antenna of the type we sce on a wdkie4alkic. O H tnuwmit, thc sotlrcc at thc basc Cccds the antelm1 wlth P mcssagc-carrying currcot d an appropriate carrier frequency. From a circuit-theory 2 THE ELECTROMAGNETIC MODEL / 1 A monopole antenna. Fig. 1-2 An electromagnetic problem. point ofview, the source feeds into an open circuit because b b e r tip of the antenna is not connected to anything physically; hence no current would flow and nothing would happen. This viewpoint, of course, cannot explain why communication can be established between walkie-talkies at a distance. Electromagnetic concepts must be used. We shall see in Chapter I1 that when the length of the antenna is an appreciable part of the carrier wavelengtht. a nonmiform current will flow :dong thc ol7cn-ended alltellna. This current radiates a tin~c-v;lryiog electrornil~netic field in space, which can induce current in another antenna at a distance. In Fig. 1-2 we show a situation where an electromagnetic wave is incident from the left on a large conducting wall containing a small hole (aperture). Electromagnetic fields will exist on the right side of the wall at points, such as P in the figure, that arc not necessarily directly behind the aperture. Circuit theory is obviously inadequate here for the determination (or even the explanation of the existence) of the field at P. The situation in Fig. 1-2, however, represents a problem of practical importance as its solution is relevant in evqluating the shielding effectiveness of the conducting wall. Generally speaking, circuit theory deals with lumped-parameter systems- circuits consisting hf components characterized by lumped parameters such ar rcsisl:~~~ccs, i~~ductancbs, :111d ~ ~ a c i l s ~ ~ c e s . VOIIP~CS and currents are the main system variables. For DC circuits, the system variables are constants and the gov- erning equations are algebraic equations. The system variables in AC circuits are time-dependent; they are scalar quantities and are independent of space coordinates. The governing equations are ordinary differential equations. On the other hand, most electromagnetic variables arefunctions of time as well as of space coordinates. Many are vectors with both a magnitude and a direction, and their representation and manipulation require a knowledge of venor algebra and vector calculus. Even in static cases, the governing equations are, in general, partial differential equations. It ' The product of the w&elength and the frequency of an AC source is the vdocity of wave propagation. i . . I , . . . A . <. .- : r NETlC MODEL .3 ' , k : , ? ' ': : the antenna wd 110tlling 11ion can be P b n t bc appr- '-ible O ~ I C I L . .rlcd I X J G C , wliidl udent from tomagnctic ire, that are inadequate e field at P. portance as ucting wall. systems - :ts such as : the main ~d the gov- circuits are oordinates. hand, most atemarly .tation and 1 s . in luations. It f $ is essential that I & arjcct must rcl;~tc LO real-world situahohs and be able to explain physlca~ phenomena; otherw~se, we would be engaged in merltal exercises for no purposc, For example, n thcoret~cal modcl could be bu~lt, from which one might obtain many mathematical relat~ons, but, ~f these relations disagree with observed results, the qbdel is of no use. The mathematlcs may be correct, Btd the hnderlying assumptions o!tfi~model may be wrong or the lmplled approximatiohs d a y not be justified. Three esskntia) steps are ~nvolved ,in bdildin$ a theory on an idealized model. First, some basic duantities germane ta the iubject of study are defined. Second. the rules of operatiod:(the mathematics) of these quantities are specified. Thud, some fundamental relatqns ore postulated. These postulates or laws are invariablj based on numerous exp&mental observatiops acquired under controlled cond~tlons and synthesized by brgliant hinds. A famiJiar example is the arcult theory built on a circuit model bf ideal sources and pure resistbnces, inductances, and oapacltances. In this case the-basic quantities are voltages (V), currehts (I), resistances (R), inductances (L), and'mpct~itadkes (C); the rules of operatiods are those of algebra, ordmary differential equations, qnd Laplace transfordation; and the fundamental postulates are Kirchhoff's voltage apd current laws. Many relations and formulas can be derived lrom this basiwlly fatho< simplc n~odcl,~q~~d the rdpoo~es of very clilborate neworhs can be determined.,The validity and valueof tli'e model have been amply demonstrated. In a like manner, qn, electronlagnetic theory can be built on a suitably chosen electrornagnctic madel. lo this section we shall take thc first step of defining the basic 4 THE ELECTROMAGNETIC MODEL I 1 ,' quantities of electromagnetics. The second step, the rules of operation, encompasses ' vector algebra; vector calculus, and partial differential equations. The fundamentals of vector algebra and vector calculus will be discussed in Chapter 2 (Vector Analysis), where C is the abbreviation of the unit of charge, co~lomb.~ It is named after the French physicist Charles A. de Coulomb, who formulated Coulomb's law in 1785, (Coulomb's law will be discussed in Chapter 3.) A coulomb is a very large unit for .electric charge; it takes 1/(1.60 x 10-19) or 6.25 billion electrons to make up - 1 C. In fact, two 1-C charges 1 m apart will exert a force of approximately 1 million tons on each other. Some other physical constants for the electron are listed in Appendix B-2. The principle of conscrvutic~n O J electric churlye, like thc principlc of conservation of momentum, is a fundamental postulate or law of physics. It states that electric charge is conserved; that is, it can neither be created nor be destroyed. Electric charges can move from one place to another and can be redistributed under the influence of an electromagnetic field; but the algebraic'sum of the positive and negative charges in a closed (isolated) system remains unchanged. he principle of conser~ution of elec- tric charge must be sutisjied at all times und under uny circum.stunccs. It is represented mathematically by the equation of continuity, which we will discuss in Section 5-4. Any formulation or solution of an electromagnetiqproblem that violates the principle of conservation of electric charge .must be incorrect. We recall that the Kirchhoff's current law in circuit theory, which maintains that the sum of all the currents leaving a junction must equal the sum of all the currentscntering the junction, is an assertion and thc techniques for solving partial clilTcrenti;~l cq~~~tioiis will hc inrroc111cc.d wl~c~i thcsc equalions wise later in thc bod,. 'l'lic / / r i d slcp, ll~c I'imlu~iic~ilal pu,\tulntcs, \till be presented in three sbbst+s in Clinpws 3, (1, and 7 as we dcal with, rcspcctivoly, static electric fields, steady magnetic fields, and electromagnetic fields. The quantities in our eleclxoi~~ngnctic niodcl ciln bc divided roughly rftd two catcgorics: source 'r"rL"lle1d qu:uilltics. I'hc source of an electronlagnctic lield is invariably electric charges at rest or in motion. However, an electromagnetic field may cause a redistribution of charges which will, in turn, change the field; hence, the separation between the cause and the effect is not always so distinct. We use the symbol q (sometimes Q) to denote electric charge. Electric charge is a fundamental property of matter and it exists only in positive or negative integral nit~ltiplcs of rllc cliatgc on an clcctroll. c.+ -. -- -. ' In 1962 Murray Gell-Mann hypothesized qtiarkJ as the basic building blocks of matter. Quarks were predicted to carry a fraction of the charge, e, of an electron; but, to date, their existence has not been veri- fied expericnentally. The system of units will be discussed in Section 1-3. e = - 1.60 x 10-l9 (C), (1-1) spectively; , into two - tic field ': c field may henye, the I charge is a ve integral (1 -1) r) XI af' the I W in . ,d5. gc u n ~ t I'ur 2 up - 1 C. nillion tons 1.4ppendix ~nscrv;ilion hat clcaric tric charges i_nfiuence of tive bharges ti011 of c4cc- rcprcscntcd ;corion 5-4. he principle Kirchhoff's ems leaving an n t i o n tompasses AamentBls Analysis), 1 . - ! [;"' ! ' Iced when ulates, will , I ' t - i .. Q~iIrhs were i not bcen veri- current law is th; 4 does not exist at a an atomic scale are unimportant of charges. In conA results. (The same is defira'i f as follows: F 'C I 'Aq p = lim - , (C/m3), A V O AV where Aq is the ambunt of charge in a very smbll volume Av. How small should Ao be? It should be small enough to represent an acctlrate variation of p, but large enough ta contain a very largk nudiber of discrete chnr$es. For exalnple, an elemental cube lbrth I sides as small as 1 micron (lo-' m or 1 pn1) has a volume of 10- l 8 m3, which wrll st111 contain about 10' (100 billion) atoms. A imoothed-out function of space coordinates. p, defined with sudh a small Aa is expecied to yield accurate macroscopic results for nearly all pctical pr<oscs. I ' In somc physical situations, an amount o T chargc Aq may be ident~fied wlth an element ofsurface As or an element ofline h/..tn such cases, it will be more appropriate to define a surface charge density, p,, or,a line charge density, p,: Except for certain bpec:al, situations, charge densities vary from point to point; hence p, ps, and p, are, ih gereral, point functions o f space coordinates. Current is the rate of change of charge dith rcspcct to time; that IS, where I itselfmay dk time-dependent. The unit of current is coulomb per second (CIS), which isihqsanie hs ampere (A). A curretlt must flow through a finite area (a coni ducting wire of a fitite ctoss section, for instafic6); hence it is not a point functlon. Iti electromagneti~s~yk define a vector poipt function uol~mw cur.r.cnr density (or simply, .current density) J, F h i ~ h measures the amouflt of current flowing through a unit area normal to the direc!ion df current flow. The bold-faced J is a vector whose magnitude is the current per uhit arei ( ~ / r n ~ ) and whose direction is the direction of current flow, We shall elnhordteon tbe.relntion between I and .I in Chapter 5. For very good 4 6 THE ELECTROMAGNETIC MODEL I 1 .
4 ,.I " ,< L I . 4 I conductors, high-frequency alternating currents are confined in the surface layer, in- stead of flowing throughout the interior of the conductor. In such cases there is a need to define a surface current density J,, which is the current per unit width on the con- ductor surface normal to the direction of current flow and has the unit of ampere per meter (A/m). I-. . There ark: fo r fundamental vector field quantities in electromgncti~~, electric /iL4/ il~(ol~si~jl I(, d ~ ~ l r i i - / / ~ ~ , ~ hwsi/y (or dwfriy (/~,Y/)/(Ic~~III~~II/) D, I I W ~ I I I P / ~ //IIY (lwsi! Y B. and ~ ~ w g ~ ~ ~ ( i c ~ i e l d i~~( i t. d : . REVIEW QUESTIONS 9 , I ., 1; q . " a - ' 1; ..w j S able 1-3 y?iversal Constints in SI Units 1 , c phenimena 1,' : k . A ux density D Universal &dnstantst ' li em, and they ich is almost 1 h e space is REVIEW QUESTIONS r; R.l-1 What is electromagnetics? '(1 -9) [ ' !
. Velocity of light in free space F (1 -7) Permeability of free bbace i R.1-2 Describqtwo phenomena or situations, otnkr than those depicted in Figs. 1-1 and 1-2, that cannot be adeqlftltely explained by circuit thedry. in &s. (1 -6) he following R.1-3 What are the three essential steps in building an idealized model for the study of a scientific subject? . , dynibof II F L - Po Value 3 x lo8 471 x lo-' R.l-4 What are the four fu?drlmental SI units in efectromagnetics? 8 (1-10) R.1-5 What are the four fundamental field quanthies in the electromagnetic model? What are / their units? Unit m/s H/m I R.1-6 What are. the hhree ukversal constants in tHe electromagnetic model, and what are their relations? I R.l-7 What are the source quantities in the electromagnetic model? 1 Instants and (? Stall., af the the ax well's ppmr i n many ICIOI I n wwld 2 / Vector Analysis 2-1 INTRODUCTION As we noted in Chapter 1, some of the quantities in clectromngnetics (such as charge, currcnt, cncrgy) arc scn1:~rs: and sotnc oll~crs (such ;is clcclric and magnetic licld intensities) are vectors. Both scalars and vectors can be functib'iis-f time and position. At a given time and position, a scalar is completely specified by its magnitude (positive or negative, together with its unit). Thus, we can specify, for instance, a charge of - 1 pC at a certain location at t = 0. The specification of a vector at a given location and time, ,on the other hand, requires both a magnitude and a direction. How do'ive specify the direction of a vector? In a three-dimensional space three numbers are needed, and these numbers depend on the choice of a coordinate system. Conversion of a given vector from one coordinate system to another will change these numbers. However, physical laws and theorems relating various scalar and vector quantities certainly must hold irrespective of the coordinate system. The general expressions of the laws of electromagnetism, therefore, do not require the specification of a co- ordinate system. A particular coordinate systctn is choscn only whcn a probl'em of a given geometry is to be analyzed. For example, if we are to determine the magnetic field at the center of a current-carrying wire loop, it is more convenient to use rec- tangular coordinates if the joop is rectangular, whereas polar coordinates (two- dimensional) will beemore appropriate if the loop is circular in shape. The basic electromagnetic relation governing the solution of such a problem is the same for both geometries. Three main topics will be dealt with in this chapter on vector analysis: 1. Vector algebra-addition, subtraction, and multiplication of vectors. 2. Orthogonal coordinate systems-Cartesian, cylindrical, and spherical coordi- nates. , Ai 3. Vector calculus-differentiation and integration of vectors; line, surface, and volume integrals; "del" operator; gradient, divergence, and curl operations. Throughout the reit of this book, we will decompbse, combine, differentiate, integrate, and otherwisc manipulate vectors. It is imperutive that one acquire a facility in vector 2-2 ' VE( AND SUE 1 i ' . i i such as charge, magnetlc field c 'ind position. lifudc (positive : q r g e of I . cation I n . l i 30 wc : numbers are n. Conversion 11f.q~r t l ~ ~ l ~ l ~ ~ ~ r ~ , -lor quanlitles expressions of itlon of a co- problem of a the magnetic nt to use rec- dinates (two- pe. The basic the same for s u d and :rations. Ire. integrate, lily in vector I ' , .
. ; g h ! ! i " algebra and Vectof cdculus. In a three-dimensional space a vector relation is, in fan, three scalar rel+t.idns The use of vectoi-analisis techniques in electromagnetics leads to concise and'eidgaht '~orrnulations. A hefieiency in vector analysis in the study Of , electromagnet& 1 ; similar to a deficiency .in algebra and calculus in the study of physics; and i f t is qbvious that these deficienilies cannot yield fruitful results. In solving prd~tica~~problems, we plwa$ deal with regions or objects of a given shape, and it is ,necessary to express gqngt.ar'fotipulas in a coordinate system appro- priate for the,$yhn geometry. For ek$m$, % $ familiar rectangll!ar (x, y, : ) cd- oqkiates arei ib$ouslg awkward to pie f@ brodems involving ; circular cylinddr or a sphere, peca4se ,t@ boundaries c $ a ckcular kylinder and a sphere cannot be describdh by kondtant ;.slues of x, y, and z. f f i this dhapter,we discuss the three most commonly~bs'ed dhhogbnai (perpendicular),coordinate systems and the representa- tion and oneratioh of'vcctors in these systems. Familaritv with these coordinate systems is essential in the solution of electrorhagnetic problems. Vector ,~2lculhs pertains to the differentiation and integration of vectors. By defining certiin differential operators, wc ,tan express ~ h c basic laws of electro~ magnetism in a cohcisefway that is invariant with the choice of a coordinate system. In this chapter wc introducc thc techniques for evalu:lting dllkrent types of integrals involvinl: Vectors, kind define and discuss the various klnds of differential operators. . , 2-2 VECTOR A D D ~ T I O ~ 4 AND SUBTRACTlClN We know that a vector has a magnitude and a direction. A vector A can be written as $ where A is the magnitude (and has the unit and dimension) of A, . ' '1 and a, is a dimendionless unit vectort with a unity magnitude having the direction of A. Thus, The vectmAxan be represented graphically by a directed straight-he segment of a length IAl = .-I with its arrnwhcnd pointing irlllhe c\irection ofa ,,as shown in Fig 2-1, .
a 1 wo vcctors are equal if Lhcy have the same hci&iitudc and the same direction, even ..though they may be displaced in space. Sin$ it is difficult to write boldfaced letters by hand, it is a common practice to use an arrow or a bar over a letter (A or A) or ---- -: : ' .. In some books the unit vector ~ r i the direction of A is arml lid) denoted by b, u , , or i,. 2-3 PRODU 1 W O vectors A B, which are nbt in the same direction nor in opposite direc- such 3s given in Fie. 2-2(.1). dctcrmine :I pl:~ne. Their slim is ils,tllt>r ycrtl>l c hlulll In the s;lmc p l ; l ~ . C = A + 15 a l l 1 b~ ubt;liled gr:~pllicllly ill ibvo w;lys, 1. BY the parallelogram rule: Tile resultant C is the di;lgonal ved&~tlle Flr;,llclo- I ' Fclm rorllld IJY A lllld 1) dr;lWll kolll lllc S:lll]c poill(, as sllot\r.ll in Fig :-2(b). It 2. BY the head-to-tail rule: The head of A connects to the tail of B. Their sum c is uct of the vector drawn from the tail of A to the head of B, and vectors A, B, and C form of twc a triangle, as shown in Fig. 2-2(c). -B = (-a,)B. The operation represented by Eq. (2-6) is illustrated in Fig. 2-3. Y" B z A. ." . A A A (a) TWO vectors. (b) Parallelogram rule. (c) Headia-rai] rule. Fig. 2-2 Vector addition, C = A + B. k B cos 1 (a) Two vectdrs. I ; (b] Subtract~on qf ., , r ' This distin- I . . vectors, A -:B. , Fig. 2-3 x Vector subtrd~Tion. rever vectors P . !
.
. A . F . ; ' i i l . ? 5 ' rposite direc- 2-3 PRODUCTS . Q F - ~ X ! ~ O ~ . . . : A her vector C 1 1 1 ~ulti~lication of k vector A by a positive scalar i . changes the magnitude of A b i k litncs wilho~lt chllnging its dircclion (k cJn be either grcatcr or less than 1). I he parallelo- I kA = a,(kA). (2 - 8) 2-2(b). 1 i It is not safficiek to say "the rn~lti~licatiotl of one vector by another" or .'the prod- c1r '.' C is 1 uct of two vectors" because there are two distihct and very diflerent types of pro,ducts :iiTform of two vectors. The$ arc (1) scalar or ddt probucts, and (2) vector or cross products. f ! These will bc defined in the foilowtng subsections. I 1 I 1 :~ulivc laws. 1 2-3.1 Scalar or ~ o t ~roduct (2-4) ! The scalar or dot produci of two vectors A and B, denoted byA . B, is a scalar, whxh (2-5) t , equals the product bf theimagnitudes 0f.A and B and the cosine of the angle between I owing way: them. Thus, i i, - (2-6) i (2-9) k as B, but i 3' In Eq. (2-9), the syhbol 4 signifies "equal by definition" and O , , 1 s the 3mallrr angle between A and B dhd ij less than rr radians (j80c), as indicated in Fig 2-4. The dot product of two vedors (1) is less than or' equhl to the product of their magnitudes; (2) can be either a positive or a negatiYe quihtity, depending on whether the angle between them is smhller or larger than 7t/2 radians (90'); (3) is equal to the product of t Fig. 2-4 1llusktfing the dbt product of A and k. 14 VECTOR ANALYSIS 1 2 , \ the magnitude of one vector and the projection of the other vector~upon the first one; and (4) is zero when the vectors are perpendicular to each other. It is evident that Equation (2-11) enables us to find the magnitude of r vector when the ~~pkessioil of ?he vector is given in any coordinate system: The dot prcduct is commutative and distrilx~tivc, -. , l a , I r , I Co~lmutative law: A B = U A . (2-12) Distributive law: A . (B + C) = A . B + A . C. (2-13) The commutative law is obvious from the definition of the dot product in Eq. 12-9), and the proof of Eq. (2-13) is left as an exercise. The associative law does not apply to the dot product, since no more than two vectors can be so multiplied and an ex- pression such as A - B . C is meaningless. - 1 . Example 2-1 Prove the law of cosines for a triangle. .
, Solutioti: The law of cosines is a scalar relationship that expresses the length of a side of a triangle in terms of the lengths of the two other sides and the angle between them. Referring to Fig. 2-5. we find the law of cosines states that ----- C = a + B2 - 2AB cos a. We prove this by considering the sides as vectors; that is C = A + B . Taking the dot product of C with itself, we have, from Eqs. (2-10) and (2-13). C2 = C . C = ( A + B ) - ( A + B ) - A - A + B - B + 2 A - B = A ' + B ' + 2AB cos U , , . \ , Fig. 2-5 Illustrating Example 2-1. Not (1 8 C and 2-3.2 Vec Hen the c Can corn , r , 1 - ; ,[ : b q the first one! Note that 8 , . is, db debition, the smdlldr Mgle between A and B and is equal t i :nt that , (180" - a); hence, b s @dB = cos (180" +,a) 4 -cos a. Therefore, , > $ , - ,: ? i s 1 (2-10) t 1 C2 = +pr - ; ~ A B Cos a, ' i . , ' and the law of coslher follo~s directly. i / ' - i I \ In Eq. (2-9), I les not apply d and an ex- i 9 The vector or ~-;~$uct 4 f two ve+tdrs A and B, denoted by A x B, is a vectdr plafle containing A and Il: its m?mitude is AB sin O , , , whr.re I).. in ' ; I .
. smclkr -!bglfix betwre:i A m d p, ,;la ~ t s c,irection follows that of t h e 9 2 3 3 of the right hand when tge fingers rotate frontA to B through the angle 8 , , (the right- hand rule.) (2-14) ! This is illustrated ifi Fig. 2-6. Since B sin 8 , , is the height of the parallelogram formed by the vectors A and 8, we recognize that the magnitude of A x B, IAB sin B , , I , which is always positive, 16 numerically equal tdthe Brea of the parallelogram. Using the dcfitlitionrin Eq. (2-14) and following the right-hand rule, we find that - I B x A = - ' A x B . (2-15) i Hence the cross prbduct is nor commutative. w e can see that the cross product obeys the distributive law, 1 Can you show this in: general without resolving the vectors'into rectangulal components? The vector prdduct is obviously noa associative; that is, ; A x ( B x C ) # ( A x B ) x C . 16 VECTOR ANALYSIS 1 2 The vector representing the triple product on the left side of the expression above is perpendicular to A and lies in the plane formed by B and C, whereas that on the right side is perpendicular to C and lies in the plane formed by A and B. The order in which the two vector products are performed is thcrcforc vital and in no case .\houlil rile parentheses be omitted. . Product of Three Vect.2~ I There are two kyds of eio i:m of l h ~ c scows: nmcly, llic ac&s iriplt /~~od~,rr and tl~c tvc~or trrpk yrotluci. 1 ' 1 1 ~ scal:ir triple product is mucli the si~nplcr of the two and has the following property: A . ( B x C ) = B . ( C x A ) = C . ( A x B). (2-18) Note the cyclic permutation of the order of the three vectors A, B, and C. Of course, --- A . ( B x C) = - A . ( C x B) = -C . (B x A). (2-19) As can be seen from Fig. 2-7, each of the three expressions in Eq. (2-18) has a magni- tude equal to'the volume of the parallelepiped formed by the three vectors A, B, and C. The parallelepiped has a base with an area equal to IB. x C I = [BC sin 8 , 1 and a height equal to \A cos O,l; hence the volume is ]ABC sin 8, cos 8 , [ . The vector triple product A x (B x C) can be expanded as the difference of two simple vectors as follows: Equation (2-20) is known as the "hack-cab'' rule and is a useful vector identity. (Note "RAC-CAR" on the right side'of the cq~~iition!) I t ; i -- - ! Fig. 2-7 . Illustrating scalar triple product A . (B x C). .
- - . . ..& . . . . . . - .
.
, ; . '." ..,7,. , , ,. ... . , < ' . , . 6 ' ' . 1 1 , .
.
I . , I ' , . . . . . .- ...A.. . , . . . b , , . ' , . . ' . . 2-3 / PRODUCTS OF VECTORS 17 :ssion above is at on the right order in which - use should the , - tripir product pler of the two i C. Of course, )) ha, .
, ,nagni- :tors A, T3, and >in 0,) and a !I'crence of two identity. (Note Example 2-2t Prove the back-cab rule of vector triple product. Solution: In order to prove Eq. (2-20), it is convenient to expand A into two components where .4,, and A, are, respectively, parallel and perpendicular to the plane containing B and C. Because the vector representing (B x C) is also perpendicular to the plane, the cross pioduct of A, and (B x C' vanishes. Let D = A x (B x C). Since only All is effective her;, ~ Y I : h;ve D =.4,1 X (B X C). Referring to Fig. 2-8, which shows the plane containing B, C, and All, we note that D lies in the same plane and is normal to A,,. The magnitude of (B x C) is BC sin (0, - 0,) and that of All x (B x C) is AllBC sin (8, - 0,). Hence, D = D a, = AllBC sin (8, - 0,) = (U sin O,)(AllC cos 0,) - (C sin 02)(AIIB cos 0,) = [B(AII C) - C(AII B)] .a,. Fig. 2-8 Illustrating the back-cab rule of vector triple product. The expression above does not alone guarantee that the quantity inside the brackets to be D, since the former may contain a vector that is normal to D (parallel to All); that is, D . a, = E . a, does not guarantee E = D. In general, we can write B(A,~ . cj - c(A~, B) = D + k ~ ~ ~ , where k js a scalar quantity. To determine k, we scalar-multiply both sides of the above e q u h n by All and obtain The back-cab ruk can be verified in a straightforward manner by expanding the vectors in the Cartesian coordinate system (Problem P.2-8). Only those interested in a general proof need to study this example. I ' r . . ' 1 ' i I $(. l' , ' % 4 18 VECTOR ANALYSIS / 2 ",,- : , f - . , . . 1 2 , < T Since All . D = 0, so k = 0 and , , I 1 wh D=B(A;II-C)-C(AIIiB), i the the which proves the back-cab rule jnapmuch as All . C = A . and All B = A B. ?
. Division by a vector is not $ejnt!d, ard expr?ssions sluch as k/A and B/A are k I ..- meaningless. qx: (b) 2-4 ORTHOGONAL COORDINATE SYSTEMS , We have indicated before that although the laws of'electromagnetism are invariant with coordinate system, solution of practical problert~s requires that the relations derived from thcsc laws he cspscwil in ;I cocrrdi~~.~lc syshm ;~ppropsi:~~c to lllc geomctry of thc given problems. I:or cxampl~., if wc are to determine the electric field at a certain point in space, we gt least need to describe the posilion of the source and the locatlon of this point in p coordinate system. In a three-dimTnsiona1 space a point can be located as the intersection of three surfaceq. Assume that the three families of surfaces are described by u, = constant, u, = coqstant, and u, = constant, where the u's need not all be lengths. (In the familiar Cartesian pr rectangular coordi- nate system, u,, u,, and u, correspond to x, y, and z respectively.) When these three surfaces are mutually perpendicular to one another, we have an orthogonal coorriinate system. Nonorthogonal coordinate systems are not used because they complicate problems. Some surfaces represented byau, = constant (i = 1,2, or 3) in a coordinate system may not be plancs; they may b~ curvkd surhccs. Let a,,, a,,, and a,,, bc the unit vectors in the three coordinate directions. They are callqI the buse vectors. In a general right-handed, orthogonal, curvilinear coordinate system, the base vectors are arranged in such a way that the following relations ace satisfied: a,, x a,, = a,, , I (2-21a) a,,, >j a,, = a,, I I (2-21 b) , aK3:x a,, = a,. ' (2-21c) These three equations are not alliqde&4dent, as the spbificaiion ofone automatically implies the other two. We have, qt cqube, I '- . 1 and Any vector A can be written as the sum of it$ cornpanents in the three orthogonal . directions, as follows: " \ ! I 1 ' I 11 perfo 2-4 / ORTHOGONAL COORDINATE SYSTEMS 19 where the magnitudes of the three components, A,,, A,,, and A,,, may change with the location of A; that is, they may be functions of u,, u,, and u3. From Eq. (2-24) the magnitude of A is = A . B . : A = /A/ = (A;$ + A ; ~ + A;~)'". (2-25) .nd S/A are Example 2-3 Given three vectors A, B, and C, obtain the expressions of (a) A . B, I (b) A x B, and (c) C (A x B) in the orthogonal curvilinear coordinate system a / ( ~ 1 , U2,"3). I ,re invariant 1. Solution: Firs? we write-A, B and C in the orthogonal coordinates (a,, u,, u3): , he relations 1 A = a,,,Aul f %,Au2 + %,A,,, ricite to the clcctric iicld \ ,re invariant I. Solution: Firs? we write-A, B and C in the orthogonal coordinates (a,, u,, u3): , he relations 1 A = a A + s A + aU3A,,, f the source I ma1 space a at the three L = AuIBul + Au2Bu2 + Au3Bu3, (2-26) in view of Eqs. (2-22) and (2-23). 1 the?- three I ~ w t r b , A x B = (%,A,,, + ~,,,A,,, 4- %1/1,,1) x (a,,,B,, + a,,,B,,, + a,,,B,,,) co~npl~cllw sx %I( A112u143 - A l l , ~ u 2 ~ 4- ~ ~ l , L ( ~ u L ~ ~ , , , - ~ ~ ~ l j , , ~ -I- ,(A,,,LY~~ - Au2Bul) mate system be the unit ecrors. In a 1 : vectors are f (2-21a) Equations (2-26) and (2-27) express, respcctivcly, the dot m d cross products of two vectors in orthogonal curvilinear coordinates. They are important and should be remembered. - i (2-21b) k c) The expression for C (A x B) can be written down immediately by combining ! (2-21c) the results in Eqs. (2-26) and (2-27). 1 - Eq. (2-28) can be used to prove Eqs. (2-18) and (2-19) by observing that a permutation of the order of the vectors on the left side leads simply to a rearrange- , ment of the rows in the determinant on the right side. (2- 24) In vector calculus (and in electromagnetics work), we are often required to perform line, surface. and'volume integrals. In each case we need to express the i 1 20 VECTOR ANALYSIS 1 2 % I I r I I I differential length-change coriesponding to .a differe'ntial ~hange in one of the co- ordinates. However, some of the coordinatesy say u,. (i = 1 , 2, or 3), may not be a length: and a conversion factor is needed to convert a differential change du, into a change in length dt, : (I/', = It, (Ill,, ( 2 20) S 1 where hi is called a metric coefic(en( aid may itrclf L a fynction of u,, u,, and u,. For example, in the two-dimensional polar coordinat& (u,, h) = (r, 4), a difrcrential change d4 (=du2) in 4 (= 7lZ) cor&onds to a differential length-ct~ange d l = r rl$ (h2 = r = u,) in the a+(=-:~~,)-~!ir~~cIio~l. A dirc~tc,l dillhc~~ii:~i I C I I ~ I ~ I ~ - I ~ : I I , ~ c i l l :III Ill1 arbitrary direction can be wrilkn i\s tbevec~or sum uflllc c.oo;pollmt lct~~tll cIlanses:t dt = a,,, dCI + aU2 dt2 + a,, d& . . = [(h, du!)' + (h2 h,)' + h, J u , ) ' ] ' ~ ~ . (2-32) a he differential volume dv formed by differential coordinate changes du,, du,, and in directions a.,, a,, and a,, respect~vely is (dll dt2 d!,), qr Later we will have occasion to express the current or flux flowing through a differential area. In such cases the crdss-sectional area perwndicular to the current or flux flow must be used, and it is convenient to consider the Pifferential area a vector with a direction normal to the surface; ihat is. ds.= a, ds. .
I 1 I I I For instance, if current de;sity J ia ndt pcrpcndiculnr to a diLrcntinl area of a ,nag- nitude ds, the current, dl, flowing through ds must be the ckponent of J normal to the area multiplied by the area. Usingthe notation i n Eq. (2-34), we can write simply ( ~ r ; = J JS : I . $ , J a,&. (2-35) In general orthogonal curvilinear coordinates, the 4iff(rentiil area ds, normal to the unit vector a,, is . : . I ,_I ' ds1 = a,,(dd, dt,) ' , J $ 4 , ' T h i s e is the symbol of the vector d. 9 , I ( ! I . , I , I . .I , 1 1 1 . 'p the 1. Thi 2-4.1 Car A r spec syst , , .
. 2-4 1 ORTHOGONAL COORDINATE SYSTEMS 21 ' me of the co- t 4t r ,Z or may not be a :$ F- I . L f nge dui into a c 1 , ; (2-36) - u,, and u,. . a differential ge dl2 = rd$ change in an gth changes:' (2- 30) ig through a 3 the current area a yector .ea of a mag- J normal to write simply n - 3 5 ) ormal to the Similarly, the differential area normal to unit vectors a,, and a,, are, respectively, and ds, = a,,(h,h, du, du,) I ds, = au,(hlh2 du, du,). I (2-37) Many orthogonal coordinate systcrns exist; but we shall only be concerned with the three that are most common and most useful: I. Cartesian (or rectangular) coordinate^.^ 2. Cylindrical coordinates. 3. Spherical coordinates. These will be discussed separately in the following subsections. 2-4.1 Cartesian Coordinates A point P(xl, y,, z,) in Cartesian coordinates is the intersection of three planes specified by x = x,, y = y,, and z = z,, as shown in Fig. 2-9. It is a right-handed system with base vectors a,, a,, and a, satisfying the following relations: a, x a, = a, \ a, x a= = a, a, x a, = a,. The position vector to the point P(x,, y,, 2,) is A v e w in Cartesian coordinates can be written as - The term "Cartesian coordinates" is preferred because the term "rectangular coordinates" is custornarlly associated with two-dirnensio'na~geomztry. I 6 . . 22 VECTOR ANALYSIS 1 2 I Fig, 2-9 Cartesian coordinates. - -.--- The dot product of two vectors A and B is, from E & (2-26), A t B = AxBx + AyBy + A&, (2-42) and the cross product of A and B i$ from Eq. (2-271, f Since x, y, and z are lengihs themselves, all three etric coefficients are unity; ill that is. = h 2 = h, = 1. The expr&ons for the differe tial length, differential area, and differential volume ark - Iroy Eqs. (2-31). (2-36), $-37). (2-38), and (2-33) - respcctivcly, . . (2-45a) ds, = a, dx dz (2-45b) ds, = a, dx dy; , (2-45c) (2-46) -, t 1.: Example 2-4 A scalar line integral of a vector field of the type jp: F . dt' is of considerable importance in both physics and electromagnetics. (If F is a force, the integral is the work done by the force in moving from P1 to P2 along a specified path; if F is replaced by E, the electric field intensity, then the'integral represents an I electromotive force.) Assume F = a2;y + aY(3x - y2). Evaluate the scalar line E, t integral from P,(5,6) to P2(3, 3) in Fig. 2-10 (a) along the direct path @, PIP2; then I. (b) along path @ , P,AP,. P s are unity; rentid area, .
- ld (2-33) - Fig. 2-10 ' Paths of integration (Example 2-4). Solution: -First we must write the dot product F . d t in Cartesian coordinates. Since this is a two-dimensional problem, we have, from Eq. (2-44), It is important to remember that dt' in Cartesian coordinates is always given by Eq. (2-44) irrespective of the path or the direction of integration. The direction of integration is taken care of by using the proper limits on the integral. Along direct path a - The equation of the path PIP2 is This is easily obtained by noting from Fig. 2-10 that the slope of the line P I P 2 is f. Hence y = ($)x is the equation of the dashed line passing through the origin and parallel to PIP,. Since line inte intersects the x-axis at x = + I, its equztion is that of the dashed line shifted one unit in the positive x-direction; it can bs obtained by replacing x yith (x - I). We have, from E ~ S . (2-47) and (2-48), Spy F .dP = Spy [xy dx + (3x - y2) dy] Path @ Path t In the integration with respect to y, the relatioq 3x = 2y + 3 derived from Eq. (2-48) was usede< b) Along path @ - This path has two straight-line segments: From P , to A : x = 5, dx = 0. From Ato P 2 : y = 3 , d y = 0 . I ; . d t = 3x d x , Hence, Path We see here that the value of the line integral hepen+ on the path of integration. In such a case, we say that the vector field F is not conservative. ', . I 2-4.2 Cylindrical Coordinates ' In cylindrical coordinates a point ~ ( r , , 4,. i,) is the intersection of a circular cylin- drical surface r = r,, a half-plane cpntaining the z-axis hnd making an angle 4 = 4, with the xz-plane, and a pl ne frallel to the xy-plane at z = 2,. As indicated in 't Fig. 2-11, angle 4 is measured from the x-iuk and the base vcctor a, is tangential to the cylindrical !vrf&e. The following right-hand relations apply: , I . d ' " '^' ' I ri integration. rcular cylin- ngle 4 = 4, indicated in v e r p a , is 1Pk 9 . ( ja) (2-49 b) (2-49~) x- 9 = 91 plane ' ' 4 F i g . 2-11 Cylindrical coordmates. Cylindrical wordinales are important for problsms wilh long line charges or curmnt,, and in places where cylindrical or circular boundaries exist. The two-dimensional polar coordinates arc a special cusc at z = 0. A vector in cylindrical coordinates is written as The expressions for the dot and cross products of two vectors in cylindrical coordi- nates follow from Eqs. (2-26) and (2-27) directly. Two of the three coordinates, r and r (u, and u,), are themselves lengths; hence h , = h3 = 1. However, q 5 is an angle requiring a metric coefficient h, = r to convert d$ to d 4 . The general expression for a differential length in cylindrical coordinates is then, from Eq. (2-31): (2-51) The expressions for differential areas and differential volume are ' and ( 1 26 VECTOR ANALYSIS / 2 ' A typical differential volume element at a point (r, 4, 3 resulting from differential changes dr, d4, and dq in t& three orthogonal coordinate dir2tions is shown in Fig. 2-12. A vector given in cylindrical Fodrdinates can be lranlformed into one in Cartesian ' coordinates, and vice versa. Spppose we want to expresi A = aJr + a d r n + a,A, in I Cartesian coordinates; that iq, w~ want to write A as a,A, i a,A, + a,A, and deter- mine A,, A,, and A,. First ofall, we note that A,, the I-component ofA, is not changed by the transformation from cylindrical to Cartesian coor inates. To find A, we equate 4 - the dot products of both expressions of A with a,, Thus, i , A x =.A . a, , 5 Ara, ax + Ada, A,. i I The term containing A, disappei)rs here because a, a, = 0. Referring to Fig. 2-13, which shows the relative position's of the base vectors q , , a,, ar, and a+, we see that ' a4.p,=L"s(:+4)=:-sis. 1 Hence, i I. 41 = A. cos 4 - Ad sin 4 . :. II 0 . i ------ . !' I !! Fig. 2-13 Relations between %,,a,, a,. and a. ji : . . 1. I I ' I 1 differential is shown in Fig. 2- 13, we see that . - 2-4 1 ORTHOGONAL COORDINATE SYSTEMS 27 f Similarly, to find A,, we take the dot products of both expressions of A with a,: A, = A a, = Arar . a, + A,a, . a,. From Fig. 2-13, we find a, . a, = cos ( : - $) f 4 and a , . a, = cos 4. It follows that It is convenient to write the relations between the components oia vector in Cartesian and cylindrical coordinates in a matrix form: Our problem is now solved except that the cos 4 and sin 4 in Eq. (2-60) should be converted into Cartesian coordinates. Moreover, A,, A,, and A, may themselw be functions of r, 0, and z. In that case, they too should be converted into functions of x, y, and z in the final answer. The following conversion formulas are obvious from Fig. 2-13. From cylindrical to Cartesian coordinates: y = r sin $J z = 2 . The inverse rilations (from Cartesian to cylindrical coordinates) are , % , 2 1 28 VECTOR ANALYSIS I 2 1 : : t r 2 - , . r Example 2-5 Express tbe yect& ' , 4 = a,/3 cos 4) - ia$r + azs in Cartesian coordinates. ,.. Solution: Using 0 or A = a,(3 cos2 q 5 + Zr sin@) + a,(3 sin d, cos Q - 2r cos 4) + n,5: 1 But, from Eqs. (2-61) and (2-62), and sin 4 = Y . , Jq' Therefore, , i 1 .
I i 3x2 I , , 1 ; which is the desired answer. i : i along the quarter-circle showp in Fig. 2- 14. I " 1 ; J , 1 : I i I . rig, f t14 Path for line integ~pl (Exgmple 2-6). - t 2-4 / ORTHOGONAL COORDINATE SYSTEMS 29 a Solution: We shall solve this problem in two ways: first in Cartesian coordinates, k - then in cylindrical coordinates. f.: C I a) In ~ a r t e & z n coordinates. From the given F and the expression for dC in Eq. I (2-44), we have I' F . dC= xy dx - 2x dy. L i t The equation or the quarter-circle ie x2 + y2 = 9(0 I x, y I 3). Therefore, k ~ ~ -d t = ~ ~ ~ x ~ = ~ d x -2 ~ ~ ~ I 1 0 = +g 3 - x 2 ) 3 / 2 1 - [ y ~ w + g s i n - ' q 1: 3 b) In cylindrical coorrlinates. Here we first transform F into cylindrical coordinates. n Inverting Eq. (2 -5), wc Il;lvc I i > i I , With the given F, Eq. (2-63) gives L i cos 4 0 I which leads to T -sin 4 cos (b (4 sin 4 cos 4 0 sin 4 cos 4 0 t F = a,(xy cos 4 - 2x sin 4) - a,(xy sin 4 + 2x cos 4). 1 For ik-present problem the path of integration is along a quarter-circle of a ; radius 3. There is no change in r or z along the path (dr = 0 and dz = 0 ) ; hence 0- Eq. (2-51) sinlplifies to i dC = a93 d4 t : and . . 30 VECTOR ANALYSIS I 2 I In this particular example, F is given in Cartesian coordinates and the path is circular. There is no compelling reasoli to solve tho problen~ in onc or the other coordinates We havc sliow~~ tlle co~lversin~l or v c c h ~ s i~nd the roccd~~rc o(sollll\on in both coordinates. r over the surface of a closed cylinder about the z-axis specified by z = & 3 and r = 2, as shown in Fig. 2-15. S o l i o ~ : In connection with Eq. (2-34) wc noted that the direction of L is normal to the surface. This statement is actuplly~imprecise because a normal to a surface can point in either of two directions. No ambiguity would arise in Eq. (2-35), since the choice of a,, simply determines the reference direction of currebt flow. In the present case, where F . ds is to be integrated over a closed surface (denoted by the circle on the integral sign), the direction of ds is always to be taken qs that of the oir~~vard normal. Our problem is to carry out the surface integral , O V l 1 : t the Fig. 2-15 jA cylindrical Surface (Exapple 2 7 ) . :ration. Along ci the path is o r the othcr c 01 mlution k I> 11or1n;il surface can 51. since the the present he circle on .he out\c.rrrd 2-4 1 ORTHOGONAL COORDINATE SYSTEMS 31 over the entire specified surface. This integral gives the net outwardpw of the vector F through the enclosed surface. , The cylinder in Fig. 2-15 has three surfaces: the top face, the bottom face, and ' the sidc wall. So, ..-.. We evaluate,the three integrals on the right side separately. a) Top fa e. z = 3, a, = a, F a, = k2z = 3k2 ds = r dr d+ (from Eq. 2-52); lop F a, ds = C" So2 3k2r dr d q 5 = 12nk2. face b) Bottom fnce. 2 = - 3, all = -az 1 7 11,~ = -I;?: = 3k2 ds = r dr d4; 5;~,~l,,l,l F. i t , , ~ = 1 2 n / < ~ . I ~ L C which is exactly thc samc as the integral over the top face. c) Side wall. r = 2, a, = a, ds = r d4 dz = 2 d4 dz (from Eq. 2-52a); F . a, ds = f : 3 So2" k, d4 dz = 12nk,. side wall Therefore, $F ds = 12xk2 + 12nk2 + 12nk, 2-4.3 Spherical Coordinates A point P(R,, dl, 4,) in spherical coordinates is specified as the intersection of the following three surfaces: aspherical surface centered at the origin with a radius R = R,; a right circular cone with its apex at the origin, its axis coinciding with the z-axis VECTOR ANALYSIS / 2 Fig. 2-16 Spherical coordinates. -1 -- and having a half-angle d = dl ; and q half-plane containing the z-axis and making an angle 4 = 4, with the xz-plane. The pose vector aR at P is radialfrom the origin and is qzrite different /ram a, in cylindrical coordinates, the fatter being perpendicular to the z-axis. The base vector a, lies in the q5 F 4, plane and is t4pge&l to the spherical surface, whereas the base vector a, is the same as that in cylindrical coordinates. These are illustrated in Fig. 2-16. Far a right-handed system we have Spherical coordinates are importanr 'for problems :involving point sources and regions with spherical boundaries. When an observer isvery far from the source region of a finite extent. the latter could be considered as the origin of a spherical coordinilte system; and, as a rcsult, suitiihlc simplifying ;~ppn)xim:ttions coi~J(l hc m;~clo. 'l.ili5 i the rcason that spllcrical ioordinatesnrc i~rcd in solving antcn~~a problems in thc far field. Thc ' ent ant A vector in spherical coordinates iswritten as - ' . , ~ F O mi = a,AR -t a,A, + a A (2-65) or ( The expressions for the dot and crgss products of two vectors in spberi~al coordinates can be obtained from Eqs. (2-26) ~nd,(2-27). In spherical coordinates, only R(u,) is a kngfh. ?he other two coordinates, B and 4 (u2 and u,), are angles. Referring to Fig. 2-17, w t q e a typical differential volume element is shown, we see that metri~ coekcients h, = Rand h3 = R sin B are required k n g an urr 'I. tc, .e >her; hate3. 2-64a) 2 -64b) 2 -64c) :s and region dilute This is he fhr 2-65) n nates es, e ume lired Fig. 2-17 A differential volume element in spherical coordinates. 'df = a, dR + a,R dO + a,R sin 0 (2-66) The expressions for differential areas and dikential volume resulting from difler- entlal changes @, do, and dm in the three coordinate directions are and F O : convenience the base vectors, metric coefficient$ and expressions for the differ- entlal volume are tabulated in Table 2-1. A ,VecfhgiWn in spherical coordinates can be transformed into one in Cartesian Or c~llndrlcal coordinates, and vice versa From Fig. 2-17, it is easily seen that (2-69a) Y = R sin 0 sin 4 (2-69b) (2-69c) I . 1 t 34 VECTOR ANALYSISIF . ! ; . . , - - , 1 \ I / I I Table 2-1 Three Basic Orthogonal Coordinate Systems < Coordinate-system Relations a . -- P - Y-ctors. - h 1 Metric Coefficients h, Conversely, measurements in Cartesian coordinates can be transformed into those in spherical coordinates: Cartesian Cylindrical Spherical Coordinates Coordinates . Coordinates (x, Y, 4 (r3 4 7 4 (R, 8 , d - ax a, a~ a~ a, . Pz a : a, 1 1 1 1 . r R 3 I 1 ' 1 i R sin 0 . Example 2-8 The position of a point P in spherical coordinates is (8, 120°, 330"). Specify its location (a) in Cartesiali coordinates, and (b) in. cylindrical coordinatcs. Solution: The spherical coordinates df the given poipt are R 8; 0 = 120°, and 4 = 330". .- Differential dc Volume a) In Carresian coordinates. We use Eqs. (2-69a, b, c): !' : . '-4 ' , .. ! . dx d y & ' r. dr d4' d; R 2 sin 0 dR dU dd x=8sin120°cp~330"=6 ,. . y = 8 sin 12 " &-'330° = - 2 8 ' ' r i P 1 . z = 8 cos 120' F .-4. 1 , Hence, the location of the point is t ( 6 , 72J?, -4), and the position vector (the vector going from the origin to the point) is , 1 . . 2 . b ) I1 ' b ! dl c;i M I t coonl n ; P C . c n ~ r d , j Esam ordin: Solutl! This i a poir all po functic definir in gcn produ Recgl; vector 0 - 6 9 , < \ - 3n a, a, 1 I- R R sin 0 i I 0 d R dO d4 fl into Iiosc 120', and i 0 v 1 > - cector (the 2-4 / ORTHOGONAL Cf30RDINATE SYSTEMS 35 ' I , b) In cylindrical coordinates. The cylindrical coordinates of point P can be obtained by applying Eqs. (2-62a, b, c) to the results in part (a), but they can be calculated a directly from the given spherical coordinates by the following relations, which can be verified by comparing Figs. 2- 11 and 2- 16: r = R sin 8 (2-71a) +=(b (2-/lb) z = Rcos0. ( 2 4 1 ~ ) We have ~(4@,330", -4); anc. its position vector in cylindrical coordinates is It is interesting to note here that the "position vector" of a point in cylindrical coordinates, unlike that in Cartesian coordinates, does not specify the position of thc point exactly. Can you write down thc position vector of the point P in spherical coordinates? Example 2-9 Convert the vector A = a,A, + a,& + agAg into Cartesian co- ordinates. a S o / f ~ ~ h ) ~ ~ : 111 111is J~I~OI)IC~II wc W ~ I I I to w r i t c A ~II llic rorm oSA ..- a,,I, i- : I ~ / I , , -t ; I ~ . & I : . This is very diflerent from the preceding problem of converting the coordinate. of a point. First of all, we assume that the expression of the given vector A holds for nll points of interest and that all three given components A,, A, and ,A4 may bz functions of coordinate variables. Second, at a given point, A,, A,, and A, will have definite numerical values, but these values that determine the direction of A will, in general, be entirely different from.the coordinate values of the point. Taking dot product of A with a,, we have Recalling that a, - a,, a,. a,, and a,. a, yield, respectively, the comDonent of unit vectors a,, a,, and a+ in the direction of a,, we find, from Fig. i-16 and Eqs. (2-69a, b, c): 1 . X - a, . a, = sin 8 cos 4 = pfy2f=:! (2-72) XZ a, a, = cos 0 cos + = (2-73) J(x2 + y2)(x2 + y2 + z2) Y :I,,, 11, = -sill ( I x -. . - . --.-. (2 - 74) Js? +-j+ ' .
v 36 VECTOR ANALYSIS / 2 ! I ) I Thus, ; i I A, ='AR sin 8 cos 4 + A, cos 8 cos ) -,A, sip 4 Similarly,. i AZ = A, cos 8 - A, sip 0 = A R ~ - - Jx2 + Y 2 + z2 J A'm . . (2-77) xZ + y2 + z2 If A,, A,, and A, are themselves functions of R, 0, and 4, they t ~ o need to be converted into functions of x, y, and z by the use of Eqs. (2-70% b, c). ~ ~ S G o n s (2-75), (2-76), and (2-77) disclose the fact that when a vector has a simple form in one coordinate system, its conversion into apotber coordinate system usually results in a more complicated expression. I ) Example 2-10 Assuming that q.cloud of electrons copfined in a region between two spheres of radii 2 and 5 ctqhag,q charge density of ' 4 find the total charge contained in the region. , . . , Solution: We have I 3 x 10-8 i = - ----1 { 4 -- c w 2 ,I), . The given conditions of the pmbled obviously point to the use of spherical coordi- nates. Using lhc crprcssioo b r du ia Eq. (2-6S), wc $crfoqn a triplc it~tcgration. Q = So2$ Sn3'S0'O5 o! 0.02 pR2 sin B ; ~ R dg d). I Two things are of importance here. First, since p is giver in units of coulombs per cubic meter, the limits of integ~ation for R must be converted to meters. Second, the full range of integration for 0 ii from 0 to n radians, not from 0 to 2 7 1 radians. A little reflection will convince up that a half-circle (not a full-circle) rotatcd about thc z-axis . I r"" pos ' -. CL, l .
4 We at a are tior poi! rep: OUS dep line con cha tllc 1 coordi- n. n mbs per :ond, the ;. P, little hc :-axis 2-5 1 GRADIENT OF A SCALAR FIELD 37 ' through 2n radians (4 from 0 to 2n) generates a sphere. We have 0.05 1 Q = -3 x So2" S : So.o2 zcosz 4 sin 8 dR dB d+ 2-5 GRADIENT OF A SCALAR FIELD In electromagnetics wc have to dcal with quantities that dcpcnd on both time and position. Since three coordinate variables are involved in a three-dimensional space. we expect to encounter scalar and vector fields that are functions of four variables: (r, i l l , 1 1 , , 11,). In general, thc fields may change as any one of the four variables changes. We now address the method for describing the space rate of change of a scalar ficld at a given time. Partial derivatives with respect to the three space-coordinate variables i~rc involvecl i ~ i ~ l . ~ I I : I S I I I I I C ~ I :IS ~ I I C r:~lc O I C I ~ ~ I I I E C III:I~'I~C tlilBrc111 i l l tlillircnt ilirsc- (ions, a vector is needcd to delinc the space rate of change of a scalar field at a gibes point and at a given time. - Let us consider a scalar function of space coordinates V(LL,, u2, u,), which may represent, say, the temperature distribution in a building, the altitude of a mountain- ous terrain, or the electric potential in a region. The magnitude of V , in general, depends on the position of the point in space, but it may .be constant along certain lines or surfaces. Figure 2-18 shows two surfaces on which the magnitude of V is constant and has the values Vl and Vl + dV, respectively, where dV indicates a small change in V . We should note that constant-V surfaces need not coincide with any of the surfaces that define a particular coordinate system. Point PI is on surface V, ; P, 38 VECTOR ANALYSIS 1 3 L I / is the corresponding point on surface Vl + dV along the normal vector dn; and P, is \ a point close to P, along another vector dP P dn. For the same change dV in V, the , space rate of change, dV/d/, is obviously greatest along dn because dn is the shortest distance between the two surfa~cs.~ Since the magnitude of dV/dL depends on the direction of d€, dV/d& is a dipqional derivative. We define the vector that represents both the ?nagnitude and the dtrection of the maximum sppcc rate of increase of a scalar ' as the gradient of that scalar, We write gradV !
A a,, -. liV' dn I (2-78) For brevity it is customary t~ employ tllc opcralor tlcl. rcprcscntcd by tllcsymbol V and write VV in place of grad 1'. Thus. ; We have assumed that dV is positive (an increase in V ) ; if dV is negative (a decrease in V from PI to P,), VV will 6e negative in the a, direction. The directional derivative along d€ is dV dVdn dV - -=--- -- (te cln d/ dn cos c? - d 1 ' -r - a,. a, = (VV) . a,, dn (2-80) Equation (2-80) states that the space rate of increase of' y in the a, direction is equal to the projection (the compor)ent) of the gradient of V iq that direction. We can also writc Eq. ( 2 80) (2-81) where ti€ = a, df. Now, dVin Eq, ( is the total di rential of V as a rcsuit of a fFe change in position (from P , to P; 2-18); it B n expressed in terms of the differentia1 changes in coordinates: Eq. the where d l l , df,, and d t , are the components of the vector differential displacement a d t in a chosen coordinate system. Ih terms of general orthogonal curvilinear coordi- ' i In a more formal treatment, changes A V ' ~ O ~ A ! would be used, and the ratio A V / N would become the ' der~vative dV/dC as hc fpproaches zero. We-avold this formality in fyor of s~mpl~ctty. . J Loc in ; defi .... .;.,3.wi? -, ; ; & + $ ; F i ; ; ; .
i ; v a ; , .
; ; .;;.,~&;.:&:y$%; . . . .". .. : , ; ; .
. i , . > . . . ' .
, - ,, . ., . . - . > - . ' ., ,,.',<, f . . . ; p $ P j; 2-5 1 GRADIENT OF A SCALAR FIELD 39 tnd P3 is in I / , the shortest s on the rprcscnts fa sculur nates (u,, u,, u,), dP is (from Eq. 2-31), dP = a,, dd, + a,, de2 + a,, dt, = aU,(h du1) + %,(h2 du2) + a U , h du3). (2-83) It is instructive to write.dV in Eq. (2-82) as the dot product of two vectors, as follows: av = + a,, - + a,, (a,, dl, + a,, df2 + a,, di,) ( , ae2 Comparing Eq. (2-84) with Eq. (2-81), we obtain av av av V V = a u , - + a - + a - at, u2 at2 U3 ad3 I I Equation (2-86) is a useful formula for computing the gradient of a scalar, when the scalar is given as a function of space coordinates. In Cartesian coordinates, (u,, u,, u,) = (x, y, z) and h , = h, = h, = 1 , we have I 4 1 is equal C311 SO In view of Eq. (2-88), it is convenient to consider V in Curreslun coordinutrs as a W I ~ vector difkercntial opcrator. Looking nt Eq, (2--Sh), onc is tc~np(ctl to dclinc V ;IS I r coordi- occoine the in general orthogonal coordinates, but one must refrain from doing so. True, this , , definition would yield a correct answer for the gradient of a scalar. However, the . .
. I I 40 VECTOR ANALYSIS I 2 same symbol V has been used cqnventionally to signify same differential operations of a vector (divq-gmcc and curl, which we will cansiber ]vier in this chapter), where . an extension of V as an operator i p g~neral orthogonal coordinates would be incorrect. Example 2-11 he electrostatic fiild intensity E is derivable as the negative gradient of a scalar electric potential V; t h a t ' i ~ , ~ ~ = - VV. Determine E at the point (1, I, 0) if '=Y . , a) V = Voe-" sip ;--, .. 4 I b) V = V o R cos 9. Solution: We use Eq. (2-86) to evaluate E = -VV in Cartesian coordinates for part (a) and in spherical coordinate6 for part (b). '=Y = (ax sin 7 - ay cos y) vOe-%. where I In view of Eq. (2-17), the rpsult :above converts very'simply to E = - azVo in Cartesian coordinates. This is riot surprising sincena careful examination of the given V reveals that V o R cos 13 is, F n fact, equal to V,r. 1n Cartesian coordinates. 216 DIVERGENCE OF A VECTOR F I C L ~ i i. h . , of ' sci by linc nit the ... . nor a fie1 wit the a nt and (\I' I m'o i J?t ,The the , surf.
w h i van that tY PC ink: d~vi "PPr n d i i t f "'x, , - .-. - differ can t i 1 . In the preceding section we con6idqed the spatial derivatives of a scalar field, which led to the definition of the gradi~nr We now turn our attention to the spatial deriv- atives of a vector field. This will lead to the definitions of T h e divergence and the curl . 1 I t I I' , ' 1 .;,., ; L , ,i&$?.jr:p'.g u.. . A ' . @ t .c"zi- ,$.'. ; , .., . ,' , .
. ,<,? . , , , ,,!.:,. - , .- +,, y & .
. :-:. , ../ ' ,A;.!-:~ .:'::,' > i;! 3~;';$5g#$g~&~&~~~g; ... . , . . . . . .- ', , ..{;;Y,$'',.:,$, "2,7 , .. ' ' . ,%,. ., . , . . , . 2-6 / DIVERGENCE OF A VECTOR FIELD 41 p $5 - perations f& r), where , @ ncorrect. . V hrj . gradient fi" 'f. > {I, 1,O) if k -a,VO in ) I 1 of the xdinates, 0 I Id, which 1x1 dcriv- 1 the curl of a vector. We discuss the meaning of divergence in this section and that of curl in Section 2-8. Both are very important in the study of electromagnetism. In the study ofvector fields it is convenient to represent field variations graphically by directed field lines, which are called Jlux lines or streamlines. These are directed , lines or curves that indicate at each point the direction of the vector field. The mag- nitude of the field at a point is depicted by the density of the lines in the vicinity of the point. In other words, the number of flux lines that pass through a unit surfxe normal to a vector is a measure of the magnitude of the vector. The flux of a vector field is analogov? to the flow of an incrinpressible fluid such as water. For a volume with an enclosed surface there 4 1 be an excess of outward or inward flow throu h 2 . - w y L t o w 2 .
i f !
.
.
, . the surficp only when tne volume contalns, respectively, a murce or a=, that IS, a net pos~tlvr, L,l v b -genre ; ~ x i i a ~ t e s the prrsence o f a sourct of fluid inside the volume, ,aid ci net negative divergence indicates the presence of a sink. The net outward flow ' of the fluid per unit volullle is therefore a measure of the strength of the enclosed source. Qra& r We dejine the divergence of a vector field A at a point, -- abbreviated div A, as the net outwurdfiux o f A per unit volume as the volume ~ b o u t the point tends to zero: $ s A ds div A lim AV-o AV ' The numerator in Eq. (2-go), representing thk net outward flux, is an integral over the entire surface S that bounds the volume. We have been exposed to this type of surface intcgral in Examplc 2 -7. Equation ( 2 -90) is thc gcncral dcfin~l~on of d ~ v A which is a scalar quuntity whose magnitude may vary from point to polnt as A itself varies. This definition holds for any coordinate system; the expression for div A, like that for A, will, of course, depend on the choice of the coordinate system. At the beginning of this section we intimated that the divergence of a vector is a type of spatial derivative. The rc:tdcr niny pcrll:qx wondcr bout thc prcbcncc of ,\n integral in the expression given-by Eq. (2-90); but a two-dimensional surface integral divided by a three-dimensional volume will lead to spatial derivatives as the volu~ne approaches zero. We shall now derive the expression for div A in Cartesian co- ordinates. Consider a differential volume of sides Ax, Ay, and Az centered about a point P(xo, yo, 2,) in the field of a vector A, as shown in Fig. 2-19. In Cartesian coordinates, A = a,A,-+a,& + a,A,. We wish to find div A at the point ( s o , yo, 2,). Since the dikrentii~l volumc has six fxcs, thc siirfucc intcgrnl in the nulncrator of Eq. (2-90) can be decomposcd into six parts. 42 VECTOR ANALYSIS / 2 . . I 1 , Fig, 2 ~ 1 9 A differential volump in Cattesp coordi~:. ;-: , i On h e front f a x , . 1.,,,,. A ' ds = A f , .
; . As front = Afro., . a.(Ay Az) face f a p face faoe ---. The quantity Ax([xo + (Ax/Z), YO, Z O ] ) can be expanded as a Taylor series about its value at (x,, yo, z,), as follows: % + higher-order term4, (2-93) where the higher-order terms (f4.q.T.) contain the factors AX}^)^, AX/^)^, etc. Similarly, on the back face, ' I . I Ax d/l, = Ax(xU, yo, z ~ ) i- - 4~ YO, ZO) - - - Ax + H.O.T. (;-95) 2 'ax (sq, yo. a , ) 'substituting Eq. (2-93) in Eq. (i-91) and Eq. (2-95); in ~ k . (2-94) and adding the contributions, we have I ( ~ 0 . Y O . Z0) Here a Ax has been factored out fromathe H.O.T. in Eqs. (2-93) and (2-95), but all terms of the H.O.T. in Eq. (2-96) still contain of A,+. . ' . ', I > I Hcr , face 91 whe Eqs. The The evalt to ar 1 Exarr ( 2 92) .ibout its fl ( 2 -93) ~ / 2 ) ' , ctc. (2 -93) . (2-95) jdi;e?\h, (2 J ) j), but all 2-6 1 DIVERGENCE OF A VECTOR FIELD 43 Following.the same procedure for the right and left faces, where the coordinate changes are +hy/2 and -Ay/2, respectively, and b s = Ax Az, we find Here the higher-order terms containthe factors Ay, (Ay)', etc. For the top and bottom faces, we have , d where the higkei-order'teims co'ntaiiibthe hctors A;; (Az)', etc. Now the results from. Eqs. (2-96), (2-Y7), and (2-98) are combined in Lq. (2-91) to obtam + higher-ordcr terms in Ax., A.Y, A;. Since Aa = Ax Ay A : , substitution of Eq. (2-99) in Eq. (2-90) yields the expression of div A in Cartesian coordinates The higher-order terms vanish as the dillerential volume Ax Ay A : approaches zero. The value of div A, in general, depends on the position of the point at which it is evaluated. We have dropped the notation (x,, yo, z,) in Eq. (2-100) because it applies to any point at which A and its partial derivatives are defined. With the vector diRerentia1 operator del, V, defined in Eq. (2-89) Ibr Currrsiao coordinates, we can write Eq. (2-100) alternatively as V A. However, the notation V . A has been customarily used to denote div A in all coordinate systems; that is, We must keep in mind that V is just a symbol, not an operator, in coordinate systems other thaqartesian coordinates. In general orthogonal curvilinear coordinates (u,, u2, u3), Eq. 12-90) will lead to Example 2-12 Find the divergence of the position vector to an arbitrary point. 3 Solution: We will find the sqlution in Cartesian i s well as$n spherical coordinates. I b a) Cartesian coordinates. Thq expiession for thd position vector to an arbitrary . point (x, y, z) is , QP=a,x.+a,y+a,z. (2-103) Using Eq. (2-loo), we have . ' 1 f . # Its divergence in spherical conttlinatcs (I<. 0, $)r';111 hc ohti~illcd fro111 tirl, (2 - 102) hy using Tablc 3- 1 as follows: r Substituting Eq. (2-104) in Eq. (2-105), we also 3btajn V (OF) = 3, as expected. I d 1 8 ; 1-.,dA, 1 V - A = z z ( R 2 ~ R ) + - - - - - (A, yin 0) + P -. R sin 0 88 . R s m 8 8 4 Example 2-13 The magnet$ fluxl density B outpide a,very long current-carrying wire is circumferential and is ipversely proportional to t e distance to the axis of the wire. Find V . B. i ' I 1' . t (2-105) 9 3 I . Solution: Let the long wire be coincident with the z-axif in a cylindrical coordinate system. The problem states that , s , . 1 B=a,;. k ! . . I . In cylindrical coordin+s (r, 9 , z), &. 12-102) rcdtks lo . , 1 i We have here a vector thqt is pgt a constant, but whose divergence is zero. This property indicates that the mqgngtic flux lines close upqn themselves and that there are no magnetic sources or sinks. A divergenceless fielq is called a solenoidal field. More will be said about this t e btfield later m tbe boqk. t YP )ordinates. i' d 2.. 1 arbitrary $. . : : . (2 - 103) \ \ t - I that there 7idal field. 2-7 1 DIVERGENCE THEOREM 45 , , Fig. 2-20 Subdiv~di volurne for proof of divergence theclrzrr,, : 2-7 DIVERGENCE THEOREM In the preceding section we defined the divergence of a vector field as the net outward . flux per unit volume. We may cxpect inluitivcly that the uolwnr inteprul o f ,/he ihrr- gence of a oecror field equals the total outward flux of the vector through the surjncr that botolds the volun~c; that is, 'I liis i d c ~ ~ l ~ l y . wl~ich will he p.ovsd ill ihc li)llowillg iii~li~g~.iipll, is ca11cd the ~ ~ I I ~ , , O C ~ I C L , theorem.' It applies lo any volume V that is bounded by surface S. The direction of ds is always that of the outward normal, perpendicular to the surface ds and directed away from the volume. For a very small differential volume element Auj bounded by a surface s,, the definition of V A in Eq: (2-90) gives directly - -, In case of an arbitrary volume V, we can subdivide it into many, say N , small dif- ferential volumes, of which Auj is typical. This is depicted in Fig. 2-20. Let us now combine the contributions ofall these differential volumes to both sides of Eq. (2-108). We have 1 [ (V A), A , = 1 [ A d . (2-109) A u J - 0 j = 1 A v J - 0 ----.- j= 1 The left side of Eq. (2-109) is, by definition, the volume integral of P . A: It is also known as Gauss's theorem. ' . ? ' 1 46 VECTOR ANALYSIS I 2 !, i ! \ ! 5 4 . The surface integrals on the right side of Eq. (2-109) are summed over all the faces of all the differential volume elements. T-he contribdtibns from the internal surfaces of adjacent elements will, however, cancel each other; bequ+ at a common internal : surface. the outward normals of the adjacent elemehts point in opposite directions. Hence, the net contribution of the right side of Eq. (2-109) is due ~ i d y to that of the external surface S bounding the volume V; that k, , i lie suustnuuon of Eqs. (2-110) and (2-111) in Eq. (2-109) yields the divergence theorem in Eq. (2-107). The validity of the limiting processes leading to the proof of the divergence theorem requires that the vector field A, as well as its first derivatives. exist and he continuous both in I/ and on S. Tllc:divcrpmx thuo~.cnl is ;ttl inlport:~~lt idc11t1(y 1 1 1 vector analysis. It converts a volimc intcgral of thc diwrgcncc of a vector to a cluscd surface integral of the vector. apd rice versa. We use it beqte~tly in establishmg . other theorems and relations in electromagnetics. We note +at, although a s~ngle integral sign is used on both sides of Eq. (2-107) fot simpticity, the volume and surface integrals represent, respectivciy, trlple and do!~bla, integrations. Example 2-14 Given A = a,' + apy + a,p, verify thedivgrgence theorem over a cube one unit on each side. The cube is situated in the first octant of the Cartesian coordinate system with one corqer at'the origin. i 1 I Solution: Refer to Fig. 2-21. Wg fir<t :valuate the &face jntegral over the six faces , , 2 , - - r face r 2. Back face: x = 0, cis = -a, 4y @; , t . . i ' A - d s = O . ( back face ! ' . L . . 1; ' , U . \ I , . I , .. . , ., - !' ! .: : , . '; 1. : . , y Fig. 2-21 A j l i ~ i t cube I . . r ). , I ' . (Example 2-14):. . 5 , . 5 5 Fig. 2-21 A'uhit cube I . (Example 2-14): , ). 5 5 Henc as be Exar fort Cdlll< n , , / ' 1;7 .!. 2-7 1 DIVERGENCE THEOREM 47 -. [;i I the faces i.5 3. Left face: y = 0 , ds = -a, dx dz; 41 surfaces r' 111 internal $ Lf, A - d s = 0 . iirections. c face to that of ! B$ $ 7 ' 6 ' 4. Right face: y = 1, ds = a, dx dz; i t ; , A ds = SO'So' x d x d z = $. ! nght (2-111) ' Y face p. 5. Top face: z = I, ds = dx dy a,; livergence F k J. -h A ds = JolJo' y d x dy = 4 . & face v . J iivergence s t and be 6. Bottom face: z = 0, ds = -aZ dx dy; dentity in o a closed L o .
.
, A . ds = O. ~ > I I \ I I I I I ~ face ! a mglr Adding the above six values, we have IL1"Pnd ( F . \ - ' / s = I + o + o + ~ + ~ + o = ? . Now the divergence of A is ~ m n over Hence, six faces. JvVs .A dv = So1 So1 So1 (3x + y) dxdy dz = 2, .- as before. $ Example 2-15 Given F = a,iR, determine whether the divergence theorem holds for the shell region enclosed by spherical surfacs at R = R , and R = R,(R, > R,) centered at the origin, as shown in Fig. 2-22. Fig. 2-22 A sph&ical shell region (Example 2-15). 1 i t i '- 48 VECTOR ANALYSIS / 2 I $ - 1 I I ' i I 7 1 Solution: Here the specified region has two surfaces, pt R = R , and R = R2. At outer surface: R = R,, $s = a R ~ : sin 0 dB d&; I .
.
, F ds = Joan 1 ; (kR,)R: sin B dm = 47ckR:. surface Actually, since the integrand is independent of 0 or.# in both cases, the integral of a constant over a spherical surface is.simply the constant multiplied by the area of the surface (47rR: for the auter sl~rfa$e and 4 7 r ~ : for the inner surface), and no intcgra- tion is necessary. Adding thc two results, we have. ---. To find the volume integral we first determinev . F for an F that has only an FR component: , : Since V . F is a constant, its voluqe integral equa1s:the product of the constant and the volume. The volume of the shall region betweed the two spherical surfaces with radii R, and R2 is 47r(~3, 1 ~ : ) / 3 . Tkrefore, !, as before. i r This example shows that the divergence t h e o r holds even when the volume has holes-that is, even when the vplume is enclosed!by a rriultiply connected surface. : s A . 1 : I 2-8 CURL OF A VECTOR FIELD , I . I In Section 2-6 we stated that a net outward flux &f a vector A through a surface bounding a volume indicates the p&ence of a souqce. This source may be called a . pow source and div A is a measure of ihe strength of ;he flow source. There is another kind of source, called vortex source, which causes a cipdation of a vector field around it. The net circulation (or simply circulation) of a veqor. field around a closed path is I defined as the scalar line integrql of the vector over &e,p<~h. We have i . I Circulation of A arbund contour I$ $ c A . dt? i . 1, Equation (2-1 12) is' a mathematical :qefinition. The meaning of circulation depends on what kind of field (be vector A represents. If A is a force acting on an object, its circulation will be thg wor done by the force jn moving the object once " F ; . !, { . + aro will boc of L'\ 9 kalr I SOU a rr In v x (1.7 t the arc; the dirt cL'1: nve ~ t lmi: whe AS" e volume d surface. a surface 3 called a s another rci~l;~tion ng on an ject once 2-8 1 CURL OF A VECTOR FIELD 49 . \ \ Fig. 2-23 Relation between a, and dP in defining curl. awund the contour; if A represen .s an I-::::t:ic field intensity, then the circulation wil! be an electromotive force ;,uund the closcd path, as we shall sce later in the. book. The familiar phenomenon of walcr whirling down a sink drain is an suarnplc of a vortex sink causing a circulation of fluid velocity. A circulation of A may exist even when div A = 0 (when there is no flow source). Sincc circulation as dclincd in Eq. (2-1 12) is a line integral of a dot product, its value obviously depends on the orientation of the contour C relative to the vector 4. In order to define point function, which is ;L ii~e;lsosc oi llic strength of2 vbrtea source, we must make C very snlall and orient it in such a way that the circulation is a maximum. We definet In words, Eq. (2-113) states that ;he curl of u vector field A, denoted b~ Curl A or V x A, is a vector whose nlugnitide is the maxin~am net circ~dation o f l per unit urea as the area rends ro zero and whose direction is the normal direcrion of the area w ! m rhe area is oriented to make the net circulation rna.uirnlu?l. Because the normal to an area can point in two opposite directions, we adhere to the right-hand rule that when the fingers of the right hand follow the direction of dP, the thumb points to the a,, direction. This is illustrated in Fig. 2-23. Curl A is a vector polnt funct~on and 1 s conventionally written as V x A (del cross A) although V 1 s not to be consdercd : 1 vector operator except in Cartesian coordinates. The component of V x A in any other direction a, is a, . (V x A), which can be determined from the circulation per unit area n q a l to a , , as the area approaches zero. - (V x A),, = a,, . (V x A) = lin1 whcrc [IN tlircction or Ihc liw in(cgr:i~ion around tl~c contour C, bounding m a As, and the direction a, follow the right-hand rule. ' In books published in Europe the curl of A is often called the rotation of A and written as rot A i . 50 VECTOR ANALYSIS / 2 ~, i 1; 1 1 $ 4 \ 1 1 1 I I I 1 -. .- -. --J, Fig. 2-24 Detemlning (V x A),. We now use Eq. (2-114) to find the three components of V x A in Cartesian coordinatca. Ilcfcr to i:ig. 2 -24 wllerc :I dillcsc~il i : ~ l I.CCI:III~III:I~ :I~C:I p:~l.i~ll~l 1 0 tllr y:-pIanc a11d h;~ving siclcs A!, [~nd A : is ~ ~ : I \ V I I ;IIIOIII :I ( ) / J ~ C ; I I 1)c)i11t I ~ ( . Y ~ ~ , jaf,, :,,), We have 3. = aT and As. = Ay LIZ and the contour C . m~sists of-thelour sides l,2,3, and 4. Thus, Ay Az-0 Ay Az (2-115) 1, 21'3. 4 In Cartesian coordinates A = a,A.=,+ a,A, + a,A, The contributions of the four sides to the line integral are I "' 3 . x,, yo + T , 3 , can be expanded as a ahl lor series: I I ' . L - :artesian Icl to the , I j.0, 4. cs I,?, a, r) (2-" 5 ) l l ~ r I L ) L I ~ ~ (2-116) (2-117) - 0 (2-118) (2-119) 2-8 1 CURL OF A VECTOR FIELD 51 , Note that dP is the same for sides 1 and 3 , but that the integration on side I is gcing upward (a Az change in z), while that on side 3 is going downward (a - Az change . in z). Combining Eqs. (2-117) and (2-119), we have (2 - 120) l k 3 The H.O.T. in Eq. (2-120) still contain powers of Ay. Similarly, it may be shown .hat ( + H.O.T.)~ Ay Az. LMc, A dt = ---? az (2--121) 2 & 4 Ih. Yo, 2%) I . Subsriru~'ng Eqs. (2-120) and (2-121) in Eq. (2-115) and notlng that the hlgher- order terms tend to zero as Ay + 0, we obtain the r-component of V x A : dA.. dA , ( V X A ) =-..-z---!. (2-123) " dy dz A close examinat~on of Eq. (2-122) wili reveal a cycl~c order in 1 . y. and : and enable us to write down the y- and ;-components of V x A. The entlre expression 1'0s tllc curl ol' .\ in C,~rlcs~a~l c.oor~lin,~(c~ 1 s dA dAy V x A = a , -2-- + ayt$ - $I.+ a=(? - $1. (2-123) 7 I -.
-- Comparcd to lllc cxprcssion for V . A ~n Eq. (2-IOO), that for V x A ~n Eq. (2-123) a morc complicalcd, as it is expected lo be, bcciiuse 11 1 s a vector w~th lhrce con~po- nents, whereas V . A is a scalar. Fortunatciy Eq. (2-123) can bc remembered rather easily by arranging it in a determinantal form in the manner of the cross product exhibited in Eq. (2-43). The derivation of V x A in other coordinate systems follows the same procedure. However it is more involved because in curvilinear coordinates not only A but also ' -A- dP changes m magnitude,as the integration of A - dP is carried out on opposite sides of a curvilinear rectangle. The expression for V x A in general orthogonal curvi- linear coordin;~tes (II,, n2. l r l ) is given l x h v . 52 VECTOR ANALYSIS / 2 1 : . . It is apparent from Eq. (2-125) td4t an operalor k r m cannot be found hcre for the symbol V in order to consider 8.x A a cross pl;oduc(, The expressions of V x A in cylindrical and spherical k8or;Bdates can be eqily obtained from Eq. (2-125) by using the appropriate u,, u,, q d y , and their metric coefRcien!s h,, h,, and h,. '.. ' Example 2-16 Show fhat V x A 1 0 if I a) A = a,,,(k/r) in cylindricalcpordjnntc~, whcrc k:is n canstanlt. or b) A = a, f ( R ) in sphepcal cpordinates, where f(R) is any fuqcion of the radial distance R. Solut iorz , a) In cylindrical coordinates th; following apply: (u,, a,. r,) = (r. 4,:):. b , = 1. 11, = 1.. and 17, = 1. Wc havc. from Eq. ( 2 .135). i I 1 /ir which yields, for the given A, , . I t b) In spherical coordinates the following apply; ru,, u,, u,) = (R. 0.4): h , = I h, = R, and h, = R siq). Hence. . . 1 > . V X A = - - R2 sm 9 dR i A, and, for the given A, I I ' c for thc L > f V x A - 125) by 3 . 2-9 / STOKES'S THEOREF. 53 A curl-free vector Iield is called an irrotutiotxd or a conservative field. We will see in the next chapter that an electrostatic field is irrotational (or conservative). The expressions for V x A given in Eqs. (2-126) and (2-127) for cylindrical and spherical coordinates, respectively, will be useful for later reference. 2-9 STOKES'S THEOREM For a very small differential area As, bounded by a contour C,, the definition of V x A in Eq. (2-1 13) leads to f (V x A), . (As,j = ( A . ( I t . .lc' ( 2 - 128) In 0ht:iiung Eq. (2-i28), we have taken the dot product of both sides of Eq. (2-! 13) with a,, Asj or Asj. For an arbitrary surface S, we can subdivide it into many, say :\I, small differential areas. Figure 2-25 shows such a scheme with Asj as a typical differential element. The left side of Eq. (2-128) is the flux of the vector V x A through the area Asj. Adding the contributions of all the differential areas to the flux, we have N - , . lim ? ( V X : ~ ) , ~ . ( A ~ , ) = ( V X A ) - ~ S . \s, . o 3 .i I J.Y Now we sum up the line integrals around the contours of all the differential elements rcprcscntcd by thc rig111 sitlc of Eq. (2- 128). Sincc the common part of thc contours ol'two i~tl.i:~ccnl clclnc~~ls is I~.;\vcrsctl in oplxwiilc tlircctio~is by two contours. thc ncl c o t ~ l ~ i l ) t ~ t i ( ~ ~ ~ or:\ll tl\c C I ) I I I I I > I I I I 1)iII'LS i l l I I K i~thxior to ~ I I C lokt1 line in~cgriii is xro, and only tht: contribu~ion I'rom he external contour C bounding the entire area S remains after the summation. Combining Eqs. (2-129) and (2-130), we obtain the Stokes's theorem: . / Js (V x A). ds = $= A . dP, 1 Fig. 2-25 Subdivided area proof of Stokes's theorem. for 54 VECTOR ANALYSIS ! 2 . ' I 4 . ? which statcs that the surfice i r y r d u i the curl oj:o aw#ur jield over w r upen surjuce is equal to the clpsed line integl;al q{!the vector don$ the contour bounding the surface. As with the divergence theorid the validity of the limiting processes leading to the Stokes's theorem requirei thqi the vector field A, as well as its first derivatives, exist and be continuous both on S and along C. Stokes's theorem converts a surface integral of the curl of a vector t~ hiline integrai af tlre vector, and vice versa. Like the divergence theorem, Stoke+ thdorern is an imdortadt identity in vector analysis, x d we will use it frequently i n sther theorems and relatrons in i .
! ' r , aagnetics. , If the surface integral of V x A is carried over a tlosed surface, there will be no surface-bounding external contour, and Eq. (2-131) tells us that gip x A ) - & = o (2-132) for any closcd surhce S. The pcmctry in Fig. 2-'25 is c h ; ~ i l dclibei:llcly to ~111- phasize the fact that 3 nontrivial application of Stvk~s's Lhcoccn~ s l ~ n y s implies ail open surface with a rim. The simplest open surface would be3wo-dimensional plane or disk with its circumference as the contour. We remind ourselves here that the directions of dP and ds ( a , ) follow the right-hand rule. Example 2-17 Given F = a,ry - a,.2x, verify SSLkes's theorem over a quarter- circular disk with a radius 3<in the first quadrant, as was shown in Fig. 2-14 (Example 2-6). , : : I I ! ' S i : Let us first find the (urfaje integral of V:x F. Prom Eq. (2-130). Therefore, (2 -sur C 1 I w / w c surfice. d i n g to ivatives, L surface x i . Like Jons in I C ill bt no 4 ',. 2-10 1 TWO NULL IDENTITIES 55 ! It is itnporlun{ lo usc thc: propcr h i t s for thc two variables of inlcgration. Wc citn interchange the order of integration as , and get the same result. But it would be quite wrong if the 0 to 3 range were used as the range of integration for both x and y. (Do you know why?) For the line integral around AB0'4 we hdve already evaluated the part around the arc from A to B in Exarilple 2-6. From R to 0 : x = 0, and F : de,- F . (11, ~ ' y ) = --'2% dy = 0. From 0 i o A': y = 0, and,F. dP = F . (a, dx) = r y dx = 0. Hence, from Example 2-6, and Stokes's theorem is verified. Of course, Stokes's theorem has been established in Eq. (2-131) as a genzral identity; there is no need to use a particular example to prove it. We worked out the example above for practice on surface and line integrals. (We note here that both the , vcclor ficltl i ~ n t l ils lirst spalii~l ilcrivalivcs nyc linitc ancl co'ntinuous on the surface ah wcll xi on LIIC contour 01'in~crest.) 2-10 TWO NULL IDENTITIES Two identities involving repeated del operations are of considerable importence in the study of electromagnetism, especially when we introduce potential functions. We shall discuss them separately below. I 2-10.1 Identity I In words,@ curl of the gradient of m y scalar Jield is identically zero. (The existence of V and itsfitst derivatives evcrywhcre is implied here.) Equation (2- 133) c;ln he p r a \ d sc:liiily in Car~csi;~n coordin:~tes by usmg 51. I ; ! 39) I'oI' .' :lntl 1x1 l ; ~ I I I I I I ~ Ll~c i ~ r c l u k d C)I)CI':L~~OIIS. 11) SCIICIYII, il. wc UJ\C 111~' s'urface integral of V x (VV) over any surface, the result is equal to the line integral of VV around the closed path bounding thi surface, as asserted by Stokes's theorem: Ss.[v x (VV)] . '1s = (2-1 34) I I 56 VECTOR ANALYSIS / 2 1 I ' a , 1 . , ' I + ?
. 1 However, from Eq. (2-81), , I I The combination of Eqs. (2-134) and-(2-135) states'that the surface integral of V x (VV) over any surface is zero. The integrand itself must therefore vanish, which leads -+ to the identity in Eq. (2-133). Since 4 coordinate system is qot specified in the deriva- tiox the identity is a gencral Qne aid is invariant jwith the choiccs of coordinatc I i t. q ; stems. F 4 converse statement r : ' Identity I can be mad$ as fqllows., I f a .:tor jeld is ; , , curl-free, then it can be expressed as the grudient of a scalar J'ieid. Let a vector field be E. Then, if: V x E = 0, we can define a scalar field V such that I The negative sign here is unimportant'as far as Identity I is concerned. (It is included in Eq. (2-136) because this relation conforms with a' basic relation between electric field intensity E and electric scalar potential V in electrostatics, which we will take up in the next chapter. At this stage it is immaterial what E and V re'$aent.) We know from Section 2-8 that a curl-free vector field is a conservative field: hence an irrota- tional (a conservative) vector jiell( ccu; ulways he expressed as the gradient o f a scalur field. I i ' 2-10.2 Identity ll in fio !
I L I In words, the divergence o j the curl ufgny vector ~ i e l i is identically zero. ur or Equation (2-137). too, can be ~rbved easily in Cartesian coordinates by using a r Eq. (2-89) for P and performing $he indicated operatipns. We can prove it in general without regard to a coordinate syst& by taking thq'volurne integral of V . (V x A) P C on the left slde. Applying the divcrgqncc thcorcm, wu.havc 2-11 HE CII Let us choose, for example, the arbitrary volume v enc~osedb~ a surface S in Fig. 2-26. . The closed surface S can be spli) into two open surfaces. 9, and S,, connected by a n bc common boundary which has bean dkywn twice as CIthrrd G , . We then apply Stokes's 1 theorem to surface S, bounded by C,; and surfacc SL boundcd by C,, and write the right side of Eq. (2-138) as . I 1, I ; \ 1 fS (V x A) . ds = Js, (v.! A) an1 b + Js (V r A) . a,,, cis 2 . ,- , , . . ! . . . , .
: . 2 .', . - 2-11 / HELMHOLTZ'S THEOREM 57 (2-135) 11ofV x ~ c h leads : deriva- ordinate r jkki :tar field (2-136) ,nciuded I clxtric I take up i c l a m ,, \ :. lJ0; (2 137) 2y using I general tv x A) (2-138) lg. 2-76. red by a Stokes's v r i t n (2-139) 1 1 Fig. 2-26 An arbitrctty volume I/ enclosed by surrace S. The nor-.als a , , and a,, to surfaces S, and S , are q:tward normals, and their relatioiia with the path directions of C, and C2 follow the &&-hand rule. Since the contours. ' C, and C , are, in fact, one and the same common boundary between S , and S,, the two line integrals on the right side of Eq. (2-139) traverse the same path in opposite directions. Their sun1 is therefore zero, and the volun~e integral of V - (V x A) on the left side of Eq. (2-138) vanishes. Because this is true for any arbitrary volume, the integrand itself must be zero, as indicated by the identity in Eq. (2- 137). A converse statement of Identity I1 is as follows: I f a vecror field is rlil;er;~rilc~li.ss. rlr('rr ir cqtrlr Iw c s p ~ ~ s s c ~ t l 11s (Irl, cir1.1 o/'trirotl~ci. UL~L'IOI j i ~ l t l . Lct ;I vcctor held be B. This converse statement asserts that if V - B = 0, we can define a vector field A such that Ii -: V x A . ( 2 - 140) In Seclion 2-6 we mcntionccl th:~t ;i divcrgcncclcss licld is also called a solenoidal ficld. S ~ l c n ~ i ~ l a l ficltls ;lrc nol :~ssoci;~~c(l will1 Ilow so~~~.ccs 01. si~~hs. Tllc net outw:~rcl llux of a solcnoidul licld L I I I U L I ~ I I :IIIY closed surl'acc is zcro, 2nd the l l ~ ~ x lincs ciosc upon themselves. We are reminded of the circling magnetic flux lines of a solenoid or an inductor. As we will see in Chapter 6, magnetic flux density B is solenoidal and can be expressed as the curl of another vector field called magnetic vector potential A. 2-11 HELMHOLTZ'S THEOREM In previous sections we mentioned that a divergenceless field is solenoidal, and a curl-free field is irrotational. We may classify vector fields in accordance with their being solenoidal and/or irrotational. A vector field F is 1. Solenoidal and irrotational if 1 . V . F = O and V x F = O . Iistr~rrpl~~: I \ sutic clccwic lidtl ill :I charge-li.ccl rcgian. 2. Solenoidal but not irrotational if V . F = O 'and V x F f O . Example: A steady magnetic field in a current-carrying conductor. 58 VECTOR ANALYSIS I 2 I -. 3. Irrotational but no,t solenoidal if V'x F = O and V - F f O . Example: A static electric field in a charged region. 4. Neither solenoidal nor irrotationnl i i , V - F # O ' rr.2 V X V # ~ . Example An electric field in ; X' - r t ; ~ .
L rnw!i~ In, it!! .i finre - ~ ~ ~ ; r y m g inagletlc field. The most general vcctor licld thcn has holh-a nonzero divcrgcncc :)lid :I wnzcro curl. and can bc considcrcd as rhc sum ol'u solcl~oitI:~l licld : ~ n r l 311 irro(;~iiol~;tl licld. Hclrl~holr:'.~ Tlrc~o,wn: rl vector ~ic/tl'.(~w~tor poirlt Jrrrlctior~) is ~I~t~~rlrirrc~d to \Ythin (211 udditiw co~~slailt I/' both its diwyerzce am/ its curl ure specijed everywlzer-k. In an unbounded region we assume that both the divergence and the curl of the vector field vanish at infinity. If the vector field is confined within a region bounded by a surface, then it is determined if its divergence and curl throughour-the region, as well as the normal componeht of the vector over the bbundirlg surface, are given. Here we assume that the vector'functfon is single-valued and that its derivatives are . , . . finite and continuous. Helmholtz's thcorem can be proved as a rnathcmaticali~hcorcm iq a gencral wayt For our purposes, we remind pupelves (see Section 2-8) that the divergence of a vector is a measure of the strenith of,the flow source and.that the curl of a vector is a measure of the strength of theg~gtex Source. When the strenphs of both the flow source and the vortex source a'% 'specified, we expect that the vector field will be determined. Thus, we can decompose a'gineral vector &ld F-intoran irrotational 6 . part F, and a solenoidal part F,: 3 : .
: , .: . A ' . with and F = Fi + F,, where g and G are assumed to be known! We have , I . I V . F = V y F , = g 1 . I (2- 144) and I . I \ C x F = ' i x F , = G . ,, (2- 145) Hrlnlhdtz's theorem asserts that when 4 and G ire s F c & d . the vector fmction F the : Exa . , See. ior instance. G. Xnlen. Afarhemoricaf Jferlt@s for Phydcisrs, .+idemic Press (1966). Section 1.15. . . 2-11 1 HELMOHLTZ'S THEOREM 59 , : h is determined. Since V. and V x are differential operators, F must be obtained by $3 ?j integrating g and G in some manner, which will lead to constants of integration. The is determination of these additive constants requires the knowledge of some boundary i? , conditions. The proccdurc for obtaining F from given g and G is not obvious at this t time; it will be developed in stages in later chapters. !t The fact that Fi is irrotational enables us to define a scalar (potential) function V, in view of identity (2- M ) , such that i . P 1 : F, = - V V . p 1 1 6 ) Sirnihdy, identity (?-137) 2nd Eq. (2- l43a) allow the definition of a vecLc; (potential) function 4 such the? Helmholtz's theorem states that a general vector function F can be written as the sum of the gradient of a scalar function and the curl of a vector function. Thus. F = - V V + V x A . (2- 14s) In following chapters we will rely on Helmholtz's theorem as a basic eiemznt in the axioinatic development of electromagnetisnl. Example 2-18 Given a vector function I: - $ I , ( 'I\, - 0 , TI I ; t , , ( < , 2 .-. 2:) -. :I ( 1 , , I , 1. z ] , 3 ) Oe~ennine the constanls c,, c2, and c3 il. 1 : is irrolaiional. M Determine the scalar potential function V whose negative gradient equals F. Solution a) For F to be irrotational, V x F = 0; that is, Each-wqlponent of V x F must vanish. Hence, c , = 0, c2 = 3, and e, = 2. b) Since F is irrotationnl. it cnu be e\presscd as the negitlve gradlent of a sc.ll;lr I'u~iclion I : ; 111:~ is, i. i.v av av F = -VV = -ax -'- ay -;-- - a= - ax oy az = ax3y + ay(3x - 22) - a1(2y + z ) . 60 VECTOR ANALYS 1 : 1 ' Three equations are obtained: r . > L . av' ' t T -= -3y a . ~ (2- 149) Integrating Eq. (2-149) partially with respect to x, we have I ' = - 3sy + ,f',(,Y. z). 2 (221 52) and Examination of Eqs. (2-152), (2--1.53); and (2-154 enables us to write the scalar potential function as - Any constant added to Eq. (2- 155) would still mkke lran answer. The constant is to be determined by a boundary condition or the condition at infinity. REVIEW QUESTIONS I R.2-1 Threc vectors A. B. ;md C. drqan ib .
I i~eod-to-tail f.l{hion. f p n ihrce sidcs of :I tri;~n$e. What 1 s A f B + C'? A + B - C ? , , . . R.2-2 Under what conditions can the dot product of two vectors be negative? I R.2-3 Write down the results of A - B aridlA x B if (a) A I( I$, and fb) A 1 B. R.2-J Which of the following produkts of vectors do not mqke sense? Explain. a) ( A . B) x C b) A@ C) , c ) A x B x C dl A/B c) Ailan f) (A x p1.C ..-. >::;:.. . . , : '.:>J$.1.:.' . , , .' . , . + . ., . . , . ' ,. . . .< > i :.>$J. !. .> ., ; -., , , 4 , ; .; , , . . !
$ REVIEW QUESTIONS 61 R.2-5 Is (A. B)C equal to A(B . C)? R.2-6 Does t i . B = A C imply B = C? Explain. R.2-7 Does A x B = A x C imply B = C? Explain. R.2-8 Given two vectors A and B, how do you find (a) the component of A in the direction of B and (b) the component of B in the direction of A? R.2-9 What makes a coordinate system (a) orthogonal? (b) curvilinear? and (c) right-handed? R.2-IC Given a idctor F in orthogonal curvilinear coordinates (u,, u,, u,), explain how to determin, (a) F an2 (b) a?. # 11~2- 1 1 WhzL arc iixlL.ic c d ~ i c i ~ , , h : ' R.2-12 Given two points Pl(l. 2. 3) and P2(- 1, 0, 2) in Cartesian coordinates, write the espres- sions or the vectors 1 7 : ; 1 1 1 d /TI. R.2-13 What are the expressions for A . B and A x B in Cartesian coordinates? R.2-14 What are the values of the following dot products of base vectors? :I) ilp ;I., 1)) :Ir . :Iy c) :I,< :hr d) a , . ;I, e) a, a, f) ar az R.2-15 What is thc physicid dcfinition orthe gradicnt of a sc~lar field? 11.2-10 Exprchs tllc space ralc ofchiunyc of a scalar in u yivcn direction in tcrms of its gradicnt. 11.2-17 What does the del operator V stand for in Cartesian coordinates? R.2-18 What is Ihc physical dcfinition of thc divcrgcncc of a vcctor ficld? R.2-19 A vector field with only radial flux lines cannot be solenoidal. True or false'! Explain. R.2-20 A vector field with only curved flux lines can have a nonzero divergence. True or false'? Explain. R.2-21 State the divergence theorem in words. R.2-22 What is the physical definition of the curl of a vector field? R.2-23 A vcctor ficld with only curvcd flux lincs cannot bc irrotational. Truc o r Salsc'? Esplain. R.2-24 A vector field with only straight flux lines can be solenoidal. True or false? Explain. R.2-25 StakS&o_kes's theorem in words. R.2-26 What is lhc dilkrcncc bclwccn a11 ilmtation:il field nud a solcnoid;il field? 1$2-~27 Slillc I lr)h,(l;'s \lrco~>cln in wcw~la. R2-28 Explain how a general vector functiomcan be expressed in terms of a scalar potential function and a vector potential function. 1 62 VECTOR ANALYSIS I 2 i 3 i L PROBLEMS P.2- 1 Given three vectors A, B, and C as follows, C A = a, + a,2 - a,3 B =,-a,4 + a, - find e) the component of A in the direction of C f) A-; C g) A . ( B x C)and(A x B ) - C 1 1 ) (A x B) y C and A x (B x C) P.2-2 The three corners of a triangle are at P,(O. 1. -2), P,(4, 1, -3), and P,(6, 2, 5). a 3) Dctcrn~iiic \vl~c[l~cr A PI lJ21'., is it ri$it [V~:III~I~-. b) Find the area of the triangle. P.2-3 Show that the two diagonals of a rhombus ire perpendicqlar to e ~ h g _ t h e r . (A rhombus is an equilateral parallelogram.) P.2-4 Show that, if A B = A C nr~d A x B = A x C, where A is pot a null vector, then B = C. P.2-5 Lnlt vectors a, and a, denote t1-;:directlons of two-dim~gsional vectors A and B that make angles u and , 6 , respectlvely. wltb a refdrence u-am, as shown ip Fig. 2-27. Obtaln a formula for the expans~on of the cosine of thp difference of two angles, cos(r - j ) , by takin, 0 the scalar product a , . a,,. Fig. 2-17 Graph for 1 ' 0 Pfbblem P.2-5. ' , ! P.2-6 Prove the law of sines for a triangle. I : P.2-7 Prove that an angle inpcgbed in a ~emicircle is a rigit in&. . i '. .. P.2-8 Verify the back-cab rule of the vepor triple product of hree vectors, as expressed in Eq. (2-20) in Cartesian coordinates. ' : 1 1 ,) - ' P.2-9 An unknown vector can be dcter&ed if both its s&l& product and its vector product . w~th 3 known vector are given, Assuming ;G is a known vector, determine the unknown vector S if both p and P are glven, where p = A X and P = A x X > P.2-10 Find the component of the vectqnA = -a,= + iz;'af the point P,(O, -2, 3), whlch is. directed toward the point ~ , ( j j . -60", 1). I : i 1 PROBLEMS 63 P.2-11 The position of a point in cylindrical coordinates is specified by (4,2n/3,3). What is the location of the point a ) in Cartesian coordinates? b) in spherical coordinates? P.2- 12 A field is expressed In spherical coordinatcs by E = a,(25/R2). a) Find [El and Ex at the point P(- 3,4, - 5). b) Find the angle which E makes with the vector B = a,2 - a,2 + a=. P.2-13 Express the base :,-ctors a,, a,, and a, of a spherical coordinate system In Carresian coordinates. , P.2-14 Givcn a vcctor function E = axy 3 a,x, evaluate the scalar line mtegral E - clP from , PL(2, 1, - 1) 10 P2(8, 2, - 1) a) along the parabola s = 2y2, b) along the straight line joining the two points. Is this E a conservative field? P.2-15 For the E of Problem P.2-14, evaluate J' E di from P3(3, 3, - 1) to P,(-l, -3, - i) by converting both E and the positions ol'l', and P, into cylindrical coordinatcs. P.2-16 Given a scalar function a) the ~nagnitudc and the dircction of the maximum rate of increase of V at the point P(1, 2, 3), b) the rate of increase of Vat P in the direction of the origin. P.2-17 Evaluate (a,3 sin 8) - ds over the surface of a sphere of a radius 5 centered at the origin P.2-18 For a scalar function f and a vector function A, prove V.(fA)= f V - A + A . V f in Cartesian coordinates. P.2-19 For vector function A = a,r2 + a,2z, verify the divergence theorem for the circular cylindrical region enclosed by r = 5, z = 0, and z = 4. -1 P.2-21 vector field I ) = a,(cos2$)/KJ cxisls in the region bctwcen two spherical shells defined by R = 1 and K = 2. Evaluate a) $ D . ds b) J'V-Ddv P.2-22 A radial vector field is represented by F = aR/(R)i What do we know about the function f ( R ) i f V - F = 0 ? i P.2-23 For two differentiable vector functions A and H, prove V o ( A x H) = H - ( V x A) - E - ( V x A). P.2-24 Assume the vector functiop A = a , 3 ~ ' ~ ' - ap3y2. a) Find 8 A . d t around the triangular contour shown in Fig. 2-28. 6) Evaluate (V . A) - ds over the triangular area. C) Can A be,expmsed as the gradient of a scalar? Explain. 1 / I + Fig. 2-28 Graph for 0 2 x Problem P.2-24. P.2-25 Gwen the rector function A = a, $in (4/2), verify Stokes's theorem over the hemispherical surface and its clrcular contour that are shown in Fig. 2-29. 1 I , Grauh for. x Problem p.2-2; ' P.2-26 For a scalar function f and a vector function G, prove v x [ f G ) r f v x G + ( v f ) . x G in Cartesian coordinates. . , P.2-27 Verify the null identities I I ' a) V x ( V V ) r O b) V.(V x A ) s O j i by expansion in general orthogonal curvilinear coordinates. I 3 / Static Electric Fields 3-1 IN I HU~UCTION In Section 1-2 wc mcntioncct th;tt three essential steps x c involvcd in cc, nstructing a deductive theory for the study of a scientific subject. They are the definition of basic quantities, the development of rules of operation, and the postulation of fundamental .--., rclntions. Wc have cic(inci1 the S O I I S C ~ ;uid field ql~;~nfitics for 1I1c cIcct~-ol~~;~g~?t:tic' ~noticl in Chapter 1 and dcvclopcd the fundamenlals of vcctor algebra and vector calculus in Chapter 2. We are now ready to introduce the fundamental postulates for the study of source-field relationships in electrostatics. In electrostatics, electric charpcs (the sourccs) are at rcst. :~nd clcctric fields do ! k t cliangc with time. Thcre are I I O I I I : I ~ I I C I ~ C liclcls; I K I I ~ C WG cIc:11 will1 :I 1t1~1tivcIy S I I I I ~ I ~ sit~1:1tio11, Al'icr wc have studied the behavior of static clectric fields and mastered the techniques for sol-. ing clcutrnstatic boi~nclury-valuc problems, wc will then go on to thc subjcct of mayxtic fields and time-varying electromagnetic fields. The development of electrostatics in elementary physics usually be, .ins with the experimental Coulomb's law (formulated in 1785) for the force between two point charges. This law states that the force between two charged bodies, q , and q 2 , that are very small compared with the distance of separation, R l 2 , is proportional to the . product of the charges and invcrscly proportion:~l to thc quart: of the dist:incc, thc dircctio~~ of the force being along the line connecting the charges. In addition, Cou- lonib found that unlike chargcs attract and likc ch;~rgcs rcpcl cach other. Using vector- notation, Coulomb's law can be written mathematically as q1Cl2 Fl2 = -T' ( 3 - 1) 1 - R12 where F12 is the vector force esertcdby (1, on q,, a,:,, is n unit vcctor in the direction from (11 to (I,, and It is :I l.rrol.rc\rtio~l;~liIy constant cicpcnding on the medium and the systcnl ol' units. Nok thi~l il'q, and r12 arc of the same sign (both positive or both ncgalivc), I ; , , is positive (repulsive); and if q , and q2 are of opposite signs, F,? is negative (attractive). Electrostatics can proceed from Coulomb's law to define electric field intensity E, electric scalar potential, V, and electric flux density, D, and then lead to Gauss's law and other relations. This approach has been accepted as "logical," 66 STATIC ELECTRIC FIELDS 1 3 , 1 2 , !
perhaps because it begins with an pxperimental laGabsered in a laboratory and not with some abstract postulates. . J We maintain, however, that Coulomb's law, though based on experimental evidence, is in fact also a postulate. Consider the two'stipulations of Coulomb's law: that the chargcd bodies be very small ~ompared wit9 the distance of separation and that the force is inversely propottianal to the squari: of the distance. The question arises regarding the first stipulatjonl,How small must! the charged bodies be in order to be considered "very small" dompared to the distahce? In practice the charged bodies cannot be of vanishing sizes (idcal point charges), and there 'is dificuity in determining the "true" distancz between two bodies ol finite dimensions. For given body sizes. the relative accuracy jn distance measurements is better when the separa- tlon is larger. However, practical cdnsiderations (weakness of force, existence of extraneous charged bodies, etc.) restrict the usable <istnpce of scparatrnn in the laboratory. and cxperlmcntal ~ I ~ ~ \ C C U S , \ C ~ C S canno[ hc cn~~rcly :~vo~tlctl. '1'111s Ic,~tl$ LO a more important question concenung the inverse-square relation of the second stipulation. Even if the charged bodies are of vanishing sizes, exp'erimental measure- ments cannot be of infin~te accuracy, no matter how skillfpl and careful an expen- mentor is. How then was it possible for Coulomb to know that the force was exnctl~ inversely proportional to the square (not the 2.000001th or the 1.999999th power) of the distance of separation? This quesfhon cannot be answered from an experimental v~ewpoint because it is not likely that durlng Coulo~rrb's time cxperiments could have been accurate to the seventh place.+ We must thei-efore conclude that Coulomb's law is itself a postulate and that [he exact relation stipulated by Eq, (3-1) is a law of nature discovered and assumed by coulomb oq the basis of his experiments of limited accuracy. a , 1 ' ' Instead of following the hisjoripal development ,of electrostatics, we introduce the subject by postulating bothUibe d&?rgence and thje curl of tht electric field inten- sity in free space. From Helmholfz's theorem in ~ectibn 2-b 1 we kcow that a vector field 1s determined ~f its divergenqe and curl are specified. We deiive Gauss's law and Coulomb's law from the divergepce afid curl relatiobs, and do not present them as separatc postulates. Thc conccptl o f sc$nr potcnlii~l fi)llr)wfi nat~~rally fro111 A VCL~OI. identity. Field behaviors 'in material media will be studied and expressions for elec- trostatic energy and forces will be developed. ! I- I ' 3-2 FUNDAMENTAL POSTULATES QF : . I \ ELECTROSTATICS IN FREE SPACE 1 . We start the study of electroma&et$fn with the co&ideration of electric fields due to stationary (static) electric charges it7, free space. ~)ect'i-ostatics in free space is the ' , ' " , < , s ' The exponent on the distance in ~ o u l o ~ b ' s ' l a w has been verifjdd by an lnd~rect experment to be 2 to wthm one part In 10" (See E. R. Wilhqfls, J.$. Faller, and H : A ~ I H ~ I ~ , ph)~.~. Rev. Lerter~, vol. 26, 1971, p. 721.) 0 i n C- c irr U n CO thi ory and .imental b's law: ion and juestion :n' order charged alty in ~r given scpara- r' ,Ids due c is . o be 2 to 26, 1971, 3-2 I FUNC-\MENTAL POSTULATES OF ELECTROSTATICS IN FREE SPACE - C simplest special case OF electromagnctics. We need only consider one of the four fundamental vector field quantities of the electromagnetic model discussed in Section 1-2, namely, the electric field intensity, E. Furthermore, only the permittivity of free space 60, of the three universal constants mentioned in Section 1-3 enters into our formulation. Electricfield intensity is defined as the force per unit charge that a very small stationary test charge experiences when it is placed in a region where an electric field exists. That is, F E = lim - (V/m). q-0 q , The electricfield iatensity F i., ;r,,mg~on;il to and in the dircction of tllc force \ F. If F is n~c;~sitrod in ~icwtons (N) ;ind cll;~lga '1 ill c o ~ ~ b i n b s (C), ~ilco C I 1 s i n ~iewlons per coulomb (NIC), which is the same as volts per meter (V/m). The test charge q, of course, cannot be zero in practice; as a matter of bct, it cannot be less tlian the charge on an electron. However, the finiteness of the test charge ivould not maks the measured E differ appreciably from its calculated value if the test charge is small enough not to disturb the charge distribution of the source. An inversc rc~atibn o f Eq. (3-2) gives the force, F, on a stationary charge q in an electric field E: The two l~~d:lin~~ititl postulates of clcctros~atics in lice sp:tcc spccify ttic divergence and curl of E. They are In Eq. (3-41, p is the volume charge density (C/m3), and so is the permittivity of free space, a universal constant.' Equation (3-5) asserts that stoiic elecrricjrlds are - irrotntB~niTt-wl~erc;~s Eq. (3-4) implics ~ l i ; ~ t : 1 st;~tic electric ficlcl ir n& roIcnoid:~l unless p = 0. Thasc iwo postulales arc concise, simple, and indepenhent of any coordinate systcm; and they can be used to derive all other relations. laws, and I ~ I T I W I B rlwl~ustillics I S u d ~ is lhe I m t ~ i y of ills dtduc~ivc, :~xio~~l;llic :~ppm:dl. 1 The permittivity of free space r;z - x (F/m). See Eq. (1-11) 36n 68 STATIC ELECTRIC FIELDS I I C 7 I ' I i f j : . I 8 Equations (3-4) and (3-5) arq polnt relations; that is, tbeyhold at every point in space. They are referred to as thq diffeiential form of fhe pqstulates of electrostatics, since both divergence and curl operations involve spatial derivatives. In practical applications we are usually interested in the total field ofan aggregate or a distribution of charges. This is more conveniently 'obtained by a ; ? integral form of Ey. (3 -4). Taking the volume integral of both sides of Eq. (3-41 over,an arbitrary volume V, we have ! c 2 I " , 1 1 " J v P- E d r = 2 . €0 J v dv. In view of the divergence theorerq in Eq. (2-104), Eq. (3 -6) becomes where Q is the total charge contained'in volume V bounded bi;ii7face S. Equa- tion (3-7) is a form of Gauss's luw, which statcs thaiithe ~ptul ourwardjlux of' the electricfield intensity over an): closed surface in fi-ee spice is equal to the total charge enclosed in the surfilce divided hy E , , . Gauss's law is one i f the most important relations in cicctrostatics. Wc will discuss it furthcr in Scctiol! 3 - 4, along with illustrative examples. An integral form can also be 06tained for the hurl relation in Eq. (3-5) by integrating V x E over an open : I . surfack , . and invokingiStokes's theorem as espressed in Eq. (2-131). We have . . %. 1 : ) - The line integral is performed over a closed contour C-bounding an arbitrary surface; hence C is itself arb~trary. As a yatte? of fact, the surface does not even cntcr into Eq. (3-8), which asserts that the . y u l ~ r line inlegrul oJ',rhe stutic clcclric Jicld rtttcrl.sily arotuzd an!. closed path vanishes. ~ h i s i s s i m ~ l ~ anothef way of saying that E is irrota- tional or conscrvative. Refcrrinito F&. . . 3-1. wc scc;,th:lt if the scalar linc intcgrnl I P 3-3 COU -ch; r ?
a , . .
.
.
- Fig 6 ry point .
k : 3. ostatics, I $ ~ractical :? ' .
-G ribution , ji 4 . (3-4). 8 , lume V, ;i-, t; 3-3 / COULOMB'S LAW 69 , of E over the arbitrary closed contour C,C, is zero, then J P p , ' E . d P = -J : E . & (3-10) Along C, Along C , or SpT E.dP=SP:' E - d P Along C, Along C, , Eq~~ation (3-1 1.1 says that the sc:ilar .linq it~lc~ra!- or the irro~:~ti&~l;~l F lic!,l I, I I I , , ~ I X I I C ~ C I I ~ tlllhe I X I ~ I I : il ~ C ~ C I I & i ~ ~ l y 0 1 1 ~ I I C CIICI l>~il~ls. i\s \ " I . sI1:111 scc i l l SCC~ioll .i 5. tile integral in Eq. (3-11) represents the work done by the electric field in moving a unit charge from point P , to point I',; hence Eqs. ( 3 4 ) ;ind ( 3 9 ) imply a st:itcmcnt of conservation of work or energy in an electrostatic iield. The two fundamental postulates of electrostatics in free space are repeated belarv because they form thc h~tnd:~tion ~ I P O I I \VIIICII wc build I I I C s ~ r ~ i ~ i i ~ i c ~ l c l ~ ~ ~ r o s ~ : ; ~ i i s . 3-3 COULOMB'S LAW We consider the simplest possible electrostat~c problem of a s~ngle polnt charge, y . at rest in a boundless free space. In order to find the electr~c ficld intanslty due to (1. we draw a hypothetical spherical surface of a radius R centered at q. Since a polnt charge has no preferred directions, its electric field must be everywhere radial 2nd - bas the si1111c intansity : ~ t :ill poiots on 1 1 1 1 : ~ l ~ l i ~ l l . i ~ i l l S U I ~ I C C , Apply~ng Eg. (3-7) to I?g, 3 - 2(:r), we l ~ v c 70 STATIC ELECTRIC FIELDS I3 (a) Point charge at the origin, (b) Point charge not at thc origin. Therefore, L Equation (3-12) tells us that ille electric jirld intensity o j u point charge is is the ourward radial direction and bqs a magairude propoytional ro the charge and incersely proportional to the square of the distance born the charge. This is a very important basic formula in electrostatics. It is readily verified:tllat V x E = 0 for the E given in Eq. (3-12). t I l the chitrgc q I \ not locutcd at the orlgln of u choseq cuord~naio \ystcnl, 5i11iablc changes should be made to the unit vector a, and thb distapce R to reflect the locat~ons of the charge and of the point at which E is to be dbtermieed. Let thc position vector of q be R' and that of a field poiht P be R, as shown in Fig. 3-2(b). Then, from Eq. (3-1 2), ) where a,, is the unit vector drawn'fiorn q to P, Since we have Example 3-1 Determine the electric fieid intensity a1 P(-0.2,0, -2.3) due to a point charge of + 5 (nC) at Q(0.2, $1, - 2.5) in air. All dimensions are in meters. I 1 . 3-3 I COULOMB'S LAW 71 Solution: The position vector for the field point P The position. vector for the point charge Q is R ' = = a,0.2 + ayO.l - a,2.5 The difference is which has a magnitude . jR - 2'1 = [(-u.4)' + (-0.1)' + (0.2)3]112 = 0.455 Substituting in Eq. (3-15), we obtain 'I'hc qwtlticy w i ~ l l i ~ ~ l l ~ c ~ ~ : ~ ~ . c ~ i ~ l ~ c s c s is thc' [Init vcctor :lu,. -= (I1 - l1')jIR -- R'/. ; I I I ~ I(:,, 11:ts :I 111:1g1i~tdc CII' 214.5 (V/III). Note: The permittivity of air is essentially the same as that of the free space. The factor 1/(4ne,) appears very frequently in electrostatics. From Eq. (1-1 1) we know that c0 = l / ( ~ ~ p ~ ) . . B ~ t pO = 4n x lo-' (H/m) in SI units; so exactly. Ifwe use the approximate value c= 3 x 10"nl/s), then l/(bc,) = 9 x 109 (m!F). When a point charge y, is placed in the ficld of another point chargc (1, 'lt the origin, a force F,, is experienccd by 4, due to electric field intensity E,, of q, at y2. Combining Eqs. (3-3) and (3-12), we have Equation(3-17) is a mathcmatical form of Coulomb's luw alrcady statcd in Scction 3- 1 in conjunction with Eq. (3-1). Note that the exponent on R is exactly 2, which is a consequence of the fundamental postulate Eq. (3-4). In SI units the proportionality constant k equals 1/(4nc,), and the force is in newtons (N). I 72 STATIC ELECTRIC FIELDS / 3 L $ 1 -" + Screen \I : Deflection -f P , , I Fig. 3-3 Electrostatic deflection system of a cathode-ray O \ C I ~ ~ ~ I ~ K I ~ ~ ( ~ \ ~ l l ~ l p ~ ~ ? 2 ) Example 3-2 The electrostatjc de1:cclion system of a cathode-my oscillograph is depicted in Fig. 3-3. Electrons from a heated cathode are given an initial velocity vo = a , ~ , by a positively charged anode (not shown). The electrons enter at r = 0 into a region of deflection plates where a uniform electric field Ed = - a,Ed is main- tained over a width w. Ignoring gravitational effects, find the vertical deflection of the electrons on the fluorescent screen at r = L. i Solutiot~: Since there is no force in the ;-direction in the z > 0 region, thc horizontal velocity v0 is maintained. The field .Ed exerts a force on the electrons each carryng a charge - e, causing a deflection in the y direction. From Newton's second law of motion in the vertical direction, we have where rn is the mass of an electron.~1~tegrating hoth.sidcy, wc obtain where the constant of integration isset to zero beCause v,. = 0 at t = 0 . Integrating again, we have The constant of integration is again zero because y = 0 at t = 0. Note that the electrons have a parabolic trajectpry between the aeflection plates. At the exit from the deflection plafes; t = w/vo, ' 3-3 I COULOMB'S LAW 73 ' and ' When the electrons reach the screen they have traveled a further horizontal distance of (L - w ) which takes (L - w)/v, seconds. During that time there is an additional vertical deflection Hence the deflection at the screen is 3-3.1 Electric Field due to a System of Discrete Charges Supposc a n clac1rost:liic liald is cra:ilcd by a group 01' ii dlscrete po~ot chiirgel 'I,, q2, . . . , q, located at different positions. Since electric field intensity is a linear func .on of (proportional to) a,q/~', the principle of superposition applies, and the tot;.] E field at a point is the vector sum of the fields caused by all the individual char: CS. From Eq. (3 -15) wc can wriic ll~c clcctric ilitcnsity at a field point whose posir on vcctor is It as Although Eq. (3-18) is a succinct expression, it is somewhat inconvenient to use, because of the need to add vectors of different magnitudes and directions. Let us consider the simple case of an electric dipole that consists of a pair of equal and opposite charges, + q and -q, separated by a small distance, d, as showc in Fig. 3-4. Let the center of the dipole coincide with the origin of a spherical coordicate System. Then the E field at the point P is the sum of the contributions due to +q Fig. 3-4 dipole. Electric field of a 74 STATIC ELECTRIC FIELDS 1 3 j . L 1 . 8 .; t : and - y. Thus. + % The first term on the right side of ~4.:(3-19) can be Simplified if d << R. We write where the binomial expansion has been used and dl terms containing thc second and higher powers of (d/R) have beeri neglected. Similarly, for the second term on the right side of Eq. (3-19), we have' Substitution of Eqs. (3-20) and (3-21) in Eq. (3-19) leads 10 I The derivation and intcrprolati.b$ of Ey. 13--22) rcqrirc thc manipulation of vcctor quantities. Wc can apprcciatcLhat tictcrrninihg thc clcctric Geld causcd by three or more discrete charges will 'bg even morc t&liouq In Scction 3 .5 wc will introduce thc conccpt or ;I scalt~r clcctric potcl~li:~l, will1 whicl~ [ l ~ c clcclric licld intensity caused by a diswibution of'cbargcs can be found more easily. The electric dipole is an irqporkant entity in tGe study of the electric field in dielectric media. We define the product of the charge q an4 the vector d (going from - q and +q) as the electric dipoff mement, p: , . - -p = yd . : . i (3-23) Equation (3-22) can then be rewritten as > . where the approximate sign (-) ovd; 'the equal sign 'has been left out for simpliaty. ! I 8 I ) ' (3-19) 2 write 3-3 1 COULOMB'S LAW 75 ' If the dipole lies along the z-axis as in Fig. 3-4, then (see Eq. 2-77) p = aZp = p(a, cos 8 - a, sin 8) - - .. (3-25) ' R . p = Rp cos 0 , (3 -76) and Eq. (3-24) becomes E = - ( a , 2 cos 8 + a, sin 6) (V/rn). 4m0R (3 -27) Equation (3127) gives the electric field intensity of an electric dipole in sphcri~n! . coordinates. We see that E of a dipole is inversely propoirional to the cube of the distance R. This is reasonable because as R increases, the fields due to the closely spaced + q and - q tend to cancel each other more completely, thus decreasing more rapidly than that of a single point charge. 7 3-3.2 Electric Field due to n Continuous Dlstrlbution of Charge The electric field caused by a continuous distribution of charge can be obtained by integrating i5kpcrposing) thc contribution of a n element of c h a r p over the charge clislrihulio~l. licfcr to 1 . i ~ . 3 -5. whurc ;I voluinc cL;irp: tlisfrih~itioi~ is sl~owi~. T ~ I C V O I U I I I ~ ~ C,~I:II,K(: I ~ ~ - I I . I I I ~ ,I I ( / I I I I ) 1-1 :I I I I I I L I i011 0 1 ~ I I C C O O I ~ ~ I I I : I ~ C : . . S I I ~ L C :I iIilli~c1111:11 clement ul' charge behaves like a point charge, the contribution of the charge p -10' in a differential volume element du' to the electric field intensity at the field point P is We have p dv' dE = a, -----. 4n.5,R2 . Fig. 3-5 Electric field due to a continuous charge distribution. h. . I I i . > I i .
' 76 STATIC ELECTRIC FIELDS I 7 I I ! ! I or, since a, = R/R, . , , 3 (3-30) ( Except for some especially sikp14 ases, the vectdr triple ibtegral in Eq. (3-29) or I Eq. (3-30) is difficult to carry out because, in general, all'three quantities in the integrand (aR, p, and R) change with the location of the differential volume dv'. If the charge is distributed on a,surface with a~urface charge density p, (C/m2), then the integration is to be carried out over the surface (not necessarily flat). Thus, 8 For a line charge, we have 1 .
. where p, (C/m) is the line charge 'density. and L' the line (not necessarily straight) along which the charge is distributed. I Example 3-3 Determine the electric field inten& ofIan infinitely long. straight. line charge of a uniform density p h air. i 4 ' - Solution: Let us assume that tlje.line charge li& alopg the zf-axis as shown in Fig. 3-6. (We are perfectly frce to dd this because t\le field obviously docs not depend . . on how wc clcsign;~tc thc linc; 11 ?,>'irrr u r w p / ( d cvrricir~ri/i,;n f r r r i : i ~ pritn!d r~r~orrlitrrc~c~.v jbr sotircc / ~ r , i t ~ / s wid ~ ~ t ~ ~ ~ r i ~ ~ ~ ~ ~ ~ ~ ~ c r ~ r ~ ~ ~ ~ ~ l ~ r ~ ~ i / ~ ~ , s / ; , I , / I v / ~ / / ) ! , ~ I I / . S ~ W I I 111m~ is (1 / r r r . $ . $ ~ / ~ ~ l i / y o f conjusion.) The problem agks us to find the e ~ & ~ i c geld intensity at a point l', which is at a distance r from the I'ine. Since the p+blern has a cylindrical symmetry (that is, the electric field is in!dep&dent of the azlmutli angle 4). it woiild bc most convenient to work with cylijdriyd coordinates. h e re writ;.^^. (3-32) as . - . . . For the problem at hand p, is co&&nt and 5 line element dLr = dz' is chosen to be at an arbitrary distance z' from the,Grigin. It is mqit important to remember that R is the d~stance vector directed fr;h the source td'thejeld point, not the other way I : i . . ii ! ' : A 1 : ,.i 1 i : i l I 3-3 1 COULOMB'S LAW 77 i - 29) or s in the iL;', , [Cim2), t). Thus, Fig. 3-6 An infinitely long straight-line charge. around. We have R = a,, - a,:'. The electric field, dE, due to the difirential line charge element p, dt' = p, d l is where and p,r dz' dEr = 4m0(r2 + 2'2)3/2 - p,zf dz' dE, = 4neO(r2 + z'2)3/2' In Eq. (3-35) we have decomposed dE into its components in the a, and a; directicns. It is easy to see that for every p, dz' at + z' there is a charge element p, d:' at - : ' , which will produce a dE with components dEr and -dE,. Hence the a= components will cancel in the integration process, and we only need to integrate the dBr in Eq. (3 -35a): 78 STATIC ELECTRIC FIELDS 1 3 , Equation (3-36) is an importa~~l i.csult for an inhiite linc chargc. Of course, no physical line charge is infinitely long; ncvcrtl~cless, Eq. (3-36) gives the approximate E field of a long straight.line charge at a point close to the line charge. 3-4 GAUSS'S LAW AND APPLICATIONS ! Gauss's law follows directly from the divergence p&tulate of elect~ostatics, Eq. (3-4), by the application of the divergence theorem, 1t;has been derived in Section 3-2 as Eq. (3-7) and is repeated here on account of its impoytance:' I --.----.--.---,- 1 Gtruss's Irr\. osscrrs rlrtrr rlrc rortrl ool\~rr~tl,/lrr.- (!I' tlrc~ IG/icItl o r w t r ~ ~ j - c~/o,scd s~rr:/ircv in p e e sptrce is oq1~11 to the ruttrl C . / I ~ I I Y I C ~ C I I ~ I O S C Y I ill tire S L I I $ I C ~ ~lil:ided by E ~ . We note that the surface S can be any hj~pothetical (nzathrr~atical) close&rJace chosen for come?lierlce: it does not have to be. and usually is not, a physical surface. Gauss's law is particularly usefui in determining he E-field of charge distributions with some symmetry conditions, such that the noynal c&lporreitt f tlze electric ,field irllcrlsil!~ is cwls~rr~i~ oocr (111 cv~c~lo,sc~l .srr~;luc.c. I n such cxcs tllc surfr~cc intcgral on thc left side of Eq. (3-37) would bc very easy to cval.u;~te, &nd Gauss's law would be a much more efficient way for fin'ding the electrik field intensity than Eqs. (3-29) through (3--33). On the ather hapd. whcn synimeti-y conditions do not exist. Gauss's law would not be of much help. TL essence of applying Gauss's law lies first in the recognition of symmetry conditions, and second in thqsdtable choice of a surface over which the normal component of E resulting fro? a given charge distribution is a constant. Such n surface is referred to as a Ga~~ssiar? surfkc. This hasic principle wiih L I S ~ t o o1mi11 l<q. ( 3 131 1'01; ;I, poi111 ~ I ; I I ~ ; , c ~ I I ; I I pos~css~s spl~c~.ical < , Y I I I I I I C I ~ Y ; consccl~~cnlly, a propcr C;a~~ssi:~lr +tirliic,: i:., ~ l i c <LII.I:L~:C!O~' ;L ~ ~ [ ~ I I ( : I ~ G C C I I I I : I ~ : I I L ~ I C point charge. Gauss's law cc)uld nbt help in the Ycrivqtion of Eq. (3-22) or (3--27) for an electric dipole; since a surfice about a sephrateb pair of equal and opposite charges over which the norqal coinponent of E-rzmains constant was not known. 1 i . t P Example 3-4 Use Gauss's law tobetermine the electric field intensity ofan infinitely long. straight, line charge of a uniform density in air. Solution: This problem was solved in Example 3-3 by using Eq. (3-32). Since the line charge is infinitely long, the resultant E field must be radial and perpendicular to the line charge (E = a,E,), aqd a component of E along the line cannot exist. With the obvious cylindrical symmetry, we construct q'cylindrical Gaussian surface of a radius r and an arbitrary lenith L with the line charge as its axis, as shown in Fig. 3-7. On t h ~ s surface, E, is cdnstant, and ds = a,r d 4 dz (from Eq. 2-52a). We I i 1 3-4 1 GAUSS'S LAW AND APPLICATIONS 79 have Infinitely long . . uniform ling charge, pp. Fig. 3-7 Applying Gauss's law to an infinitely long line charge (Example 3-4). There is no contribution from the top or the bottom-face of the cylinder because on Ihc top f x c 11s ---- :I2r tlr r l $ but li, has no z-component there, making E ils = 0. S~III~I:IIIY I'or Ilic I I I I I ~ O I I I I ~ I ~ G , ' l l ~ ~ O I J I I ~ I I I I . ~ C C I ~ ~ : I C ~ + C L I i ~ i ~ I I C C ~ I ~ I I C I C I ~ is Q -. ,,, b . Solution: First we recognize that the given source condition has spherical symmetry. The proper Gaussian surfaces must therefore be concentric spherical surhces. We must find the E field in two regions. Refer to Fig. 3-9. A hypothetical spherical Gaussian surface S, with R < h is constructed with111 rhc electron cloud. On this surface, E is radial and has a constant magnitude. E = a;,E,, . - rls = a, d s . The total outward E flux is 6, E - ds = ER J s dr = ER4nR2. The total charge enclosed within the Gaussian surface is e = J , p I (Example 3-6). I . Substitution into Eq. (3-7) yields i f We see that within the uniformelectron cloud-the J3 field is directed toward the center and has a magnitude pfdportiona~ to the distance from the center. . . , , b) R 2 b I , .I P For this case we construpt a fpherical Gaussiaq &face So with R > h outside the electron clouil. We obtain ihe same expression lor jL0 E ds as in case (a). The total chargc e~closed is , h ' . which follows the inverse qquare law and could have been obtained directly from Eq. (3-12). We observe that aurside the charged claud the E field is exactly the same as though the total chafige is concentrated on a single point charge at the center. T h ~ s is true, in general, for a spherically symmetrical charged rcglon even though p is a function of R. The variation of ER versus R is plotted in Fig. 3-9. Note that the formal solution of this problem requires only a fe% lines. If ~ a u s s s law is not used. it is necessary (1) to choose a differential volume dement arbitr&ily located in the electron cloud, (2) to express its vector distance R: to a field poi4t in a chosen coordinate,syitem, and (3) t6 perform a triple integration as indicated'in Eq, (3-29). This ir a hopelevly involved process. The moral is: T!) to apply Gdhssls.law if symmetry conditions exist for the given charge distributicn. , t h i: , .
a 3-5 ELECTRIC POTENTIAL: I .. < In connection with the null identhy in Eq. (2-134 we poted that a curl-free vector field could always be expressed ,zs the gradient of'a sc, jar field. This induccs us to 1 ' define a scalar electric potentiql, c, such that , . ! ' (3- 38) , because scalar quantities are aasidr ;to handle thad vect r uantities. If we can deter- 5 mine V more easily, then E cap b& fbund by'a gradient operation, which is a straight- forward process in an orthogonal~coordinate system. The reason lor the inclusion of a negative sign in Eq. (3-38) $ill be explained pr&eerltly, t a - )In faces 'ig. 3-8. :d sheet 3-4 / GAUSS'S LAW AND APPLICATIONS 81 , Example 3-6 Determine the E field caused by a spherical cloud of electrons with a volume charge density p = -p, for 0 5 R i b (both po and b are positive) and ' , p = O f o r R > b . Solution: First we recognize that the given source condition has spherical symmetry. The proper Gaussian surfaces must therefore be concentric spherical surfaces. We must find the E field in two regions. Refer to Fig. 3-9. A hypothetical spherical Gaussian surface Si with R < h is constructed within ihs electron cloud. On this surface. E is radial and has a constant magnitude. E = a,$,, . -rls = a, d s . The total outward E flux is 6, E . ds = E, J s ds = ER4nR2. The total charge enclosed within the Gaussian surface is e = j ' , p du 4n. = - I = - - 3 l;i&, 3-0 lilcclric licld Inlcnslty of' a sphcrical electron cloud (Example 3-6). aard the I -. 1 outside case (a). f- 11) ,..?rn L!LYl, c at the ion cvcn \olutloll eccssary In cloud, 's) stem, ipclessly )nditions :e vector 2es us to , r (3- !I\ tlclw ~trn~ght- luslon of 3-5 1 ELECTRIC POTENTIAL 83 Electric potential does have physical significance, and it is related to the work done in carrying a charge from one point to another. In Section 3-2 we defined rhe , electric field intensity as the force acting on a unit test charge. Therefore, in moving a unit charge from point P , to point P, in an electric field, work must be done against the,jield and is equal to ' 'Many paths may be followed in going from P , to P, . Two such paths are drawn in ~ i g . 3-10. Since the ~ a t h between P , and P, is not specified in Eq. (3-39). the question naturally arises, d o ~ i iilc work dcpund on the p t h i:lkcot! h liitlc il~ou$it i i i ; i lead Gs'to conclude fhat CV/q in Eq. (3-39) should not depend on the path; for. l f it did, one would be able to go from P1 to P, along a path for which W is smaller and then to come back to PI along another path, achieving a net gain in work or energy. This would be contrary to the principle of conservation of energy. We have nlrcndy alluded to the path-independence nature of the scalar line integral of the irrotationni (conservative) E field when we discussed E q (3-8). Analogous to the concept of potential energy in mechanics, E q (3-39) represents the difference in electric potential energy of a unit charge between point P, and point P I . Denoting the electric potential energy per unit charge by V, the electric potenrial, wc have Mathematically, Eq. (3-40) can be obtained by substituting Eq. (3-38) in Eq. (3-39). Thus, in view of Eq. (2-81), -Spy E . dP = SP'' (VV) . (al d l ) Fig. 3-10 Two paths Icading from P, to P, in an electric. field. 84 STATIC ELECTRIC FIELpS /:J' ! , . : 4 1 Direction of iwrcming V ' Fig. 3-1 1 Rclutivc direction\ u of and increasing V. . ; j ,. . , , What we have defined in Eq, (3-40) is a ptentid difference (elcrrrorroric i.oltoge) between points P, and PI. It makes no moresense l ; o talk about the absolute potential of a point than about the absolpte phasc of a, phasor ar thc absolutc altitude of a geographical location: a refercpceqro-potential point. a reference zero phase (usmlly :I( r : ; 0). or :I rclixc~~cc zc1.11 ;~lli!h!c (IISII:III~ ;I( S Y IwuI) I U \ I S ~ lirst hb sl)chyiliccl. 111 most (1~1t not i l l ) G I S ~ S , tlic ~ c ~ ~ o - p i ~ l ~ ~ ~ ~ i ; ~ l l;o1111 is l4Lc11 ;I! i11Ii11ity. WII~II (lie I V I ~ ~ L - I I C ~ zero-potential poinl is 1101 ;\I inlini!y, it shoulri bc-spcci/icully statctl. We want to make two mqre about Eq;(3-38). First, the inclusion of the negative sign is necessary in ordoe to copform with tile conve&on that in going oyoiirst the E field the electric potenlial V ina.eo.ser. For instance. when a DC battery of a voltage Vo is connected between two parallel qondu~ting plates. as in Fig. 3-1 1, positive and negative charges;cu$ulate, respectively,: 011 the top and bottom plates. The E field is directed from pdsitive to negative chitrges. whiic thc potential incrcnscs in the opposilc direction. Sccqn?.'dx know from &&tiqp 2 --5 when we dclincd the gradient of a scalar field that the dirkction of V V isiiormyl to the surfaces of constant V . Hence, if we use directed flili'(iljir~e.s or .strennili+s~to indicate the direction of the E field. they are everywhere perp&diculitr to qiiipotu@iul lines and ~qsiporoliid . . : !' . S U I ~ ~ J C ~ ' S . ' I ! 1 3-5.1 Electric Potential due to a I . Charge Distribution ! I I ' The electric potential of a at 2 distance R fr a point charge q referred to that at infinity. can be obtained rGdily from Eq. (3-43: , I - , d . . which gives I (3 - 42) ' I r , This is a scalar quantity and on, besid~sq,~only the distance R. The potential difference between gny two poiqts'~, and PI at bistalfes R, and R,, respectively. , ? I ' . ! : , I I. I . . i ; i t : 1 , ' a ! i . ! 4 ; I 8 3-5 / ELECTRIC POTENTIAL 85 f ,,---\ / ' / /' ,/-- $ ' \ 1 \ I / I \ 1 I I \ 9 R~ I I \ I I / \ '\ /' I \ '---/ / / '\ I $ / '\ / , ' Fig. 3-12 Path of integration '%---- about a point charge. .
$ from q is 'll~is rcsult m ~ y ;~ppc:~r : I little siirprisi~~g a1 first, si~lcc lJ2 ;~nd 1 1 ~ y no1 lie on tile same radial line through q, as illustrated in Fig. 3- 12. However, the concentric circles (spheres) passing through P, and P, are equipotential lines (surfaces) and Vp2 - 1 .
, is tllc S : I I ~ :IS I<,, - V,.,, Fro111 IIIC poi111 d v i c i of l<q, (3 40) wc G I I I c l ~ ~ ~ o s c 111c p ; ~ t l ~ 01' i ~ l l c p y ~ ~ i o ~ l I ' I ' O I I ~ l ' , 111 I , i l l i d ~ I I C I I liu111 lbi 10 /Ii. NU W O ~ L is ~ I C ~ I I C I ~ U I I I 1': 10 P, bccause E is perpendicular to dP = a,R, dq5 along the circular path (E - clC = 0). The electric potential due to a system of n discrete point charges q,, q,, . . . . qn located at R'I, R;, . . . ,R: is, by superposition, the sum of the potentials due to the individual charges: Since this is a scalar sum, it is, in general. easier to determine E by taking the negative gradient of V than from the vector sum in Eq. (3-18) directly. As an example, let us again consider an electric dipole consisting of c h a ~ e s t q and - q with a small separation d. The distances from the charges to a field point P are designated R+ and R-, as shown in Fig. 3-13. The potential at P can be written down directly : '- If d < < R, we have V P 1 ! 5 I 1 1 I 71 S 1 1 , .Fig. 3-13 :An elq~trid dipole. I and 1 d ' - 1 (1 cos 0 . R_z(R+jms:O) Z R - ' ( I ; ~ ~ ) qd cos 8 Y!= --- -1 4ncOR2 I where p = qd. (The "approxima!e" sign (-) has bee0 dropped for simplicity.) The E field can be obtaineq fro'rn - VV. In spherical coordinates we have : L = --. ( a , ~ cos o + a ; sin 4 ) . (3 491 47T601<3 - L~ a Equation (3-49) is the shme as eq.~3-27), but has Feen bbtained by a simpler pro- cedure without manipulating psit ti on vectors. I : Example 3-7 Make a twq-d@en;ional sketch of the q~uipotential lines and the electric field lines for an electrii; dipole. : . T I . ' Solution: The equation of an eqiipotential surfade of q charge distribution is ob- tained by setting the expressiop for V to equal a constant. Since q, d, and E, in Eq. (3-48) for an electric dipole afe fixe'd quantities, a constant V requires a constant ratio (cos 0/R2). Hence the equation' for an equipotential surface is 1 : , Pi= c , & a , ' - ' , (3 -50) 3 . ' t , 9 ;: I I. : . 1. , . i a I . .. . I ( 3 - 37) /? ( 3 38) I \ f2 ( 3 -19) ~Icr pro- and the P\ m is ob- - 0 111 constant (3 -50) 3-5 / ELECTRIC POTENTIAL 87 , F where c, is a constant. By plotting R versus 8 for various values of c,, we draw the solid equipotential lines in Fig. 3-14. In the range 0 i 8 I n/2, V is positive; R is maximum at 8 = 0 and zero at 8 = 90". A mirror image is obtained in the range ' n/2 I 8 S n where V is negative. -4 3 The electric field lines or streamliiks represent the direction of the E field in space. We set -, fig. 3-14 Equipotential andrlectric field lines of an electric dipole (Example 3-7). . I , i 1 :., 8 I s J . 88 STATIC ELECTRl 1 . , b . i d 1 2 : where k is a constant. In spheri 1 &?ordinates, Eq.43-51) becomes (see Eq. 2-66). r , L a, dR + aeR d B + aJ? pih8 d+ = k ( g R f 4 ~ Ee + a$.&, (3-52) 5 . . which can be written I dR R.d8 R sin B d4 -= I - - (3-53) . iE8 IE, Eq, ' For an electric dipole, there is na E4 :component, arld I i .
. 'dR Rd8 " : = - 2 cos 8 sin 0 or 4 6 : 2 (/(sin a) -= 1 .
1 -54) p sin 0 Integrating Eq. (3-54). we obtajp : ; -1 . R h cE sinZ 8, (3-55) where c, is a constant. The electriq field lines, having m a x p at 8 = n/2, are dashed in Fig. 3-14. They are rotqtion@lly sfmmetrical abdut thq z-axis (independent of 0) and are everywhere normal to !fie equipoteptial l$?s, u 1 The electric potential due to a ; c p h u o u s qistributiqn of charge confined in a I eiven'region is obtained by intdgrafirlg the contribpdon of an element of charge over ;he charged region. We have, fqr q ldlume charge &itrib tion, I 8 I 1 '" For a surface charge distributi~g, I- , Example 3-8 Obtain a foqultiFor the electljo~field4intensity on the axis of a circular disk of radius b that clrrieb a uniform surface charge density p,. , : I I t !; . ' ;, ': i 1 i ij. !; 1, $ ). ,i ! I , !, : . 2-66). (3-52) # (3-53) (3 -54) (3-55) r d a s i d nt c led in a -ge over (3 -56) (3 -57) t n (3--""' xis of a ' t 3-5 / ELECTRIC POTENTIAL 89 , 3 I Solution: Although the disk has circular symmetry, we cannot visualize a surface around it over which the normal component of E has a constant magnitude; hence Gauss's law is not useful for the solution of this problem. We use Eq. (3-57). Working I with cylindrical coordinates indidated in Fig. 3-15, we have I ds' = r' dr' d#' and R = Jmi. The electric potential at the point P(0, 0, z) referring to the point at infinity is r ' I dr' d4' 1 P n =- [(z2 + b2)'I2 - 1211. 260 (3 -59) Therefore, E = -VV= - av a, - az The determination of E field at an off-axis point would be a much more dificult problem. Do you know why? For very large z, it is convenient to expand the second term in Eqs. (3-60a) and (3-60b) into a binomial series and neglect the second and all higher powers of the ratio (b2/z2). We have Fig. g l 5 A uniformly charged disk (Example 3-8). 1 ' ' I . ; ; . , . , Substituting this into Eqs. (3-Pa) &d I 4 ' I E=a,- ~ K E ~ Z ~ . where Q is the total chargp on ihe disk. Hence, when the point of observation is very far away from the charged disk, thpE field approximatelj follows the inverse square law as if the total charge were coqcentrated at a point. I, - Example 3-9 Obtain a fo;rnuln b r the electric field iltensity along the axis of a uniform line charge of length 4. The uniform Iioe-dmrgfi'densit is p,. --L - Solution: For an infinitely lopg line charge, the E fieldtpn be determined readily by applying Gauss's law. B s iq the solution to J3xamplp 3-4. However, for a line charge of finite length, as showq in Fig. 3-16, we cannot cqnstruct a Gaussian surface over which E . ds is constant. qpuss's law is therefore not pseful here. Instead, we use Eq. ( 3 -5 8 ) ' by taking an element dlt = drf at il. The distance R from the charge elebent .to the p o i ~ t P(0, the axis of the line charge is . . , i . r . , L . 6" R : = (Z z ' ) , Z > T . .
, I Here it is extremely important distinguish the podition bf the field point (unprimed coordinates) from the position ~f tpe source point (prime4 coordinates). We integrate 4 $ .
t ' 3-6 ( ELECT :3-61a) 3-61b) 1 s very squarr c1s of li / ' - r e t a 1 1 .
- \L!Y~.. J ' . The. hc l111e ~ n m e d regrate n 3-6 / CONDUCTORS IN STATIC ELECTRIC FIELD 91 I? . ' , , . over the source region . , pc ~ 1 2 dz' V = - 47E0 S-LIZ I - zf z + (L/2) L ="In[: 1, 2 , -. - 4nc0 z - (L/2) 2 (3 -62) The E field at P is the negative gradient of V with respect to the unprimed field coordinates. For this problem, The preceding two exampies iiiustrate the procedure for determining E by first finding V when Gauss's law cannot be conveniently applied. However, we emphasize that, if~ymmetry conditions exist such that a Gaussian surfrrce cun be constructed over which E . ds is constant, it is always easier to determine E directly. The potential V , if desired, may be obtained from E by integration. 3-6 CONDUCTORS IN STATIC ELECTRIC FIELD So far we have discussed only the electric field of stationary charge distrlbut~ons in frcc spacc or air. Wc now cxaminc: thc licld bchavior in matcnal mcdia. In gcncral, we classify materials according to their electrical properties into three types: con- hctors, setniconductors, and insulators (or dielectrics). In terms of the crude atomic model of an atom consisting of a positively charged nucleus with orbiting electrons, thc clcctrons in thc outermost shclls of Lhc atoms of~conductors arc vcry loosely held and migrate easily from one atom to another. Most metals belong to this group. The electrons in the atoms of insulators or dielectrics, however, are held firmly to their . orbits; they cannot be liberated in normal circumstances, even by the applicatlorl of an external elect~ic field. The electrical properties of semiconductors fall between those of conductors and insulators in that they possess a relatively small number of freely movable charges. 111 kcrms of khc band theory of solids, we find that there are allowed energy bands for electrons, each band consisting of many closely spaced, discrete energy states. Between these energy bands there may be forbidden regions or gaps where no eiec- trons of tbwlfd's atom can reside. Conductors have an upper energy band partially filled with electrons or an upper pair of overlapping bands that are partially filled so that the electrons in these bands can move from one to another with only a small change in energy. Insulators or dielectrics are materials with a completely filled upper band, so conduction could not normally.occur because of the existence of a large energy g?p to the next higher band. If the energy gap of the forbidden region is relatively small, small amounts of external energy may be sufficient to excite the electrons in the filled upper bapd to jump into the next band, causing conduction. Such materials are semiconductors. I 92 STATIC ELECTRIC FlE+PS 3 ; > I ? ;.I The macroscopic electri 1 ptoperty of a mattrial r(lcdium is characterized by a constitutive parameter calle~cdnhctitlity, which we Mil define in Chapter 5. The definition of conductivity, holueger$ is not importbt id this ;chapter because we are not dealing with current flow and are now interested anly in the behavior of static electric fields in material media,'In this sectiop we exgmi e the electric field and i charge distribution both insiqe t\iq bulk and on t4c surfi~cc qf a conductor. Assumc for the prcscnl tkul yoiilc positive (oi qg+v@ichargc~ are introduced ' in the interior of a conductofi. An electric field will be let up in the conductor, the field exerting a force on the charges and making them nji,ve away from one another. This movement will continua'until all the charges reach the conductor surface and redistribute themselves ip suck a way that both thacharge and tfie field inside vanish. Hence, I ', I (tJnCfer Static Copditions) ' 1 p = o -1 (3 -64) 1 ' I I When there is no charge in the inteiior of a conquctgr (p ; 0ik must be zero because, according to Gauss's law, the lot@ putward electric flu thropgh ony closed surface constructed inside the conduc or &st vanish. : t ,' The charge distribution ob thesurface pf i , conductPr depends on the shape of the surface. Obviously the chqges ~ ~ o u l d not be in b state of eqdfjbrium if there were a tangential component of the electtic field intepsity that pro@ces k tangential force and moves the charges. Therefork; hnder sraric co~ditio'~ the E3eld on a conductor surfice is everywhere normal tq th4sbrfaee. In othet:wor$, rh$ surface of a conducror is (In equrpofentiul surface un$r ydtic contlition.s.!A~ n,pqttct of fact, since E = O cvcrywlicrc inside o conclucior, f l ~ c whole col~d\rclor llasYthc w n c clcctro\tat~c potential. A finite limc is rc irql for thc chqrgcjs to f;pdistributc on a conductor surface and reach the eguilibri m State. This time qepenqp on the conductivity of the material. For a good c~pduct r s ~ c b as copper, this tim is in the order of 10- l 9 (s), J ? a very brief transient. (This ppinh will be elaborated in Section 5-4.) Figure 3-17 shows an int fa between a c~ncl$ctor $nd bee space. Consider the contour ubcdu, which hasew7ih'# = cd .
= Aw and = Ah. Sides ab and cd are parallel to ihe int&fq$d Applying that E in a conductor is'tero, ' e bpiain immecjjavly ' - ?' Y : & - : ; E ; .<' r' $ 1 1 4 dl' = Et Av:= 0 : + J .
, or . . - L. (? E , = O , ,,! i !:a (3 -66) which says that the tengential ~ot$nent of the $ fikld oLIh a ocnddq~toor surfoce is zero. In order to find E,,, the norm 1 cfiflponent of E ailthe fprfacc of the conductor, we ! ? , I > , !,' ;i --' : l i . b ' . L ' ( 4 I 1; i. t ; 1 3 !! ! ; : d by a 5. The we are c f static :Id ayd oduced or, the nother. ~ce and vanish. , . . (3 -64) (3 -p, \ :cause, surface !
a l p of' : e were 1 1 force lducror ~;luc.tc,r E = O O b l d t l C lductor of the - 1 9 (4, der the des ub noting n I (3 -66) is zero. tor, we 3-6 1 CONDUCTORS IN STATIC ELECTRIC FIELD 93 , 1 r. 1 6 , ) . A I h 1 I . ; .
.
.
I I I I i Ps I 1 . ' I I Fig. 3- 17 A conduct&-free space interface. I construct a Gaussian surface in the form of a thin pillbox with the top face in free space and the bottom face in the conductor where E = 0. Using Eq. (3-7), we obtain P E,, = -2. € 0 Hence, the normal component of the E field at a conductor-free space boirilbry is equul to the surfuce churoe density on the conductor divided by the permirtioitp of j x e spucc. Summarizing the buundury condifions at the conductor surhcc, we have When an uncharged conductor is placed in a static electric field. the external field will cause loosely held electrons inside the conductor to move in a direction opposite to that of the field and cause net positive charges to move in the direction of the f i e l q h e s e induced free charges bill distribute on the conductor surface and create an induc2d field in such a way that they cancel the external field both inside the conductor and tangent to its surface. When the surface charge distribution reaches an equilibrium, all four relations. Eqs. (3-64) through (3-67). will hold: , a'nd the conductor is agdin an equipotential body. Example 3-10 A positive point charge Q is at the center of a spherical conducting shell of an inner radius Ri and an outer radius R.. Determine E and V as functions of the radial distance R. I 94 STATIC ELECTRIC FIELDS ( 3 : , . 1: t: ,' 1 : 1; 1 : , L i Fig. 3-14 J E I W ~ ~ field intensity and potential vdiatio$ of a point charge +Q at the knier flf a conducting shell ( ~ x a m ~ l e 3 , 0). i , I 1 ! . li 1 : Solution: The geometry qf the is shown I id' ~ g . j3- 18(a). Since there is spherical symmetry. it is sirnplest,jo usi Gauss's law io determine E and then find V by integration. There are three p t i n a regions: (I) R > a, (b) R, S R 5 R,, and (c) R < R,. Suitable spherical Gaupiqn qurfaces will be cons{ructed in these regions. Obviously, E = a,E, in all fhree rbgiohd. - !
I 6 r ' - j 3 a) R > R, (Gaussian surface S,): i! i 1' ii! I! i $ $ i = E R , 4 n R 2 = - : jEO: B . or t 1 - 3- I 3-7' RL ELEC~RIC Idc e x t ma1 eve effe I 1 I 1 i / ' - J here is sn lind to, and .egions. r. (3 -68) 3-7 1 DIELECTRICS IN STATIC ELECTRIC FIELD 95 ' . The E field is ths kame as that o f a point charge Q without the presence of the shell. The potential referring to the'point at infinity is , -.. I 1 b) R, 5 R < Ro (Gaussian surface S,): Because of Eq. (3-65), we L o w ER2 = 0. (3 -70) Since p = 0 in the conducting shell and since the total charge enclosed in surface S2 must be zero, an amount of negative charge equal to - Q must be induced on the inner shell'surface at R = Ri. (This also means an amount of positive. charge, eqcl?! LO t Q is induced on the outer shell surface at R = R,.) The cull- ducting shell is an equipotential body. Hence, c) R < Ri (Gaussian surface S,): Application of Gauss's law yields the same formula for ER3 as ERl in Eq. (3-68) for the first region: Q ER3 = - 4 x e , ~ ~ ' ' (3 -72) The potential in this region is & = -s ER3 dR + C = --- +c, 4m, R where the integration constant C is determined by requiring V, at R = R, to equal V2 in Eq. (3-71). We have and The variations of E i and V versus R in all three regions are plotted in Figs. 3-18(b) and 3-18(c). 3-7 DIELECTRIC~N STATIC ELECTRIC FIELD Ideal dielectrics do not contain free charges. When a dielectric body is placed in an external electric field, there are no induced free charges that move to the surface and make the interior charge density and electric field vanish, as with conductors. How- ever, since dielectrics contain bound charges, we cannot conclude that they have no effect on the electric field in which they are placed. i F (6. !' , , I . 96 STATIC ELECTRIC FIELDS / 3 ' , I I dielectric material. more dissimilar which do not have permanent cules arc of lhe order o f dipoles in a polar on thelindividual that shown in Fig. 3-19. Int PO] anc is Her Rec.
whe: of tt 1uclcus lectrlcs 3 f c p 30511, , 1 S 1 T ! crc;llc; dlpoles !la1 the ~ d e the t'lCII Ill two or decules, r mole- hv~dual .lament livldual l o w in n I l'LJcll- (3 -74) 3-7 1 DIELECTRICS IN STATIC ELECTRIC FIELD 97 , : where n is the number of atoms per unit volume and the numerator represents the vector sum of the induced dipole moments contained in a very small volume Av. . The vector P, a smoothed point function, is the volume density of electric dipole -S moment. The dipole moment dp of an elemental volume dv' is dp = P du', which produces an electrostatic potential (see Eq. 3-48) P a , d V = - dv' . 4m0RZ (3-75) Integrating over the volume V' of the dielectric, we obtain the potential due to the polarized dielectric. a dv' , (3-76)+- L - where R is the distance from the elemental volume do' to a fixed field polnt. In Cartesian coordinates, RZ = ( X - x')' + ( y - y')2 + ( Z - zl)', ( 3 -77) and it is readily verlfred that the gradient of 1/R with respect to the primed coordiacres is (3-78) Hence, Eq. (3-76) can be written as Recalling the vector identity (Problem 2-18), V 1 . ( f A ) = f v ' . A + A . v'f, . (3-80) and letting A = P and f = l/R, we can rewrite Eq. (3-79) as .1 V' - P V = -[J, 4xc, V 1 - ( : ) d u r - sV, ~ d v ' ] . (3 -8 I ) The first volume integral on the right side of Eq. (3-81) can be converted into a closed surface integral by the divergence theorem. We have where a. is the outward normal to the surface element ds' ofthe dielectric. Comparissn of the two integrals on the right side of Eq. (3-82) with Eqs. (3-57) and (3-56), , ' We note here that V on the left sjde of Eq. (3-76) represents the electric potenriol at a field polnt. and V' on the right side is the volume of the polarized dielectric. I . , 1 ' , . I i c $ $' . 1 . , . I 98 STATIC ELECTRIC FIELD^ 3 . . 11; . " . : ; ; J :.:-.,:. .;:... . . " - I I f , , . 1 : " :. , i # - respectively, reveals that the eleqplc pbtential (and thkrefoi the electric field intensity also) due to a polarized dielectriq can be calculated fiom the contributions of surface and volume charge distributions haying, respectivsld densities , and (3-84)' These are referred to as polariz(ltion,.charge densities or ((pund-charge densities. In other words, a polarized dieleetrio cnn,he rcplncrd hy oy rqsi~olent polorizntior~ a,$,ce dargc (lensity p, end UII rquiuq~e~~t~polurizatioi~ voluflle charge density pp for field calculations. i r (3 -85) with the aid of a vector identity, a charge distributions. . The sketch in Fig. 3-19 the ends of similarly oriented dipoles exist on fpolarization. Consider an imaginary elemental Tl~e application of an external electric field the bound charges: positive charge +q move a t h ~ field and negative charges - q move an equal thd field. The net total 4f2 F nq(d . anKAs). , , I ' (3 -86) ' tt But the dipole moment per 4 it volume, is by dehnitioR the polarization vector P. We have . f ?, . ' t 9Q = P . an(As) ;+ 9 ; JI (3-87) and t : , a,,, ! I -. ' as given in Eq. (3-83). nbnnal. This relation -w . . correctly gives a positive surface in Fig. 3-19 and a negative surface charge on the I . - - I" - 1 % ' The prune ~ g n oh a. m d V h n heen drop@ b r smplic~ty. rjnd'e' , 9s. 1;)-83) and (3-84) involve only source coordinates and no confuqiqn ~)J'@u[t:! i ' r 4 - ;, , , : 3 - i p ; : f 1 -r t i 1 . 1 > . $ 4 '.J i 5 i - wit wl1 whi to t the vcrl whe 3-8 ELEC DIELECTRIC Bea the 1 diffe ' mils tensity surface (3-83)+ (3-84)' 11ca 111 surjace or field (3 -rx . ;I1 1111 ton$. llilldl.ly on\~der II :\I) ll.ll;~cb: 2gallve et total .vhere 11 :ma1 to ld (3-86) I vector (3-87) p, ' r ~ I i \ t i \ 9 and a olve only 3-8 / ELECTRIC FLUX DENSITY AND DIELECTRIC CONSTANT 99 For a surface S bounding a volume V , the net total charge flowing out of V as a result of polarization is obtained by integrating Eq. (3-87). The net charge remaining ' within the volume V is the negative of this integral. L 0 Q = - $ s P . a,, ds I - . which leads to the expression for the volume charge density in Eq. (3-84). Hence, when the divergence of P does not vanish, the bulk of the polarized dielectric appears ; ; !
x chdrged. However, since we started with an electrically neutral dielectric body, ' thc total charge of the body after polarization must remain zero. This can be readily verified by noting that Total charge = $ p,,, (1s + Jv p, du = $ s ~ . a , , d s - J v v . ~ d ~ = ~ , (3 -89) where the divergence theorem has again been applied. .8 ELECTRIC FLUX DENSITY AND DIELECTRIC CONSTANT Ih.c:~\~sc :I ~xil;~ri/.ctl tliclcc~ric ~ivcs rise t i ) :I vol~~nlc c h ; ~ r ~ c tlcnsily I),,. wc espcct ihc electric lield inicrlsity duc to a given sourcc distribution in a ciiclcctric to be different from that in free space. In particular, the divergence postulated in Eq. (3-4) must be modified to include the effect of p,; that is, 1 . -5 ' v.32 = d P + p,). , (3-90) €0 Using Eq. (3-841, we have Wc now define a new fundamental field quantity, the electricJlux density, or e1ectr.i~ displacement, D, such that The use of the vector D efiiiblcs us to write il divergence rc1:rtion hctwccn the electric field and the distribution of j i v e charges in any medium without the necessity of dealing explicitly with the polarization vector P or the polarization charge density p,. Combining Eqs. (3-91) and (3-92), we obtain the new equation I !:i !' ; / i'.. { , . f ' 1 , I r. 1 .i where P is the voiume depsity C $ file; char& ~ ~ u a k i & s 1 3 -; !
I and (3 -5) are the two fundamental governing diffqreqfial equation9 ' f ~ f &ctrostaics in any medium. Note that thel~ermittivity af fred space, ro, does not! appaar e$licitly in these two equations. $ The corresponding integral fyrrq ok Eq. (3-93) is obtaiqed by taking the volume integral of both sides. We have : ' ; ' i I ' Equation (3-951, another f o p of G&SS'S law, states! that ;he total outward pk ql the e/ecfric displacone~~ (or, simply, tlfe totol of~t~varif e1ee;~ic~us) ouer ally closed suguce is equul to the total pee charg4 +enclosed in the~surjaqe. As-b,s been indicated in Section 3-4, Gauss's law is rqos{ju eful in dete&jninajthe elect& field due to charge distributions under symmetry c nditions. :' 6 When the dielectric properfp b ( the medium $re linear and isotropic, the polariztion is directly proporticjpa] to the electric geld iqensity, and the propor- tionality constant is independent of the direction of the field. We write :i , , , # = EOXA (3 -96) where xi is a dimensionless dielectric medium is linear if X. is of space is a dimensionless constant know as t& relotiye or the dielectric constunt of the medium. The coefiicient c gsdil the absoluIe &@it jvity (often called simply I permittiuit~) of the medium and 1s p,+sured in meter (F/m). Air has a dielectric constant of l.WQ59; habcei,tb ennittivity h us$lly taken as that of free space. The dielectric constants gf s f ! tther ma.r&i a r included in a table in Appendix B. : f ! 4 ' . L , . .. : I ; ! .!, ' 1 , 111 . ' A tensor would be required to represent the d&tric swceptibility~ the I i ' t - 'f i , : , . > : is anisotropic. So CO I sol >fin, v 1 ,the - 5 t the :irc 111c ledium. ' cse two volume (3 -94) (3-95) flu o f close~f t11c;llod d u i p IIC. t' )rap . (3 96) I ~ ~ I U J I I f spacc 1 (3-97) (3 -98) )rlstallf s i n I L L of fsw ~ble 1 . 3-8 / ELECTRIC FLUX DENSITY AND DIkLECTRIC CONSTANT 101 . , . . Nolo 1h;il r, can h a fur~clion ofspnm coordinnlcs. Ilr, is indcpcndcnt ofposition, the medium is said to be homogeneous. A linear, homogeneous, and isotropic medium is called~a simple medium. The relative permittivity of a simple medium is a constant. I Example 3-11 A positive point charge Q is at the center of a spherical dielectric shell of an inner radius Ri and an outer radius R.. The dielectric constant of the shell is c,. Determine E, V, D, and P as functions of the radial distance R. Solution: The geometry of this problem is the same as that of Example 3-10. The conducting shell has now been replaced by a dielectric shell, but the procedure of solution is similar. Because of the spherical symmetry, w e apply Gauss's law to . 5nJ E and D inthiee regions: (a) R > R.; (b) Ri I R 5 and (cJ X <.A,. Potential V is found from the negative line integral of E, and polarization P is determined by the relation P = D - cOE = E ~ ( E , - 1)E. (3 -99) The E, D, and P vectors have only radial components. Refer to Fig. 3-20(a), whcre the Gaussian surfaces are not shown in order to avoid cluttering up the figure. a) R > R, The situation in this region is exactly the same as that in Example 3-10. We have, from Eqs. (3 -68) and (3 -69), Q v1 = -. 4nc,R From Eqs. (3-97) and (3-99), we obtain and P,, = 0. The applicaiion of Gauss's law in this region gives us directly Dielectric shell F i g . 5-20 'Vicld yuri:itions of a point charge +& at the . center of a dielectrjc shell (4 : : , (Example 3-1 1,) a . . t 7 .
. I have a ,m. 3-8 I ELECTRIC FLUX DENSITY AND DIELECTRIC CONSTANT 1 J3 c) R < Ri Since the medium in this region is the same as that in the region R > R., the application of Gauss's law yields ;he 'same expressions for E,, DR,and PR in .... both regions: To hnd v,, we must add to V2 at R = Ri the negative line integral of E,,: The variations of e,E, and DR versus R are plotted in Fig. 3-?O(b). The difference (DR - roE,) is PR and is shown in Fig. 3-PO(c). The plot for V in Fig. 3-20(d) is a composite graph for V,, V , , and V , in the three regions. We note that DR is a con- tinuous curve exhibiting no sudden changes in going from one medium to another and that PR exists only in the dielectric region. It is instructive to compare Figs. 3-20(b) and 3-20(d) with, respectively, Figs. 3-18(b) and 3-18(c) of Example 3-! 1. From Eqs. (3-83) and (3-84) we find on the inner shell surface; Q --. on the outer shell surface; and = - I ( ' i . j aK.(K'l'R2) = 0. (3-lW) Equations (3-107), (3-108);and (3-109) indicate that there is no net polarization volunle charge inside the dielectric shell. However, negative polarization surface charges exist on the inner surface; positive polarization surface charges, on the outer !i - < 3 1 t 1 \ i 1 t I 1 104 STATIC ELECTRIC ~ 1 @ \ 0 ~ 1 ' 3 ' , ' : I I 3 ' . I1 I r , 14 ! Table 3-1 q i c l w f Strengths of ~ o ~ c ! ~ & n ~ o n Materials t . -+ . Material li ~ i c l k c t r i c ~ t r c n ~ t l ~ ( V / I ~ ) ----..------I ) Air (atma~pher/c jyessure) . 3 x 10" 2 Mineral oil I ; l$ix lo6 Polystyrene . , i 2 0 x 1 0 ~ Rubber 25 )( lo6 Glass 1 L i 30 x 106 Mica , 200 x' 10" 7 I , surface. These surface charges produce an electric fisld intensity that is directed radially inward, thus reducing tbe.E field in region 2 due to the point charge + Q 3 > . . at the center. - - - I -- t 3-8.1 Dielectric Strength We have explained that an g\ecf$c field causes small Pisplacements of the bound charges in a dielectric material, teiulting in pola&ation. If the electric field is very strong, it will pull electrons ~oppletely out of the m lecules, causing permanent dislocations in the molecular strtidure. Free char'ges w 1 1 appear. The material will 4 become conducting, and larle cbirents may res$lt. Tbis phenomenon is called a dielectric breakdown. Thi mqiqhA electric figld hten$Ity that a dielectric material can withstand without breakdqwn is the dielecttic st ength of the material. The approximate dielectric streng hs f some common8'$ubst~ce~are given in Table 3-1. f 9 The dielectric strength of a mqtendmust not be c&nfustid with its dielectric constant. A convcnicnt number t o ~ c ~ k i n h c r i\ that tlsc iliu cctric strcnpth of ;i~r a t thc 1- atmospheric prcwirc 1s. 3 k Y / i ~ j ~ n . WIJCII L ~ I C S I C C ~ I I C I , I I C I C & I J I L C I I ~ A L Y cx~cccls Lll~s value, air breaks d o w n .
; , + 4 a s s ' i v i v g ~ i s n i z a t i o n taked'place, and sparking (corona dis- charge) follows. Charbe tendglto concentrate at sharp ~oints. In view of Eq. (3-67). the electric field intensity in tl~c immediate vicinitxd shgrp points is higher than that at points on a surface with ij small curvature. This is 'the principle upon which a . lightning arrester works. Disgbay$e through the s h a ~ p oinL of a lightning arrester prevents damaging discharg~e thmugh nearby ~biect-, 1 The fact that the electric field intensity tends to be point near the ~su of a charged conductor with a larger curvature is in the follo$ng : I . I -i 7 Example 3-12 Consider twp sph:erical conductsjrs~ wjfh rqdii b, and b2 (b, > b,), which are connected by a co-ducdng wire. The ~$stange of,separation between the P conductors is assumed to be,yeqIarge cornpafed to ' so-that the charges on the spherical conductors may berco@iered as u a i f ~ ~ ~ l y ~ i s t r i b u t e d . A total charge Q I J I I ( I ii ; \ ! ! . 3-9 6 ELECTF directed rg,: + Q r--- : b~,,,d ! is vfry i'nlancnt :rial will cailcd a Inatcrial -id. The :ble 3-1. :onstant. ir at the :cds this ona dis- . (3-67), han that which a arrester : e l e p inductor bz > bl), ween the :s on the :harge Q 3-9 1 BOUNDARY CONDITIONS FOR ELECTROSTATIC FIELDS Fig. 3-21 Two connected conducting spheres (Example 3-12). is deposited on the spheres. Find (a) the charges on the two spheres, and (b) the electric field intensities at the sphere surfaces. Solution a) Refer to Fig. 3-21. Since the spherical conductors are at the same potential, we have Hence the charges on the spheres are directly proportional to their radii. But, since we find QI=- b1 Q and b2 b l + b2 2 - - - bl + b2 Q. b) The electric field intensities at the surfaces of the two conducting spheres are The electric field intensities are therefore inversely proportional to the radii, being higher at the surface of the smaller sphere which has a larger curvature. 3-9 BOUNDARY CONDITIONS FOR ELECTROSTATIC FIELDS Electromagnetic problems often involve .hedia with different physical properties and require the knowledge of the relations of the field quantities at an interface between two media. For instance, we may wish to determine how the E and D vectors / w h ; ill!( L , 0 , t I Fig. 3-22 'An inttrface or between two medi~. 1 l ady know the bouqdary conditions that must interface. These copditi~ns have been given I - whe ln Eqs. (3-66) and (3-67). w e now consider an interface bl)tween&-general media shown in Fig. 3-22. i 4 ' lnte Let us construct a smqll path abc% with sidesab and c m media 1 and 2 respec- the tlvely, both being parallel to t& inte'face and equal tb AW; ~ ~ u i t i o n (3-8), whlch is , becc assumed to be valid for regions containing discoqtihuouq~medja, is applied to this path.' If we let s~des bc =ilo -Ah ,Tpproach zero, .their pontribufions to the line integral of E around the path can beiqglected. We hbve i w h ~ " 6 , -' f $ ebcde E . ~ ~ = ~ ~ . A w + ~ ~ . ( - A w ) = E ~ , A w ~ E ~ , I ; w = O . I we 1 Therefore i i I, t t ! or (3-1 10) i t i I( cc i I elec which states that the tanyenrid cQppotaent o f un E je4d i s c~ntinuous acrosr an inter- face. Eq. (3-110) simplifies t o &. (3-66) if one of the media is a conductor When media 1 and 2 are dieleftr&'$ith'.permittivities r; and r, respectively, we have 1 , ' 1 4, , , ;-=-. I . ) I (3-1 11) i I --\ if1 € 2 , .- i In order to find a relation betdecn the norrnal.com~pnems of the fields at a - ,xa boundary, we construct a small illldx with its top ?ace ip medium 1 and bottom elec face in medium 2, as was ia Fig. 3-22. Thdfaces ave an area AS, and the :i ,, . , fi height of the pillbox Ah is vanish nglj( small. ~ ~ ~ l ~ i n k Gayfs's law Eq. (3-95) to the Soil1 I $1 . L 1 . on: I I See C. T Tal. -On the presentation of axwe$ theory: ~ r o c e e & ~ o f (he IEEE, vol. 60. pp 936-945. " 1 intc A U ~ U S ~ 1972. r, '1 necc lat must .n given :I m @ $ respec- \ h1c i to this the line t i n infur- r. When w e have (3-111) /- :Ids a bottom .and (j) to the I ~ I 936-945, 3-9 / BOUNDARY CONDITIONS FOR FLECTROSTATIC FIELDS 137 4 i pillbox, we have , I $Dado = (Dl .an2 + D, .anl) AS i ' =an2 .(Dl - D,) AS = pS As, (3-112) where we have used the relation a,, = -a,,. Unit vectors a,, and a,, are, respectively, outward unit normals to media 1 and 2. From Eq. (3-112) we obtain -1 (3-113a) 1 or . . . t (3-113b) a where the reference unit nornml is omvard from rnediurn 2. Eq. (3-1 13) states that the norntal component of D field is discontinuous across an interface where a surface charge exists-the amount of discontinuity being equal ro -. the surface charge density. If medium 2 is a conductor, D, = 0 and Eq. (3-ll3b) becomes Dln = 6lEln = P s ' (3-1 14) which simplifies to Eq. (3-67) when medium i is free space. When two dielectrics are in contact with no free charges at the interface, p, = 0, we have Recapitulating, we find the boundary conditions that must be satisfied for static electric fields are as follows: Tangential components, El, = E,,; Normal components, a,, (Dl - D,) = ps. Example 373- A lucite sheet ( E , = 3.2) is introduced perpendicularly in a uniform electric field E, = a,E, in free space. .Determine Ei, Di, and Pi inside the lucite. Solution: We assume that the introduction of the lucite sheet does not disturb the original uniform electric field E,. The sitnation is depicted in Fig. 3-23. Since the interfaces are perpendicular to the electric field, only the normal field components need be considered. No free charges exist. t . .d\ ( 108 STATIC ELECTRIC F Lw;/ 3 lr 1 , ' . r . . i I i :I, 1 i t Boundary condition E ~ : (3-1;14) at the left ikt&rfagc gives Di = a,D, = a,D, or .
$ 1 D, = axcoEo: There is no change in electric )?@ density rcrpss thp int&& The electric field intensity inside the ludte shbet i v ',I 1 , $ 4 F i = - D i = - D i = f l 3. ,e cOE, . +2 The polarization vector is zbro oqtside the lucitd sheet(Po = 0). Inside the sheet, 11 1 ~ , - ~ , i ~ ~ ~ ~ ~ = a . ( l $j)LO~O =p a,b.k875r0~, (q/rn2) I ! 2 Clearly, a similar applicati~n dffhe boundqry @ondipon Eq, (3-114) on the right interface will yield the prigifid 4 and D,, in ihdfree /paceon the right of the lucite sbcct. Dacs thc solution ofi(hi$ firohlcrn clli~n if t b dpinal clcctric field i v not uniform, that is, if E, = q,E y)') , , . $ , , . :t ?' , i 1 ;:. f < i r . : . ? ;. . ( ( ; . 1% : , I . i i : 1 " [; i ' ; . Fig. 3-24 ~dund+ conditions at the interfwe b e t w two dielectric media (E~a$~lc.?/!cl). , 1 4 , I : . I I It t 8 1 r , I t 1 , t I , i ; r 4 . . . 1 : i t ! hc ight J ' . 7~ luclte d is not I 3-10 / C~PACITANCE AND CAPACITORS 109 I ,I ,!, Example 3-14. do dielectric media with Permittivities c, and c, are separated by a charge-fred boundary as shown in Fig. 3-24. The electric field intensity in mediuni 1 at the point F, has n magnitude E l and makks an angle cr, with the normal. Deter- mine the magnitude and direction of the electric field intensity at point P, in medium 2. 1 Solution: ~ w o , k ~ & t i o n s are needed to solve'for two unkpowns E2, and E,,,. After E2, and E,,, have been found, E, and I, will follow directly. Using Eqs. (3-1101 and (3 - 1 15), we have . . . $ ,. E, sin a, = El sin ai (3-117) and I n h i ," ' c2E2 COS at = €,El cos bl. (3-118) Division of Eq. (3-1 17) by 5q. (3-1 18) gives -- tan a, The magnitude of E, is By examinink Fig. 3-24, can you tell whether r, is larger or smaller than s,? From Section 3-6 we understand that a conductor in a static electric field 1 s an equipotential body and that charges deposited on a conductor will distribute them- selves on its surface in such a way that the ilectric field inside vanishes. Suppose the potential due to a charge Q is V. Obviously, increasing the total charge by some factor k would merely increase the surface charge density p, everywhere by the same factor, witlrout-affecting the charge distribution because the conductor remains an equipotential body in a static situation. We may conclude from Eq. (3-57) that the potential of an isolated conductor'is directly proportional to the total charge on it. This may also bd seen from the fact that increasing V by a factor of k increases E = -VY by a factor oft. But, from Eq. (3-67), E = anp,/c,; it follows that p, and con- sequently the total charge Q will also increase by a factor of k. The ratio Q/V therefore 11 0 STATIC 'ELECTRIC FIELDS 1 3 remains unchanged. We write where the constant of proportionality C is called the capacitance of the isolated conducting body. The capacitance is the electric charge that must be added to the body per unit increase in its electric potential. Its SI unit is coulomb per volt. or farad (F). Of considerable importance in practice is the capacitor which consists of two conductors separated by free space or a dielectric medium. The conductors may be of arbitrary shapes as in Fig. 3-25. When a DC voltage source is connected'between the conductors, a charge transfer occuqs, resulting in a charge + Q on one conductor and -Q on the other. Several eiectric field lines originating from positive charges and terminating on negative charges are shown in Fig. 3-25. Notc tha! the field lines are perpendicular to the conductor S L I ~ ~ ~ I C C S . which arc cq~iipotcntiill sti~f;lces. Equation (3-121) applies here if V is taken to mean the potential ditrerence between the two conductors, V, ,. That is, The capacitance of a capacitor is a physical property of the two-conductor system. It depends on the geometry of the conductor6 and on the permittivity of the medium between them; it does not depend on either the charge Q or the potential difference V,,. A capacitor has a capacitance even when no voltage is applied to it and no free charges exist on its conductors. Capacitance C can be determined from Eq. (3-122) by either ( I ) assuming n V,, and dotermining Q in terms of V,,. or (2) assuming a Q and determining V,, in terms of Q. At this stage, since we have not yet Fig. 3125 A two-conductor . capacitor. i (3-121) 1 ' .
I r , isolated I d to the I volt, or ! of two may be 3etween nductor ch wgcs he field urfaccs. )elween r ( 3 - 1 ~ lductor y of the ~tcnrial :d to It d from . or (2) not yet P i ' 3 studied the methods lor solving boundary-value problems (which will be taken up in Chapter 4), we find C by the second method. he procedure is as follows: 1. Choose an apprdpriate coordinate system for the given geometry. 2. Assume chargks 4 Q and --Q on the conductors. 5 3. Find E from Q by Eq. (3-114)' Gauss's law, or other relations. 4. Find V I 2 by evaluating from the condbctor carrying - Q to the other carrying + Q. 5. Find C by taking the ratio Q/V,,. . -.- Example 3-15 A parallel-plate capacitor consists of two parallel conducting plates of area S separated by a uniform distance d. The space between the plates is filled with a dielectrlc of a constant permittivity E. Determine the capacitance. Solution: A cross section of the capacitor is shown in Fig. 3-26. It is obvlous that the appropriate coordinate system to use is the Cartesian coordinate system. Follow- ing the procedure outlined above, we put charges +Q and - Q on the upper and lower conducting plates respectively. The charges are assumed to be uniformly distributed over the conducting plates with surface densities +ps and -p,, where " From Eq. (3-1 14), we have which is constant within the dielectric if the fringing of the electric field at the edges of the plates is negiectcd. Now Fig. 3-26 Cross sectlon of a parallel-plate capacitor (Example 3-15). Therefore, for a parallel-plate capacitor, . I I - which is independent of Q or V, , . For this problem we-could have started by assuming a potential difference V12 between the upper and lower plates. The electric field intensity between the plates is uniform and equals The surface charge densities at the upper and lower conducting plates are +p, and -p,, respectively, where, in view of Eq. (3-67), , . 1'; 2 p., = eb, = c--. d Therefore, Q = psS = (cS/(i)Vl2 and C = Q/V12 = sS/d. as before. - ---.
.
- Example 3-16 A cylindrical capacitor consists of an inner conductor of radius a and an outer conductor whose inner radius is b. The space between the conductors is filled with a dielectric of permittivity c, and the length of the capacitor is L Deter- mine the capacitance of this capacitor. Solutioti: We use cylindrical coordinates for this problem. First we assume charges + Q and - Q on the surface of the inner conductor and the inner surface of the outer conductor, respectively. The E field in the dielectric can be obtained by applying Gauss's law to a cylindrical Gaussian surface within the dielectric a c r < b. (Note that Eq. (3-1 14) gives only the normal comporlent of the E field at a conductor surface. Since the conductor surfaces are not planes here, the E field is not constant in the dielectric and Eq. (3-1 14) cannot be used to find E in the u < r < b region.) Referring to Fig. 3-27, we have Fig. 3-27. A cylindrical capacitor (Example 3- 16). t p ; and :hargcs c outer ~pijing ' . (Note ,urfacc. in the ferring 3-10 1 CA~ACITANCE AND CAPACITORS 113 i Again we neglect the fringing effect of the field hear the edges of the conductors. The potential difference Between the inner and outer conductors is Therefore, for a cylin+ical capacitor, We could not soive this problem from an assumed Vuh because the electric field is not uniform between the inner and outer conductors. Thus we wouid not know how lo cxprcss li and Q in lcrms of Vuh until wc lcmcd how to solve such a boundary- value problem. Example 3-17 A sphcriwl capacitor consists of an inncr conducting spllcrc nl radius R i and an outer conductor with a spherical inner wall of radius R,. The space in-between is filled with a dielectric of permittivity s. Determine the capacitance. Solution: Assume ciiarges + Q and - Q, respectively, on the inner and outer con- ductors of the spherical capacitor in Fig. 3-28. Applying Gauss's law to a spherical Gaussian surface with radius R(Ri < R < R,,), we have 'Fig. 3-28 A spherical (Example 3- 17). ' capacitor > . i 11 4 STATlC ELECTRIC FIELDS 1 3 tiC?---+2y = fq Fig. 3-29 Series connection of + - 4- - capacitors. Therefore, for a spherical capacitor, 3-10.1 Series and Parallel Connections of Capacitors Capacitors are often combined in various ways in electrk-hcuits. The two basic ways are series and parallel connections. In the series, or head-to-tail. connection shown in Fig. 3-29.' the external terminals are from the first and last capacitors only. When a potential difTerence or electrostatic voltage V is applied, charge cumulauons on the conductors connected to the external terminals are + Q and - Q. Charges will be induced on the internally connected conductors such that +Q and -Q will appear on each capacitor independently of its capacitance. The potential differences across Ihc individual capacitors arc Q / C , , Q/Cz,. . . , (LIC,,. anti where C , is the equivalent capacitance of the series-connected capacitors. We have In the parallel connection of capacitors, the external terminals are connected to the conductors of all the capacitors as in Fig. 3-30. When a potential difference V is applied to the terminals, the charge cumulated on a capacitor depends on its capacitance. The total charge is the sum of all the charges. - ' Capacitors, whatever their actual shape, are conventionaily represented in circuits by pairs of parallel bars. on of (3- 127) / - \ L O >IC lnnef 'n mrs ",.,y. nuldt~ons rirgc\ w 1 1 1 -Q will ~ITtrences \it Iiavc (3- 128) nected to krence V cis l r r s of pdrallel ,I 3-10 / CAPACITANCE AND CA'PACITORS I F I 1 -Y - d - b Fig. 3-30 Parallel connection + - of capacitors. Therefore the equivalent capacitance of the parallel-connected capiicitors is C , , = C , + C2 + . . . + C,,. We note that the formula for the equivalent capacitance of series-connected capacitors is similar to that for the equivalent resistance of parallel-connected resistors and that the formula for the equivalent capacitance of parallel-connected capisitors is similar to that for the equivalent resistance of series-connected resistors. Can you explain this? Example 3-18 Four capacitors C, = 1 pF, C2 = 2 pF, C, = 3 pF, and C, = 4 / I F are connected as in Fig 3-31. A DC voltage of 100 V is applied to the external terminals a-b. Determine the following: (a) the total equivalent capacitance between terminals u-b: (b) the charge on each capacitor; and (c) the potentla1 difirence across each capacitor. Fig. 3-31 A cornblnatlon of 100 ( V ) capacitors (Exxmple 3- 18). \ ' 11 6 STATIC ELECTRIC FIELDS I 3 Solution a) The equivalent capacitance C,, of C , and C , in series is The combination of C12 in parallel with C3 gives C l Z 3 = C12 + C3 = 9 (PF). The total equivalent capacitance C , , is then b) Since the capacitances are given. the voltages can be found as soon as the charges have been determined. We have four unknowns: Q,, Q,, Q,, and Q,. Four equa- tions are needed for their determination. Series connection of C , and C 2 : Q I = Qz. Q i Qi'Q-3 Kirchhoff's voltage law, Vl + V2 = V3: - + - = -. c, c z c 3 Q3 Q4 Kirchhoff's voltage law, V3 + V4 = 100: - + - = 100. c, c 4 Series connection at d : Q 2 + Q 3 = Q4. Using the given values of C,, C,. C,. and C4 and solving the equations. we obtain c) Dividing the charges by' the capacitances, we find Q4 V4 = - = 47.8 (V). c 4 These results can be checked by verifying That V, + V2 = V, and that V3 + V4 = 100 (V). 3-1 1 AND F I. -, - \ - d ~ h c charges Four equa- r ,. \ r: obtain 1 i 3-11 1 ELECTRdSTATlC ENEI'GY AND FORCES 117 4 I , . I I In Section 3-5 +. ihdicated that electrlc potential at a polnt in an eiectrlc field 1 s 11 the work req$& rb bring a unit positive charge from mfinlty (at reference rero- L t potential) to that bolnt. In order to bring a charge Q, (slowly, so that kinetlc energy and radiation effects may be neglected) from lnfinlty againrt the field of a charge Q, in free space to a distance R12, the amount of work required is This work is stored in the assembly of the two charges as potential energy. Combining Eqs. (3- 130) and (3-131), we can write Now suppose adother charge Q, is brought from.infinity to a point that is R , , from Q1 and R,, from Q,; an additional work is required that equals The sum of AW in hq. (3-133) and W, in Eq. (3-130) is the potential energy. W,, stored in the asserhbiy of the three charges Q,, Q,, and Q,. That is. We can rewrite W, ifi the following form: --.- - + Q3 (" 4 7 r ~ ~ R , ~ ~ T ~ E ~ R ~ ~ = f(Qlv1 + QZV2 + Q3V3). (3- 135) III Eq. (3-135), V,, tHe potential at the position of Q,, is caused by charges Qz and , Q,; it is different frbm the V, in Eq. (3-13n in the two-charge case. Similarly. V2 and V, are the potentials, respectiyely. at Q2 and Q, in the three-charge assembly. Extending thii procedure of bringmg in ddditional charges, we arrive at the followins general expression for the potential energy of a group of N discrete point I charges at rest. (The purpose of the subscript a on CC; is to denote that the energy is 1 1 of an electric nature.) We have I I where V,, the electric potential at Q , , is caused by all the other charges and has the following expression: V,=- -. (3-137) 1 ( I h l j Two remarks are in order here. First, We can be negative. For instance, W2 in Eq. I 0-130) will be negative if Q, and Q, are of oppos~te signs. In that case, work is done by the field (not against the field) established by Q , in moving Q, from Infinity. Second, & , in Eq. (3- 136) represents only the lnteractlon energy (mutual energy) and does not ~nclude the work requlred to assemble the ~ndividuai polnt charges them- selves (self-energy). Example 3-19 Find the energy required to assemble a uniform sphere of charge of radius b and volume charge density p. Solution: Because of symmetry. it is simplest to assume that the sphere of chake is assembled by bringing i ~ p a si~cccssion of sphsric;lI i:~yers of thicl\i>ess ,/R, ~ c t l!~ili)l-ln v o l ~ l l l ~ C I I ; I ~ ~ C dcnrity he p, I[ : I r:ldila H shorn in Fig. 3-32, the potential is Q V R =-.!L-. 4 7 i ~ ~ R where QR is the total charge contained in a sphere of radius R: Q,~ = pinl< 3 . Fig. 3-32 Assembling a uniform sphere of charge (Example 3-19). e energy is (3- 136) nd Ips the (3- 137) 1 1 ; in Eq. ) A 1 s done m ~nfinity. ~lcrf- IIIJ rgcs u~cm- f charge of ;f charge is K . Let the e potential n . , I ' The differential ch;rg& in a spherical layer of'thlckness dR is and the work or energy in bringing up dQ ii Hence the total work or energy rcquired to assbmble a uniform sphere of charge of radius b and chkrge dpnsity p is ' l In terms of the total tharge 4 n ~ Q = p 7 b3, 3 we have Equation (3-139) shows that the energy is directly proportional to the square of the total charge and inversely proportional to the radius. The sphere of charge in Fig. 3-32 could be a clodd of electrons, for instance. For a continuchs charge distribution ofdensity p the formula for l.V, in L q (3- 136) for discrete charges must be modified. Without going through a separate proof, wz replace Q, by p dv and the summation by an integration and obtain In Eq. (3-140), V is the potential at the point where the volume charge density is 17. and V' is the volume of the region where p exists. Example 3-20 Solvc the problcm in Example 3- 19 by using Eq. (3 - 140). Soheion: In ~xamble 3- 19 we solved the problem of assembling a sphere of charge by bringing up a succession of spherical layers of a dilkrential thicknesb. Now we assume thatJte_ sphere of charge is already in place. Since p is a constant, it can be taken out of the integral sign. For a spherically symmetrical problem, where V is the potehtial at a point R from the center. To find V at R, we must find the negative of the line integral of E in two regions: (1) E, = aRER, from R = % to 120 STATIC ELECTRIC FIELDS I 3 R = b, and (2) E, = aRER, from R = b to R = 0. We have and Consequently, we obtain Substituting Eq. (3- 142) in Eq. 13- 141). we get which 1 s the same as the result in Eq. (3- 138). Note that y, in Eq. (3- 140) includes the work (self-energy) required to assemble the distr~bution of macroscopic charges, because it e the energy of interaction of every infinitesimal charge element w~th all other infinitesimal charge elements. As a matter of fact. we have used Eq. 13- 140) in Emmpie 3 -20 to find the wif-energy of a un~form sphcr~c.~l charge. A, the radius h approaches zero, the self-energy of a (mathematical) point charge of a given Q is infinite (see Eq. 3-139). The self-energies of pomt charges Q, are not included in Eq. (3-136). Of course, there are, strictly, no I point charges inasmuch as the smallest charge unit, the electron, is itselfa distrrbution I of charge. 3-11.1 Electrostatic Energy in Terms of Field Quantities In Eq. (3-140), the expression of electrostatic energy of a charge distribution contains 1 the source charge density p and the potential function V. We frequently find it more I , convenient to have an expression of We in terms of field quantities E and/or D. , without knowing p explicitly. To this end, we substitute V . D for p in Eq. (3-140): W, = isv, (V.D)Vdu. Now. using the vector identity (from Problem P.2-18) (3- 142) r I .isscmble :raction of lents. As a r-cncrgy of ncrgy uf a :If-energies strictly, no .istribution t . jn contams ind it inore m f ' ~ D, :q. (3-140): (3 -43) (3- 144) we can write Eq. (3-143) as w . = + J V , v - ( v ~ ) d u - + J v , ~ . v v d u - f $ s , V D . a , , d s + ~ , D - E d " , " s v (3 - 145) -, where the divergence theorem has been used to change the first volume integral Into a closed surface integral and E has been substituted for - V V In the second volume integral. Since V' can be any volume that includes all the charges, we miiy choose it to be avery large sphere with radius+k. As we let R -+ m, electrlc potential V and the magnitude of electric displacement D fall off at least as fast as, respect~vely, 1/R and l/RZ.+ The k e a of the boundmg surfate S' mcreaqes as R'. Hence the surface integral in Eq. (3-145) decreases at least ' 1 s fast as 1/R and will v a u h ds K -+ x We are then left with only the second integral bn the right s~da of Eq. (3-145) 1 / W = ; J D - E d i ( J ) 1 ( 3 - 146'1) Using the relation D = EE for a linear medium. Eq. (3-146a) can be wntten in two other forms: and We can always define an elecirostoiic energy density we mathematically, such that its volume integral equals the total electrostatic energy: i q = Sv, we d c . (3-14;) We can, therefore, write ul', = D . E (J/m3) (3 - 1 4Sn) or ---- ' For point charges V c c 1/R and D cr 1/R2; for dipoles' V cc 1/R2 and D cc l/R3 \ 122 STATIC ELECTRIC FIELD? / 3 Fig. 3-33 A charged'parallel- - plate capacitor (Example 3-21). However, this definition of energy density is artificial because a physical justification has not been found to localize energy with an electric field; all we know is that the volume integrals in Eqs. (3-146a, b, c) give the correct total electrostatic energy. Example 3-21 In Fig. 3-33, a parallel-plate capacitor of area S and separation d is charged to a voltage V. The perhittivity of the dielectric is E . Find the stored electro- static energy. Soli~tion: With the DC source (batteries) connected as shown, the upper and lower plates are charged positive and negative, respectively. If thpf~inging of the field at the edges is neglected, the electric field in the dielectric is uniform (over the plate) and constant (across the dielectric), and has a magnitude tl Using Eq. (3-146b). wc havc . , The quantity in the parentheses of the last expression, aS/d, is the capacitance of the parallel-plate capacitor (see Eq. 3-123). So, Since Q = CV, Eq. (3-149a) can be put in two other forms: and It so happensthat Eqs. (3-149a, b, c) hold true for any two-conductor capacitor (see Problem P.3-35). I I stlfication .
S that the energy. raration d d electro- md I ~ e r e 6 ; . dt he p' 7 ) ce of the 3-149a) t 1-149b) 0 1-149~) pacitor 3-11 1 ELECTRil.'TATIC ENERGY AND FORCES 123 3-1 1.2 Electrostatic ~ o i c i s Coulomb's law governs the force between two point charges. In a more complex system of charged qodies, using Coulomb's law to determine the force on one of the bodies that is caused by the charges on other bodies would be bery tedlous. This would be so even in [he simple case of fihding the force between the plates of a charged parallel~plkte capacitor. We will now discuss a method for calculating the force on an object iii a charged system from the electrostatic energy of the system. This method is based on the principle o f virtudl displacement. We will consider two cases: (1) that of an,isolated system of bodies with fixed charges, and (2) that of a system of conducting bodies with fixed . System of Bodies with Fixed Charges We consider an isolated system of charged conducting, as well B s dielectric, bodies separated from one another with no con- nection to the outside world. The charges on the bodies are constant. Imagine that the electric forces Hkve displaced one of the bodies by a differential distance d t (a virtual displacement). The mechanical work done by the system would be where F, is the total hectric force acting on the body under the condition of constant charges. Since we have an isolated system with no external supply of energy, this mechanical work mhst be done at the expense of the stored electrostatic energy: that is, Noling f r o m Eq. (2-LI I ) is Section 2--5 that the difkrential'change oia scalar resulting from a position change dP is the dot product of the gradient of the scalar and l i t , we write d W, = (V We) dl'. (3-152) Since dP is arbitrary, con~parison of Eqs. (3-151) and (3-152) lends to Equation (3-153) is & very simple formula for the calculation of FQ from the electro- static energy of the system. In Cartesian coordinates, the component forces are -1- 2 we ( F ~ ) x = -- 3s (3 - 1 54a) -, 124 STATIC ELECTRIC FIELDS 1 3 ! If the body under consideration is constrained to rotate about an axis, say the z-axis, the mechanical work done by the system for a virtual angular displacement ; d4 would be dW = (T,): d 4 , i (3-155) i where (TQ), is the z-component of the torque acting on the body under the condition of constant charges. The foregoing procedure will lead to I System of Conducting Bodies ivith Fixed Potentials Now consider a system where conducting bodies are held at fixed potentials through connections to such external sources as batteries. Uncharged dielectric bodies lnny also be present. A di'splacement de by a conducting body would result in a changc in total elcctrostatic energy and require the sources to transfer charges to the conductors in order to keep them at their fixed potentials. If a charge dQ, (which may be posiiive-or negative) is added to the kth conductor that is maintained at potential I / , , the work done or energy supplicd by the sources is V, dQ,. The total encrgy supplicd by the yoimxs to thc system is The mechanical work done by the system as a consequence of the virtual displace- ment is d W = F V . dt, (3-158) where Fv is the electric force on the conducting body under the condition of constant ' potentials. The charge transfers also change the electrostatic energy of the system by an amount dWe, which, in view of Eq. (3-136), is Conservation of energy demands that Substitution of Eqs. (3-1571, (3-I%), and (3-159) in,Eq. (3-160) gives FV. d i ? = dWe = (V We) - dB or uis, say the splacement (3-155) -. condition (3-156) 1~111 hhcrc :h cxtcrnal placement :nergy and ,p them at :, ,.%cd or emorgy cc the (3-157) ' di$place- (3-158) ~f constant he system (3-159) ( -60) (3-161) 3-11 / ELECTROSTATIC ENERGY AND FORCES 125 1 Comparison of E ~ S . (3-161) and (3-153) rekalr that the only difference between the formulas for the electric forces in the two cases is in the sign. It is clear that, if the conducting body 'is' constrained to rotate about the z-axis, the I-component of the electric torque will e b which differs fronl Bq. (3-156) also only by a sign change. Example 3-22 Determine the force on the conducting plates of a char, -ed parnilel- pliltc capacitor. Thc~plate\ have an ;ma S iind arc separated i n air by a d i ~ t r ~ i ; ~ \ . . Solutiot~: We solve the problem in two ways: (a) by assuming fixed charpcs: and then (b) by assummg fixed potentials. The fringing of field around the edges of the plates will be neglected. Fixed ci1orge.s: With fixed charges 2 Q on the plates, an electric 6c!d intsnsity E, = Q/(coS) = V/.Y exists in the air between the plates regardless of their separa- tion (unchanged by a virtual displacement). From Eq. (3-149b). by,= i Q V = LQE 2 XI, where Q and Ex are constants. Using Eq. (3-154a), we obtain 1 "(' ) 2 o2 : (Fy), = -- 5 QE,x = -- QE, = -A (3-163) L X 2e0s1 where the negative signs indicate that the force is opposite to the direction of increasing x. It is an attractive force. Fixed porenrials: With fixed potentials it is more convenient to use the expression in Eq. (3-149a) for bxY,. Capacitance C for the parallel-plate air capacitor is eOS/x. We have, from Eq. (3-161), How different are (FQ), in Eq. (3-163) and ( F , ) , in Eq. (3-lM)? Recalling the relation -\ ---- we find The force is the same in both cases, in vite of the apparent sign difference in the formulas as expressed by Eqs. (3-153) and (3-161). A little reflection on the physical problem will convince us that this must be true. Since the charged capacitor has fixed dimensions, a given Q will r'esult in a fixed V, and vice versa. Therefore there 1 s I i 126 STATIC ELECTRIC FIELDS I 3 ! t a unique force between the plates regardless of whether Q or V is given. and the 1 force certainly does not depend on virtual displacements. A change in the conceptual I constraint (fixed Q or fixed V ) cannot change the unique force between the plates. I The preceding discussion holds true for a general charged two-conductor capaci- tor with capacitance C. The electrostatic force F, in the direction of a virtual displace- ment d t for fixed charges is i Q2 dC For fixed potentials, V 2 iC QZ JC (3-167) It is clear that the forces calculated from the two proccdurcs. which assumed different constraints imposed on the same cb;~r& c;lp;lcil~r. ;ire equal. -.
- . REVIEW QUESTIONS R.3-I Write the dillcrential form of the fundamental postulatcs of clcctrostatics in free sp;lcc, RJ-2 Under what conditions will the electric field intensity be both solenoidal m d irrotntional? RJ-3 Write the integral form of the fundanlental postuiates of electrostatics in free space. and state their meaning in words. R 3 - 4 When the formula for the electric field intensity ofa point charge, Eq. 13-12). w;ls derived, a) why was it necessary to stip~llate that q is in a boundless free space? b) why did we not construct a cubic or a cylindric~~l surface ;,round y! R.3-5 In what ways does the electric field intensity vary with distance for a) a point charge? b) an electric dipole? R.3-6 State Coulomb's law. . R.3-7 State Gauss's law. Under what conditions is Gauss's law especially useful in determining the electric field intensity of a charge distribution? R.3-8 Describe the ways in which the electric Reid intensity of an infinitely long, straight line . charge of uniform density varies with distance? R.3-9 Is Gauss's law useful in finding the E field o l a finite linichrrges! Explain. R.3-10 See Example 3-5, Fig. 3-8. Could a cylindrical pillbox with circular top and bottom faces be chosen as a Gaussian surface? Explain. R3-11 Make a two-dimensional sketch of the electric field lines and the equipotential lines of a point charge. 1, and the :onceptual : plates. or capaci- 1 displace- (3-166), lO7) I dill'crent f-, ii'cc .p3ce. o:~tional? spacc, and IS deri\ ed. .sight line I r' lines of a I REVIEW QUESTIONS 127 I . i R.3-12 At what valuk of 6 is the E field of a z-dirdcted electric dipole pointed in the negative z-direction? R3-13 Refer to Eq. (3-59). Explain why the absoldte sign around 2 is required. R.3-14 If the electric potential at a point is zero, does it follow that the electrical field intensity is also zero at that poiht? Explain. R3-13 If the electric held intensity at a point is zero, does it follow that the electric potentiai is also zero at that point? Explain. R.3-16 An uncharged spherical conducting shell af a finite thickness is placed in an external electric field E,, what is the electric field intensity at the center of the shell? Describe the charge distributions on both the outer and the inncr surfxu of the shell. .. !<.' . 17. ( ':ill V'( l/lt) 111 Iiq. (3 79) lx i~c~)laccd by V( i,/1(jt! LxpIai~i. R.3-18 Define polariztltion vector. What is ~ t s SI unit? R.3-19 What are polarizat~on charge dcnsities? What are the SI units Tor P . a,, and V . P ? R.3-20 What do we rflean by simple inerlium? R.3-21 Define electrrc displucei~ei~t wcior. What is its SI unit? R.3-22 Define electric susceprihility. What is its unit? K.3-23 What is the differcncc bctwecn thc permittivity and the dielectric coilmnt of a mechum'! R3-24 Does the electric flux density due to a given Charge distribution depend on the properties of the medium? Does the electric field intensity? R.3-25 What is the difference between the riielectric constunt and the dielectric strengrh of a dielectric material? R.3-26 What are the kenera1 boundary conditions for electrostatic fields at an interface between two different dielectric media? R.3-27 What are the boundary conditions for electrostatic fields at an interface between a conductor and a dielectric with permittivity c ? R.3-28 What is the boundary condition for electrostatic potential at an interface between two different dielectric media'! 113-29 Does a force exist between a point charge and a dielectric body? Explain. R.3-30 Define cupacitance and cupucitor. - -. R.3-31 Assume that the permittivity of the dielectric in a parallel-plate capacitor is not constant. Will Eq. (3-123) hold if the average value of permittivity is used for E in the formula? Explain. R.3-32 Given three lbpF capacitors, explain how they should be connected in order to obtain a total capacitance of 128 STATIC ELECTRlC FIELDS I 3 R3-33 What is the expression for the electrostatic energy of an assembly of four discrete point ' charges? R3-34 What is the expression for the electrostatic energy of a continuous distribution of charge in a volume? on a surface? along a line? R.3-35 Provide a mathematical expression for electrostatic energy in terms of E and/or D. R.3-36 Discuss the m~aning and use of the principle of virtual displacement. R3-37 What is the relation between the force and the stored energy in a system of stationary charged objects under the condition of constant charges? under the condition of fixed potentials? .' PROBLEMS , P.3-1 Refer to Fig. 3-3. a) Find the relation between the angle of arrival, a, of the electron beam at the screen and the deflecting electric field intensity Ed. b) Find the relation between wand L such that d , = d0,'2O. -, .
- 1'3-2 Tllc ca[liotlc-v:ly oscillogl.;~ph (CNO) sliown in Fig. 3 3 is used to nicasurc tlic \:OII;IFL' applied to the paralid deflection plates. a) Assuming no breakdown in insulation, what is the maximum voltage that can be mea- sured if the distance of separation between the plates is l ~ ? b) What is the restriction on L if the diameter of the screen is D? c) What can be done with a fixed geometry to double the CRO's maximum measurable voltage? P3-3 Calculate the electric force between the electron and nucleus of a hydrogen atom. as- suming they are separated by a distance 5.28 x lo-'' (m). P3-4 Two point charges, Q, and Q,, are located at (1,2,0) and (2,0, O), respectively. Find the relation between Q, and Q,, such that the total force on a test charge at the point P(- 1 , 1,O) will have a) no x-component, b) no y-component. P3-5 Two very small conductiig spheres, each of a mass 1.0 x (kg) are suspended at a common point by very 'thin nonconducting threads of a length 0.2 (m). A charge Q is placed on each sphere. The electric force of repulsion separates the spheres, and an equilibrium is reached when the suspending thread makes an angle of 10'. Assuming a gravitational force of 9.80 (Nfig) and a negligible mass for th'e threads, find Q. P3-6 A line charge of uniform dcnsity p, in free space forms a semicircle of radius b. Determine the magnitude and direction of the electric field intensity at the center of the semicircle. P.3-7 Three uniform line charges-p,,, p,,, and p,,, each of length L-form an equilateral triangle. Assuming p,, = 2p,, = 2pt3, determine the electric field intensity at the center of the a triangle. h hscrete point on of charge and/or D )f 5tstiunary i potentials? : 5~:ucn and ~ h e p p g e 1 1 L 1- measurable 1 atom, as- ). Find the 1, 1, 0) will )pended at pi.iced on !r redchcd 1 ' . t - i r n D r t ~ le quliaL ..I ~ter of the PROBLEMS 129 P.3-8 .4ssuming that h e electric field density is E 4 a,100x (V/m), find the t a l e c t r i c charge contamed insidc ' 1 a) a cubicai v h n c 100 (mm) on a side centerdd at the ongin. b) a cylindricii vdlume bf radius 50 (mm) and height 100 (mm) centered at the origin. ? 1 . P.3-9 A spherlcai didtributlon of charge p = p,/[l - (R2ib')] exlsts in the regron 0 5 R 5 b Thls charge distrlbutldn is concentrically surrounddd by a conductmg shell wlth Inner radlus R, (> b) and outer radibs R,. Determine E everywhet'e. P.3-10 TWO infinilely.long coax~al cylindrical surfdces, I- = a and r = b (b > a), carry surface charge densities p,, ~ n d p,, respectively. I bctwccn o and b in order that E van~>hc\ for r > b'! - P.3-I I At what va!uus ot' 0 docs the electric field inten,lty of a xiirccted d~polr have no z- component? a) Determine V d ; d E at a distant point P(R, 0, (b). b) Find the equations for equiporential surfaces and streamlines C) Sketch a famil? of equipotential lines and streamlines. (Such an arrangement of three charges is called a linear electrostatic qimdrupolr.) I P.3-13 A finite line chsrge of length L carries a uniform line charge density p , , a) Determine V iil thr plane bisecting the line charge. b) Determine E from p, directly by applying Coulomb's law. c) Check the answer in part (b) with -VV. P.3-14 A charge Q is distributed uniformly over an L x L square plate. Determine I/ and E at a point on the axis perbendicular to the plate, and through its center. P.3-15 A charge Q is distributed uniformly over the wall of a circular tube of radius b and height h. Determine V and E on its axis 4 ) at a point outside the tubs, then b) at a point inside the tube. P.3-16 A simple clasdcsl model of an aton1 consists of a nucleus of a posit~ve charge k ~ r ~ surrounded by a spherical electron cloud of the same total negative charge. ( N is the atomic number and e is the electronic charge.) An external electric field E,, will cause the nucleus to be displaced a distance r,, from the center of the electron cloud, thus polarizing the atom. Assuming a uniform ch~~e-distribution within the electron cloud of radius b, find r,,. P.3-17 Determine the work done in carrying a -Z(pC) charge from P,(2, 1. - 1) to P,(8. 2, - 1) in the field E = a,y + a,,x a) along the parabola s = ~ J J ~ , b) along the straight line joining P, and P2 130 STATIC ELECTRIC FIELDS I 3 P.3-18 The polarization in a dielectric cube of side L centered at the origin is given by P = P,(a,x + a,y + a,z). a) Determine the surface and volume bound-charge densities. b) Show that the total bound charge is zero. P3-19 Determine the electric field intensity at the center of a small spherical cavity cut out of a large block of dielectric in which a polarization P exists. P.3-20 Solve the folloiving problems: a) Find the breakdown voltage of a parallel-plate capacitor, assuming that conducting plates are 50 (mm) apart and the medium between them is air. b) Find the breakdown voltage if the entire space between the conducting plates is filled with plexiglass, which has a dielectric constant 3 and a dielectric strength 20 (kV,'mm). c) If a 10-(mm) thick plexiglass is inserted between the plate.;, what is the maximum voltage that can be applied to the platCs without a breakdown? P3-21 Assume that the z = 0 plane separates two lossless dielectric regions with E , , = 2 and E , ~ = 3. If we know that E, in region 1 is a,2y - a,.3.u + a,(5 + z), what do we also know about E, and D, in region 2? Can we determine E, and D, at any point ip region 2? Explaln. -. _ - , P.3-22 Dctcrtninc ti~c bound:~ry condirions for thc tangential and the normal components of P at an interface between two perfcct diclcctric tncdia with dielcctric constants E , , 2nd E , ? . P.3-23 What are the boundary conditions that must be satisfied by the electric potential at an interface between two perfect dielectrics with dielectric constants E,, and E,, ? P.3-24 Dielectric lenses can be used to collimate electromagnetic fields. In Fig. 3-34, the left surface of the lens is that of a circular cylinder, and the right surface is a plane. If E, at point P(r,, 45", 5 ) in region 1 is a,5 - a,3, what must be the dielectric constant of the lens in order that E, in region 3 is parallel to the x-axis? Fig. 3-34 Dielectric lens (Problem P.3 -24). P3-25 The space between a parallel-plate capacitor of area S is filled with a dielectric whose permittivity varies linearly frome, at one plate (y = 0) to E , at the other plate (y = d). Neglecting fringing effect, find the capacitance. ,by P = out of a nducting ,lit(\ with 11:11j. n vduge = 2 and .w about r - .>ncnt,- -f 1 d Er %ti31 ar an 4. the left , st point ; in order n xric whose Keglecting I PROBLEMS 131 , P.3-26 Consider the earth as a conducting sphere of radius 6.37 (Mm). a) Determine its ~a~acitance. b) Detcrminc thimhximum charge that can exist on it without cawing a breakdown of the air surroundirlg it. P.3-27 Determine the capacitance of m isolated cbnducting sphere of radius h that is coated with a dielectric layer ~f uniform thickness d The dielectric has an electric susceptibility %. . P3-28 A capacitor cbnrists of two concentric spnerical shells of radii R, add Ro. The space between them is filled with a dielectric of relative pkrmittivity c from R, to b(R, < b < R.) ard another dielectric of relative permittivity 26, from b to R,. a) Determine E and D everywhere in terms of A n applied voltage I/. h) Dctcrminc thc c:~paci tancc. P.3-29 Assume that the outer conductor of the cylindrical capacitor in Example 3-16 is grounded, and the inner conductor is maintained at a potential Vn. a) Find the electiic field intensity, E(a), at thcsurface of the inner conducror. b) With the irrner radius, b, of the outer conductor fixed, find a so that E(n) is minimized. c) Find this minimum E(ul. d) Detcrrhine the capacitance under the conditions of part (b). P.3-30 The radius of the core and the inner radius of the outer conductor of a very long coaxial transmission line are ri and r, respectively. The space between the conductors is filled with two coaxial layers of dielectrics. The dielectric constant4 of the dielectrics are E,, for ri < r < b and E , ~ for b < r < ro. Determine its capacitance per uriit length. P.3-31 A cylindrical capacitor of length L consists of coaxial conducting surfaces of radii ii and r,,. Two diebctricmcdii~ of dillcrcnl diclvctric copstants r,, and r,, fill the space between the conducting surfaccs as shown in Fig. 3-35. Detcrmihe its capacitance. -A Fig. 3-3 I - with two dielectrl (Problem P.3-31). 5 A cylindrical capacitor ic media P3-32 A capacitor consists of two coaxial me@llic cylindrical surfaces of a length 30 (mm) and radii 5 (mm) and 7 (mm). The dielectric material between the surfaces has a relative permittivity E, = 2 + (4/r), where r is measured in mm. Determine the capacitance of the capacitor. 132 STATIC'ELECTRIC FIELDS I 3 t P.3-33 Calculate the amount of electrostatic energy of a uniform sphere of c h a m with radius b and volume charge density p stored in the following regions: a) inside the sphere, b) outside thesphere. Check your rekits with those in Example 3-19. [ P.3-34 Find the electrostatic energy stored in the region of space R z 6 around an electric dipole o T moment p. I P.3-35 Prove that Eqs. (3-149) for stored electrostatic energy hold true for any two-conductor , : capacitor. i P.3-36 A parallel-plate capacitor of width s, length L. and separation d is partially filled with a dielectric medium of dielectric constant t.?m shown in Fig: 3-36. A battery of I,, volts is con- nected between the piates. a) Find D. E, and p, in each region. b) Find distance r such that the electrostatic cnergy storcd in each region ir the same. Fig. 3-36 A parallel-plate capacitor (Problem P.3-36). P.3-37 Using the principle of virtual displacement, derivc m expression for the force between two point charges +Q and -Q separated by a distance x in free space. 1 P.3-38 A parallel-plate capacitor of width IV. length L. and separation d has a solid d~electric ' slab of permittivity r in the space between the plates. The capacitor is charged to a voltage Yo by a battery, as indicated in Fig. 3-37. Assuming that the dielectric slab is withdrawn to the position . shown, determine the force acting on the slab a) with the switch closed, b) after the switch is first opened. \ Switch . . .
L .. . x Fig. 3-37 A partially filled parallel-plate capacitor (Problem P.3-38). 1 i " I < , with rad~us I an electric )-conductor filled w ~ t h a 011s IS con- .ce between d diclectric ~Itagc V, by h i position 4-1 INTRODUCTION Electrostatic problems are those which deal with the effects of electric charges at rcst. These problem,d can present themselves in several different ways according to what is initially kiwwn. Thc solutioli usually calls for ~ h c determination of electric potential, electric field intensity, and/or electric charge distribution. If the charge distribution is given, both the electric potentilll and the electric field inlensity can - be found by the formulas developed in Cha ter 3. In many practical problems, however, the exact qharge distribution is not everywhere, and the formulas in Chapter 3 cannot be applied directly for finding.the potential and field inten- sity. For instance, if the charges at certaln discrete points in spaco and the potentials of some conducting boclics arc givcn, it is rathcr difficult LO find the distribution of surface charges on the conducting bodies and/or the electric field intensity in space. When the cobducting bodies have bohdaries of a simple geometry, the method of images may be used to great advantage. This method will be discussed in Section 4-4. In another type: of problem, the potentials of all conducting bodies may be known, and we wish to find the potential and field intensity in the surrounding space as well as the distribution of surface charges on the conducting boundaries. Differential equations must be solved subject to the appropriate boundary condi- tions. The techniques far solving partial aiifer-ctjai equations in the various co- ordinate systems.wil1 be discussed in Sect;& 4-5 through 4-7. . . , 4-2 POISSON'S AND LAPLACE'S E Q U A T I O ~ ~ In Section 3-8, we pointed out that Eqs. (3-93) and (3-5) are the two fundamental gpverning dificrcntirll cquations for clectrostatics in any medium. These equations are repeated below for convenience. , Eq. (3-93): Eq. (3-5): 134 SOLUTION OF ELECTROSTATIC PROBLEMS 1 4 "' ' ' ' " ? The irrotational nature of E indicated by Eq. (4-2) enables us to define a scalar electric potential V, as in Eq. (3-38). In a linear and isotropic medium, D = rE, and Eq. (4-1) becomes V . c E = p . (4-4) Substitution of Eq. (4-3) in Eq. (4-4) yields where E can be a function of position. For a simple medium$ that is, for a medium that is also homogeneous, E is a constant and can then be taken out of the divergence operation. We have , - - - . . In Eq. (4-G), we have introduced a new operator, V2, the Luplnciun operutor, which stands for "the divergence of the gradient of," or V V. Equation (4-6) is known as Poisson's equution; it states that the Laplacian (the divergence of the gradient) of V equals - p/e ji,r u simple tnediutn, where e is the permittivity of the medium (which is a constant) and p is the volume charge density (which may be a function of space coordinates). . Since both divergence and gradient operations involve first-order spatial deriva- tives, Poisson's equation is a second-order partial difkrential equation that holds at every point in space where the second-order derivatives exist. In Cartesian coordi- nstcs, and Eq. (4-6) becomes Similarly, by using Eqs. (2-86) and (2-102). we can easily verify the following ex- pressions for V2V in cylindrical and spherical coordinates. Cylindrical coordinates: (4-3) (4-3) (4-5) 1 mcdiLm livergence (4-6) / - tor. \, ..dl known as -adient) of urn (which tl of space :la1 Jeriva- at holds at an coordi- !
' I ; - / (4-7) n Ilowh k- (4-8) 4 % ' ' ~4-2' 1 PO IS SON^ AND LAPLACE'S EQUATIONS 135 ' r I - < . . 1 d'V (4-9) The solut~on of Poidon's equation in three dimensions subject to prescribed bound- ary conditions is,.ih beneral, not an easy task. At points ifi a kimple medium where there is no free charge, p = 0 and Eq. (4-6) reduces to Y which is known as Laplace's eqr~atiotr. Lapiace's equation occupies a very important position in electrom~netics. It is the governing equation for problems involving a set of conductors, sbch as capacitors, maintained at different potentials. Once V is found from Eq. (4-1 o), E can be determined from - VV, and the charge distribution on the conductor sdrfaces can be determined from p, = €En (Eq. 3-67). . I Example 4-1 The .two plates of a parallel-blate capacitor are separated by a distance d and mainiained at potentials 0 and V,, as shown in Fig. 4-1. Assuming negligible fZlnging effect at the edges, determine (a) the potential at any point between the plates, and (b) the surface charge densities at the plates. a) Laplace's equation is the governing equatidn for the potential between the plates since p = 0 therq. Ignoring the fringing effect of the electric field is tantamount to assuming that tHi field distribution between the plates is the same as though the plates were infidtely large and that there is no variation of V in the x and z directions. Equation (4-7) then simplifies to where d2/dy' is used instead of d-,;!'. since y is the only space variable. -1 - ' t Fig. 4-1 A parallel-plate capacitor (Example 4-1). t 136 SOLUTlON OF ELECT .ubr''!C PROBLEMS / 4 i f Integration of Eq. (4-11) with respect to y gives , . ; dV - = C1, dy I where the constant of integration C, is yet to be determined. Integrating again. we obtain v = Cly + C , . (4-12) I 1 Two boundary conditions are required for the determination of the two constants of integration: I Substitution of Eqs. (4-13a) and (4-13b) in Eq. (4-12) yields immediately C , = V,/d and C , = 0. Hcnce thc potmllid at m y point I . bctwern 111~. plates is. from Eq. (4-12), The potential increases linearly from y = 0 to y = d. b ) In order to find the surface charge densities, we must first find E at the conducting , plates at y = 0 and y = d. From Eqs. (4-3) and (4-14), we have \ which is a constant and is independent ofy. Note that the direction of E is opposite to the direction of increasing V. The surface charge densities atihe conducting plates are obtained by using Eq. (3-67)' 1 ' I(, -> :I,, . 1 1 ; - E At the lower plate, ' -7 At the upper plate, Electric field lines in an electrostatic field begin from positive charges and end in negative charges. g again, (4-12) mstants (4- 13a) (4-4 3b) .ly C, = IS, from ( @ P I ) ~duct~ng (4-15) opposite nductmg n \ ; and end t 1 Ekample 4-2 Detehine the E field both in~ide hnd outside a spherical cloud of electrons with a unibrm volume charge density p = -p, for 0 i R 5 b and p = 0 for R > b by solving~Poisson's and Laplace's equations for V. I Solution: We re$ that this problem was dblveh in Chapter 3 (Example 3-6) by applying Gauss's aw. We now use the sam problem to illustrate the solut~on of one-dimensional Poisson's and Laplace's equ tions. Since there are no variations 3 in 0 and 4 directio?s,'tue are only dealing with functions of R in spherical coordinates. J a) Inside the cloud,; -1. t O S R I b , p y -PO. In this region, Poisson's equation ( V 2 y - p h o ) holds. Dropping i l Z O and a/@ terms from kq. (4-9), we have which reduces to Integration of E4, (4-16) gives The electric field intensity inside the electron cloud is Since Ei cannot be infinite at R = 0, the integration constant C, in Eq. (4-17) must vanish. We obtain P 0 Ei=-a,--R, 0 1 R s b . (4-18) 360 b) Outside the cloud, R 2 b , p - 0 . Laplace's equaiion holds in this region. We have V2 Y. = 0 or Integrating Eq. (4-191, we obtain 138 SOLUTION OF ELECTROSTATIC PROBLEMS 1 4 i The integration constant C, can be found by equating Eo and E, at R = b, where t there is no discontinuity in medium characteristics. i c 2 Po -- -- b2 kO b, i from which we find pOb3 C2 = -- (4-22) 1 3e0 4 -3 and ELECT P ' O b 3 R R ~ . Eo = -a R- 3c,R2 ' (4-23) Since the total chargc coniaincd in the clcctron cloud is Equation (4-23) can be written as . which is the familiar expression for the electric field intensity at a point R from a point charge Q. Further insight to this problem can be gained by examining the potential as a function of R. Integrating Eq. (4-17), remembering that C , = 0, we have It is important to Aote that C; is a new integration constant and is not the same as C,. Substituting Eq. (4-22) in Eq. (4-20) and integrating, we obtain However, C; in Eq. (4-26) must vanish since V, is zero at infinity (R -+ a ) . As electro- static potential is continuous at a boundary, we determine C; by equating l/i and Voat R = b: p0b2 ; b'+C - pob2 660 1 - -- 3c0 (4-22) (4- 23) n (4-24) t R from itial as a (4-25) same as (4-26) r' elec~ : T/T anu 1 4-3 1 UNIQUENESS Q+ ELECTROSTATIC SOLUTIONS 139 4 i' , . 3 ..or r . ' 5 . .. .
. C : podZ. - c ; = --, 2% (4-27) 'and, from Eq. (4-25), We see that C: in &q. (4-28) is the same as V in Eq. (3-i42), with p = - p o . P I 1 1 4-3 UNIQUENESS OF irh ELECTROSTATIC SOLUTIONS 111 L I I C L w u rcla~ivcl~. siniplc cxa~nplcs ~n tl~e Ids[ section, wc ob1.1' ' lned t i x solutions by direct integration. In more complicated situations other methods of solution must be used. Befoie these methods are discussed. it is important to know that n solutioti o j Poissoft's eq~ution (of which Laplace's equation is s special cxci r h r rurisjies rile giaen hoiiniiaq. c~oiiilirioi~s is ii wiipzrr .solsiioti. This statement is cnlied the u~liqtieness theorem. The implication of the uniqueness theorem is that a solution of an electrostatic problem with its bou~ldsry conditions is tile o~li! poisihle . s ~ / ~ i t i o t i irrespective of the method by which the solution is obtained. A solotion obt:lined even by intelligent guessing is the only correct sollrtion. The import:lnce of this theorem will be appreciated when we discuss-the method of ima; ~ e s in Section 4-1. To prove the uniqueness theorem. suppose a volume r i s bo~inded outside by a surface So, which may be a surface at infinity. Inside the closed surhce So there are a number of charged conducting bodies with surfaces S,, S,, . . . ,Sn at specified potentials, as depicted in the two-dimensional Fig. 4-2. Now :lssumr th:ll. contrnry to the uniqueness theorem, there are two solutions, V, and V,, to Poisson's equation in 5 : 140 SOLUTION OF ELECTGO7F-TIC PROBLEMS I 4 ..-, \ Also assume that both Vl and V, satisfy the same boundary conditions on S,, S,, . . . , S, and So. Let us try to define a new difference potential From Eqs. (4L29a) and (4-29b), we see that V, satisfies Laplace's equation in s On conducting boundaries the potentials are specified and I / , = 0. Recalling the vector identity (Problem 2-18), V . (,/'A) = ,/'V . A + I \ . Vj'. (4-32) and letting f = V, and A = VV,, we have where, because of Eq. (4-31). the first term on the right side vanishes. Integration of Eq. (4 -33) ovcs n vnlumc r yiclds ---. where a, denotes the unit normal outward from r. Surface S consists of So as well as S,, S,, . . . , and S,. Over the conducting boundaries, v, = 0. Over the large surface So, which encloses the whole system, the surface integral on the left side of Eq. (4-34) can be evaluated by considering So as the surface of a very large sphere with radius R. As R increases, both V, and V, (and therefore also I / , ) fall off as 1/R; consequently. VV, falls off as 1/R2, making the integrand (5 VV,) fall off as 1/R3. The surfacc area So, however, increases as R2. Hence the surface integral on the left side of Eq. (4-34) decreases as 1/R and approaches zero at infinity. So must also the volume integral on the right side. We have Jr jVv,12 do = 0. (4-35) Since the integrand IVV,I2 is nonnegative everywhere, Eq. (4-35) can be satisfied only if IVI/,I is identically zero. A vanishing gradient everywhere means that T/, has the same value at all points in z as it has on the bounding surfaces, S,, S ,,.. . . , S,, where V, = 0. It follows that I/, = 0 throughout the volume z . Therefore V, = V2, and there is only one possible solution. It is easy to see that the uniqueness theorem holds if the surface charge distri- butions (p, = EE, = -E JV/Jn), rather than thc potentials, of the conducting bodics . . are specified. In such a case, VV, will be zero, which in turn, makes the left side of Eq. (4 -34) vanish and leads to thc same conclusion. In fact, thc uniqucncss thcorcrn applies even if an inhomogeneous dielectric (one whose permittivity varies with position) is present. The proof, however, is more involved and will be omitted here. ,IS [cell as ge burface Eq. (4--34) i radius R. sequently, rfa& area Eq. (4-34) le integral e satisfied hat V, has ( 2 : . . . , s,,, e V, = V,, eft SL.. of ;s theorem xies with ~itted here. L ' F i 4-4 1 METHOD OF IMAGES 141 - 1 $- ! I t 4-4 METHOD OF I M A ~ E S b . .I i - There is a cladi(o[%l~ctro~tatic problems with boundary conditions that appear to be difficult to satis? if the governing Laplace's quation is to be solved directly, but the conditions oh the bounding surfaces in these can be set up by appropriate image (equiva1ent)kharges and the potential distributions can then be determined in a straightforward danner. This method of replac{hg bounding surfaces by appropriate image charges in lieu of a formal solution of Laplace's equation is called the method , o f images. . , Consider the case of a positive point char&, Q, located at a distance d above a . large grounded (zero-potential) conduc~ing plane, as shown in Fig. 4-3(a). The problem is to find thk. potential at every point above the conducting plane ( y > 0). ' I Ilc formal proccdurc for doing 50 would b~ LO sdvc Ldplacc's cq~~utlon in C:rrtca~,m . coordmates which must hold for i, > 0 except at the point charge. The solution V ( s , J., z) ahould satisfy the following conditions: 1. At all points on the grounded conducting plane, the potential is zero: that is, 2. At points very close to Q, the potential-approaches that of the pomi charge alone; that is 0 v-+- ~ ? C E ~ R ' where R is the distance to Q. 3. At points very far from Q(x - j CQ, y - + cc, or z - & a), the potential ap- proaches zero. Grounded plane conductor - - - -\ (a) Physical arrangement. - 0 (Image charge) . I (b) Image charge and field lines. Fig. 4-3 Point charge ;rnd grounded plilne condt~ctor. 142 SOLUTION OF ELECTROSTATIC PROBLEMS / 4 c . I . i 4. The potential function is even with respect to the x and z coordinates; that is, It does appear difficult to construct a solution for V that will satisfy all of these conditions. From another point of view, we' may reason that the presence of a positive ; charge Q at y = d would induce negative charges on the surface of the conducting ; plane, resulting in a surface charge density p,. Hence the potential at points above : the conducting plane would be # where R, is the distance from ds to the point under consideration and S is the surface of the entire conducting plane. The trouble here is that p, must first be determined fro111 tlic houndary condilion I.(.. 0. : ) = : 0. Mmcovcr, the indic~tcd surface intcp,il is dillicult to cvaluatc cvcn after p, ha> bccn tictcrmlncd at cvcry point on tlic con- ducting plane. In the following subsections, we demonstrate how the method of images greatly simplifies these problems. 4-4.1 Point Charge and Conducting Planes The problem in Fig. 4-3(a) is that of a positive point charge, Q, located at a distance d above a large plane conductor that is at zero potential. If we remove the conductor and replace it by an image point charge - Q at y = - d, then the potential at a point P(x, y, z) in the y > 0 region is where R+ = [x2 + ( y - d ) 2 + z 2 ] l i Z , I R- = [x2 + ( ~ + d ) ~ + z 2 ] l i 2 . It is easy to prove by direct substitution (Problem P.4-5a) that V(x, y, z) in Eq. (4-37) satisfies the Laplace's equation in Eq. (4-36). and it is obvious that all four conditions listed after Eq. (4-36) arc satislicd. Thcrcforc Eq. (4-37) is a solution of this problem; and, in view of the uniqueness theorem, it is the only solution. Electric field intensity E in the y > 0 region can be found easily from - V V with . Eq. (4-37). It is exactly the same as that between two point charges, i- Q and - 0, i - .- . ! $; that is,': ,' . 8 2 / ' 1 of these 1 pos~tive mducting nts above, ie surface Termined e Integral tb-3n- let,. of I r k l ~ ~ c e onductor .t a point (4-37) q. (4-37) ) n + n ? s 3rotr&. < vv with illd -Q, I I I I / / - I 1---- p----• I +Q 1 4-Q + t ? -Q (a) Physical arrangemeht. (b) Equivalent image-charge (c) Forces on charge Q. arrangement. - Fig. 4-4 Point chard and perpendicular conduct& planes. spaced a distance ?d apart. A few of the field lines are shown in Fig. 4-3(b) The solution of this elec ostatic problem by the method of images is extremely simple; but it must be asized that the image charge is located ourside the region in which the field is to be determined. In this problem the point charges +Q and - Q cutmor bu used to ca cula~c the V or E in the y < 0 region. As P matter of fact, both V and E are zero in he y < 0 region. Can you bxplain that? t It is readily seed that the electric field of a line charge p, above nn infinite con- ducting plane can be found from p, ;ind its imiigc -p, (with tile condticting plane removed). : I - 3 A pusilric p o i ~ ~ l d1i11.g~ Q is I O C : L L C ~ .LL d~hliillcc~ d l 1 1 1 1 d i f i , rei- pectivel~, from two grounded perpendicular conducting half-planes, as shown in Fig. 4-4(a). ~etermihe the force on Q caused by the charges induced on the planes. Solution: A formal solution of Poisson's equation, subject to the zero-potential boundary condition at the conducting half-plahes, would be quite difficult. Now an image charge - Q in the fourth quadrant would make the potential of the horizontal half-plane (but not that of the vertical half-plane) zero. Similarly, an image charge -Q in the second uadrant would make the potential of the vertical half-plane 2 (but not that of the brimntal plane) zero. But if a third image charge + Q is added in the third quadrant, we see from symmetry that the image-charge arrangement in Fig. 4-4(b) satisfies the zero-potential boundary condition on both half-planes and is electrically-eqyivaknt to the physical arrangement in Fig. 4-4(a). Negative surface charges will be induced on the half-planes, but their effect on Q can be determined from that of the three image charges. Referring to Fig. 4-4(c), we have, for the net force on Q, 1144 SOLUTION OF El ';tTRCJ$TATIC PROBLEMS 1 4 where F, = Q - 4ns,(~d,)~' F2 = -ax Q2 4 ~ 6 ~ ( 2 d ~ ) ~ ' F, = Q2 a 2d1 + a,2d2). I 4rrc0[(2d,)' + (2d2)2]31' ( " Therefore, Q2 F=- d2 ' The electric potential and electjic field intensity at points in the first yuodt-at~t and the surface charge density induced on the two half-planes can also be found from the system of four charges. - 1-4.2 Line Charge and Parallel 1 . Conducting Cylinder We now consider the problem of a line charge p, (Clm) located at a distance d from the axis of a parallel, conducting, circular cylinder of radius a. Both the line charge and the conducting cylinder are assumed to be infinitely long. Figure 4-5(a) shows a cross section of this arrangement. Preparatory to the solution of this problem by the method of images, we note the following: (1) The imagcmust be a parallel line charge inside the cylinder in order to make the cylindrical surface at r = u an equipotentiai surface. Let us call this image line charge pi. (2) Because of symmetry with respect to the line OP, the image line charge must lie somewhere along OP, say at point P,, which is at a distance di from the axis (Fig. 4-5b). We need to determine the two unknowns, pi and d,. ,--I / I \ \ (a) Line charge and Parallel conducting cylinder. (b) Line charge and its Image. Fig. 4-5 Cross section of line charge and its image in a parallel conducting circular cylinder. c l r ~ f and nd from : d from : charge shows a n by thc c chargc mtcntial respect ?oint Pi, . , : .
. i 4-4 1 METHOD OF IMAGES 145 , , i ! r :, . .i - 8 .- . : : let u$ assume that i I "i (4-38) At t h i ~ stage, Fq. (4-181 is just a trial solution (an intelligent guess), and we are not sure that it will hold tbue. We will, on the one hand, proceed with this trial solution until we find that it $ils to satisfy the boundary conditions. On the other hand, if Eq. (4-38) leads to a iolutian that does sitisfy dl1 boundary conditions, then by the uniqueness theored 14 is the only solurion. Our next job will be to see whether we can determine d;. . 3 The electric poiedtial at a distance r from a line charge of density p, can be obtained by integrating the electric field intensity E given in Eq. (3-36). Note that the refeihce point for zero potential, r,, cannot be at infinity becais: . setting ro = in ~q.(4-39) would make V infinite everywhere else. Let ui leave r0 unspecified for the time being. The potential at a point on or outside the cylindricnl surface is obtained by addin: the contributions ofp, and pi. In particular, at a point .
I on the cylindrical surface shown in Fig. 4-5(b),.we have In Eq. (4-40) we have chosen, for simplicity, a point equidistant from p, and p, as the reference point for zero potential so that the In ro terms cancel. Otherwise. s constant term should be included in the right side of Eq. (4-40), but it would not affect what follows. Equipotential surfaces are specified by r i - = Constant. r (4-41) If an equipotential surface is to coincide with the cylindrical surface (OX = , l ) , ihc point P, must be louted in such a way as to mate triangles OMP, and OPM s~mllar. Note that t-two triangles already have one common angle, L MOP,. Point P, should be chosen70 make , L OMP, = L: OPM. We have rt di a . - =-- --- d - Constant. r q , . i 1 t 146 SOLUTION OF ELECTROSTATIC PROBLEMS I 4 , ! A % I r , ) I r . t From Eq. (4-42) we see that if the image line charge -p,, together with p,, will make the dashed cylindrical surface in Fig. 4-5(b) equipotential. As the point M changes its location on the dashed circle, both ri and r will change; but their ratio remains a constant that equals ald. Point Pi is called the inverse point of P with respect to a circle of radius a. The image line charge -p, can thcn replace the cylindrical conducting surface. and V and E at any point outside the surface can be dcter~nincd from thc line charges p, and -p,. By symmetry, we find that the parallel cylindrical surface surrounding the original line charge p, withadius a and its axis at a distance 0, to the right of P is also an equipotential surface. This observation enables us to calculate the capaci- tance per unit length of an open-wire transmission line consisting of two parallel conductors of circular cross section. -1 . Example 4-4 Determine the capacitance per unit length between two long, parallel, circular conducting wires of radius a. The axes of the wires are separated by a distance D. Solution: Refer to the cross section of the two-wire transmission line shown in Fig. 4 6. Thc cquipotcnti:ll surfxcs of the two wircs can bc considcrcd to hnvc been generated by a pair of line charges p, and,-p, separated by a distance (D - 24) = d - di. The potential difference between the two wires is that between any two points on their respective wires. Let subscripts 1 and 2 denote the wires surrounding the equivalent line charges p, and -p, respectively. We have, from Eqs. (4-40) and (4-421, Fig. 4-6 Cross section of two-wire transmission line and equivalent line charges (Example 4-4). t (4 -43) 11 surface led circle, ~ / d . Point g surface, ie charges rounding right df I ' le capaci- o parallel P J , p.~. llel, ;?WC a ,hewn in have been 3 - ?dl) = two points unding the 4-40) arld 0 1 4 1 <r.- - - ' and, similarly, . - 1 . rc a I t V, = -L in 2nco , -. i ' - 1 ' Hence the capacitarib per unit length is d = f (n + V r ~ 2 - 4 2 ) . Using Eq. (4-45) in kq. (4 -44). yc havc Since In [x + fiZq] = cash-I x for x 1 1, Eq. (4-46) can be written alternativefy as 4-4.3 Point Charge dnd Cbnducting Sphere . t The method o k images can also be applied to solve the electrosrat~c problem of a point charge in the,prerence of a spherical conductor. Referring to Flg. 4-7(a) where a positive poidt charge Q is located at a distanced from the center ofa grounded conducting spher6-okradius a (a < d), we now proceed to find the V and E at polnts external to the sphere. By reason of symmetry, we expect the image charge Q, to he a negative poi6t Ch:lrpu .sitwitrcl iwidc IIIC S ~ I I C I C : 1 1 1 d 011 iiw lint joltling 0 ,111i1 Q. Let it be at a dialunce di l i m 0. It is obvious that Ql cannot bc equal to -0, since - Q a>dbthe oribinal Q do not make the spherical surface R = a a zero-potential surface as required. (Whd would the zero-potential surface be if Ql = -Q?) We must, therefore, treat both di and Q1 as unknowns. ' The other solution. d = i ( ~ - 4 -1 , is discarded because both D and d arc usually much larger than a. L t , ,/--- , , .' / ' I. I I \ i v = o \ . - - 1 . (a) Point charge and grounded conducting sphere. (b) Point charge and its image. Fig. 4-7 Point charge and its image in a grounded sphere. In Fig. 4-7(b) the conducting sphere has been replaced by the image point cliiirgc Pi. wliicll makes thc po(e1itii11 :it ;dl points on the sphcric:ll s~irfxc R = il zero. At a typical point M , the p)(csli:~l c;wed by Q and Qi is which requires r i - - - --= r Constant. Q (4-49) Noting that the requirement or. the ratio rJr is the same as that in Eq. (4-41). we conclude from Eqs. (4-42), (4-43), and (4-49) that and , The point Q, is, thus, the inverse point of Q with respect to a circle of radius a. The V and E of all points external to the grounded sphere can now be calculated from the V and E caused by the two point charges Q and - aQ/d. -A " Example 4-5 A point charge Q is at a distance d from the center of a grounded conducting sphere of radius a (a < d). Determine (a) the charge distribution induced on the surface of the sphere, and (b) the total charge induced on the sphere. nage. , , - agc point 'acc R = a u n h e d frolr 'he I grounded 'n induced 4 Solution: he-bhysicb problem is that shown id Fig. 4-7(a). We solve the problem ' by the method of imagks and replace the grounded sphere by the image charge Q, = -uQ/d at a d i s h c c d = u2/d ldrom the center of the sphcre, as shown in Fig. 4-8. The electric potential 1 at an arbitrary point P(R, 0) is Q V(R, 0) = - - - - 4 ~ 6 0 (iQ , d i Q ) ' where, by the law of cobines, RQ = [R2 + d2 - 2Rd cos 81'1' and (4-52a) Using Eq. (4-52) in Eq. (4-53), we have ER(R, 8) L - I< - J cos 0 + d2 - 2Rd cod 8)3n - a[R - (a3/d) cos 81 d[R2 + (a2/d)' - 2R(a2/d) cos 8]312 1. (3-54) : i a) In order t-find the'induced surface charge od the sphere, we set R = o in Eq. (4-54) and evaluate 1 . ,s, Ps = €oE~(a, 6) 9 (4-55) which yields the following aftet simplificat!on: ' e ( d 2 - nd) PJ = - 4na(s2 + d2 - 2ad cos 8)3/2' (3-56) . b) The total charge induced on the sphere is obtained by integrating p, over the surface of the sphere. We have Total induced charge = $ p ds - - So2" J : p g 2 sin 0 do d$ a = -- Q = Qi. - % I ' ,I d , : $ .. (4-57) . \'., . . - 2 . , We note that the total induced charge is exactlj equalto the image charge Qi that replaced the sphere. Can you explain this? If the conducting sphere is'electrically neutral and is not grounded, the image of a point charge Q at a distance d from the center of the sphere would still be Qi at di given, respectively, by Eqs. (4-50) and (4-51) in order to make the spherical surface R = u equipokntial. However, an additional point charge at the center would be needed to make the net charge on the replaced sphere zero. The electrostatic problem of a point charge Q in the presence of an electrically neutral sphere can then be solved as a problem with three point charges: Q' at R = 0, Qi at R = a2/d, and Q at R = d. 4-5 BOUNDARY-VALUE PROBLEMS IN CARTESIAN COORDINATES We have seen in the preceding section that the method of images is very useful in solving certain types of electrostatic problems involving free charges near conducting boundaries that are geometrically simple. However, if the problem consists of a system of conductors maintained at specified potentials and with no free charges, it cannot be solved by the method of images. This type of problem requires the solution of Laplace's equation. Example 4-1 was such a problem where the electric potential was a function of only one coordinate. Of course, Laplace's equation applied to three dimensions is a partial differential equation, where the potential is, in general, a func- tion of all three coordinates. We will now develop a method for solving three- dimensional problems where the boundaries, over which the potential or its normal derivative is specified, coincide with the coordinate surfaces of an orthogonal, curvi- linear coordinate system. In such cases the solution can be expressed as a product of three one-dimensional functions, each depending separately on one coordinate variable only. The procedure is called the method of separation of vuriubles. " (4-57) .
; . 3: charge QL ' - . i : I. Ime zero. lectrically 'at R = 0, I i useful in onducting I lslsts of a charged, it le solution : potential : d to three ral, a func- its ingEi . ~nal, c- L l i - I prodlr-of Laplace's equatibn for scalar electric potedlial V in Cartesian coordinates is ' aZv a2v azv - , + 7 + - - = 0 . ax- ay- ai2 (3-58) To apply the method of separation-of variables, we assume that the solution V ( s , y, r) can be expressed as d product in the following form: 1 where Xb), Y(Y), a& Z(Z) are functions, respectively, of x, y, and z qnly Substituting Eq. (4-59) in Eq. (498), we have .I VVIIILU, wnen alvlaea, through by the product X(x)Y(y)Z(z), yields Note that each of@hthree terms on k? i ; : rid4 of Eq. (4-60) is a function of only one coordinate variabie And that only ordine7 ierjvztives are involved. In order for Eq. (4-60) to be satis@dFr a 1 1 values of x, y, z, eacd 0 : the three terms must be a constant. For instance, if wiidtRersntiate Eq. (4-60) with hspect to x, we have ? since the other typ tkrms are independent of x. lati ti on (4-61) requires that . - 1 .-. .I 1 1 d2X(x) X ( x ) d x l = - k:, (4-62) ,. ,3?$ +'&\:.>: 3 ,, . where k : is a constant of integration to be detehined fib the boundary conditions . of the problem. The negative sign on the right side of Eq. (4-62) is arbitrary, just as the square sign on k , is arbitrary. The separation constant k, can be a real or an imaginary1 number.'lf k, is imaginary, k j is a negative real number, making - k,2 a positive real number. It is convenient to rewrite Eq. (4-62) as and - where the separation constants k, and k, will, in different from k,; but, because of Eq. (4-60), thc following condition must be satisfied: Our problem has now been reduced to finding the appropriate solutions-X(x), Y ( y ) , and Z(z)-from the second-order ordinary differential equations, respectively, Eqs. (4-63), (4-64), and (4-65). The possible solutions of Eq. (4-63) are known from our study of ordinary differential equations with constant coefficients. They are listed in Table 4-1. That the listed solutions satisfy Eq. (4-63) is easily verified by direct substitution. The specified boundary conditions will determine the choice of the proper form of the solution and of the constanrs A and B or C and D. The solutions of Eqs. (4-64) and (4-65) for Y ( y ) and Z(z) are entirely similar. Table 4-1 Possible Solutions of X"(x) + k f X ( x ) = 0 kf 2, x ( x ) Exponential formst of X(x) + k A , sin kx + B, cos kx C,elk + Die-Jk - jk A, sinh kx + B2 cosh kx .' C2@ + D2e-L The exponential forms of X(x) are related to the trigonometric and hyperbolic forms listed in the third column by the following formulas: eih = cosh kx f sinh kx, cosh kx = f(ek + e-"), sinh kx = f (d" - e-'"). I . ;- .
r (4-64) (4-65) - n L : ~ t , (4-66) 1s X(x), ,pect~vely, ,own from They ,are wlfied by a choice of :solutions n , , i Example 4-6 Two grounded, semi-infinite, parallel-plane electrodes are sepamted ' O y ;I dihtancc h. A lh rd clectlodc pcrpcndiculqr to both is mamtalned at a uunLtant potential Vo (see Fig. 4-9). Determine the potential distribution in the region enclosed by the electrodes. 1 Solution: Referring, to the coordinates i n Fig. 4-9, we write down the boundary conditions for the tentlal function V(x, y, z) as follows. 9 With V independent of z: !
b v(x, Y , 4 = v(x, Y). . (4-67a) In the x-dircctioi: V(O, Y ) = Vo (4-67b) t V(cf4 Y) =3 0. (4-67c) In the y-directiog: F V(S, 0) = 0 (4-67d) 1 -1 V(x, b) = 0. (4-67e) Condition (4-67a) idpliea k, = 0 and, from Table 4-1, I Z(Z) = B,. (4-68) The constant A. vanlshes because Z is independent of z. From Eq. (4-66), we have 2 I I ky = -k2 = k2, (4-69) where k is a real nudber. This choice of k implies that k, is imaginary and that ky is real. The-uwpf kt = jk, together with the condition of Eq. (4-67c), requires us to choose the exponehtialiy decreasing form for X(s), which is I X(x) = ~ , e - ~ . (4-70) In the y-directio& k , = k. Condition (4-67d) indicates that the proper cholce for Y( y) from Table 4-1 is . I Y,(y) = A, sin ky. (4-71) Combining the solutions given by Eqs. (4-68), (4-70), and (4-71) in Eq. (4-59), we obtain Vn(x, y) = (B,D,A,)~-~ sin ky - - = Cne-k" sin ky, .
+ , (4-72) where the arbitrary constant C, has been written for the pr~duct B,D,A,. Now, of the five boundary conditions listed in Eqs. (4-67a) through (4-67e), we have used conditions (4-67a), (4-67c), and (4-67d). In order to meet condition (4-67e), we require K(x, 6) = Cne-kx sin kb = 0, which can be satisfied, for all values of x, only if , sin kb = 0 kb = nx 1111 I< =-, 11 = 1,2,3,. , ,. .% + (4 -74) h Therefore, Eq. (4-72) becomes n x Vn(x, y) = Cne-nnx'b sin - y. (4-75) .
. b Question: Why are 0 and negative integral values of n not included in Eq. (4-74)? We can readily verify by direct substitution that K(x, y) in Eq. (4-75) satisfies the Laplace's equation (4-58). However, Vn(x, y) alone cannot satisfy remaining boundary condition (4-67b) at x = 0 for all values of y from 0 to b. Using the technique of expanding an arbitrary function within a specified interval into a Fourier series, we form the infinite sum In order to evaluate the coefficients C,, we multiply both sides of Eq. (4-76) by mn sin - y and integrate the products from y = 0 to y = 6: b nx mn S : mx 2 Ji c,, sin - y sin - y dy = V , sin - y dy. (4-77) n = 1 b .
b b The integral on the right side of Eq. (4-77) is easily evaluated: mn if m is odd J,b V , sin - y dy = b if m is even (4-72) : h (4-67e), t condition (4-73) f3 -74) r (4-75) q. (4-74)? 5 ) satisfies rctnaming technique rier series, (4-76) (4-76) by P 7 7 ) , (4-78) . L Each-integral on the ltft side of Eq. (4-77) is I, f nn kmn C, b ({-m)x ~ J ~ ~ , , s i n ~ ~ s ~ b Y d y = - J 2 0 [COs y - cos .I b b ' - 7 f m = n , I 'r I (4-79) If rn + n. Substituting Eqs. (bk) and (4-79) in Eq. (4-7$), we obtain v (4-80) if n is wen. The desired ~otential distribution is, then, a suberposition of V,(x, y) in Eq. 14-75),' c c I..(s, y~ = 7 c,,~ - m m sin - y J ,IL 1 h Equation (4-81) is a rather complicated expression to plot In two dimensions. hut. since thc amplit dc of tho sinc tcrms in the series dccrcares very rapldly as increases, only the fir t fe v terms are needed to obtain a good approxlmation. Several k equipotential lines at skztched in Fig. 4-9. 1 Example 4-7 Con der the region enclosed on three sides by the grounded con- ducting planes show , in Fig. 4-10. The end plate on the left has a constant potential f Vo. All planes are as8unxd to be infinite in extent in the r-direction. Determine the potential distributioi within this region. Solution: The boundary conditions for the potential function V(x, y. z) are as follows. , 1 With V independent of 2 : -1- V(s, y, 2) = tqu, 4 ' ) . In the s-directioh: Since Laplace's equatioa is a limar partial di8erential equatron, the supcrpos~tron of soluirons IS also r solution. 156 SOLUTION OF ELECTROSTP: IC PRQBLEMS 1 4 ' ' Fig. 4-10 Cross-sectional figure for Example 4-7. , In the y-direction: V(X, 0) = 0 (4-82d) V(x, b) = 0. (4-82e) r Condition (4-82a) implies k, = 0 and, from Table 4-1, As a conscqucncc, Ey. (4-66) rcduccs to .
. .
.
. -- which is tlw same as Eq. (4-69) in Examplc 4-6. The boundary conditions in the y-direction, Eqs. (4-82d) and Eq. (4-82e), are the same as those specified by Eqs. (4-6713) and (4-67e). To make V(x, 0) = 0 for all values of x between 0 and a, Y(0) must be zero, and we have Y(y) = A , sin ky, (4-85) as in Eq. (4-71). However, X(x) given by Eq. (4-70) is obviously not a solution here, because it does not satisfy the boundary condition (4-82c). In this case, it is convenient to use the general form for k, = jk given in the third column af Table 4-1. (The exponential solution form given in the last column could be used as well, but it would not be as convenient because it is not as easy to see the condition under which the sum of two exponehtial terms vanishes at x = a as it is to make a sinh term zero. This will be clear presently.) We have X(x) = A , sinh kx + B, cosh kx. (4-86) A relation exists between the arbitrary constants A, and B, because of the boundary condition in Eq. (4-82c), which demands that X(a) = 0; that is, 0 = A, sinh ka + B, cosh ka sinh ka B2 = -A,.------. cosh ka --Ue), are' I) == O for (4 -85) tion here, onvenient 4-1.-(The ell, but it ion under sin11 term (4-86) boundary n , . I- 4v0 if n is odd G7:, = nn sinh (nnulb)' 4 A , sinh X(s - tr), 1. (4-87) i i where A, has been krllten fbr A,/cosh ka. It is bvident that Eq. (4-87) satisfies the condition X(a) = 0, experience, we shouldf,be able to write the solution given in Eq. (4;87) directly, the steps leading 1 b it, as only a shift in the argument of the sinh functibn is make it vanish tit x = a. Collecting Eqs. (4483), (4-85) and (4-87), we obtain the product solutlon V,(x, y) = BoA,A3 sinh k(x - a) sirl ky hn nn,, C k sinh - (x - a) sin - y, n = 1, 2, 3, . . . , b b (4-88) where C. = B,A,A,, Bnd k has been set to equdl nn/b in order to satlsfy boundary condition (4- 82e). i . We have now urbd all $the boundary condiiions except Eq. (4-82b), whlch may be satisfied by a ~odrierdseries expansion of v(0, y) = Vo over the interkal from y = 0 to y = b. We hdbe I t 1177 1 1 n <, = Y,((), y) = -.- 1 C.:, sinh - o sin - - y , 0 < v < h . (4-89) 11 I> . 1 1 T- I ' 11 I We note that Ecji (Hb) is of the same form as Eq. (4-76), except that C, is replaced by - C n sinh (nna/b). he values for thr, coefficieht Ci can then be written down from Eq. (4-80). I 0, if 11 is cvcn. The desired potentihl istribut~oa v ; ; Z : , s ; Z : enclosed region in Fig. 4- 10 is a summa- tion of K(x, y) in Eq. 158 SOLUTION OF ELECTROSTATIC PROBLGMS / 4 I s -, . .'<, ' . , 4-6 BOUNDARY-VALUE PROBLEMS IN CYLINDRICAL COORDINATES For problems, with circular cylindrical boundaries we write the governing equations in the cylindrical coordinate system. Laplace's equation for scalar electric potential V in cylindrical coordinates is, from Eq. (4-8), A general solution of Eq. (4-92) requires the knowledge of Bessel functions, which we do not discuss in this textbook: In situations where the lengthwise dimension of the cylindrical geometry is large compared to its radius, the associated field quantities may be considered to be appro5imately independent of z. In such cases, d2V/Sz' = 0 and Eq. (4-92) becomes a two-dimensional equation: --- -r Applying the method of separation of variables, we assume a product solution where R(r) and @(4) are, respectively, functions of r and 4 only. Substituting solution (4-94) in Eq. (4-93) and dividing by R(r)(D(+), we have In Eq. (4-95) the first term on the left side is a function of r only, and the second term is a function of 4 only. (Note that ordinary derivatives have replaced partial derivatives.) For Eq. (4-95) to hold for all values of r and 4, each term must be a constant and be the negative of the other. We have where k is a separation constant. Equation (4-97) can be rewritten as dZ@& / + kz@(4) = 0. ' d42 (4-98) 1 ' . This is of the same form as Eq. (4-63), and its solution can be any one of those listed in Table 4-1. For circular cylindrical configurations, potential functions and therefore potential (4-92) which we on of the ' luantities Vilz2 = 0 (4-93) i o n P (4 1 wlution (4 - 95) : second ;:y-r; (4-96) (4- 97) r (4-98) se listed . lerefore I 4-6 1 B O U N ~ ~ ~ Y ~ A L U E PROBLEMS IN ljYLIND&:7AL COORDINATES 159 1 . 1 ! . fi i:' ; ! i t , @()) are periodtk id ) and the hyperbolic fhctiohs do not apply. In fact, if the range ,' o f ) is unrestricted, k bust be an integer. Let k equal n. The appropriate solution is , 4 , @()) = A, sin n q 5 + Bi m s n), (4- 99) where A, and B, are dbitqry constants. We now turn our httention to Eq. (4-96), which can be rearranged as 1 'i , where idteger n has b k n written for k, implying a 2n-range for 6 . The solution of Eq. (4-100) is t R(r) = A,P + B,r-". (4- 101). This can bc vcrilicd S, direct substitution. Taking the product of the solut~ons in i (4-99) and (4-101), , e obtain a general solution of the .--independent Laplace's equation (4-93) for cikcular cylindrical regions with an unrestricted range for 4 : K(r, k) = rn(A, sin ng + in cos n d ) , +r-"(A; sin np -i Bi cos ,I&), i~ # 0. (4-107) Depending on the bobndary conditions, the complete solution of a problem may be a summation of the rms in Eq. (4-102). It is bseful to note that, when the region of interest includes e cylindrical axis where ). = 0;the terms contnlnlng the r-" factor cannot exist. the other hand, if the region of interest includes the point at zero as r-r co. infinity. thc terms cohiaining thc P factor cannot exist, since the potential must be When the potentktl is not a function of ), ii = 0 and Eq. (4-98) becomes 1 . I d2@(4;) -- d q 5 - 0 , (4- 103) '2 The general solution O f Eq. (4- 103) is 8(4) = A,@ + B,. If there is no circumferential variation, A, vanished,' and wt? have 8 ( 4 ) = B 0 , . k = 0 . (4- 104) The equation for ~ ( r j also becomes simpler whkn k = 0. We obtain from Eq. (4-96) , ' The term A& should d retained i f there is circumferential variation, such as in problems rnvolwng a wedge. The product of Eqs. (4-104) and (4-106) gives a solution that is independent of either zor 4 : .V(r) = Cl In r + C,, (4- 107) where the arbitrary constants t, and C2 are determined from boundary conditions. Example 4-8 Consider a very long coaxial cable. The inner conductor has a radius a and is maintained at a potential Vo. The outer conducjor has an inner radius b and is grounded. Determine the potential distribution i g ~ e space between the conductors. Solution: Figure 4-11 shows a cross section of the coaxial cable. We &ume no =-dependence and, by symmetry, also no +dependence (k = 0). Therefore, the electric potential is n function of r only and is givcn by Eq. (4- 107). . . The boundary conditions are Substitution of Eqs. (4-108a) and (4-108b) in Eq. (4-107) leads to two relations: . C l l n b + C 2 = 0 , (4- 109 a) . Cl Ina + C2 = Vo. (4- 109 b) Expressions C; and C2 are readily determined: Therefore, the potential distribution in the space's I r I b is i - Obviously, equipotential surfaces are coaxial cylindrical surfaces. , i- ,-I of either (4 107) ilditions. , radius rac'f' b ween . rle ,sume ti0 ic electric (4-108a) (4- lO8b) tions: (4- 1 O9a) ( 4 - l o b ) / I n f (4- 110) 6 p 4-6 t OUND&~-Y-WUE PROBLEMS IN I ~ ~ ~ ~ E I ~ ~ COORDINATES 161 ' 1 ; . . ?' . ' :!;<is < . . tJ , , -b I . : 1 I I I , tl Fig. 4-12 Cross section of split \ \ / A ' circular cylir~bcr and cqu~potcnt~:ll '----/ i: lines (Example 4-9). Example 4-9 An i%finirely long, thin, pnducting circular tube of rndiub b is split i n ~ w o halves. The er half is kept at a potential V =. V, and the lower half at V = -Vo. Determine distribution both inside and outside the tube. - Solution: A cross section of the split circular tkbe is shown in Fig. 4-12. Since rhe tube is assumed to b ! infjnltely long, the potential is independent of 2 and the two- dimensional Laplace'b equation (4-93) applies. The boundary conditions are: These conditions are plotted in Fig. 4-13. Obviously V(r, 4) is an odd function of 4. We shall determine. P(r, @i inside and outside the tube separately. a) Inside the tube, I I I . I 1 ! I o T 2n i +' I 3 I. L----vo d L- , Fig. 4113 Boundary condition for . Example 4-9. Because this region includes r = 0 , terms containing the r-" factor cannot exist. Moreover, since V(r, 4) is an odd function of 4, the appropriate form of solution V,(r, 4) = A,? sin n4. However, a single such term d ~ e s not satisfy the boundary conditions specified in Eq. (4- 11 1). We form a series solution m . - W . 9 4) = 1 K(rY 4 n = 1 = 2 A.? sin n4, (4- 1 13) I n = 1 and require that Eq. (4- 11 1) be satisfied at r = b. This amounts to ex'panding the i rectangular wave (period = 2x), shown in Fig. 4-13, into a Fourier sine series. 1 The coefficients An can be found by the method illustrated in Example 4-6. As a matter of fact, because-we already have the result in Eq. (4-80), we can directly write if n is odd 4-7 (4-115) 1 IN St The potential distribution inside the tube is obtained by substituting Eq. (4-115) h I !
' in Eq. (4- 1 13). b) Outside the tube, - In this region, the potential must decrease,to zero as r -r m. Terms cbntaining the factor r" cannot exist, and the appropriate form of solution is = 2 Bnr-" sin n4. ?it i r ! 1, t - ;I' ' (4-113) inding the inc scries. (4-1 14) P 4-' , a in directly (4-115) Zq. (4- 1 15) (4- 116) tontaining P I' (4-1 17) not exist. r i rso~utio? . : '1'. . , . i i , (4-112) a specified 1 I The coefficients $,, in Eq. (4-1 18) are analogous to A,, in Eq. (4-114). From Eq. . (4-115) we obtaih 'I 6 Y' I - !
, ' 4 ' .n I if n is odd ' B d = (4-1 19) 1 0 7 if n is even. Therefore, the pdtcntial distribution outside the tube is Several equipotefitial lines both inside and outside the tube have been sketched in Fig. 4-12. 1 The general equation in spherical coordinates is a very involved discussion to cases where the electric potential is independenteof the llzirnuthal angle 4. Even with this limitation we will need to introduce some new functions. From Eq. (4-.9) we have Applying the method of separation of variables, we assume a product soiutlon I' V(R, 0) = T(R)O(B). (4-112) Substituti-)his Bolution in Eq. (4-121) yields, after rearrangement, I In Eq. (4-123) the first term on the left side & a idfiction of R only, and the second term is a function of 0 onb. If the equation is to hold for all values of R and 0, each term 164 SOLUTION OF ELECTROSTATIC PROBLEMS 1 4 must be a constant and be the negative of the other. We write - where k is a separation constant. We must now solve the two second-order, ordinary differential equations (4-124). and (4-125). Equation (4-124) can be rewritten as which has a solution of the form r,,(R) = AnRn + B,R-'"'I). - In Eq. (4-127). A,, and B,, are arbitrary constants. and the fitbwjng relation between 11 and k can be verified by substitution: n ( / t + I ) = I?, ( 4 - 1 3 ) i i where n = 0, 1 , 2,.. . . is a positive integer. k 1 With the value of k2 given in Eq. (4-128), we have, from Eq. (4-125), ! + n(n + I)@(@) sin 8 = 0, r which is a form of Legendre's equation. For problems involving the full range of 0, I t from 0 to n, the solutions to Legendre's equation (4-129) are called Legendre j~ncrions. usually denoted by P(cos 8). Since Legendre functions for integral values of n are t E polynomials in cos 0, they are also called Legendre polynomiuls. We writc 1 @,(O) = P, (COS 0). (4-130) Table 4-2 lists the expressibns for Legendre polynomialst for several values of n. Combining solutions (4-127) and (4-130) in Eq. (4-122). we have, for spherical boundary-value problems with no azimuthal variation, Depending on the boundary conditions of the pivcin problem, the completc solution may be a summation of the terms in Eq. (4-131). We illustrate the application of ' Actually Legendre polynomials are Legendre functions of the first kind. There is another set of solutions to Legendre's equation, called Legendre functions ofxhe second kind; but they have singularities at 0 = 0 and n and must, therefore, be excluded if the polar axis is a reglon of interest. (4-125) xdinary : . I / ' (4 - 120) (4- 127) b e t y n (4-128) (4-1 29) nge of 0, fuitctions, s f n are (4- 130) ues of n. spherical ( 4 ~ 1 ) :soil, . cation of of solutions ies at 0 = 0 f P " (cos 0) 4 t . , , : . ' '\ " . , t 1 cos Q 2 A(3 cos2 0 - 1) 3 :: +(3 C O S ~ e - 3 cos 8 ) Lcgcndre po~yno$ids in thc Solution of a sidple boundary-v;lluc problem in the , following example. . I Example 4-10 An Uncharged conducting sphere of radius b is placed in an initially uniform electric field $ , = azEo. Determine (a) the potential distribution V(R. 0) and (b) the electric field idtensity E(R. 0) after the introduction of the sphere. Solution: After the conduct~ng sphere is introdsccd into the clectrlc lield, : l sep;lr:i- llon and redistributiqp of charges will take place in such a way that the surface of thc sphere is The electric field intensity within the sphere 1 s will intersect the surface normally, and the the sphere will not be affected appreclably. The geometry of thistproblem is depicted in Fig. 4-14. The potentla1 is, obviously, independent of the adnutbal angle 4, and the solhtion obtained in this section applies. k ' B.1 . . Fig. 4-14 Conducting sphere electric field (Example 4-10). in a uniform a) To determine the potential distribution V(R, 6) for R 2 b, we note the following boundary conditions : . Equation (4-132b) is a statement that the original Eo is not disturbed at points very far away from the sphere. By using Eq. (4-131), we write the general solution as However, in view of Eq. (4-132b), all An except A, must vanish, and A , = - E,. We have, from Eq. (4-133) and Table 4-2, -1 = B,R- ' + (B,R-' - E,R) cos f3 + BnR-(ntl)P,,(cos 8), R 2 b n = 2 (4- 134) Actually the first term on the right side of Eq. (4-134) corresponda to the potential of a charged sphere. Since the sphere is uncharged, Bo = 0, and Eq. (4-134) becomes (4-13.5) Now applying boundary condition (4-132a) at R = b, we require 0 = ( -2 .
- E 0 b ) cos 6 + 2 Bnb-'n+l)P,,(cos e), n = 2 from which we obtain B, = Eob3 and Bn=O, n 2 . 2 . ' For th~s problem it is convenient to assume V = 0 in the equatorial plane (0 = 7~12). which leads to V(b.0) = 0 , since the surface of the conducting, sphere is equipotential. (Sec Problem P.4-21 for - V(b, O) = V , .
) . 1 solution potential 1 . (4-134) w h ~ h ldds P.4-21 for t 1 i 4 . , p. ,, - j i 1 , We hive, tli$aliy,;~rom'~q. (4-135), . , 0 4 . j ri )(R,~)=-E, p R z ~ . (4-136) . ; i . # , , b) The electric f i e l ~ intensity E(R, 0) for R'F b can be easily determined from -VV(R,8): 1 . F E(R, 0) = aREi. + aoE,, where ' (4-137a) f ,: dV I p and !a= - a = ~ o [ l + 2 ( ' ~ ] c o s 0 , R k b (4-137b) The surfac; charbe density on the sphere can be found by noting which is p;oportional to cos 8, bemg zero at O = n/2. Some equipotent~al and - field lines are gketched in Fig. 4-14. In this chapter wd'ha~e discussed the analytical solution of electrostatic problems by the method ~f imQes and by direct solution of Laplace's equation The method of images is useful when charges exist near conductlng bodies of a simplc and com- patible geometry: a point charge near a conducting sphere or an infinite conductlng plnnc; and it linc chafgc I G I ~ iL pat~ilcl wiiriucling cylindcr or .I p:~r:iIlcl conductitlg plmc. Tlic aolillioii a/' Laplacc'a cquauon by the method of separation of variables requires that the bouhdaries coincide with cooidinate surfaces. These requirements restrict the usef~ln~ss~of both methods. In practical problems we arc often faced with more complicated bqpdarics, which are not amenable to neat analytical solutions. In such cases, we must resort to approximate grkphical or numerical methods. These methods are beyond the scope of this book.+ I . > REVIEW QUESTIONS I , ! R.4-1 Write Poisson's lequation in vector notation -1 . j a) for a sinipte b) for a linciw and isotropic, hut inl~omo~cncous ti~cdiunl. R.4-2 ~epeablfi cartdsian coordinates both p&ts of R.4-1. ' See, for instance, B. D. Popovit. Introductory Engineering ~ k ~ t r o m a ~ n e t i c s , Addison-Waley Publishing Co. (1971), Chapter 5. ' ,; " R.4-3 Write Laplaa's equation for a simple medium t . > 3 . '; a) in vector notation, b) in Cartesian coordinates. R.4-4 If V ' U = 0, why does it not follow that U is identically zero? R.4-5 A fixed voltage is connected across a parallel-plate capacitor. a) Does the electric field intensity in the space between the plates depend on the permittivity of the medium? P b) Does the electric flux density depend on the permittivity of the medium? Explain. 1 i i ' I ' R.4-6 Assume that fixed charges +Q and - Q are deposited on the plates of an isolated parallel- I plate capacitor. !
' a) Does the electrlc field intensity in the space between the plates depend on the permittlvlty of the medlum'? , b) Does the clectric flux density depend on the perm~ttivlty of the medium'? ! Explain. I R.4-7 Why is the electrostatic potential continuous at a boundary? - R.4-8 State in words the uniqueness theorem of electrostat~cs. -- R.4-9 What is the image of a spherical cloud of electrons with respect to an infinite conducting : plane? R.4-10 Why cannot the point at infinity be used as the point for the zero reference potentlal for 1 an infinite line charge as it is for a point charge? What is the physical reason for this difference? I R.4-11 What is the image of an infinitely long line charge of density p, with respect to a parallel , conducting circular cylinder? i R.4-12 Where is the zero-potential surface of the two-wire transmission line in Fig. 4-6? 1 I R.4-13 In finding the surface charge induced on a grounded sphere by a point charge, can we 1 set R = a in Eq. (4-52) and then evaluate ps by -ro aV(a, QWR? Explain. i i R.4-14 What is the method of separation of variables? Under what conditions is it useful in , solving Laplace's equation? R.4-15 What are boundary-varue problems? I L I R.4-16 Can all three separation constants (kx, k,, and k , ) in Cartesian coordinates be real? Can they all be imaginary? Explain. i R.4-17 Can the separation constant k in the solution of the two-dimensional Laplace's equa- ! , ' 7 tion (4-97) be imaginary? I 1 i , R.4-18 What should we do to modify the solution in Eq. (4-'110) for Examplc 4-8 if the inncr i , - i conductor of the coaxial cable is grounded and the outer conductor is kept at a potential Vo? . .
. ' 1 ; z I . . . R.4-19 What should we do to modify the solution in Eq. (4-1 16) for Example 4-9 if the con- . . , ducting circular cylinder is split vertically in two halves, with V = Vo for - n/2 c 4 n/2 and V = - Vo for x/2 < 4 < 3n/2? i 1 "I ui parallel- xrmittivlty / -' conductmg otent~al for d~fference? o a paallel 1-6? rge, can we it useful in e real? Can a t P q u a - I if the inner ntial Vo? I if the cod: c lr/2 and - ,,?+.: - 2 "' < .
. .. . ., . A b. r ; .: .". ' i PROBLEMS 169 , , . i !; { I . . R.4-20 Can functibns ' k ( ~ , = cos 0, where C, and C, are , '? arbitrary constants, be db1utid.m of Explain. , " 7 r 1 .
. 4 . PROBLEMS I! ' [ P.4-1 The upper and lower conductmg plates of a k g e parallel-plate capacitor are seprrated by a distance d and mai ttiin~d at potentials Vo and respct/vely. A dielectric slab of dielectric constant r, and u n i f o d hiekoess 0.8d is placed over the lower plate. Arsumlng negligble fringing t effect, determine , r a) the potential adti eleqtric field distribution inihe dielectric slab, b) the potcntial aHd electric field distribution i H the air space between the dielectric slab and the upper plate, C) the surface chdfge densities on the upper +d lower plates. P.4-2 Prove that the scalar potential V in Eq. (3-56) satisfies Po~sson's equation; Eil. 14-6). . , , PA-3 Prove that a 60tential function satisfying Laplace's equation in a given region possesses no maximum or minimum within the region. ' P.4-4 Verify that > r V, = C,/R and V, = C2z/(.y2 + yZ + i2)'I2, where C, and C, are arbitrary constants, are solutions of Laplace's equation P.4-5 Assume a point charge Q above an infinite cdnducting plane at J> = 0. a) Prove that V(x,'): i) in Eq. (4-37) satisfies Laplace's equation ~f the conduct~ng plane is maintained at tero uotential. b) What should t ~ k expression for V(x, y;z) be i6the conducting plane has a nonzero poten- tial V,'? 4 c) what-is ths el~~trostatic force of attraction between the charge Q and the conducting plane? .I: PA-6 Assume that spkce between the inner and outer conductors of a long coaxial cylindrical structure is filled with ad electron cloud having'a volume density of charge p = /I/r for a < r c b, where a and b are, respdctively, the radii of the inner and outer conductors. The inner conductor is maintained at a potential V,, and the outer conductor is grounded. Determine the potential distribution in the reg& a < r < b by solving Poisson's equation. P.4-7 A point charge' Q exists a a distance d above a large grounded conducting plane. Determine a) the surface chatke density p,, b) the lmal_qharge indqced on the condxiillg $erz. I ,+ P.4-8 Determine the &ystems of image chi\;ib (hat ; 1 1 1 1 replace the conducting boundaries that are maintained at Zero potential for ' a) a point charge located between two large, grounded, parallel conducting planes as shown in Fig. 4-15(a), + b) an infinite line charge p; located midway between two large, intersecting conducting planes forming P 60-degree angle, as shown in Fig. 4-15(b). 3. '< , , . . :.L'.,f.,r . ., , >,f..l .? , . , , ' : " , .: .~ . , . . , t j ~ ? . i i ~ + " ~ : ~ : ; , ~ E E 2 ~ \ ~ ~ ~ i j . 5 ~ !
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. ' . I.. - ' . 170 , %OLU- '2.N OF ELECTROSTATIC PROBLEMS 1 4 , . . . - (a) Point charge between grounded parallel planes. ' ( 1 G a Fig. 4-15 Diagrams - - for Problem P.4-8. (b) Line charge between grounded intersecting planes., 1 : P.4-9 Two infinitely long, parallel line charges with line densities pc. and -p, are located at . - - b b & - + - and z = - - i f 2 2 respectively. Find the equations for the equipotential surfaces, and sketch a typical pair. i i P.4-10 Determine the capacitance per unit length of a two-wire transmission line with parallel I conducting cylinders of dillkrent radii u , and o , , tllcir axcs being scp;wutcd by n distancc D (where D > a, + a,). i 1 . - PA-I 1 A straight conducting wire or radius tr is p;u;~llcl to and nt height 11 from thc surbce of I the earth. Assuming that the earth is perfectly conducting, determine the capacitance per unit length between the wire and the earth. i P.4-12 A point charge Q is located inside and at distanced from the center of a grounded spherical conducting shell of radius b (where b > d). Use the method of images to determine a) the potential distribution inside the shell, b) the charge density p, induced on the inncr surface of the shell. P.4-13 Two dielectric media with dielectric constants E , and e2 are separated by a plane bound- ary at x = 0 , as shown in Fig. 4-16. A point charge Q exists in medium 1 at distance d from the boundary. a) Verify that the field in medium 1 can be obtained from Q and an image charge -Q,, both acting in medium 1. (Image charge) I (Image charge) Medium 2 (62) Medium 1 (el) Fig. 4-16 Image charges in dielectric x = O media (Problem P.4- 13). 7f1ir. r parall~l stance D : bound- : d from ge -QI: , . . ' . Verify that the field in'medium 2 can be obtained $om Q and an image charge +Q,, both acting in mediud 2. c) Determine Q, and Q , . [ ~ i n t : Consider neighboring boints P, and P , In media 1 and 2 , respectively and requird the continuity of the tangential component of the E-field and of the normal coapon&t of the D-field.) P.4-14 in what way should we hodify the solution in E ~ . 14-91) for ~ x a r n ~ l e 4-7 rfthe boundary conditions on the top, bottbm, and rlght planes in Fig. 4-10 are dV/an = O? PA-15 In what way shollld idmodify the solution in Eq. (4-91) lor Example 4-7 if the top, bottom, and left planes in F& 41-10 are grounded ( V = 0) and an end plate on the right is rnain- tained at a constant potential to? P.4-16 Consider the rcctangulab region shown in Fig. 4.- 10 as thc cross scction of an cnclosurc formcd I~y'rour contlucli~~g pli~t~?i. I'hc Icl't mid right plntcs arc yroundcd, and the top and bottom pixto i ~ l r in;~imiiincd a 1 constnht potentials v, and V, respstively. Determine the potential distribution insidc the enclosurq; P.4-17 Consider a metallic rektrihgular box with sidcs a and b and height c. The side walls and thc bottom surface are grounded, The top surface is isolated and kept at a constant potentla1 Vo. Determine the potential distribution inside the box. PA-18 An infinitely long, thin, c nrluctinp circular cylinder of radius b is split in four quartcr- cylinders, as shown in Fig. 4-17. he quarter-cylinderi in the second ;~nd foilrtli L ~ I I I : \ it< r \ l \ t < : \ S t grounded. and those in the first and third ,l~iitdru~,[< 4rc i c j l ( ;I( i~o~c~tials i; ; ~ s d - I; respcc- tivdy, Dclcrn\iw tl~c pukalid ilbtvibutial both inr/de md outside the cylinder. C ' . Fig. 4-17 Cross section of long circular cylinder split in foir I quarters (problem PA-18). P.4-19 A long, grounded conductink cylinder of radius b is placed along the z-axis in an initially uniform electric field E, = a,E,. Deterpine potential distribution Y(r, (6) and clectric field in- 'tensity E(r, $) outside the cylinder. ' P.4-20 A long dielectriccq4nder of tadius b and dielectric constaht 6, is placed along the z-axis in an initially uniform electric field ko = a,Eo. Determine V(r, #J) and E(r, 4) both inside and outside the dielectric cylinder. P.4-21 Rework Example 4-10, asshmipg V(b, 0) = Vo in,Eq. (4-132a). P.4-22 A dielectric sphere of radius b and dielectric constant 6, is placed in an initially uniform electric field, Eo = aZEo, in air. Detmniqe V(R, B) and E(R, 0) both inside and outside the di- electric sphere. -.
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: .,, , . , . . , , . 1 . . .< , 5 / Steady Electric Currents . [ 1 :,. , . . . . . , . , , 5-1 INTRODUCTION i In Chapters 3 and 4 we dealt with electrostatic problems, field problems associated i with electric charges at rest. We now consider the charges in motion that constitute \ current flow. There are several types of electric currents caused by the itloti011 of pee i charges.+ Conduction currents in conductors and semiconductors are caused by drift motion of conduction electrons and/or holes; e l e c t r o l ~ ~ i i ~ ~ r c i ~ t s are the result of ! migration of positive and negative ions; and convection currents result from motion of clcctrons :rntl/or ions in a v:uxum. In this chnpter wc shi~ll pay spccinl attention to 3 conduction currents that are governed by Ohm's law. We will proceed from the point form of Ohm's law that relates current density and electric field intensity and obtain the V = IR relationship in circuit theory. We will also introduce the concept I principle of conservation of charge, we will show how to obtain a point relationship I of electromotive force and derive the familiar Kirchhoff's voltage law. Using the i between current and charge densities, a relationship called the equation of continuity 1 from which Kirchhoff's current law follows. When a current flows across the interface between two media of different conductivities, certain boundary conditions must be satisfied, and the direction of t currcnr llow is ch:~ngcd. Wc will discu\s thcsc boundary contlitions. Wc will :11w i show that Sor a homogeneous conducting medium, the current density can be 1 expressed as the gradient of a scalar field, which satisfies Laplace's equation. Hence. j an analogous situation exists between steady-current and electrostatic fields that is the basis for mapping the potential distribution of an electrostatic problem in an electrolytic tank. i 1 . The electrolyte in an electrolytic tank is essentially a liquid medium with a low F conductivity, usually a diluted salt solution. Highly conducting metallic electrodes are inserted in the solution. When a voltage or potential difference is applied to the j electrodes, an electric field is established within the solution, and the molecules of the electrolyte are decomposed into oppositely charged ions by a chemical process I .,A , .. called electrolysis. Positive ions move in the direction of the electric field, and negative ! I ' In a time-varying situation, there is another type of current caused by bound charges. The time-rate of change of electric displacement leads to a displacement current. This will be discussed in Chapter 7. t e 5-2 1 CI ,nt. ' 1ENSIlY AND OHM'S LAW 173 . i 11 4 -. ) ions move in a directid opposite td the field, both contrib'uting to a current-flow in 1 , I the direction of tkb k fiel, . An experimental model ban be set up in an electrolytic tank, ' with electrodes dqpibpkr geometdcal shapes sirndating the boundaries in electrostatic 4 problems. The deasurkd potential distribution iii the electrolyte is then the solution to Laplace's eqlYation b r difficult-to-solve analytic problems having complex bound- aries in a homogeneoys medium. associated constitute iou o f free ed by drift ie r e t of xn . 'on tier. .-.. to 1 frca the tensity and he w x e p t rJymg the cld~mnship f conrinuity 3 f different lirection of le will also jity can be ion. Hence, fields that problem, in I with a low c c l p y d e s p i $3 the noli of I: , process m i \ negative 'he rime-rate of 'hapter 7. Convectioq. curreyts ark the result of the motion of positively or negatively charged particles in ajacuum or rarefied gas. Fahiliar examples are electron beams in a cathode-ray tube, >nd the violent motions of dharged particles in a thunderstorm. Convection curtents, t e result of hydrodynamic motion involving a mass transport, are not governed by Iinl's law. d The mechanism o conduction currents is diRerent from that of both electrolytic currents and convecttbn currehts. In their normal state, thc atoms of a conductor occupy regular positions in u crystalline structdre. The atoms consist of positively charged nuclei surrounded by electrons in a shell-like arrangement. The electrons in the inner shells are tightly bound to the nuclei and are not free to move away. The electrons in the outerrflost shells of a conductor atom do not completely fill the shells: they are valence or corlduction electrons, and are only very loosely bound to the nuclei. These latter electrons ,may wander from one atom to another in a random manner. The atoms, on the ~Gerage, remain electrically neutral, and there is no net drift motion of electrons. ,when an external~electric field i's applied on a conductor, an organized motion of the conduct.ion electrdns will result, producing an electric current. The average drift velocity of the clcctrdns i s vcry low (on the order of 10-' or lo- m/s) even foi"vcry good conductors, because they collide with the atoms in the course of their qotion, dissipating part of their kinetic energy as heat. Even with the drift motioh of conduction electrons, a conductor remains electrically neutral. Electric forces prevent excess electrons from accumulating at any point in a conductor. We will sdow gnalytically that the charge density in a conductor decreases exponentially with tide. In a good conductor the charge density diminishes extremely rapidly toward zero as the state of equilibrium is approached. 1 5-2 CURRENT DENSITY ND OHM'S LAW Consider the sttady rhotion of one kind of charge carriers, each of charge q (which is negative for electrons), across an element of surface As with a velocity u, as shown in Fig. 5-1. If N is the h m b e r of charge carriers per unit volume, then in time At each 1 charge carrler moves a distance u At, and the amount of charge passing through the surface As is AQ = N q u a, As At (C). . (5-1) Since current is the tlme rate of change of~harie, we have I . . i ' . P 174 STEADY ELECTRIC CURRENTS I 5 1 . : 1 I Fig. 5-1 Conduction current due to drift motion of charge carriers across a surface. . I \ 3 In Eq. (5-2), we have written As = a,As as a vector quantity. It is convenient to define a vector point function, yolurne current density, or simply current density, J, in amperes per square meter, J = Nqu (A/m2); (5-3) so that Eq. (5-2) can be written as - . . -. A1 = J . As. (5 -4) The total current I flowing through an arbitrary surface S is then the flux of the J vector through S: Noting that the product Nq is in fact charge per unit volume, we may rewrite Eq. (5-3) as which is the relation between the contiection current density and the velocity of the charge carrier. In the case of conduction currents there may be more than one kind of charge carriers (electrons, holes, and ions) drifting with different velocities. Equation (5-3) should be generalized to read As indicated in Section 5-1, conduction currents are the result of the drift motion of charge carriers under the influence of an applied electric field. The atoms remain neutral ( p = 0). It can be justified analytically that for most conducting materials ivcnicnt to 2111 ciolsiry, (5-3) -4, ax 01 me J (5-5) nay rewrite (5 -6) ocity of the d 'of charge lation (5-3) n - 7 ) : t motlon of 3ms remain ~g materials , \ k~ DENSITY AND OHM'S LAW 175 4 ' i I ' I , the average drift directly propo;tioml to the electric field intensity. Con- sequently, we (5-3) or Eq. (5'7) i s where the proportioba~ity coastant, o, is a mhcroscopic constitut~ve parameter of - the medium called e nductiuity. Equation is a constitutive relation of the con- 1 1 ducting medium. Is ropld materials the linear relation Eq. (5-8) holds are called ohmic meqba. The unit for o is ampep per volt-meter (AP-m), or siemens per meter (S/m;). Ca per, the most commonly used conductor, has a conductivity d 5.80 x lo7 (S/h). 0nihe other hand, hard rubber, a good insulator, has a conductivity o f only 1 0 -I ' IS/m). hppcnciix H 4 11\ts thc conduct~vlt~cs of some other frequently . uacd materials Howpver, note that, unlike the dielectric constant, the conductlvlty of materials valies dver an extremely wlde range. The reciprocal of conductlv~ty 1 s called rrsi~tiuity, in oHrn meter (i2.m~. We prefer to use conductivity; there is really no compelling need to ube both conductlv~ty m d resistivity. . We recall Ohm's iua from circuit theory that the voltage V,, across a resistance R, 111 which a current I flows from point 1 to point 2, is equal to RI; that is, V12 = RI. (5-9) Here R is usually a of conducting materid of a giyen lengh: V, is the voltage bctwecn two lcrn~inals I i11Id 2 ; i~nd I is the total current [lowing iron1 terminal 1 to tcrm~nal 2 through a frnitc cross 4cctlon. Equation (5-9) 1s not a point reiatlon. Although there is little resemblance between Eq. (5-8) anti Eq.'(5-9), the former is gknerally referred to as the point form of Ohids law. it holds at all points in space, and D can be a function of space co- ordinates. Let us'use the poiht form of Ohm's law to dirive the voltage-current relationship of a piece of homogWleous material of conductivity c, length L and uniform cross- section S , as shown i H Fig. 5-2. Within the conducting material, J = oE where both J and E are in the dirktion of current flow. The potential difference or voltage between Y A :'.,'.' " ' 8 . r , . < , .
, . . , $> - t - .' terminals land 2 ist ' ' "' , . ' . , - - ., , v,, = E d or Vl 2 E=-. IP (5-10) The total current is I = J J . ~ ~ = J s _ a , . , or t ,. , - , .k+rs I . 7 ,: +, I - < , . , , 1 J = ~ . . (5-1 1) Using Eqs. (5-10) and (5-11) in Eq. (5-8), we obtain which is the same as Eq. (5-9). From Eq. (5-12) we have the formula for the resistance of a straight piece of homogeneous material of a uniform cross section for steady current (DC). We could have started with Eq. (5-9) as the experimental o h m s law and applied it to a homogeneous conductor of length C and uniform cross-section S. Using the formula in Eq. (5-13), we could derive the point relationship in Eq. (5-8). Example 5-1 Determine the DC resistance of 1 (km) of wire having a I-(mm) radius (a) if the wire is made of.copper, and (b) if the wire is made of aluminum. Solution: Since we are dealing with conductors of a uniform cross section, Eq. (5 -13) applies. a) For copper wire, a,, = 5.80 x lo7 (S/m): G = lo3 (m), S = 7 ~ ( l O - ~ ) ~ = lo-% (m2). We have . , .
, ,~";;; R , = - = lo3 G -- . = 5.49 (n). a,S 5.80 x lo7 x lo-% - ' We will discuss the significance of V,, and E more in dctail in Section 5-3. (5-li) /-- -12) 2 rem~unce for steady J (5-13) d applied it Using the mm) radius I : II b) For a l u k n u h ilre, ual = 3.54 x .lo7 mi: 1 - . . 1 I The conductahcej{G, or the reciprocal of ri%istance, is useful in combining resis- tances in parallel : 1 S + ' (9. : , G=-=b.- % I R e (5-14) d ft From circuit theory %e know the following: ', a) When resistakeb R, and R, are connectod in series (same current), the total resistance R is I (5 - 15) b) When resistances R , and R, arc connected in parallel (same voltage), we have 5-3 ELECTROMOTIVE FORCE AND KIRCHHOFF'S VOLTAGE LAW . In Section 3-2 we &inted out that static electric field is conservative and that the scalar line integral a ! static electric i?+o-Gty around any closed path 1s zero; that IS, $ ~ - d t ' = 0 . (5-17) For an ohmic material J = oE, Eq. (5-17) becomes Equation (5-18) t1119hs that a steady current cannot be maintained in the same direction in a closed circuit .by an electrostaticjield. A steady current in a circuit is the result of the motion of c~arge carriers, which, in their paths, collide with atoms and dissipate energy in the circuit4 This energy must cdme from a nonconservative field, since a charge carrier completing a closed circuit in conservative field neither gains nor 178 STEADY ELECTRIC CURRENTS I 5 . , 1 + + 2 - - ++02 - - - - + - + & - + - +E'- . Fig. 5-3 Electric fields inside an Electric battery electric battery. loses energy. The source of the nonconservative field may be electric batteries (con- version of chemical energy to ,electric energy), electric generators (conversion of mechanical energy to electric energy), thermocouples (conversion of thermal energy to electric energy), photovoltaic cells (conversion of light energy to elect'ric energy), or other devices. These electrical energy sources, when connected in an electric circuit, provide a driving force for the charge c~rriers. TllisJorcc mmifcsts itself as . an equivalent irnpressed electric field irtterisity E;. Consider an electric battery with electrodes 1 and 2, shown schematically in Fig. 5-3. Chemical action creates a cumulation of positive and negative charges at electrodes 1 and 2 respectively. These charges give rise to an electrostatic field in- tensity E both outside and inside the battery. Inside the battery, E must be equal in magnitude and opposite in direction to the nonconservative E, produced by chemical action. since no current flows in the open-circuited battery and the net force acting on the cli;~rgc carriers must vanish. Thc line inlcgrnl of thc impressed ficld intensity E, from the negative to the positive electrode (from electrode 2 to electrodc 1 in Fig. 5-3) inside the battery is customarily called the electromotive forcet (emf) of the battery. The SI unit for emf is volt, and an emf is not a force in newtons. Denoted by V , the electromotive force is a measure of the strength of the nonconservative source. We have Inside the source The conservative electrostatic field intensity E satisfies Eq. (5-17). Outside Inside the source the source ' Also called electromotance. - :rics (con- w u o n of la1 energy ic energy), in electric ts itself as n a t d in char- at IC field in- :e equal in ; chemical me actlng 1 ldtensity [rode 1 in ' (emf) of ,. Denoted riservative (5-19) -0) combining Eqs. (5-119) and (5-20), we have . . , " . ' " , = J : E ~ (5-21) Outs~de ' the source or ir )I "= v,,= v, - v,. (5-22) In Eqs. (5-21) and @-22) we have expressed the emf of the source as a line integral of the conservative & and interpreted it as a uqltage rise. In spite of the nonconserva- tive nature of E,, the emf can be expressed as a potential difference between the positive and negative terml?als. This was what we did in arriving at Eq. (5-10). Whcn a resisto? in tile form o f Fig. 5-2 i.p conncctcd bctwecn tcrm~nals 1 '~nd 2 . ol l l x battery, com~lcling the circu~l. lie lotul clcar~c field intcnbity (c1ectroat:rtic E caused by charge cumulation, as well as impressed Ei caused by chemical sctlon) must be used in the point form of Ohm's law, We have, instead of Eq. (5-8), where E, exists inside the battery only, while has a nonzero value both inside and . outside the source. From Eq. (5-23), we obtain The scalar line integkal of Eq. (5 -24) around the closed circuit yields, in view of Eqs. (5-17) and (5-19), r- = $(E + E,) . dP = - J . LIP. I I Equation (5-25) should be compared to E q (5-18), which holds when there IS no source of nonconse~vative field. If the resistor has a conductivity o, length /, and uniform cross-section S, J = I/S and the right side of Eq. (5-25) becomes RI. We havef = X I . (5-26) If there are more than x c soarc; illc!c~.;;,,.~otive force and more than one resistor (including the interdal resistances 01 the sources) in the closed path, we generalize Eq. (5-26) to - ' We assume the battery to have a degligible 1ntern:l resistance; otherwlu is elfcvt must be lncluded in Eq. (5-26). An idcul witaye solrm in one whosc tchinal Voltage is equal to ltr c n ~ l and is ,ndcpcndc~ of the current flowing through it. This Implies that an ideal voltage source has a zero internal resistance. L ' I L 4 . , I : Equation (5-27) is an expression of Kirchhoff s voltuge l& It states that around a I, I closed path in an electric circuit the algebraic sum of the emf's (volhige rises) is equal : , . to the algebraic sum of the voltage drops across the resistances. It applies to any closed ' path in a network. The direction of tracing the path can be arbitrarily assigned. and the currents in the different resistances need not be the same. Kirchhoff's voltage law is the basis for loop analysis in circuit theory. I i i I . I t 5-4 EQUATION OF CONTINUITY AND . I I 1 KIRCHHOFF'S CURRENT LAW \ < . L The priwiplc ofmnrrruotion of cbergr is one of the fundamental postulates of physics. Electric charges may not be created or destroyed; all charges either at rest or in : motion must be accounted for at dl times. Consider an arbitrary volume V bounded by surface S. A net charge Q exists within this region. If a net current I flows across i Illc surfhce oirl oS Illis region, the cll;~rge in 1I1e volume inus( ( / ~ ~ c ~ . ~ ~ r a ;I( ;I r;~le tI1;11 t I equals the current. Conversely, iSa net currunt llows across the surhce illto the region, - the charge in the volume must i~~creusr at a rate equal to tht-current. The current leaving the region is the total outward flux of the current density vector through the surface S. We have Divergence theorem, Eq. (2-107), may be invoked to convert the surface integral of , J t o the volume integral of V . J. We obtain, for a stationary volume, ap S V v . ~ d v = -Svzdu. (5 -29) 7 In moving the time derivative of p inside the volume integral, it is necessary to use : partial dinerentiation because p may be a function of time as well as of space co- I ordinates. Since Eq. (5-29) must hold regardless of the choice of V, the integrands ; must be equal. Thus, we have. I i h S! t u (5-30) t -1 i -7 # - This point relationship derived from the principle of mnxrvation of charge is called the equation of continuity. : I -.
. T For steady currents, charge density does not vary with time, ap/& = 0. Equation , C( (5-30) becomes I . 1 V . J = O . (5 -3 1) " . -Thus, steady electric currents are divergenceless or solenoidal. Equation (5-31) 1 U is a point relationship and holds also at points where p = 0 (no flow source). It means : t in ; A 4 1 I , , < . < ' I urourid u . , J~seyttul , , rtly closed , , jned, and r s voltage I fphysics. est or in . bounded h s across rate that ic rcgion, : current 0 ug'P,2 i t 5 -28) ilegral of (5 -29) ry to use pace co- tegrands (5-30) is c 6 d :qui (5-31) n (5-31) It means 5-4 I EBUA~ON:?F CONTINUITY A/I KIRCHHOFF'S CURRENT LAW 181 d ! i . 4 that the field lines df stryimlines of steady cdrrents close upon themselves, unlike tho& of electrosiatit field intensity that orig!hate and end on charges. Over any enclosed surface, kq. (5-31) leads to the follo~ing integral form: I $ J - d s - 0 , I (5-32) ' which can be writted as (5-33) I 1 . , Equation (5-33) is a9exb;kssion of ~ i r c / & f l ' s currrnr iuw. It states that rile o/~ehr?iic sum o/ull the out ($a junction ill on electric circuit is zero.' KirchholT's for node analysis in circuit theory. In Section 3-6 ~e stilted that charges introduced in the interior of a coiiductor will move to the cod uctor surface and redistribute themselves in such a way as to make p = 0 and E ='! inside h d e r equilibrium conditions. We arc now in a position to prove this statem&nt and to calculate the time it takes to reach a n equilibrium. Combining Ohm's I&, Eq. (5-8), with the equation of continuity and assuming a constant n, we have In a simple medium, V E'= p i t and Eq. (5-343 becomes The solution of Eq. (3-34) is where po is the initial charge density at r = 0. Both p and po can be functions of the space coordinates, add Eq. (5-36) says that the charge density at a given location will decrease with time cxponentially. An initial charge density po will decay to l/e or 36.8% of its vdue in a time equal to 1- .-. The time constant T is cdled the re/avo: 3,. : i r ~ -or a good conductor such as copper-a = 5.80 x 10' (S/m), r 2 so = 8.85 x lo-'' (F/m)- r equals 1.52 x 10-19(s), a very short time indeed. The transient time is so brief that for all practical ' This includes the currcnta of current generators at the junction. if any. An ideal a n e n t genemior s one whose current is independent of its terminal voltage. This implies that an ideal current source h n an Infinite internal resistance. I purposes p can be considered zero in the interior of a conductor-see Eq. (3-64) in Section 3-6. The relaxation time for a good insulator is not infinite, but can be hours or days. , 5-5 POWER DISSIPATION AND JOULE'S LAW In section 5-1 we indicated that under the influence of an electric field, conduction electrons in a conductor undergo a drift motion macroscopically. Microscopically these electrons collide with atoms on lattice sites. Energy is thus transmitted from the electric field to the atoms in thermal vibration. The work An done by it11 clectric field E in moving a charge q a distance A( is qE . (At). which co~.responds LO ;L power where u is the drift velocity. The tot;~l power delivered to all the charge carriers in a volume dv is which, by virtue of Eq. (5-7), is Thus the point function E ..I is a power dm,siry rtndcr rtcztdy-currcnt conditions. For a given volume V, the total electric power converted into heat is This is known as Joule's law. (Note that the SI unit for P is watt, not joule, which is the unit for energy or work.) Equation (5-39) is the corresponding point relationship. In a conductqr of a constant cross section, dv = ds dt, with d t measured in the direction J. Equation (5-40) can be written as P = S ~ E ~ / ~ J ~ ~ = V I , where I is the current in the conductor. Since V =,RI, we have (5-41) Equation (5-41) is, of course, the familiar expression for ohmic power representing the heat dissipated in resistance R per unit time. 1 I ) . 8 . i j3-64) in ' . . 1 t can be ' a . I nduction topically . ;ted from' n electric ) 3 power (5-38) ricrs in a ' P (5-39) ions. or' . - (5 -40) \~hich is tionship. ed in the . - . : I > I f I . a - P I 2. (-6 /BOUNDARY CONDI~~ONS F ~ R CURRENT DENSITY 183 , ~ & - I I . I A ; 5-6 BOUNDARY COND~TI~NS FOR CURRENT DENSITY r: I , L ' A . ' k ' When current .obliqiaely posses an interface4;between two media with difierent conductivities, the cuhent density vector changis both in direction and in magnitude. A set of boundary cohditions can be der~ved fur J in a way similar to that used in Section 3-9 for obt~ining the boundary conditions for D and E. The governing equations for steady currcpt density J in the absence of nonconservative energy sources are I, - dovernjng Equations for $tea& Current Density Differential Form lhtegral Form The divergence equation a the same as Eq. (5-31), and the curl equation is obtalned by combining Ohm'sd lau (J = oE) with V x E = 0. By applying Eqs. (5-42) a d (5-43) at the interface between two ohmic media with bonductivities o, and oi, we obtain the boundary conditions for the normal Bnd tangential components of J. Without actually bonstructing a pillbox at the interface as was done in Fig. 3-22. we know from Section 3-9 that the normal rowq>onont o f 'l.u diuergence1er.s oectarJe1~1 i s continuous. Hence, ham V J = 0, we have Equation (<-2vstatei that the ratio of rhr rangenrial componel~ts of J at tbvo sides of an interface is equal to the ratio of the conductivities. Example 5-2 Two cbnducting media with contluctivities o, and o , are separated by an interface, as shown in Fig. 5-4. The steady current density in medium 1 at point PI has a magnitude J, and makes an angle a, with the normal. Determine the magnitude and direction of the current density at point P i n medium 2. 184 STEADY ELECTRIC CURRENTS I 5 f? , . . - "- .
.
! 1 I i I Fig. 5-4 Boundary conditions at interface between two conducting media (Example 5-2). 1 . ' Solution: Using Eqs. (5 -44) and (5-49, we have I 1 ' J, cosa, = J, cosa, (5 -46) and (5 -47) a 2 4 sin sr, = alJ2 sin a,. Division of Eq. (5-47) by Eq. (5-46) yields -- (5 -48) tan a, If medium 1 is a much better conductor than medium 2 (a, > > a, or a,/a, 2 0), a, approaches zero and J, emerges almost perpendicular to the interface (normal to the surface of the good conductor). The magnitude of J, is J2 = J G J Z = J(J, sin a,)' + (J, cos a,), = [ ( : Jl sin c x l r + (Jl cos all2 I , ' , By examining Fig. 3-4, can you tell whether medium 1 or medium 2 is the better conductor? i . I r7 For a homogeneous conducting medium, the differential form of Eq. (5-43) 1 simplifies to t h v x J = O . (5-50) i s &'; . . / . - i ': 1 . . . From Section 2-10 we know that a curl-free vector field can be expressed as the r 1 G gradient of a scalar potential field. Let us write ., , J = -Vt//. (5-51) t i , 1 1 ; 1 .ce 5 -2). (5-46) r.5 -47) ( F 8 ) I + b), g2 lormal to 4 (5 -49) the bettef Jq. (5-43) n (5-50) ied as the (5-si) , Substitution of Eq. (4-51)~hto V . J = 0 yields a Laplace's equation in (I; that is, 1 A problem in steadGburrent flow can therefor& be solved by determining (1 (Aim) from Eq. (5-52), ct to"appropriate boundary conditions and then by finding J from its negative in exactly the same way as a problem in electrostatics is 4 . solved. As a matter 01 fact, ) and electrostatic dotential are simply related: (1 = GV. As indicated in similarity between electrostatic and steady-current electrolytic tank to map the potential distribution of difficult-to-solve boundary-value problems.' When a stcady cllrrent flows ilcross thc boundary hetwcen two dillcrent lossy ' dicleclrics (diclcctrics with per~nittivitics e l and r , ilnd finite conductivi[ics n , ;ind n2), tlic L : I I I ~ C I ~ ~ ~ : I I C O I I I ~ O I I C I ~ L 01' tI1c clcct~ic Iicld is C O I I ~ ~ I I L I O U S ;~crobs the intct-hce ;is usual; that is, E,, = El,, which is equivalent to Eq. (5-45). The normal component of the electric field, however, must simultaneously satisfy both Eq. (5-44) and Eq. (3 - 1 13). We require J l r l - .JZ,, - El,, = qZE2,, (5 53) Dln - D 2 n = P A + E ~ E I , ~ - E z E ~ ~ = P s , (5-54) where the reference &it normal is outward frqm medium 2. Hence, unlsss o,jo, = c2k,, a surface charbe must exist at the interface. From Eqs. (5-53) and (5-54). we find Again, if medium 2 is a much better conductor than medium 1 (a, > > a , or ~ , / o , - 0). Eq. (5-55) becomes abproximately ps = clEln = dl,, which is the same as Eq. (3 - 1 14). Example 5-3 An emf Y is, applied across a parallel-plate capacitor of area S. The space between the conductive plates is filled with two different lossy dielectrics of thicknesses dl and d,, permittivities cl and c,, and conductivities ol and G , respec- tively. Deteriiiine (a) the current density between the plates, (b) the electric field intensities in both dielectrics, and (c) the surface Charge densities on the plates and at the interface. ' See, for instance, E. Webet, Elecmrqagnefic Fields, Vol. I : Mapping o f Fields, pp. 187-193. John Wilcy and Sons, 1950. , i I . t . , 1 ; A : ; .
c , .I -. \'I; : . . "$..'.I' , \ . . . , / r, 1 1 '- Y I ' 5 - j , - ,[ Fig. 5-5 Parallel-plate I ; a capacitor with two lossy I r dielectrics (Example 5-3). I Solution: Refer to Fig. 5-5. a) The continuity of the normal component of J assures that the current densities and, therefore, the currents in both media are the same. By Kirchhoff's voltage law we have and (5-58) o l E l = ozE2. (5 -59) , Equation (5-59) comes from J , = J,. Solving Eqs. (5-58) and (5-59), ive obtain and f ' 6 - r -- Ps2 = -c2E2 = - E2019'- ( c / m 2 ) . (5-63) 62dl + ~ l d 2 I " i'! <- f . ,". The negative sign in Eq. (5-63) comes about because E2 and the outward normal I , < at the lower plate are in opposite directions. F I voltage (5-58) (5 -59) e obtain (5-621 ' normal 5-7, / RESISTANCE CALCULATIONS 187 \ ' I J . ~~uatiod{i-5$) can be used to find the ikface charge density at the interface of the dielectria $e have a . ! I ' c 7 From these resultl, we see that p,, # - p ,,,,, but that p,, + p,, + p,, = 0. In Example 5-3 wb encounter a situation w i r e both static charges and a steady current exist. As we dball see in Chapter 6, a steady current gives rise to a steady magnetic field. We hde, thcn, both a static electric field and a steady magnetic field. . They constitute an efdctromugnetostuticfield. The electrlc and magnetlc fields of an electromagnetostatic;fjeld are coupled through the constitutive relation J = aE of the conducting m e d i ~ h . In Section 3-10 we diicussed the procedure for finding the capacitance between t i o conductors separated by a dielectric medium. ~ h e s e conductors may be of arbitrary shapes, as was shown in Fig. 3-25, which is reproduced here as Fig. 5-6. In terms of electric field quarltitieb, the basic formula for capacitance can be written as $ s D ds $ s EE. ds .I c=-= - - I. (5-65) V - J , E . ~ < -J,E.&' A whcrc the surhcc i d ral in thc numerator is carried out over a aurhce enclosing the positive conductor, an the line integral in the denominator is from the negatlve (lower t potential) conductor tb the positive (higher potential) conductor (see Eq. 5-21). 4 1 I 1 d0 fig. 5-6 Two cnndiictors in r bssy v 1 2 dielectric medium. When the dielectric medium is lossy (having a small but nonzero a current will flow from the positive to the negative conductor and a current-density field will be established in the medium. Ohm's law, J = cE, ensures that the stream- lines for J and E will be the same in an isotropic medium. The resistance between the conductors is where the line and surface integrals are taken over the same L and S as those in Eq. (5-65). Comparison of Eqs. (5-65) and (5-66) shows the following interesting relationship: -1 Equation (5-67) holds if r and u of the medium have the same space dependence or if the medium is homogeneous (independent of space coordinates). In these cases, if the capacitance between two conductors is known, the resistance (or conductance) can be obtained directly from the €/a ratio without recomputation. Example 5-4 Find the leakage resistance per unit length (a) between the inner and outer conductors ofa coaxial cable that has an inner conductor of radius a, an outer conductor of inner radius b, and a medium with collductivity 0; and (b) ofa parallel- wire transmission line consisting of wires of radius a separated by a distance D in a medium with conductivity a . Solution a) The capacitance per unit length of a coaxial cable has been obtained from Eq. (3-126) in Example 3-16. Hence the leakage resistance per unit length is, from Eq. (5-67), I ( ) = n ) CI (Q/m). - (5-68) ' The conductance per unit length is GI = l/R,. ' T T' It A1gtl. Agtk . Ir a . -!
@ ! curre1 comp be cor const: h frinplr TI betwe~ 1. C i 2. 4. 3. Fi ge e ' I E 4. Fl P i-P ? - Fl It is ir; contlu hold. circurr he stream- ttween the (5 -66) lose in Eq. interesting .s/' 1denG or if :ase. he ncc) can be : inner and a, an outer T a parallel- m e D inla d from Eq. n. I , . ' , . i ; ' b) For the parallel-~ire transmission link, Eq;(4-47) in ~ x a m ~ l e 4-4 gives the . capacitance per Lihit length. ' I , I ' '# C. nE c; = (F/m). cosh- l (&- b Therefore, the ledcage resistance per unit length is, without further ado, 'I'hc cunductuncc per unit Icnytli is C;, -: 1/1( ,. It must be empHasized here that the resistance betweer1 the conductors for a length L of the coaxidl cable is R J t , not LR,; similarly, the leakage resistance of a length L of the parallel-wire transmission line is R'JL, not CR',. Do you know ~vhy.? In certain situations, lcctrostatic and steady-current problems are not ex:<cily ;m~logous, even wheh the gcotnctrical conljgurations are the same. This is becatise current tlow can be tonlin?d strictly within a conductor (which ius a wr.!, l o r q ~ ri compared to that of'the surrounding medium), whereas electric flux usually cannot be contained within EL dielectric slab of finite dimensions. The range of the dielectric constant of available materials is very limited (see Appendix B-31, and the flux- fringing around conductor edges makes the computation of capacitance less accurate. The procedure for computing the resistance of a piece of conducting material between specified eqbipotential surfaces (or terminals) is as follows: 1. Choose an apprdpriate coordmate system for the given geometry. 2. Assume a potential diRerence V, between conductor terminals. 3. Find elects~c ficld ultcnsity E w1t11111 I ~ C conductor. (lf thc m,iter~'ll 1 s homo- geneous, having a cousratlt conductivity, the gcncral method 1 s to mlve Lapl'lcc's equation V2V = 0 for V in the chosen coordinate system, and then obtam E = -VV.) 4. Find total currefit where S is the cross-sectional area over whir b : I flows. n ) 5. Find resistance R by taking the ratio V,,/I. . It is important to note that if the conducting material is inhomogeneous and if the conductivity is a function of space coordinates, Laplace's equation for V does not hold. Can you expfhin why and indicate how E can be determined under these circumstances? L , J, < b # , . ;r-, ti.'- . . a . ., . When the given geometry is such that J can be determined easily from a total ." , + .: r current I, we may start the solution by assuming an I. From I, J and E = J b are . 4 ; found. Then the potential difference V o is determined from the relation , , where the integration is from the low-potential terminal to the high-potential terminal. The resistance R = Vo/I is independent of the assumed I, which will be canceled in the process. I - Example 5-5 A conducting material of uniform thickness h and conductivity u , h has the shape of a quarter of a flat circular washer, with inner radius a and outer radius b, as shown in Fig. 5-7. Determine the resistance between the end faces. I i Solution: Obviously the appropriate coordinate system to use for thi~'~roblem is f the cylindrical coordinate system. Following the foregoing procedure, we first assume a potential difference V, between the end faces, say V = 0 . on the end face at y = 0, and V = V o on the end face at x = 0. We are to solve Lapluce'Sequ:lfion in V subject to the following boundary conditions: Since potential V is a function of 4 only, Laplace's equation in cylindrical coordinates simplifies to d2V -- d4' - O' (5 -7 1) The general solution of Eq. (5-71) is V= c14 + C2, which, upon using the boundary conditions in Eqs. (5-70a) and (5-70b), becomes .-" ,W<! . . - $ . - . Fig. 5-7 A quarter of a flat circular 0 X washer (Example 5-5). terminal. lceled in ! . I I (. , < l ctivity a i' ~d outer ' :es. ~blem is assume it y = 0, ' aubject P 15-- (5-70b) .dinates (5 -7 1 3 :comes (5,-72) n I \ . i I : RFVlEW QUESTIONS 191 . . . . The total current, I b n be found by integrating J over the 4 = n/2 surface at which ds = - a6h dr. We have J Therefore, Note that, for this problem, it is not convenient to begin by assuming a total current I because it is not obvious how J varies with r for a given I without' J, E and Vo cannot be ddtermincd. REVIEW QUESTIONS I ? R.5-I Explam the dikerewe between co~duct~on and convect~on currents. R.5-2 Explain the dbera:ion of on electrolytic tank. In what ways do electrolytic currents differ from conduction and convection currents? R.5-3 What is the point f x m for Ohm's law? R.5-4 Define conductivity. What is ~ t s SI unit? R.5-5 Why does the Rsisiance fornlula in Eq. (5-13) require that the material be homogeneous and straight and that it hav: a uniform cross section? R.5-6 Prove Eqs. (5-15) and (5-16b). R.5-7 Define electroHtotive Jorce in words. R.5-8 What is the dikrencc between impressed and electrostatic field intensities? R5-9 ~t'a~irchhoff's vdtage law in words. R5-10 What are the charactiristics of an ideal voltage source? 115-11 Can the currehts in different branches (resistors) of a closed loop in an electric network flow in opposite directions? Explain. R.5-12 .What is the pHpica1 significance of the equation of continuity? 192 STEADY FLECTRIC CURRENTS I 5 R5-13 State Kirchhoff's current law in words. R5-14 What are the characteristics of an ideal current source? R5-15 Define relaxition time. R.5-16 In what ways should Eq. (5-34) be modified when a is a function of space coordinates? R.5-17 State Joule's law. Express the power dissipated in a volume a) in terms of E and a , b) in terms of J and a . R5-18 Does the relation V x J = 0 hold in a medium whose conductivity is not constant? Explain. I R.5-19 What arc the boundary condilions of the normal and tangential components of steady i current at the interface of two media ~ 5 t h different conductivities? R.5-20 What is the basis of using an electrolytic tank to map the potential distribution of elec- i [rcwttic hound;~ry-v:dw problen~s? ' '-\ H.5-21 WIi;tt is tllc rclation bclwcen the rcsis[;~ncc and the capacitace fortil~d by tivu con- --. ductors immersed in a lossy dielectric medium that has permittivity E and conductivity n? R5-22 Under \ ~ h i ~ t situations will the relati011 between Rand C in R.5-21 be only approrimatcly correct? Give a specific example. PROBLEMS P S I Starting with Ohm's law as expressed in E q (5-12) applied to a resistor of length t. conductivity a, and uniform cross-section S, verify the point form of Ohm's law represented by i Eq. (5-8). P.5-2 A long, round wire of radius a and conductivity 0 is coated with a material of conduc- tivity 0. la. a) What must be the thickness of the coating so that the resistance per unit length of the uncoated wire is reduced bv SO%? b) Assuming a total currenf I in the coated wire, find J and E in both the core and the coating material. Fig. 5-8 A network problem (Problem PJ-3). 1 t constant? J tts ut' steady oy two coh- it? a P of length C, presented by length bf the core and the r PROBLEMS 193 i , I P.5-3 Find the current h d the heat dissipated in each of the five resistors In the network shown " A . - in Fig. 5-8 if 5 I 4 J R l = 4 (a), ; R 2 = 20 (Q), R, = 30 (Q), R, = 8 (Q), R 5 = 10 (a), and if the source is ad idebl D C voltage generator of 0.7. (V) with its positive polarity at terminal 1. What is the total resistdnce seen by the source'at terrhinal pair 1-2? P .
5 -4 Solve problem PS-3, assumlng the source IS hd ideal current generator that supphes a direct current of 0.7 (A) out of terminal 1. q 7 P.5-5 Lightning strike&,a lossy dielectric sphere-€ 1.2 E,, a = 10 (S/m)-of radius 0.1 (m) at time t = 0, depositing pnifo;mly in the sphere a totkl charge 1 (mC). Determine, for all r, a) thc electric ficld htensiiy both inside and outsihc the sphere, 1)) the current dcnslty in tlie sphere. P.5-6 Ilcfer to I'roblcm 1'5-5. a) Calculate the t h e it takes for the charge denshy in the sphere to diminish to 1 7 , of its initial value. 1 1 b) Calculate the change in the electrostatic energy stored in the sphere as the charge density diminishes from the nitial value to 1 % of its value. What happens to this energy'? . c) Determine the e1ect:osratic energy stored in the space outside the sphere. Does this energy change with time? P.5-7 A DC voltage af 6 (V) applied to the ends of 1 (km) ?fa conducting wire of 0.5 (mm) radius results in a curred of 116 (A), Find a) the conductivity of the wire, . b) the electric field hteilsity in the wire, c) the power dissiphted in the wire. a) Draw the eq~ivhlent circuit of the two-layer, barallel-plate capacitor with lossy dielec- trics, and identify the magnitude of each component. b) Determine the Ijbwer dissipated in the capacitdr. P .
5 -9 An emf Y' is appliec across a cylindrical capacitor of length L. The radii of the inner and outer conductors are a a i d 11 ruspectively. The space between thc conductors is fillcd with two different lossy dielectric.; having, respectively, permittivity E , and conductivity a, in the region a < r < c, and permittivity €2 and conductivity a, in the region c < r < h. Determine l a) the current density in each region, b) the surface charge densities on the inner and outer conductors and at the interface between the two dielectrics. ---. P5-10 Rekr to the flat quaricr-circular washer In Example 5-5 and Fig. 5-7. Find the resistance between the curved sldes. P .
5 -1 1 Determine the resi~tance between concenttic spherical surfaces of radu Rl and R2 (R, < R,), assuming that a material of conductivity a = a,(l + k/R) fills the space between them. (Note: Laplace's equation for V does not apply here.) 194 - STEADY ELECTRIC CURRENTS 1 5 , t., 1 /,.& > . P5-12 A homogeneous material of uniform conductivity t ; is'shaped like a truncated conical block and defined in spherical coordinates by + R , I R S R , and 0 1 0 ~ 8 , . Determine the resistance between the R = R, and R = R, surfaces. P.5-13 Redo problem P.5-12, asguming that the truncated conical block is composed of an inhomogeneous material with a nonuniform conductivity a(R) = aoRl/R, where R, 5 R I R,. I P5-14 Two conducting spheres of radii b, and b, that have a very high conductivity are immersed i A % in a poorly conducting medium (for example, they are buried very deep in the ground) of con- ductivity a and permittivity e. The distance, d, between the spheres is very large compared with the radii. Determine the resistance between the conducting spheres. H i n t : Find the capacitance between the spheres by following the procedure in Section 3-10 and using Eq. (5-67). / ' I P.5-15 Justlfy the statement that the steady-current problem associated with a conductor t buried in a poorly conducting medium near a plane boundary with air, as shown in Fig. 5-9(a), can be replaced by that of the conductor and ~ t s image, both immersed in the poorli conducting ! medium as shown in Fig. 5-9(b). Q Boundary removed o = o ----------- I u d Fig. 5-9 Steady i 0 current problem with a plane boundary -a A.e .. . -. .-.. .
a .
4 , .
.
.
.
. . . I I .
U (Problem P.5-15). I ! (a) Conductor in a poorly (b) Image conductor in conducting. conducting medium near medium replacing the a plane boundary. plane boundary. P5-16 A ground connection is made by burying a hemispherical conductor of radius 25 (mm) in the earth with its base up, as shown in Fig. 5-10. Assuming the earth conductivity to be S/m, find the resistance of the conductor to far-away points in the ground. Fig. 5-10 Hemispherical conductor in ground (Problem P .
5 -1 6 ) . P .
5 -1 7 Assume a rectangular conducting sheet of conductivity u, width a, and height b. A - , potential difference V, is applied to the side edges, as shown in Fig. 5-11. Find .-.. a) the potential distribution b) the current density everywhere within. the sheet. Hint: Solve Laplace's equation in Cartesian coordinates subject to appropriate boundary conditions. 1 i sed of an i R 5 R 1 . ~mmerscd d) of con- lared with ' I . spacitance 1 . conductor -ig. 5-9(4, zonducting /-' :ad y em with a lry -15). %us 25 (mm) ~ctivity to be f- d heirL+ b. A $ . ' s equation in j r' v = 0 -- = an I I Fig. 5-11 A conducting sheet 1 - a ---7 (Problem P.5-17). - , P .
5 -1 8 A uniform curreht density J = a,Jo flows in a k r y large block of homogeneous material of conductivity a. A hole of radius b is drilled in the material. Assuming no var~ation in the =-direction: find the nevJ current density J' in the cdllducting material. H~nt: Solve Laplace's equation in cylindrical &ordinates and note that V approaches -(Jor/a)cos$ as r -+ a. 6 / Static M ngnetic Fields 6-1 INTRODUCTION f In Chapter 3 we dealt with static electric fields caused by electric charges at rest. We saw that electric field intensity E is the only fundamental vector fikld quantity required for the study of electrostatics in free space. In a material medium, it is con- venient to define a second vector field quantity, the electricJlx density D, to account for the effect of polarization. The following two equations'form the basis of the electrostntic model : V .U=,, (6- 1) The electrical property of the medium determines the relation between D and E. If the medium is linear and isotropic, we have the simple cortstit~itiue rclurion D = EE. ' When a small test charge q is placed in an electric field E, it experiences an electric force F,, which is a function of the position of q. We have When the test charge is in motion in a magnetic field (to be defined presently), experi- ments show that it experiences another force, F,,,, which has the following character- istics: (1) The magnitude 'of I ?
, is proportional to q ; (2) the dircction of I;, at any point is at right angles to the velocity vector of the test charge as well as to a fixed direction at that point; and (3) the magnitude of F, is also proportional to the com- ponent of the velocity at right angles to this fixed direction. The force F, is a magnetic force; it cannot be expressed in terms of E or D. The characteristics of F, can be described by defining a new vector field quantity, the magnetic Jux density B, that specifies both the fixed direction and the constant of proportionality. In SI units, the magnetic force can be expressed as ?
: : J $ . . , (6-4) 6 -2 MAGE r s at rest. i quantity . it is con- o account .sis @he (6-1) \ .2) ) and E. If )n D = EE. :riences an tly), experi- , : character- I ' F, at any s to a fixed to the com- 4 a magnetic : F, can be 3sityFthat ;I u r . , the . 1 I: I where u (m/sJ is thk vdocity vector, and B is mkasured in webers per square meter (Wb/m2) or t e d d 0,' The total eleetrohag$etie force on a charge q is, then, F = F e + F m ; t h a t i s ,; (6-5) which is called Lotdm's jhrce equation. Its kilidity has been unquestionably cstabiishcd by experinlcnts. We may tonbider Fe/q for a small q as the definition for electric field intensity ,k (as we did in Eq. 3-2) and Fm/q = u x B as the defining relation for magneiic tux density B. Alternative y, we may consider Lorentz's force I equation as a [undadental postulate of our elktromagnetic model; it cannot be derived from ofher postulates, We begin the stlldy of static magnetlc fields in free space by two postulates specifying the divergedce and the curl of B. From the solenoidal character of B. a vector magnetic potential is defined, which is shown to obey a vector Poisson's equation. Neit we deiive the Biot-Savart law, hich can be used to determine the magnetic field of a curf'eni-carrying circuit. The postulated curl relat~on leads d~rectly to Ampire's circuital law which is particularly useful when symmetry exists. The macroscopic bffe-t of magnetic materials in a magnetic field can be studled by defining a magnetihation vector. Here we introduce a fourth vector field quantity, the magnetic field intensity H. From the relation between B and H, we define the permeability of the nl$terial, following which .we discuss magnetic clrcults and the microscopic behaviorlof magnetic materials. We then examine the boundary con- ditions of B and H at the ~pterface of two different magnetic media; self- and mutual inductances; and magfieti; energy, forces, and torques. , 6-2 FUNDAMENTAL POST~JLCTES OF MAGNETOSTATICS IN FRE$ SPACE To study magnetostatjcs (;;teady magnetic fields) in free space, we need only consider the magnetic flux dendity iector, B. The two fundamental postulates that specify the divergence and the curl of B in free space are - ' One weber per square meter or one I& yuals la4 yuss in CCS units. The earth rnagnetlc field IS about 4 gauss or 0.5 x lo-' T. (t, weber is the same as a volt-second.) 198 STATIC MAGNETIC FIELDS 1 6 In Eq. (6-7), po is the permeability of free space / (see Eq. 1-9),'and J is the current density. Since the divergence of the curl of any vector field is zero (see Eq. 2- l37), we obtain from Eq. (6-7) which is consistent with Eq. (5-31) for steady currents. Comparison of Eq. (6-6) with the analogous equation for electrostatics in free space, V . E = p/eO (Eq. 3-4), leads us to conclude that there is no magnetic analogue for electric charge density p. Taking the volume integral of Eq. (6-6) and applying divergence theorem, we have . where the surface integral is carried out over the bounding surface of an arbitrary volume. Comparing Eq. (6-8) with Eq. (3-7). we again de$the existence of isolated magnetic charges. There are no magnetic Jow sources, and the magnetic Jux lines always close upon themeloes. Equation (6-5) is also referred to as an expression for the law of conservation of rnagnetic Jux, because it states that the total outward magnetic flux through any closed surface is zero. The traditional designation of north and south poles in a permanent bar magnet does not imply that an isolated positive magnetic charge exists at the north pole and a corresponding amount of isolated negative magnetic charge exists at the south pole. Consider the bar magnet with north and south poles in Fig. 6- l(a). If this magnet is cut into two segments, new south and north poles appear and we have two shorter magnets as in Fig. 6-l(b). If each of the two shorter magnets is cut again into two segments, we have four magnets, each with a north pole and a south pole as in Fig. 6-1(E). This process could be continued until the magnets are of atomic dimensions; but each infinitesimally small magnet would still have a north pole and a south pole. Successive division ics in free1 analogue ~ P P ~ Y ~ Q -(6 -8) arb~trary >f ~srficd flux ..,ies ession for I or .d ar magnet 1 polc and outh pole. magnct is v o shorter I into two as in Fig. msnsions; outh pole. n flux lines follow er end outside the magnet, and then The designation of north and south ends of a bar magnet freely and south directions. in Eq. (6-7) can be obtained by integrating Stokes's theorem. We have " , . i ( V x B ) - d ~ = ~ , , f J.ds . . or ! s B - dP= poi, (6-9) I whcrc thc path C for !he line ihtcgral ia thc contour boundlng the iurlicc S, m d I is the total current t S The sense o f tracing C and the dl~ection of current flow follow the Equation (6-9) is a form of Ampere', c~rcs~tnl lniv, which states that the ~irculation o f the lnayrietic flex doisltj! in jrrc space nroiinii oiijl I closed path is equal to po tunes the rotol current jawing through die utjace bounded by the path. -4mpkre9J circuital law is very usefbl in determining the maqnrtlc flux density B caused by LI cuirent I when there is a closed path C around the current - such that the magnitdtle cf R is constant over the path. Thc followihy is a'sunrmary of the two fundamental postulates of magnetostat~cs in free space: Free Space Example 6 4 An infinitely long, straight conductor with a circular cross section --.
. of radius b carrles a 3teady current I. Determine the magnetic flux density both inside and outside the contiuctor. Solution: First we note tliat this is a problem with cylindrical symmetry and that Ampkre's circuital law can beused to advantage. If we align the conductor along the "axis, the magnetic fldx density B will be &directed and will be constant along any t circular path around the z-axis. Figure 6-2 shows a cross section of the conductor ! and the two circular paths of integration. C , and C,, inside and outside, respectively, the current-carrying conductor. Note again that the directions of C , and C , and the : direction of I follow the right-hand rule. (When the fingers of the right hand follow the directions of C , and C,, the thumb of the right hand points to the direction of I.) a) Inside the cortductor: -1 . B l = a g B g l , de=a,,,r,drl, $c, B , - d f = So2" B,,r1 dg = 2nrlBgi. The current through the area enclosed by C , is Therefore, from Ampkre's circuital law, f liorlI rl 5 b. t Bl = .
, B , l = a , j -& . (6-10) 1 C b) Outside the conductor: , ! c B2 = a,Bg2, d t = a,r2 d4 $B2-dP=Znr,B,,. Path C2 outside the conductor encloses the total current I. Henc? Examination of Eqs. (6-10) and (6-1 1) reveals that the magnitude of B increases i 1 - linearly with r , from 0 until r, = h, after which it decreases inversely with r,. Example 6-2 Determine the magnetic flux density inside a closely wound toroidal coil wilh an air core having N turt~s a n d cilrryillg ; I c~trrcI11 I. Tllc 101.0itl I I ~ I S a lncilrl radius b and the radius of each turn is u. i I 1 4 . 1 . I . conductor :spectively, 4 1 C2 dnd the dnd follow aion of I.) n 1 1 (6- 10) i 1 I i r'. (6-1 1) 3 Increases -I r 2 . I d toroidal as a mean i I 1 i 1 1 6-2 1 FUI: AMENTAL POSTULATES OF'MAG~~ETOSTATICS IN :< FREE SPACE Fig. 6-3 Current-carrying toroidal coil (Example 6-2). I Solution: Figure 6-3 drpicts the geometry of this problem. Cylindrical symmetry ensures that B has only a +-component and is cbnstant along any circular path about the axis of the toroid; We constructa circular contour C with radius r as shown. For (b - a) < r < b + a, kq. (6-9) leads diredtly to $ B - d~ = 2nr~,= ~ , N I , where we have assumed that the toroid has an air core with permeability p , . Therefore, I 'oNI, (b - o) < r < (b + a). - B = asB4 = a4 - (6-12) 2zr It is apparent that = 0 for r < (b - a) and r > (b + a), since the net total current enclosed by a contour constructed in these two regions is zero. Example 6-3 Determine the magnetic flux density inside an infinitely long solenoid with an air core having n closely wound turns per unit length and carrying a current I. Solution: This problem can be s'olved in two ways. a) As n dire&application ofAmpere'scircuita1 law. It is clear that there is no magnetic field outside of the holenoid. To determine the B-field inside we construct a rectangular contour C of length L that is pdttially inside and partially outside , the solenoid. By reason of symmetry, the B-field inside must be parallel to the axis. Applying Ampkre's circuital law,,we have 202 STATIC MAGNETIC FIELDS I 6 la Fig. 6-4 Current-carrying long solenoid (Example 6-3). The direction of B goes from right to left, conforming to the right-hand rule with respect to the direction of tjle current I in the solenoid, as indicated in Fig. 6-4. b) As a speciul cux o f toroid. Thc straight solenoid may be regarded .as a special case of the toroidal coil in Example 6-2 with an infinite radius (b + cc). In such a case, the dimensions of the cross section of the core are very small compared with b, and the magnetic flux density inside the core isa~proximately constant. We have, from Eq. (6-12). which is;the same as Eq. (6-13). The &directed B in Fig. 6-2 now goes from right to left, as was shown in Fig. 6-3. 6-3 VECTOR MAGNETIC POTENTIAL The divergence-free postulate of B in Eq. (6-6), V - B = 0, assures that B is solenoidal. As a consequence, B can be expressed as the curl of another vector field, say A, su'ch that (see Identity 1 1 , Eq. (2-137), in Section 2-10) The vector field A so defined is called the vector magnetic potential. Its SI unit is weber per meter (Wb/m). Thus, if we can find A of a current distribution, B can be obtained from A by a differential (or curl) operation. This is quite similar to the introduction of the scalar electric potential V for the curl-free E in electrostatics (Section 3-S), and the obtaining of E from the rclation 15 = - V V . Ilowcvcr, the definition of a vector requires the specibtion of both its curl and its divergence. Hence Eq. (6-14) alone is not sufficient to define A; we must still specify its divergence. How do we choose V A? Before we answer this question, let us take the curl of B in Eq. (6-14) and substitute it in Eq. (6-7). We have rule with Fig. 6-4. special 111 such :mpared : o n s p t . les from c~loidal. A. such unit is can be to the Jsq- .cr, ~..e rgence. rgel,. curl of I , 6-3. I V&TOR MAGNETIC POTENTIAL 203 I Here we digress t6 inhoduce a formula for the curl curl of a vector: . f v 2 ~ = v ( v . l i ) - V X V X A . (6-16b) Equation (6-16a)' or (6-16b) can be regarded as the definition of V'A, the Laplacian -, " of A. For Cartesian &ordinates, it can be rekdily verified by direct substitution (Problem P.6-10) that , 2 V2-4 = ax VIA, + a,,'v2~, + az VIA=. ; .
! (6-17) Thus for Cartesian coordinates, the ~ a ~ l a c i a n of a vector field A 1 s another vector field whose componehts are the Laplaclan (the divergence of the gmdlent) of the corresponding components of A. This, howevbr, is not true for other coordinate systems. We now expand V x V x A in Eq. (6-15) according to Eq. (6-16a) iind obtun V(V.A) - V% = poJ. (6-13) Wlth the purpose oidmpl~fying Eq. (6-18) ta the greatest extent possible, we choose: / V . ~ = O , / and Eq. (6- 18) becomes This is a vector IJoissn's quutiotz. In Cartesian Coordinates, Eq. (6-20) is equivalent to three scalar Poisson's zquations: V'A ,. = - 11, J,, , (6-21b) V2A, = -poJ,. (6-21~) Each of these three equations is mathematically the same as the Poisson's equation, Eq. (4-6), in electrostatics. In free space, the equation ' Equation (6-Iba) can also be obtained heuristl~llly from the vector triple product formula in E q (2-20) by $onsidering the del operator, V, a vector: V X (V x ~j = V(V . A) - ( V .
. V)A = V(V . A) - V2A. : Equation (6-19) holds 1 0 ; static magnetic fields. Modification is necessary for time-varylnp electm- magnetic fields (see Eq. 7-46), 204 STATIC MAGNETIC FIELDS 1 6 .,- s , has a particular solution (see Eq. 3-56), Hence the solution for Eq. (6-21a) is We can write similar solutions for A, and A:. Combining the three components, we have the solution for Eq. (6-20): Equation (6-22) enables us to find the vector magnetic potential A from the volume current density J. The magnetic flux density B can then beobtained from V x A by differentiation, in a way similar to that of obtaining the static electric field E from - vv. Vector potential A relates to the magnetic flux @ through a given area S that is bounded by contour C in a simple way: @ = B - d s . b (6 -23) The SI unit for magnetic flux is weber (Wb), which is equivalent to tesla-square meter (T.m2). Using Eq. (6-14) and Stokes's theorem, we have ( D = 1 7 ( V x A). ds = $ A . dP (Wb). (6-24) 6-4 BIOT-SAVART'S LAW AND APPLICATIONS In many applications we are interested in determining the magnetic field due to a current-carrying circuit. For a thin wire with cross-sectional area S, dv' equals S dd', and the current flow is entirely along the wire. We have J dv' = J S dP' = I dt', (6-25) and Eq. (6-22) becomes where a circle has been put on the integral sign because the current I must flow in ments, we (6- 22) l i . .olume - ' " "bb' m j t' is (6-23) !d-4 luare & (6-24) due to a als S dL', (Yv) (6-20) : flow in ' L ' ', 6-4 1 6101-Si VART'S LAW AND APPLICATIONS 205 v .' 4 .', < 4 ' i: a closed path,' whicn is designated C'. The mdgnetic flux density is then , B = V x A = V x - 5 [kt%, % " I (6-27) I It is very important 10 note in Eq. (6-27) that the unpiinwf curl operation implies differentiations with respect to the space coordjnates of the field point, and that the integral operation is'hith :iespect to the brimed source eoordmates. The integrand in Eq. (6-27) can be dipanrted into two terms by using the follow~ng identity (see Problem P.2-26): ' ' i X ( f G ) = f V x G + ( V f ) x G . (6-38) Wc liavc, with f = ILH and G = df', I 3 = $ [LV x dP1 + /In: ( ' R 16-29) Now, slnce the unprlhed .ind primed coordinates are independent. V x dPf equals 0 . and the first term on the rizht side oSEq. (0-29) vanishes. The d~stance R IS me~buied from dt" at (x', y', z') to the field point ilt (x, y. : ) . Thus we have i = [(x - .q2 + ( y - yi)' + ( z - ,)?I - 1 2; R - - - ax(" - x') + a,(y - y') + a,(? - 3') [(x - 7' + 0' - y')' + ( z - z')']~'~ R -= 1 R j - -a, 7, R (6-30) where a, is the unit vector directed jmn the source point to the field point. Substituting Eq. (6-30) in Eq. (6-19), me get 4 R ce (6- 3 1) Equation (6>N).is known as Biot-Souart's law. It is a formula for determining B caused by a current I in a closed path C', and is obtained by taking the curl of A in Eq. (6-26). Sometimes it is convenient to write Eq. (6-31) in two steps. ' w e are now dealing with direct (non-time-varying).currehts that give rise to steady magnetlc fields. Circuits containing time-vdrying sources may send time-varying currents along an open wire and deposit charges at its ends. Antennas are examples. 206 S T A ~ C MAGNETIC FIELDS 1 6 t , (6- 32) with which is the magnetic flux density due to a current element I dt". An alternative and sometimes more convenient form for Eq. (6-33a) is Solution: Currents exist only in closed circuits. Hence the wire in the present problem must be a part of a current-carrying loop with several straight sides. Since we do not know the rest of the circuit, Ampere's circuital law cannot be used to advantage. Refer to Fig. 6-5. The current-carrying line segment is aligned with the ;-axis. A typical element on the wire is dP' = a, dz'. The cylindrical coordinates of the field point P are (r, 0, 0). Fig. 6-5 Current-carrying straight wire (Example 6-4). /(,)I tic' x N = ( 3 ) m i I Comparison of Eq. (6-31) with Eq. (6-9) will reveal that Biot-Savart law is, in general. more difficult to apply than Ampere's circuital law. However. AmpCre's circuital law is not useful for determining from I in a circuit if a closed path cannot -. be found over which B has a constant magnitude. Example 6-4 A direct current I flows in a straight wire of length 2L. Find the magnetic Hux density B at a point located at a distance r from the wire in the bisecting plane: (a) by determining the vector magnetic potential A first, and (b) by applying Biot-Savart's law. (6-32) (6-33a) itive and (0--3ib) t i v i\, in '-FS CJ. c IhL .
; :lbeCilllg ipplying ~ o b l e m , : do not vantage. -axis. A P I ! 6-4 I B I O T - ~ V A ~ T ' S LAW AND APPLICATIONS 207 r - a) By jinding B frod V x A. Substituting R , / - into Eq. (6-261, we have I L dr' i A = ~ Z K J - ~ , / - Cylindrical symmetry around the wire assures that BA,/d$ = 0. Thus, When r < < L, Eq. (6-35) reduces to which is the expression for B at a point located at a distance r from an infinitely long, straight wire carrying current I. b) By applying Biot-Gauart's law. ~ r o ; Fig. 6-5, we see that the distance vector from the source ekment dz' to the field point P is R = a,r - a:=' ; + dt" x R = a= dz' x (arr - aZr1) = a,r dz'. Substitution in Eci, (6-33b) gives . which is the same as Eq. (6-35). Example 6-5 Find the magnetic flux density at the center of a square loop. with side w carrying a direct current I. . Fig. 6-6 Square loop carrying Y current I (Example 6-5). a Solution: Assume the loop lies in the xy-plane, as shown in Fig. 6-6. The magnetic flux density at the center of the square loop is equal to fo#,r times that caused by a single side of length w. We have, by setting L = r = w/2 in gq. (6-35). A where the direction of B and that of the current in the loop follow the right-hand rule. Example 6-6 Find the magnetic flux.density at a point-&he axis of a circular loop of radius b that carries a direct current I. Solution: We apply Biot-Savart's law to the circular loop shown in Fig. 6-7. Again it is important to remember that R is the vector from the source element dl" to the field point P. We have Y Fig. 6-7 A circular loop carrying x currcnt I (Example 6-6). nagnetic \cd by a (6-37) nd rule. r circular 7 . .ent de' . 'i < , . 8-5 1 THE MAGNETIC DIPOLZ 209 -: , , i ! ' Because of c$ihdr!cal symmetry, it is easy to see that the a,-component is canceled by the contribution ohthe element located dianletrically opposite to dt", so we need only consider the a,-cbmponent of this cross product. We write, fford EQ. (6-jl), 6-5 THE MAGNETIC DIPOLE Wc begin this scction with an cxamplc. Example 6-7 Find the magnetic flux density at a distant point of a small circular loop of radius b that carries current I. I Solution: It is appaIent from the statement of the problem that we are inter- ested in determining B at a point whose distahce, R, from the center of the loop satisfies the relation R > > b; that being the case, we may make certain simplifyin- approximations. We select the centgt of the loop to be the origin of spherical coordinates. as shown in Fig. 6-8. The soutb coordinates are primed. We first find the vector magnetic potential A and then determine B by V x A . A small circular (Example 6-7). loop carrying 210 STATIC MAGNETIC FIELDS I 6 Equation (6-39) is the same as Eq. (6-26), except for one importart point: R in ! Eq. (6-26) denotes the distance between the source element d t ? at P' and the field point P; but it must be replaced by R , in accordance with the notation in Fig. 6-8. Because of symmetry, the magnetic field is obviously independent of the angle $ of , the field point. We pick P(R, 0, n/2) in the yz-planc. Another point of importance is that a , . at de' is not the same as a, at point P In fact, a, at P, shown in Fig. 6-8 is -a,, and I dl" = (-a, sin 4' + a, cos 4')b d@. (6 - 40) I i For every I dP' there is another symmetrically located differential current element on the other side of the y-axis that will contribute an equal amount to A in the -a, direction, but will cancel the contribution of I de' in the a, direction. Equation (6-39) can be written as d4' 1 or The law of cosines applied to the traingle OPP' gives R f = R~ + b2 - 2bR cos +, where R cos + is the projection of R on the radius OP', which is the same as the , projection of OP" (OP" = R sin 6) on OP'. Hence, R f = R2 + b2 - 2hR sin 0 sin 4' and 1 1 b2 2b - = - (1 + -T - - sin 0 sin 4' Rl R R R >-'I2 When R2 > > b2, b 2 / ~ ' can be neglected in comparison with 1. 1 b g - (1 + - sin 0 sin 4' R R Substitution of Ey. (G-42) in 13q. (6. 41) yiclcls b A = a, Jni2 (1 + - sin 6 sin 4' 2nR R , oint: R in d the field 1 Fig. 6-8. angle 4 of it point P. (6 - 40) lement on ! the -a, 1011 (0 .39) r 1) TL' ,is the (6-42) 0 (6-43) C, I . , + 6-5 / THE MAGNETIC DIPOLE 211 I The magnetic flux density is B = V x A. Equation (2-127) can be used to find /f lb2 B=O 4R3 (a, 2 cos 0 + a, sin O), (6-43) , which is our answer. At this point we recognize the sin~ilarity between Eq. (6-44) and the expression for the electric field intensity in the far field of an electrostatic dipole as given in Eq. (3-49). To exanline the similarity further, we rearrange Eq. (6-43) as is defined as the rniy(t1eti~ dipole inotncric, whlch is a vector whose ma, onltude 1 s the product of the current in m d the area of the loop and whose direction is the direction of the thumb as the fingars of the right hand follow the direction of the current. Comparison of E q (6-45) with the expression for the scalar electric potential of an electric dipole in Eq. (3-48), reveals that, for the fwo cases, A is analogous to V. We call a small current-carrying , loop a magnetic dipole. The analogous quantities are as follows: . - Electric Dipole Mngnet~c Dipole , In a similar manner we can also rewrite Eq. (6-44) as 212 STATIC MAGNETIC FIELDS 1 6 Except for the change of p to m, Eq. (6-48) has the same form as Eq. (3-49) does for the expression for E at a distant point of an electric dipole. Hence, the magnetic flux lines of a magnetic dipole lying in the xy-plane will have the same form as that of the electric field lines of an electric dipole positioned along the z-axis. These lines have been sketched as dashed lines in Fig. 3- 14. One essential difference is that the electric field lines of an electric dipole start from the positive charge + q and terminate on the negative charge -q, whereas the magnetic flux lines close upon themselves.+ 6-5.1 Scalar Magnetic Potential In a current-free region J = 0, Eq. (6-7) becomes The magnetic flux density B is tfxm curl-free and can bc exprcssed as thc gradient of a scalar field. Let B = -1'0 VT/;,,, (6-50) where I / ; , is called the scular. r~ltrgrretic polc~r~ti~il (esprcssed il~iqmperes). The negative sign in Eq. (6-50) is conventional (see the definition of the scalar electric potential V in Eq. 3-38), and the permeability of free space p, is simply a proportionality constant. Analogous to Eq. (3-40), we can write the scalar magnetic potential differ- ence between two points. P2 and PI, in free space as If there were magnetic charges with a volume density p, (A/m2) in a volume V', we would be able to find V, from The magnetic flux density B could then be determined from Eq. (6-50). However, isolatcd magnetic charges have never been observed experimentally; they must be considered fictitious. Nevertheless, the consideration of fictitious magnetic charges in a mathematical (not physical) model is expedient both to the discussion of some magnetostatic relations in terms of our knowledge of electrostatics and to the estab- lishment of a bridge between the traditional magnetic-pole viewpoint of magnetism and the concept of microscopic circulating currents as sources of magnetism. The magnetic field of a small bar magnet is the same as that of a magnetic dipole. This can bc vcrilied expcrimcntally by obscrving'thc contours of iron Jilings around a magnet. The traditional understanding is that the ends (the north and south poles) QT ' Although the magnetic dipole in Example 6-7 was taken to be a circular loop, it can be shown (Problem P.6-13) that the same express~ons-Eqs. (6-45) and (6-48)-are obtalned when the loop has a rectangular shape, w~th m = IS. as given in Eq. (6-46). does for ~ a l c f ux .
; that of :se lines that the :rminate ~selves.' i o do) dicnt of (6- 50) ; e g ; P : ; t e n d :on I ; (i~llcr- (6- 5 1) ume V', (6-52) - owever, xust be xges in ,f some e estab- getism . n d i ~ , 4. are\;- ' h POL-, Problem &kngular 6-6 1 MAGNET' ITION AND EQUIVALENT CURRENT DENSITIES 213 .
$ . a of a magnet are the location of, respectively, positive and negative magnetic charges. For a bar mhgnet', the fictitious magnetic charges + q,, and - q,, are assumed to be separated by a pistance d and to form an equivalent magnetic dipole of moment , ! m = q ; d = a,lS. (6-53) The scalar magnetic potential v,, caused by this magnetic dipole can then be found by following the procedure used in subsection 3-5.1 for findin, 0 the scalar electric potential that is caufed by an electric dipole. We obtain, as in Eq. (3-48), II Substitution of Eq. (6 -54) i n Eq. ( 6 50) yiclds thc same R as given in Eq. (6-48). We note that th,c ex~rcssions of the scalar magnetic potential V,, in Eq. ( 6 ~ 5 4 ) for a magnetic dipole are cxactly analogous to those for the scalar electric potential V in Eq. (6-47) for an electric dipole: the likeness between the vector magnetic potential A (in Eq. 6-35) and b' in Zq. (6-17) is nor as exact. Holvever, since magnetic ch:~rgcs do not exist in practical problems, I / ; , rilust be determined from a givzn current distribution. This determination is usually not a simple process. Moreover. the curl- free nature of IZ indicatccl in Ey. (6--49), from which the scalar magnutic potential V,, is defined, holds only ~t points with 110 currcnts. In a region whcrc currents csis:. the magnetic field is rlot ~ c i l s ~ u ~ i i i c e , and,the scalar magnetic potential is not a single- valued function; hence tile magnetic potential difference evaluated by Eq. (6-51) depends on the path of il~tegration. For these reasons, we will use the circularing- current-and-vector-pbten;ial approach, instead of the fictitious magnetic-charge- and- scalar-potential approach. for the study of magnetic fields in magnetic materials. We ascribe the macroscopic properties of a bar magnet to circulating atomic currents (Ampkrian currents) caused by orbiting and spinning electrons. 6-6 MAGNETIZATION AN^ EQOIVALENT CURRENT DENSITIES According to the elerhentary atomic mcdel of matter, all materials are composed of atoms, cach with a positively chargcd nuclcus and a numbcr of orbiting negatively charged electrons. The orbiting electrons cause circulating currents and form micro- scopic magnetic dipoles. In addition, both the electrons and the nucleus of an atom rotate (spin) on their own axes with certain magnetic dipole moments. The magnetic dipole moment. of a spinriing I I L I C ~ C ~ I S is u~u;tIly negligible compared to that of an orbiting or spinqing electron hecause of the much largcr nx~ss and lower anp1:ir velocity of the nucleds. A camplete understanditig of the magnetic effects of materials requires a knowledge of quantum mechanics. (We give a qualitative description of the behavior of different kinds of magnetic materials later in Section 6-9.) In the absence of an external magnetic field, the magnetic dipoles of the atoms of most materials (except permanept magnets) have random orientations, resulting 214 STATIC MAGNETIC FIELDS I 6 in no net magnetic moment. The application of an external magnetic field causes both an alignment of the magnetic moments of the spinning electrons and an induced magnetic mom'ent due to a change in thc orbital motion of clcctrons. 111 ortlcr 10 obtain a formula for determining the quantitativc change in the magnetic llux dcnsity caused by the presence of a magnctic matcrisl. we kt in, hc thc ~nagaetic d ~ p u l s moment of an atom. If there are H atoms per unit volun~e, we define a inugi~etizution vector, M, as k = l M = lim - A W O AV (A/m) 9 which is the volume density of magn'etic dipole moment. The magnetic dipole moment dm of an elemental volume dti' is'dm = XI da' that. according to Eq. (6-451, will produce a vector magnetic potential Using Eq. (3-75). we can write Eq. (6-56) a s Thus, where V' is the volume of the magnetized material. We now use the vector identity in Eq. (6-28) to write . and expand the right side of Eg. (6-57)hto two terms: The following vector identity (see Problem P. 6- 14) enables us to change ;he volume integral of the curl of a vector into a surfacc integral. jv, P x F d d = -$s, F x ds', (6- 60) where F is any vector with continuous first derivatives. We have, from Eq. (6-59) \es both induced srdcr to . density ; . dipole 'ti:ation d (6-55) noment Is), will 16 56) f- volume P (6--60) 6-6 1 MAGNETIZATION AND E ~ I V A L ~ N T CURRENT QENSITIES 21 5 I . where a : is the unit outward normal vector from d s ' and St is thc surface bounding the volume V'. I, , . ,' I A comparison ofthe expressions on the right side of Eq. (6-61) with the form of A in Eq. (6-22), expdssed in terms of volume current density J suggests that the effect of the magnetizatio$vectqr is equivalent to b&th a volume current density I and a surface current density J,, = M x a,, (Ajm). (6-63) , In Eqs. (6-62) and (6-6.i) we hare omimd the primes on V and a , , for simplicity, since it is clear that both refer to the coordinates of the source point where the map- nctization vector M exisis However. the primes should be retained when there is n possibility of confusing tile coordinates cf the source and field points. The problem of finding the magnetic flux density E caused by a given iaiume . density of magnetic dipvls moment i \ .
i is then reduced to finding the equiiaicnt tnaqnerizoiion current iie.&rirs J, and J,,,, by using Eqs (6-61) m d (6-631, deter- mining A from Eq. (6-611, and then obtaining B from the curl of A. The externally applied magnetic field. if i t also exists, must be accounted for separately. The mathematical derivation of Eqs (6-62) and (6-63) is straightforivord. Tbs equivalence of a volume density of magnetic dipole moment to a volume current density and a surface cuxent density can be appreciated qualitatively by rekrring to Fig: 6-9 where a cross scction of a magnetized material is shown. It is assumed that an externally applied magnetic field has caused the atomic circulating currents to align with it, thereby magnetizing the material. The strength of this magnetizing Fig. 6-9 A cross section of a magnetized material. 216 STATIC MAGNETIC FIELDS 1 6 effect is measured by the magnetization vector M. On the surface of the material, there will be a surface current density J,,, whose direction is correctly given by that of the cross product M x a,. If M is uniform inside the material, the currents of the neighboring atomic dipoles that flow in opposite directions will cancel everywhere, leaving no net currents in the interior. This is predicted by Eq. (6-62), since the space derivatives (and therefore the curl) of a constant M vanish. However, if M has space variations, the internal atomic currents do not completely cancel, .esulting in a net volume current density J,. It is possible to justify the quantitative relationships between M and the current densities by deriving the atomic currents on the surface a i d in the interior. But as this additional derivation is really not necessary and tends to be tedious, we will not attempt it here. Example 6-8 Determine the magnetic flux density on the axis of a uniformly magnetized circular cylinder of a magnetic material. The cylinder his a radius h, length L, and axial magnetization M. Solution: In this problem concerning a cylindrical bar rnagqt. let the axis of the magnetized cylinder coincide with the z-axis of a cylindrical~oordinatc system, as shown in Fig. 6-10. Since the magnetization 1 C I is a constant within the magnet, J, = V' x M = 0, and there is no equivalcl;t volume current density. The cquivulent magnetization surface current density on the side wall is Jms = M x a; = (aJ4) x a, = a,M. (6-64) The lnagrlet i s then like a cylindrical sheet with a lined clrrrerlt density of M (A/m). There is no surface current on the top and bottom faces. In order to find B at P(0, 0, s), we consider a differential length dz' with a current a,M dz' and use Eq. (6-38) to Fig. 6-10 . A uniformly magnetized circular cylinder (Example 6-8). atcrial, by that of the where, e space s space ~g In a mhips surface 3 tends formly dius b, , oi the 3 . ~ 5 t i r .?asnet, !I\ d' ih 64) ( A m). (0, 0, I ) , -38) to n, obtain , j , i l o ~ b 2 dzl dB = a, -- 2[(z - z'JZ + b 2 F and , , +, ' B = - J ~ B ' = a, J L 0 p o ~ b 2 dz' 2 [(z - z')' + b2] 3 / 2 6-7 MAGNETIC FIELD INTENSITY AND RELATIVE PERMEABILITY Because the application af an txternal magnetic fieid ciloses both :In :~ligilmeor ai the internal dipole moments and an induced magnetic moment in 3 magnetic material. we expcct that the resultant magnetic flux density in the presence of Y magnetic material ~ i i l be different from its value in free space. The n~acroscopic etTect ai mag- netization can be studied by incorporating the equivalent volume current densit!. J, in Eq. (6-62). into the basic curl equation, Eq. (6-7). We have We now define a new f~mdamental field quantity, the inagnetic field intensity, H , such that The use of the vector H enables us to write a curl equation relatlng the magnetic field and the distribthon of free'currents in any medium. There is no need to deal explicitly with the ninpnetization vector M or the equivalent volume current density .I,,,. ~ o m h i i i T i ~ ~ Eqs. ((660) and 36-67). we ohlain the ncw cqu:ition (6-68) where J (A/m2) is the volume density olfiee ci~rlrnt. Equations (6-6) and 16-68) are the two fundamental governing differential equations for magnetostatics in any 218 STATIC MAGNETIC FIELDS 1 6 medium. The permeability of free space, pO, does not appear explicitly in these two equations. The corresponding integral form of Eq. (6-68) is obtained by taking the scalar surface integral of both sides. or, according to Stokes's theorem, where C is the contour (closed path) bounding the surface S, and 1 is the total current passing through S. The relative diiections of C and current flow I follow the right- hand rule. Equation (6-70) is another form of Ampc'.re's circuital law: It states that the circulation of the ntugneticjeld intensity urounil any closed path is equal to the Jrcc c l i l - r e n r jlor.inc/ T / I I . O L I ~ / ~ ~ / I C SIII./~KC' h l l l 7 d d I.!. p ~ l t . / ! ~ ~ - \ s \ve iildic,ltcd in Section 6-3. Amptre's circuital law is most useful in determi'ning the magnetlc field caused by a current when cylindrical symmetry exists-that is, when there is a closed path around the current over whichthe magnetic field is constant. When the magnetic properties of the medium are linear and isotropic, the mag- netization is directly proportional to the magnetic field intensity: where zm is a dimensionless quantity called magnetic susceptibility. Substitution of Eq. (6-71) in Eq. (6-67) yields (6-72b) where is another dimensionless quantity known as the relative permeability of the medium. b < . ) L I 6-7 / M A ~ N E T ~ C FIELD INTENSITY AND RELATIVE PERMEABILITY 219 Fig. 6-11 Coil on air gap (Example 6- ferromagnetic toroid w ~ t h -9). The parameter p = u,p. is the absoliite perazeu8ilit)~ (or, sometimes. just pemiubilitj 1 of the medium and is measured in H/m; z,,,, and therefore p,, can be a function of space coordinarcs. For a simple mcdium - linear, isotropic, anct homogeneous -- z,,, m d p, are consttlnts. The permeability of nost materials is very close to that of free space (p,,). Fdr ferromagnetic materials such as iron. nickel, and cobalt. p, could be very large (50-5000, and up to 106 or more for special alloys): the permeability depends no[ only on lhc magnitude ol 11 but also on liic prcvious hibtory of the material. Section 6-8 contains some quali~ative discussions of the macroscopic behavior of magnetic materials. Examplc 6-9 Assunw illat iZl turns ol'wire arc wo~inci around a roroid~ll care ol'.i ferromagnetic material with permeabiilty p. The core has a mean radius r,], 3 circular cross section of radiUs u iu I< r,,), and a narrow air gap of length I,, as shown in Fig. 6-11. A steady currdnt I,, Rows in the wire. Determine (a) the magnetic flux denhity. B f , in the ferromagnetic core; (b) the magnetic field intensity, H,., in the core; and (c) the magnetic field intensity, H,, in the air gap. Solution a) Applying Ampkke's circuital law, Eq. (6-70), around the circular contour C, which has a mean radius r,, we have lilliir lr:~k:~ga is ncglc~k~i. illr S : I ~ C w h l 11~1s will lloa iii 170111 ilia Scrron~ayctic corf ;md in ihc air g:lp: I f lhc fringing clkul ol. lllc llur in the air gap is also ncg- lected, the magnetic Hux density B in both the core and the air gdp will also be . the same. However, Zccause of the different permeabilities, the magnetic field intensities in both parts will be different. We have 220 STATIC MAGNETIC FIELDS / 6 In the ferromagnetic core, and, in the,air gap, Po Substituting Eqs. (6-75), (6-76), and (6-77) in Eq. (6-74), we obtain and B I U O P N I ~ ; ' po(2ni-,, - + pi, ' b) From Eqs. (6-76) and (6-78) we get c) Similarly, from Eqs. (6-77) and (6-78). wc havc Since H,/H, = p/p0, the magnetic field intensity in the air gap is much stronger than that in the ferromagnetic core. Why do you think the condition a cc r, is stipulated in this problem? 6-8 MAGNETIC CIRCUITS The problem in Example 6-9 is, essentially, one of a magnetic circuit in which the current applied to the winding causes a magnetic flux to flow.in the ferromagnetic core and thc air gap in series. We define the line integral of magnetic field intensity around a closed path, as magnetomotiw force, mmf. Its SI unit is ampere (A): but, because of Eq. (6-74), mmf is frequently measured in ampere-turns (A t). An mmf is not a force measured in newtons. Assume 1'; = N1,denotes a magnetomotive force that causes a magnetic flux, 0, to flow in a magnetic circuit. If the radius of the cross section of the core is much ' Also called magnetomotance. r 6 -76) (6-77) (6-78) i 0 -, P l (6 SO) .rongcr ich the lgnetic .tensity (6 p, ' J X I I , , J x H u . s much I L : 6-8 1 MAGNETIC CIRCUITS 2 2 1 . : smaller than the mean radius of the toroid, the magnetic flux density B in the core is approximately co&ant,'and cf, E BS, (6-81) wlicrc S is thc cross-scclionnl area of the cute. Combination of Eqs. (6-81) and (6-78) yields 4 Equation (6-82) carfbe rewritten with where tl = 2nr, - fg is the length of the ferrorhagnetic core, and Both 3 , and 2, have the rime form :is the formula. Lq. (5-1 3). for ihc DC res~st:incc -. o 1 s : p i c f 1 o 1 1 c 1 o s i i I a i cross s i S. U l l h arc called rcliiaunr~: 2 , of the fcrromiignctic corc; and r?,,, of the ;iir gip. Tile SI unit for reluctance is reciprocal henry (H-I). The fact that Eqs (6-84) and 16-85] are as they are, even though the core is not straight, is a consequence of assuming that B is approximately constant over the core cross section. Equation (6-83) is analogous to the expression for the current I in an electric circuit, in which an idealvoltage source of emf Y is connected in series with two resistances Rf and R,: (a) Magnctic circuit, (b) Elcctric circuit. Fig. 6-12 Equivalent magnetic circuit and analogous electric circuit for toroidal coil with air gap in Fig. 6- 1 1. 222 STATIC MAGNETIC FIELDS / 6 , , respectively. Magnetic circuits can, by analogy, be analyzed by the same techniques . we have used in analyzing electric circuits. The analogous quantities are Magnetic Circuits Electric Circuits mmf, Y C , , (= N I ) emf, -/' magnetic flux, 0 electric current, I reluctance, 9 4 resistance, R permeability, p conductivity, a In spite of this convenient likeness, an exact analysis of magnetic circuits is inherently very difficult to achieve. First, it is very difficult to account for leakage fluxes, fluxes that stray or leak from the main flux paths of a magnetic circuit. For the toroidal coil in Fig. 6-11, leakage flux paths encircle every turn of the winding; they .partially transverse the space around the core, as illustrated, because the perrneabilitu1ncs ich as J y not : sohe H, 1~ 4 7 a ) / 1 a of the r 11nc. ' rrecqo. _ 6. h . 1 MAGNETIC ClRCUlTS 223 . 4 Second, if we asstime no leakage flux, the total flux C D in the ferromagnetic core and in the air gap must be the same, or B, = B,,:+ \ Substitution of Eq. (6-87b) in Eq. (6-87a) yields an equation relating BJ and H, in the core: This is an equatio$fbr'k straight line in the B-H plane with a negative slope- , u , C , / & , . The intersection of this line and the given B-H curve determines the operating point. Once the operating has been found, p and H , and all other quantities can be obtained. The similarity between Eqs. (6-83) and (6-86) can be extended to the writing of two basic equations for magnetic circuits that correspond to Kirchhoff's voltage and current laws for electr~c circuits. Simi;nr to Kirchhofl's voltage law in Eq. (5-37), we may write, for any closed path in a magnetic circuit, Equation (6-89) states that around a closed pu~h in a magnetic circuir rhe algebraic sum of umpere-turns i s e q t d to the algebraic sum of the products of the re2ucti;nce.s und fluxes. Kirchhoff's current lsw for a junction in an electric circuit, Eq. (5-33), is a consequence of V . J = 0 . Similarly, the fundamental postulate V . B = 0 in Eq. ( 6 4 ) leads to Eq. (6-8). Thus we have which states that the dgebraic sum of all the magneticJuxes jlowing out of a juncrion in a magnetic circuit is zero. Equations (6-89) and (6-90) form the bases for. respec- tively, the loop and ndde, analysis, of magnetic circuits. Example 6-10 Consider the magnetic circuit in Fig. 6-13(a). Steady currents I, and I, flow%Twindings of, respectively; N , and N 2 turns on the outside legs of the ' Th~s assumes an equal crobs-sectional area for the core and the gap. If the care were to be constructed of ihsulated laminations of ferromagnetic material, the effective area for flux passage in the core would be smaller than the geometrical cross-sectional area, and B, would be larger than B, by a factor. This factor can be determined from tho datq on the insulated laminations. 1 224 STATIC MAGNETIC FIELDS / 6 (a) Magnetic core with current-carrying windings. (b) Magnetic circuit for loop analysis. Fig. 6-13 A magnetic circuit (Example 6-10). ferromagnetic core. The core has a cross-sectional area Sc and a permeability p. Determine the magnetic flux in the center leg. Solurioiz: The equivalent magnetic circoit Sor loop ;mllyis is.sIlown in Fig. b- l3lb). Two sources of mmf's. N , I l and N212. arc shown with prop~~ol:lritics is scrics with reluctances A, and .f12 r~sp~~IivcIy. This is til~vioilsl~ ii two-iaop IIC~\V(>I.~. Since wc arc determining mapoc~ia llua i ~ i the center leg I',P,, it is crpediellt to choose the two loops in such a way that only one loop flux (a),) flows through the center leg. The reluctances are computed on the basis of average path lengths. These are, of course, approximations. We have The two loop equations are, from Eq. (6-89). Loop 1 : N I I l = (!MI + g 3 ) ( b 1 + LVlO2, (6-32) Loop 2: N I I l - N212 = g l Q l + ( 9 1 + g2)02. (6-93) Solving these simultaneous equations, we obtain which is the desired answer. 6-9 1 B E H A V ~ R OF MAGNETIC MATERIALS 225 4 . Actually sincd t ,e magnetic fluxes and thhrefore the magnetic flux densities in the three legstare ! ifferent, different permeabilities should be used in computing the reluctances in E$s. (6--!,la), (6-91b) and (8-91c). But thc value of permeab~llty, in turn, depends dn the magnetlc flux density. The only way to improve the accuracy of the solution, prokided thC B-H curve of the core material IS given, is to use procedure of ducces8ive approximation. For instance, Q,, Q2, and cD, (and therefore B,, B2, and B3) are first solved with an assumed p and reluctances comp~~ted from the three parts of Eq! (6-91). From B,. B,, anQ B, the corresponding ,ul, p 2 , and p, can bc found from'lthe B-H curve. These will modify the reluctances. A second approximation for dl, B2; and B, is then obtained with the modified reluctances. From the new flux dgnsities, new permeabilities and new reluctances are determined This procedure is repeated until further iterations brmg little changes in the computed values. 6-9 BEHAVIOR OF MAGNETiC MATERIALS In Eq. (6-71), Section 6 l 7 , we dsscribed the macroscopic magnetic property of i linear, isotropic med;lum by defining the magnetic susceptibility x,,, a dimensionless coefficient of proportionality between magnetization M and magnetic field intensity H. The relative pernleabiljty p, is simply 1 + x,. Magnetic materials can be roughly classified into three main.groups in accordance with their , u , values. A materiai is said to be Diamagnetic, if p, 5 I :(.y, is a very small negative number). Purarnuyneric, if , u , 2 L,(x, is a very small positive number) Ferromugnetic, if p, >,:I (2, is a large positive number). . . As mentioned before, a thorough understanding of microscopic magnetic phenomena requires a knowledge of quantum mechanics. In the following we give a qualitative description of the behavior of the various types of magnetic materials based on the classical atomic modkl. In 3 cli(i~ti(l{]l!~ficmatcri:ll thc nct magnclic ~~iomcnt due to thc orbital and spinning motions of the electrons in any particular atom is zero in the absence of an externally applied magnetic field. As predicted by Eq. (6-4), the application of an external magnetic field to this material produces a force on the orbiting electrons, causing a perturbation in the angular velckities. As a consequence, a net magnetic moment - , is created. This is a hrocess of induced magnetization. According to Lenz's law of electromagfie'etic induction (Section 7.-2), the induced magnetic moment always opposes the applied field, thus reducing the magnetic flux density. The macroscopic effect of this process-is equivalent to that of a negative magnetization that can be described by a negative rnagnetic susceptibility. This effect is usually very small, and x,,, for most known diamagnctic materials (bismuth, copper, lead, mercury, germanium, silver, gold, diam.ond) is in the order of - 226 STATIC MAGNETIC FIELDS 1 6 Diamagnetism arises mainly from the orbital motion of the electrons within an atom and is present in all materials. In most materials it is too weak to be of any practical importance. The diamagnetic effect is masked in paramagnetic and ferro- magnetic materials. Diamagnetic materials exhibit no permanent magnetism, and the induced magnetic moment disappears when the applied field is withdrawn. In some materials the magnetic moments due to the orbiting and spinning electrons do not cancel completely, and the atoms and molecules have a net average magnetic moment. An externally applied magnetic field, in addition to causing a very weak diamagnetic effect, tends to align thc molecular magnctic moments in tlzc direction of the applied field, thus increasing the magnetic flux density. The macro- scopic effect is, then, equivalent to that of a positive magnetization that is described by a positive magnetic susceptibility. The alignment process is, however, impeded by the forces of random thermal vibrations. There is little coherent interaction and the increase in magnetic flux density IS quite small. Materials with this behavior are said lo bc />tn.trrlrtr!/r~c,/ic.. P;~r;~~n;~piclic m;ltcri;~ls pcrally haw vcry slnall positive v;~Iucs 0 1 ' 111;1g11clic s~~sccplilGlily. ill lhc 01dcr ot I0 ' ; ~ I u t l l i ~ l u ~ l l , 111;1$11csiutll. titanium, and tungsten. - --. Paramagnetism arises mainly from the magnetic dipole moments of the spinning electrons. The alignment forces, acting upon molecular dipoles by the applied field, are counteracted by the deranging effects of thermal agitation. Unlike diamagnetism. which is essentially independent of temperature, the paramagnetic elrect is tem- perature dependent, being stronger at lower temperatures where there is less thermal collision. . The magnetization of ferromagnetic materials can be many orders of magnitude larger than that of paramagnetic substances. (See Appendix B-5 for typical values of relative permittivity.) Ferromcrgnctism can be explained in terms of magnetized tlort~rrirr.~. According l o this ~l~otlcl, w i ~ i c l ~ 11:~s I x x r i cxpcrit~\cnt:~lly co~l(i~.ll~ctl, ;I ferromagnetic mutcrial (such as cobalt, nickel, and iron) is composcd of many small domains, their linear dimensions ranging Srbm a few microns to about 1 mm. 'These domains, each containing about 10'"r 1016 atoms, are fully magnetized in the sense that they contain aligned magnetic dipoles resulting from spinning electrons even in the absence of an applied magnetic field. Quantum theory asserts that strong coupling forces exist between the magnetic dipole moments of the atoms in a domain, holding the dipole moments in parallel. Between adjacent domains there is a transition region about 100 atoms thick called a clonzailz wall. In an unmagnetized state, the magnetic moments of the adjacent domains in a ferromagnetic material have different directions, as exemplified in Fig. 6-14 by t h e polycrystalline specimen shown. Viewed as a-whole, the random nature of the o;ientations in the various domains results in no net magnetization. , . ,. When an external magnetic field is applied to a ferromagnetic material, the walls . . of those domains having magnetic moments aligned with the applied field move in such a way as to make the volumes of those domains grow at the expense of other thin an of any 3 ferro- :m, and sn. pinning :vcrage lislng a :s in the macro: wrihed xpeded oil and /lor ;ire ,mji\ive ;:csium, .!nlF'- s i fic.1~. net! 1s tcm- i ! l c ~ r ~ > ~ l gnitude ! values metized med, a iy small I. These .le sense ns everi strong hmain, ansition .ate, the liirerent s h q :om,,,ls hc \va. . :nave in of other 6-9 1 BEHAVIOR OF MAGNETIC MATERIALS 2?? , Magnetized -domain Domain -wall 1:iy. 0-1 J Do~nain SLSLIC~LII-C 01' a polycrystalline ferromagnetic specimen domains. As a result magnetic flux density is increased. For weak applied fields, say up to point P, in tig. 5-15, domain-wall movements are reversible. But when an applied field becorned stronger (past P,). domain-wall movements are no !onger reversible. and domah n)tation towitrd the direction of thc irpplicd ficid will rdso O ~ L I ~ . For cr;wpic, if all ipplicil licld is rcduced lo zero at point I),, the B-H rcla- tionship will not foliop th: solid curve P,P,O, hut will go down from P, to P;. ;iionp the iines of the broken curve in the ligure. This phenomenon of magnetization lagzing behind the field producing it is called A?;stcizsis, which is derived from a Greek word meaning "to lag." 4 s ihe applied field becomes even much stronqer (past PI to P,). domain-wall motion and domain rotation will cause essentially a total ah, nnment of the microscopic magnaic moments with the applied field. at which point the magnetic material is said to have reached rorilrcsioii. The curve O P , P,P, on the R I 1 plunc is callcd thc irc~i.t,rtrl mr~picliztilioti clrl-vc. ' If +he applied magnetic field is reduced to zero from the viilue at P,. the magnetic flux density does not go to zero but assumes the value at B,. This value is called the residaal or reinanent flux &i?sir~~ (in Wbp') and is dependent on the maximum applied field intensity. The existence of a remanent flux density in a ferromagnetic material makes permahent magnets possible. Fig.6-15 Hysteresis loops in B-H plane for ferromagrletic material. ' 228 STATIC MAGNETIC FIELDS 1 6 In order to make the magnetic flux density of a specimen zero, it"is necessary to apply a magnetic field intensity H , in the opposite direction. This required H , is called coersive force, but a more appropriate name is coersivefield intensity (in Aim). Like B,, H , also depends on the maximum value of the applied magnetic field intensity. It is evident from Fig. 6-15 that the B-H relationship for a ferromagnetic material is nonlinear. Hence, if we write B = pH as in Eq. (6-72a), the permeability p itself is a function of the magnitude of H. Permeability ,LL also depends on the history of the material's magnetization, since-even for the same H- we must know the location of the operating point on a particular branch of a particular hysteresis loop in order to determine the value of hi exactly. In some applications a small alternating current may be superimposed on a large steady magnetizing current. The steady magnetizing ficld intensity locates the operating point. and the local slope of the hysteresis curve at the operatins point determines the incre~ilentcll prrnwahilit!.. Ferromagnetic materials for use in electric generators, motors, and transformers should have a large magnetization for a very small applied field: they should have tall :~nd narrow Iiystercsis loops. As the applied magnctic ficld intensity varics pcrio~lically hciwcc~l 5 11,,,:,,. the I~yslcluis loop is \raced w_cc per cycle. 'l'llc ;Ire:\ of the hysteresis loop corrtxponds to energy loss (h!~sr~w.si.s los.;jper unit volume pcr cycle (Problem P. 6-21). Hysteresis loss i? the energy lost in the form of heat in over- coming the friction encountered during domain-wall motion and domain rotation. Ferromagnetic materials, which have tall, narrow hysteresis loops with small loop areas, are referred to as "soft" materials; they are usually well-annealed materials with very few dislocations and impurities so that the domain walls can move easily. Good permanent magnets, on the other hand, should show a high resistance to demagnetization. This requires that they be made with materials that have large coercive field intensities H , and, hence, fat hysteresis loops. These materials are referred to as "hard" ferromagnetic materials. The coercive field intensity of hard ferromagnetic materials (such as Alnico alloys) can be lo5 (A/m) or more, whereas that for soft materials is usually 50 (A/m) or less. As indicated before, ferromagnetism is the result ofstrong coupling effects between the magnetic dipole moments of the atoms in a domain. Figure 6-16(a) depicts the atomic spin structure of a ferromagnetic material..When the temperature of a ferro- magnetic material'is raised to such an extent that the thermal energy exceeds the couplicg energy, the magnetized domains become disorganized. Above this critical temperature, known as the curie temperature, a ferromagnetic material behaves like a paramagnetic substance. Hence, when a permanent magnet is heated above its curie temperature it loses its magnetization. The, curie temperature of most ferro- magnetic materials lies between a few hundred to a thousand degrees Celsius, that of iron being 770°C. 2: . .
. .. : . < a , , .
. Some elements, such as chromium and manganese, which ire close to ferro- magnetic elements in atomic number and arc neighbors of iron in the pcriodic table, also have strong coupling forces between the atomic magnetic dipole moments; 6-9 / BEHAWOR OF MAGNETIC MATERIALS . '9 1 : I y i ~ . 6-16 Schcm:llic atomic spin struclurcs I'or la) ferromagnetic, (b) mtikrromagnetic, and ic) fcrrimagnctic materials. but their coupling forces produce antiparailel alignmen~s of electron spins. 2s illus- trated in Fig. 6--16(b). The s$ns alternaic in direction from atom to atom and rcsult in no net magnetic moment. A material possessing this Eroperty is said to be mri- ferrontagnetic. ~ntiferronla~netism is also temperature dependent. When an anti- ferromagnetic material is hated above its curie temperature, the spin directions suddenly become random and the material becomes paramagnetic. There is another class of inagnetic materials that exhibit a behavior between ferromagnetism and antiferromagnetism. Here quantum mechanical effects make the directions of the magnetic moments in the ordered spin structure ‘ 1 , 1 ternate and the magnitudes unequal, resulting in a net nonzero magnetic moment, as depicted in Fig. 6-16(c). These materials are said to be ferrimagnetic. Because of the partial cancellation, the maximum magnetic flux density attained in a ferrimagnetic substance is substantially lower than that in a ferromagnetic specimen. Typically, it is about 0.3 Wb/m< approximately one-tenth that for ferromagnetic substances. Fcrrites are a subgroup of fcrrimagnetic material. One type of ferrites, called ~i~uy~leric spinels, crystaliize in a complicated spinel strucure and I1;~ve the formula XO . Fe203, where denotes a divalent metallic ion such as Fe, Co, Ni, Mn, Mg, Zn, Cd, etc. These are ceramic-like compounds with very low conductivities, (for instance, lO-%d_(S/m) compared with lo7 (S/m) for iron). Low conductivity limits eddy-current losses at High frequencies. Hence ferrites find extensive uses in such high-frequency and microwave :ipplicntions as corcs for FM antennas, high-frequency trnnsformurs, and phase shii'lcrs. Other fcrritcs include magnetic-oxide garnets, of which Yttrium-Iron-Garnet ("YIG," Y,Fe,D,,) is typical. Garnets are used in microwave multiport jut~ctions. . ' 4 230 STATIC MAGNETIC FIELDS I 6 I I I I/ 1 . I 6-10 BOUNDARY CONDITIONS FOR I MAGNETOSTATIC FIELDS In order to solve problems concerning magnetic fields in regions having media with different physical properties, it is necessary to study the condition: (boundary con- ditions) that B and H vectors must satisfy at the interfaces of different media. Using techniques similar to those employed in Section 3-9 to obtain the boundary con- ditions for electrostatic fields, we derive magnetostatic boundary conditions by applying the two fundamental governing equations, Eqs. (6-6) and (6-68), respec- tively, to a small pillbox and a small closed path which include the interface. From the divergenceless nature of the B field in Eq. (6-6), V B = 0, we may conclude directly, in light of past experience, that the normal component of B is continuous across an interface; that is, For linear media, B, = p1Hl and B2 = j1,H2, Eq. (6-95) becomes 1 - (6-96) The boundary condition for the tangential components of magnetostatic field is obtained from the integral form of the curl equation for H, Eq. (6-70), which is repeated here for convenience: We now choose the closed path rbcda in Fig. 6-17 as the contour C. Applying Eq. (6-97) and letting bc = da = Ah approach zero, we have where J,, is the surface current density on the interface normal to the contour C. The direction of J , is that of the thumb when the fingers of the right hand follow Fig. 6-17 Closed path about the interface " of two media for determining the boundary condition of H,. dia with lry con- . 3. Using , iry con- ~ons by rc\r,cc- c. From onclude nrinuous (6-98) xbur C. d follow P rface ldary 6 -10 ~OUNDARY CONDITIONS COR MAGNETOSTATIC FIELDS 231 : I G ? the direction of the &h. In Fig. 6-17, the positive direction of J," for the chosen path is out of the paper. ~ i l e following is a more concise expression of the boundary + , condition for the tangential components of H, which includes both magnitude and direction relations (Problem P. 6-22). a,l2 x ( H I - H,) = J, (A/m), (6-39) whom a,,, is the olrlwut+ uililnortt~sl j,l)rtt t~~adiun~ 2 :it thc in~crf:lcc. Thus, the tongcntiiri ... , component o j the HJi~ld is discontis~u,u.s across un interjoce where o surjbce cilrrerlt exists, the amount of.disccntinuity being determined by Eq. (6-99). When the condu~kvities of both media ire finite, currents are defined by voiume current densities and free surface currents do not exist on the interface. Hence. J, equals zero, and the tut.qenrio1 conlponcnt of H is continuous ucioss the boiiridury h . . ' . . o 1 h y i c I ~ I ~ I : 11 1 s discontinuoos only whcn an intcrfacc with an ideal pcrfcct conductor or a ~up~xconductor is assumed. i Example 6-11 Two magnetic media with permeabilities ji, and p , have a common boundary, as shown in FIG. 6-18, The magnetic field intensity in medium 1 at ;hi. point l', has a magnihde 11, and m:~kcs an angle a , with the normal. Determine he magnitude and the directic.n of the magnetic field intensity at point P, in medium 1 . Solution: The desired un1,ooyn qu:intities 'lrc Hz and a,. Continuity of the normal component of B field requires, from Eq. (6-96), p2H2 cos z2 = p l H 1 cos a,. (6-1OC) Since neither of the medic is a perfect conductor, the tangential component of H field is continuous. We have H , sin a, = H 1 sin a l . Division of Eq. (6-101) by Eq. (6-100) gives + Fig. 6-18 Boundary conditions for magnerostatic field at an interface (Example 6-1 1). . .- 232 STATIC MAGNETIC FIELDS 1 6 which describes the refraction property of the magnetic field. The magnitude of Hz is H2 = , / - = ,/(H~ sin,^,)^ +(Hz cos a?)' We make three remarks here. First, Eqs. (6-102) and (6-104) are entirely similar to. respectively. Eqs. (3-119) and (3-120) for the electric fields in dielectric media- except for the use of permeabilit'ies (instead of permittivities) in the case of magnetic fields. Second, if medium 1 is nonmagnetic (like air) and medium 2 is feriomagnetic (like iron), then ~ 1 , > > p, and. from Eq. (6-102), a, will be nearly ninety degrees. Thw means that for any arbitrary angle a, that is not close to zero. the magnetic field in a ferromagnetic n~ediun~ runs almost parallel to thc intc;fa~c: Third, if medium I is ferromagnetic and medium 2 is air (p, > > p2), then a, will be nearly zero; that is. if a magnetic field originates in a ferromaghetic medium, the flux lines will emerge into air in a direction almost normal to the interface. In current-free regions the magnetic flux density B is irrotational and can be expressed as the gradient of a scalar magnetic potential Vm, as indicated in Sec- tion 6-5.1. B = -$i'V,. (6-105) Assuming a constant p, substitution of Eq. (6-105) in V B = 0 (Eq. 6-6) yields a Laplace's equation in Vm: VZVm = 0. (6-106) Equation (6-106) is entirely similar to the Laplace's equation, Eq. (4-lo), for the scalar electric potential V in a charge-free region. That the solution for Eq. (6-106) satisfying given boundary conditions is unique can be proved in the same way as for Eq. (4-10)-see Section 4-3. Thus the tcchniqucs (method of images and method of separation of variables) discussed in Chapter 4 for solving electrostatic boundary- value problems can be adapted to solving analogous magnetostatic boundary-value problems. However, although electrostatic problems with conducting boundaries maintained at fixed potentials occur quite often in practice, analogous magnetostatic problems with constant magnetic-potential boundaries are of little practical impor- tance. (We recall that isolated magnetic charges do not exist and that magnetic flux lines always form closed paths.) The nonlinearity in the relationship between B and H in ferromagnetic materials also complicates the analytical solution of boundary- value problems in magnetostatics. . , . " I I 6-11 1 INO~CTANCES AND INDUCTORS 233 i ? i I . t : $ \ 6-11 INDUCTANCES AND I ~ D U C T ~ R S f ! (6-103) Consider two neighboring closed loops, C, add C, bounding surfaces Sl and S2 i respectively, as shown in Fig. 6-19. If a current I, flows in C,, a magnetic field B, : ofH, is will be created. Some of the magnetic flux due to B, will link with C , - that is, wlll pass through the surf$ce S2 bounded by C,. Let us designate this mutual flux a,,. We have ecs. This i ? can be i in Sec- 1 , for the . (6-106) :ty as for elhod of ~undary- :ry-'PY unci ,s .etostafic 11 imf xtic flux : n B and mndary- From physics we knok thiit a time-varying I, (and therefore a time-varying 81,) wll produce an induced e~ectromotive force or voltage in C2 as a result of Foraday's low of electromagnetic i$uction. (We defer the discussion of Faraday's law untll the , next chapter.) I Iowcdor, ( I , , , cxists cvcn ~f I , is a steady DC current. From ~iot-~avar! law, Eq. (6-31), we see that B , is directly proportional to I, ; hence ( D l , is also proportional to I,. We write where the proportionslity constant L,, is called the mutual inductance between loops C , and C,, with SI unit henry (H). In case C2 has N , turns, thejux linkage A,, due to Q12 is A12=N2@12 (Wb), and Eq. (6-108) generalizes to Fig. 6-19 Two magnetically coupled loops. 234 STATIC MAGNETIC FIELDS / 6 j ! ' The mutual inductance between two circuits is then the magneticjux linkage with one circuit per unit current in the other. In Eq. (6-108), it is implied that the permeability of the medium does not change with I,. In other words, Eq. (6-108) and hence Eq. (6-11 1) apply only to linear media. A more general definition for L,, is f t v Some of the magnetic flux produced by I , links only with C, itself, and not with C,. The total flux linkage with C, caused by I , is ! 1 The self-inductance of loop C, is defined as the magneticflux linkage per: unit current in the loop itself; that is, ! L for a linear medium. In general, , The self-inductance of a loop or circuit depends on the geometrical shape and the 1 f physical arrangement of the conductor constituting the loop or circuit, as well as on I the permeability of the medium. With a linear medium, self-inductancc does not depend on the current in the loop or circuit. As a matter of fact, it exists regardless of I whether the loop or circuit is open or closed, or whether it is near another loop or circuit. A conductor arranged in an appropriate shape (such as a conducting wire wound ' as a coil) to supply a certain amount of self-inductance is called an inductor. Just as t a capacitor can store electric energy, an inductor can storage magnetic energy, as we 1 shall see in Section 6-12. When we deal with only one loop or coil, therc is no need to carry the subscripts in Eq. (6-1 14) or Eq. (6-1 15), and inductunce without an adjective 1 will bc taken to mean self-inductance. The proccdurc for determining the self-in- f ductance of an inductor is as follows: 5 ' ' 1. Choose an appropriate coordinate system for the given geometry. 2. Assume a current I in the conducting wire. 3. Find B from I by Ampire's circuital law, Eq. (6-9), if symmetry exists; if nbt, ! ' Biot-Savart law, Eq. (6-31), must be used. ; 1 \vith orle meability nd hence S I (6-1 12) I not with (6-113) 1t curreut ( ( 0 4) (6-115) e and the file11 as on does not ardless of :r loop or re wound Ir. Just as rgy, as we !o need to . ad/.e le sell-in- ts; if not, 6-11 / INDUCTANCES AN3 INDUCTORS 235 t s 4. Find the flux linking with each turn, Q, from B by integration, B = B a d s , I Ss ' where S is the area over which B exists and links with the assumed currenr. 5. Find the flux linkkge A by multiplying B by the number of turns. 6. Find L by taking !he ratio L = All. Only a slight modification of this proceduk is needed to determine the mutual inductance LIZ betden two circuits. After choosing an appropriate coordinate system, proceed as fd~lows:.Assume I, + find B, - find B,, by integrating B, over surface S2 -. find fluk linkage A,, = N,(P,, -i find L, , = A, ,/I, Example 6-12 Assdme I? turns of wire are tightly wound on n toroidal frame of a rectangular cross section with dimensions as shown in Fig. 6-20. Then assuming the permeability of the medium to be ,uo. find the self-inductance of the toroidal coil. Sulution: It is clear jhat the cylindrical coordinate system is approprlate for th~s problem because the toroili is symmetrical about its axis. A s s u m q a currenr I in the conducting wire, we find. by applying Eq. (6-9) to a circular path wlth radius r(u < r < h): This result is obtained because bath B, and rare constant around the circular path C, Since the path encircles a total current NI, we have Fig. 6-20 A closely wound toroidal ,coil (Example 6-12). 236 STATIC MAGNETIC FIELDS / 6 and , .-,, , ,. ,. . . i . . PoNI - B, = -. 2nr Next we find The flux linkage A is N@ or poN21h b A=- In -. . 2n u Finally, we obtain A poN2h b L=-=- In- (H). I 2 a (6-116) We note that the self-inductance is not a function d?ffor a constant nmiium permeability). The qualification that the coil be closely wound on the toroid is to minimize the linkage flux around the iliriividual turns of the wire. Example 6-13 Find the inductance per unit length of a very long solenoid with air core having n turns per unit length. Solution: The magnetic flux density inside an infinitely long solenoid has been found in Example 6-3. For current I we have, from Eq. (6-13), B = ,uonI, which is constant inside the solenoid. Hence, where S is the cross-sectional area of the solenoid. The flux linkage per unit length is Therefore the inductance per unit length is Equation (6-1 19) is an approximate formula, based on the assumption that the length ol' Ihc solcnoid is vcry n~uch yraltcr th;m thc 1inc;lr dimcnsions of its crow scction. A more accurate derivation for the magnetic flux density and flux linkage per unit length near the ends of a finite solenoid will show that they are less than the values given, respectively, by Eqs. (6-13) and (6-118). Hence, the total inductance of a finite solenoid is somewhat less than the values of L ' , as given in Eq. (6-1 19), muiti- plicd by the length. (6-116) n : meuium :oic' to d with air een found (6-1 17) . t length is (6 - 1'1 8) the 1,- lth m se~-.~n. ;e per unit the values tance of a 19), multi- . q , ; .
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, , .!a; , ! . ~ , . 6-11 ,'I I ~ D U C T A ~ L C S AND INDUCTORS 237 , .: , , C 4 , The following isia significant obser&ion hbout the results of the previous two examples: The self4 ductance of wire-wound ihductors is proportional to the square ~f the number of tu ps. 1 t Example 6-14 ~ n ' k i r cdaxial transmission lide has a solid inner conductor of radius a and a very thin outer conductor of inner rahius b. Determine the inductance per unit length of the lide. !' ' . Solution: Refer to i g . 6721. Assume that a &rent I flows in the inner conductor and returns via the h e r donductor in the othet direction. Because of the cylindrical symmetry, B has onl) a $-component with diherent expressions 111 ihc two resions: (a) ins~de the inner cbndu;tor, and (b) between the inner and outer conductors. Also ' assume that the currlk!nt I is liniformly distributed over the cross section of the Inner conductor. a) Inside the inner cbndu :tor, O l r l a . From Eq. (6-lo], b) Between the inner and outer conductors, u ~ r ~ h . From Eq. (6-1 11, Now consider an:annclar ring in the inner conductor between radii r and r i cir. The current in a unit length of this annular ring is linked by the flux that can be obtained by integrating Eqs. (6-1 20) and (6-121). We have Fig. 6-21 Two views of a coaxial transmission line (Examplz 6-14). 238 STATIC MAGNETIC FIELDS / 8 But the current in the annular ring is only a fr&tion (2nr dr/na2 = 2r dr/a2) of the total current I.' Hence the flux linkage for this annular ring is 2r dr dA' = - a2 dW. (6-123) The total flux linkage per unit length is , . . The inductance of a unit length of the coaxial transmission line is therefore The first term po/8n arises from the flux linkage internal to the solid inner conductor; it is known as the internal inductance per unit length of the inner conductor. The second term comes from the linkage of the flux that exists between the inner and the outer conductors; this term is known as the external inductance per unit length of the coaxial line. --. Before we present some examples showing how to detkrmine the mutual indue- . tance between two circuits, we pose the following question about Fig. 6-19 and Eq. (6-111): Is the flux linkage with loop C2 caused by a unit current in loop C, equal to the flux linkage with C , caused by a unit current in C,? That is, is it true that L I Z = L2,? (6-125) We may vaguely and intuitively expect that the answer is in the affirmative .because of reciprocity." But how do we prove it? We may proceed as follows. Combining Eqs. (6-107), (6-109) and (6-Ill), we obtain N2 L I Z = - L2 B, . ds,. I , (6-126) ' .I J . It is assumed that the current is distributed unifohnly in the inner conductor. This assumption does not hold for high-frequency AC currents. . > I , . . i ' j; 4 ' n2) of the . I.,!: I 1 ' . I (6-123) r r ' i \nductor; <tor. The .r and the length of i;il induc- 5-19 and i loop C 1 . is it true : "because ornbining n ion does not a But, in $ew of Eq. (6,14), B1 can be written as the curl of a vector magnetic potential A,, B1 = V x A,. Welhave < : : ! N2 ' L12 = - (V x Al). ds2 ' f . 11 I In Eqs. (6-127) and (6-128), the contour integrals are evaluated only once over the periphery of the loogs C , and C , respectively-the effects of multiple turns habinp been taken care of sebarately by the factors N , and N,. Substitution of Eq. (6-128) in Eq. (6-127) yields where R is the distance between the differential lengths dt', and d&. It 1 s customary to write Eq. (6-129aj as ' I whcre N , and N 2 h d ~ c been absorbed in the contour integrals over the circzilt~ C, and C2 from one end to the other. Equation (6-129b) is the Neumann formela for mutual inductance. U is a general formula requiring the evaluation of a double line integral. For any givdn problem we always first look for symmetry conditions that may simplify the dekrmination of flux linkage and mutual inductance without resorting to Eq. (6-129b) directly. It is clear from Eq. (6-1296 that mutual inductance is a property of the geo- metrical shape and tilt physical arrangement of coupled circuits. For a linear medium mutual inductance is proportional to the medium's permeability and is independent of the currents in the circuits. It is obvious that interchanging the subscripts 1 and 2 docs not31ange thd value of ille double integral: hence an slfirmative answer to thc qitcstiot~ poscd in Eq. (0-125) followvs. This is ;In ini1wrt:lnt conclusion bccnuss it allows us to use the simpler of the two ways (flnding L,, or L,,) to determine the mutual indu~tance.~ ' In circuit theory books the symbol M is frequently used to denote mutual inductance. . . .
( ? I , . , , ', rk b , 9 $ ~ d - ; f . , I . Y 8 r Fig. 6-22 A solenoid with two windings (Example 6-15). - Example 6-15 Two coils of turns N , and N2 are wound concentrically on a straight nonmagnetic cylindrical'core of radius a. The windings have lengths C, and C , respectively. Find the mutuil inductance between the coils. Solution: Figure 6-22 shows such a solenoid with two concentric windings. Assume current I, flows in the inner coil. From Eq. (6-117) we findthat the flux Q I 2 in the solenoid core that links with the outer coil is - - Since the outer coil has N2 turns, we have Hence the mutual inductance is L12 = - - - 5 NI NZxa2. (H). , (6-130) 11 el Example 6-16 Determine ihe mutual inductance between a conducting triangular loop and a very long straight wire as shown in Fig. 6-23. Solution: Let us designate the triangular loop as circuit 1 and the long wire as circuit 2. If we assume a current I, in the triangular loop, it is difficult to find the magnetic flux density B, everywhere. Consequently, it is difficult to determine the mutual inductance L I Z from A, JI , in Eq. (6-1 11). We can, however, apply Amre's circuital law and readily write the expression for B2 that is caused by a current I , in the long straight wire. ly on a i d, and Assume i r h . A big. 6-23 A conducting triahgular loop and a long straight wire (Example 6-16). The flux linkage A,, = @,, is where d s l = a + z , d r . (6-133) The relation between z and r is given by the equation of the hypotenuse of the triangle: l z = - [r - (d + b)] tan 60" = -a [ . - (dm+ b)]. (6-134) Substituting Eqs. (6-1311, (6-133), and (6-134) in Eq. (6-132), we have - - ~ [ ( d + b ) 1 h ( l + ~ - b ] . 2 7 7 Therefore, the mutdd inductance is 6 t (H). (6-135) 6-12 MAGNETIC ENERGY t So far we have dishssed self- and m u t ~ a l ihductances in static terms. Because inductances depend on the geometrical shape and the physical arrangement of the conductors constitutlhg the circuit?, and, for a linear medium, are independent of the 242 STATIC MAGNETIC FIELDS 1 6 currents, we were not concerned with nonsteady currents in the defining of induc- tances. However, we know that resistanceless inductors appeat as short-circuits to steady (DC) currents; it is obviously necessary that we consider alternating currents when the effects of inductances on circuits and magnetic fields are of interest. A general consideration of time-varying electromagnetic fields (electrodynamics) will be deferred until the next chapter. For now we assume quasi-static conditions, which imply that the currents vary very slowly in time (are low of frequency) and that the dimensions of the circuits are very small compared to the wavelength. These condi- tions are tantamount to ignoring retardation and radiation effects, as we shall see when electromagnetic waves are discussed in Chapter 8. In Section 3-1 1 we discussed the fact that work is required to assemble a group of charges and that the work is stored as electric energy. We certainly expect that work also needs to be expended in sending currents into conducting loops and that it will be stored as magnetic en'ergy. Consider a single closed loop with a self- inductance L, in which the current is initially zero. A current generator is Lonnected to the loop. which increases the current i , from zero to I ,. From physics we know that an electromotive forcc (eml) will bc induced in the loop that opposes the current change.+ An amount of work must be done to overcome thisiiitluced emf. Let o, = L, di,/dt be the voltage across the inductance. The work required is Since L, = @,/Il for linear media, Eq. (6-136) can be written alternatively in terms of flux linkage as which is stored as magnetic energy. Now consider two closed loops C1 and C , carrying currents i, and i,, respectively. The currents are initially zero and are to be increased to I , and I,, respectively. To find the amount of work required, we first keep i , = 0 and increase i , from zero to I ,. This requires a work W, in loop C,, as given in Eq. (6-136) or (6-137); no work is done in loop C,, since i, = 0. Next we keep i, at 1, and increase i , from zero to I,. Because of mutual coupling, some of the magnetic flux due to i2 will link with loop C,, giving rise to an induced emf that must be overcome by a voltage v,, = L,, di,/dt in order to keep i, constant at its value I , . The work involved is At the same time, a work W,, must be done in loop C , in order to counteract the induced emf and increase i , to I,. WZ2 = 4L21$. (6-139) ' The subject of electromagnetic induction will bc diadussed in Chnpter 7. ics) will a t ,- r, which that the : condi- ' ( hall see \ 3 group :cl t h ; ~ t r ~ c l that a self- nnccted ow hhat c u r 7 7 rt t', - ;I terms xtlvely. :ely. To 70 to I,. work is 0 to 1,. 30p c1, diJdt i , ; : - ?rj3. .. -The total amount of kork done in raising'the dlrfents in loops Cl and C2 from zero to I, and I,, respectitrely, is then the sum of wj Wz,, and W2,. ' C , k2 = f L J : + L ~ J ~ ~ ~ + fL21: Generalizing this reiplt to a system of N lo&s carrying currents I,, I,, . . . . I,,, we obtain which is the energy stbretl in the magnetic field. For a current I flowing in a slnglz inductor with inductahce L, the stored magnetic energy is It is instructive to1 dclivc Eq. (6-141) in an altcrnativc way. Conbider a typlcal " kth loop of N magnetically coupled loops. Let v, and i , , be respectively, the voltage across and the current in the loop. The work ddne to the kth loop in time tlt 1 s where we have used the relation u, = dlp,/dr. Note that the change. dg,. in the flux +k linking with the kth loop is the result of the changes of the currents in all the coupled loops. The djfferential work done to, or the differential magnetic energy stored in, the system h then ! ' N N I dWm = 1 dWA = i,d+,. (6 - 143) k = 1 k = 1 The total stored energy is the integration of rlI.Yv, and is independent of the manner in which the final valttes of the currents and fluxes are reached. Let us assume that all the currcnts and ll,&txcs are brought to thcir final values in concert by an equal fraction cc that increases from 0 to 1; that is, i , = crl,, and 4, = a@, at any instant of time. We obtain the .tdtal tnag~telicr energy: N 244 STATIC MAGNETIC FIELDS 1 6 which simplifies to Eq. (6-137) for N = 1, as expected. NO^^ that, for linear media, N 3 - @k = L,kI,, we obtain Eq. (6-141) immediately. . F 6-12.1 Magnetic Energy in Terms of Field Quantities Equation (6-145) can be generalized to determine the magnetic energy of a continuous distribution of current within a volume. A single current-carrying loop can be con- sidered as consisting of a large nimber, N, of contiguous filamentary current elements of closed paths C , , each with a current AI, flowing in an infinitesimal cross-sectional area Aa; and linking with magnetic flux a , . 1 where S, is the surface bounded by C,. Substituting Eq. (6-146) in Eq. (6-145), we have Now, As N + a, Av', becomes dv' and the summation in Eq. (6-147) can be written as an integral. We have I Wm = f 6, A J dv' (J), ( (6 - 148) where V' is the volume of the loop or the linear medium in which J exists. This volume can be extended to include all space, since the inclusion of a region where J = 0 does not change W , . Equation (6-148) should be compared with the expression for the electric energy We in Eq. (3-140). It is often desirable to express the magnetic energy in terms of field quantities . B and H instead of current density J and vector potential A. Making use of the vector identity, V ~ A X H ) = H . ( v x A ) - ' A ~ V x H), (see Problem P.2-23), we have A . ( V x H ) = H - ( V x A ) - V . ( A x H ) or A . J = H . B - V ( A X H). (6-149) volume x J = O sion for 4 [:I , ; . , l' 6-12 / MAGNETIC ENERGY 245 ; 1 5 , substitution ,of ~ h . (4-14$ in Eq. (6-148), we Obtain 4 4~ = 4 yv, H B du' - n t 6 . 1 ~ X H) . a. ds'. (6-150) 11 , " i In Eq. (6-150) we hare applied the divergence theorem, and S . is the surface boundlng V'. If V' is taken to be sufficiently large, the pdints on its surface S' will be very far from the currents. ~b those far-away points, tHe contribution of the surface integral in Eq. (6-150) tends. to zero because Blls bfT as l/R and /HI falls OK as 1/R2, as can be seen from E ~ $ (6-22) and (6-31) respectively. Thus, the rnagnirude of (.4 x H) decreases as 1 / ~ ' , vqhereas at the same time, the surface Sf increases only as R2 When Rapproaches ihfinity. Illcsarf:~;~ i n ~ c p ~ ~ ; ~ l i l l EL,. (6 - 150) Y:II~~SIICS. \lit hii\c 1lic11 Noting that H = B/p, we wn write Eq. (0-151h) in two nlternatiile form,: and The expressions in E ~ S . (6- lSla), (6-151 b), and (6-15 lc) for the magnetlc energy W, in a linear medium are analogous to those of electrostatic energy U: in, reipcc- tively, Eqs. (3-146a), (3 -l46b), and (3-146c). a If we define a m g n e r k energy demiry, w,, such that its volume integral equals the total magnetic energy , we can write w,,, in tHtee diiferent forms: w,,, " F-I . B (J/m3) or By using Eq. (6-142), w can often determine self-inductance more easily from stored magnetic energy calculated in terms of B and/or H, than from flux linkage. 246 STATIC MAGNETIC FIELDS / 6 . . We have . , . , . I Example 6-17 By using stored magnetic energy, determine the inductance per unit length of an nir coaxial transmission line that his a solid inner conductor of rtldius e and a very thin outer conductor of inner radius h. Solution: This is the same problem as that in Example 6-14, where the self-induc- tance was determined through a consideration of flux linkages. Refer again to Fig. 6-21. Assume a uniform current.1 tlows in the inner conductor and returns in the outerconductor. The magnetic enprgy per unit length stored in the inner conductor is. from Eqs. (6- 120) and (6- 15 1 b), 1 W;, =-J," B;, 2nr,lr 3 1 0 - -1 - J : r3 dr = - 4na4 '(o" (Jim). . 16n (6-155a) The magnetic energy per unit length stored in the region between the inner and outer conductors is, from Eq. (6-121) and (6-151b), 1 Wk2 = - [ ~;,2nr dr 2'(0 - pol2 b - g [ ; d r = - - I n - (J/m). 4n 4n a (6 - i 5513) Therefore, from Eq. (6-154), we have 2 L ' = - (Wm, + Wk,) I2 which is thc same as Eq. (1-124). Tli: pruccdurc used in this solution is compnr;llively simpler. MAGNETIC FORCES AND TORQUES In Section 6-1 we noted that a charge q moving with a velocity u in a magnetic field with flux density B experiences a magnetic force F,,, given by Eq. (6-4), which is repeated below: F, = qu x I 3 (N). (6- 156) , , .( , , . . . , ... . , , ~. I , .
. ' unit I ~ U C - ) Fig. n the tor is. r' 155. I out dP with a rross-sectional area S. If there are u in the direction df dP, ,' (6- 157) where 9, is the charge bn ea,ch charge carrier. The two expressions in Eq. (6-157) are equivalent since u d)d cl< have the same dirwtion. Now, since NqlSlul equals the current in the condiietor,,,we can write E ~ . (6-157) as 1. The magnetic force on a conlplete (closed) circuit of contour C that carrles a current ' I in a magnetic field B ill then (6- 159) 1 . - - - - - - - - . When we have two circuits carrying currents Ir and I , respectively, the situatmn is that of one current-cafryingcircuit in the magnetic field of the other. In the presence of the magnetic field Bll, which was caused by the current I, in C , , the force F,, on circuit C, can be written + I I where B,, is, from the hiot-bavart law in Eq. (6-31), Combining Eqs. (6-1608) and (6- 160b), we obtain which is AmpPre'slaw o j force between two current-carrying circuits. It is an inverse- square relationship and should be compared with Coulomb's law of force in Eq. (3- 17) between twaaationary charges, the latter being much the simpler. The force ~y;on circuit C , , in the presence of the magnetic field set ua br the . . current I , in circuit C , , can be written from Eq. (6-16fa) by interchanging the subpipts 1 and 2. , 1 248 STATIC MAGNETIC FIELDS 1 6 I . . . I , However, since dt', x (dl', p a.,,) # -dCl x.(d& x a.,,), we inquire whether this means Fzl # -F,,- that is, whcther Nowton's third law governing thc l'orcrs of action and reaction fails here. Let us expand the vector triple product in the integrand of Eq. (6-161a) by the back-cab rule, Eq. (2-20). Now the double clbsed line integral of the first term on the right side bf E ~ . (6-162) is In Eq. (6-163) we havemade use ofEq. (2-81) and the relatio3q.(l/R2,)= - aR2,/RZl. The closed line integral (with identical upper and lower limits) of d(l/R,,) around circuit C1 vanishes. Substituting Eq. (6462) in Eq. (6-161a) and using Eq. (6-163), we get Po FZ1 = -- ' dP2) 4 7 1 l 2 c c R:, (6- 164) which obviously equals - F , , , inasmuch as a,,, = -aR2,. It follows that Newton's ihird law holds here, as expected. Example 6-18 Determine the force per unit length between two infinitely long parallel conducting wires carrying currents I, and I , in the same direction. The wires are separated by a distance d. Solution: Let the wires lie-in the yz-plane and be parallel to the z-axis, as shown in Fig. 6-24. This problem is a straightforward application of Eq. (6-160a). Using F;, to denote the force per unit length on wire 2, we have .- F;2 =I,(% x BIZ), (6-165) where B12, the magnetic flux density at wire 2. set up by the current I, in wire 1 , is constant over wire 2. Because the wires are assumed to be infinitely long and cylin- drical symmetry exists, it is not necessary to use Eq. (6-160b) for the determination , , . . of BIZ. We apply Amptre's circuital law, and write, from Eq. (6-ll), 4- around 6-I/ :ly lpng 2e wires i . .2" , . , . 6-13 1 %GN~TIC FORCES AND TORQUES 249 . . . , : .
; ; ! x ' ! A 3 .. , Fig. 6-24 Force bitween two parallel I current-carrying wires (Example 6-18). Substitution of Eq. (6-16t) in Eq. (6-165) yields I We see that the force on w~re 2 pulls it toward wire 1. Hence, the force between two . wires carrying currents i u the some direction is one of attractioll (unl~ke the force between two charges df the same polarity, which is one of repulsion). It is trivial to prove that F;, = - f l : i 2 - - a,(p,1~1,/2nd) and that th.e force between two wires carrying currents in opeosite directions is one of repulsion. f Let us now considir a small circular loop of radius b and carrying a current I in a uniform magnetic field, of flux density B. It is convenient to resolve B into two com- ponents, B = B, + Blitwhere B, and Bll are,,respectively, perpendicular and parallel to the plane of the loo& As illustrated in Fig. 6-25a, the perpendicular component B, tends to expand the lobp (or contract it, if the direction of I is reversed), but B, exerts nc net force to move tlie loop. The parallel component Bll produces an upward force (a) (W Fig. 6-25 A circular loop in a uniform magnetic field B = B, + BIl. 250 STATIC MAGNETIC FIELDS 1 6 i ' I 1 1 1 dF, (out from the paper) on element dt, and a downward force (into the paper) dF2 = -dFl on the symmetrically located element d t 2 , as shown in Fig. 6-25b. I > Although the net force on the entire loop caused by BII is also zero, a torque exists I that tends to rotate the loop about the x-axis in such a way as to ulign the magnetic I field (due to I) with the external BII field. The differential torque produced by dF, and dF, is I ! dT = ax(dF) 2b sin 4 = a,(I dt BII sin 4j2b sin 4 = a,21b2BII sin2 4 d4, (6-168) If the definition of thc rnagnctic dipolc momcnt in Eq. (6--46)-is~!;cd, where a, is a unit vector in the direction of the right thumb (normal to the plane of the loop) as the fingers of the right hand follow the direction of the current, we can write Eq. (6-169) as The vector B (instead of B,,) is used in Eq. (6-170) because r n x (BL + BII) = rn x Bll. This is thc torque that aligns thc microscopic mclgnclic dipolcs in magnctic matcrials and causes the materials to be magnetized by an applied magnetic field. It should be remembered that Eq. (6-170) does not hold if B is not uniform over the current- carrying loop. Example 6-19 A rectangular loop in the xy-plane with sides b, and b, carrying a current 1 lies in a unijorm magnetic field B = a,B, + ayBy + a$,. Determine the force and torque on the loop. Solution: Resolving B into perpendicular and parallel components B, and B,,, we have BL = a$, (6-171a) BII = a,B, + ayBy. (6-171b) Assuming the current flows in a clockwise direction, as shown in Fig. 6-26, we find that the perpendicular component a$, results in forces Ib,B, on sides (1) and (3) and 8 : paper) I I 6-25b. - , i ' IC exists , .
, lagnetic a by dF, ' ( ' i I m x Bil. naterials lould be current- .rrying a the force . we find d (3) and - Fig. 6-26 A rectangular loop in a ,, uniform ~nagnetlc field (Example 6-19). i forces Ib,B, on sides (2) and (41, all directed toward the center of the loop. The vector sum of these four contracting forces is zero, and no torque IS produced. The parallel combonsnt of the magnetic flux denslty. Bll, produces the following ' forces on the four sidb: Again, the net force bn the loop, F, + F, + F,'+ F,, is zero. However, these forces result in a net torque' that can be computed as follows. The torque TI,, due to forces ' F, and F, on sides (1) and (3), is TI, = axIblb2B,; (6- 173a) the torque T2,, due to forces F, and F, on siddk (2) and (4), is T2, = -a,lb,blBx. , The total torque on the rectangular loop is, the4 1 Since the magnetic(domenl of the loop is r n = - aJb,b,, the result in Eq. (6- 174) isexactly T = r n % (a$x -j- aYBJ = r n x I % en&, in spite of the fact that Eq. (6- 170) was derived for a iircular loop, the torque fodula holds for a p!anar loop of any shape as long as it i s lbcated in a uniform magnetic field. , . 6-13.1 Forces and Torques in Terms of . . a .
. Stored Magnetic Energy All current-carrying conductors and circuits experience magnetic forces when situated in a magnetic field. They are held in place only ifmechanical forces, equal and opposite to the magnetic forces, exist. Except for special symmetrical cases (such as the case of the two infinitely long, current-carrying, parallel conducting wires in Example 6-18), determining the magnetic forces between current-carrying circuits by Ampere's law of force is a tedious task. We now examine an alternative method of finding magnetic forces and torques based on the principle o f virtual displacement. This principle was used in Section 3-1 1.2 to determine electrostatic forces between charged conductors. Here consider two cases: first, a system of circuits with constant magnetic flux linkages; and, second, a system of circuits with constant currents. , System of Circuits with Constant Flux Linkages If we assume that no changes in flux linkages result from a virtual differential displacement d f of one of the current- carrying circuits, there will be no induced emf's and the sources will supply no energy to the system. The mechanical work, F , df. done by the system& at the expense of a decrease in the stored magnetic energy, where F, denotes the force under the constant flux condition. Thus, F , . d f = -dW,= -(VWm).df, from which we obtain pzGk7q In Cartesian coordinates, the component forccs arc If the circuit is constrained to rotate about an axis, say the z-axis, the mechanial . work done by the system will be (T,), dm and (TaL = - - (6-178) X . , . which is the z-component of the torque acting on the circuit under the condition of constant flux linkages. indirl$ . Thid ' I large8 t , ~gnetik! anical n dition b Fig. 6-27 An eldttromagnet (Example 6-20). ! Example 6-20 Codbider the electromagnet in Fig 6-27 where a current I in an N-turn coil producck a llux O in thc magnetic circuit. Thc cross-sectional area of thc core is S. Dctcrdinc 1.11~ lifting forcc on the armature. Solution: Let the armature take a virtual diyplacement ily (a differential increase . in y) and the source.be adjusted to keep the llux (b constant. A displaccment of the armature changes oqly the length of the air aps; consequently, the disphcement changes only the magnetic energy stored in t e two air paps. We have. from Eq, (6-151b), 1 b An increase in the length (a positive dy) increases the stored magnetic energy if (r, is constant. ~ s i t l i Eq. ,(6-177b), we obtain Here, the negative sign indicates that the force tends to reduce the air-gap length: that is, it is a force 01 attraction. -. I' --I - Systcm of Circuits w l L Constant Currcnts In this case the circuits arc connected to current sources that :coullter:ict the induced emf's resulting from changes in flux linklgcs that are caUsed by a virtual displaccment dP. The work done or energy supplied by the sour& is (see Eq. 6-144), - 254 STATIC MAGNETIC FIELDS / 6 . . This energy must be equal to the sum of the mechanical work done by the system dW (dW = F, -dt, where F, denotes the force on the displaced circuit under the - constant current condition) and the increase in the stored magnetic energy, dW,,. That is, dW, = dW + dWm. From Eq. (6-145), we have Equations (6-182) and (6-183) combine to give dW.= F,. d e = dWm , = (vw,) dt or which differs from the expression for F, i .
n Eq. (6-176).only by a sign change. If the circuit is constrained to rotate about the z-axis, the z-component of the torque acting on the circuit is The difference between the expression above and (T,), in Eq. (6-178) is, again, only in the sign. It must be understood that, despite the difference in the sign, Eqs. (6-176) and (6-178) should yield the same answers to a given problem as do Eqs. (6-184) and (6-185) respectively. The formulations using the method of virtual displaccment unclcr conscant llux linkagc :mcl conhtanl currcrlt col~ditions arc simply two means of solving the same problem, Let us solve the electromagnet problem in Example 6-20 assuming a virtual displacement under 'the constant-current condition. For this purpose, we express W m in terms of the current I : W, = ~ L I ~ , (6-186) where L is the self-inductance of the coil. The flux, @ , in the electromagnet is obtained by dividing the applied magnetomotive force (NI) by the sum of the reluctance of the core (9,) and that of the two air gaps (2y/p,S). Thus, . If the tori 6-185) n. only 6-176) 6-184) cement means , , virtual rcss Wm 1 " ~nduct&e L ib'kqtt~1 tdkux linkage per ynidcurrent. 1 2 .; r: N@ ' s ' ,N2 \ 4 ; L=-= "! I with currents I, and I , , self-inductances L, and L2, and mutual inductance t,,, the magnetic energy is, from Eq. (6-140), I: W, = +L,I: +'L,,I~I, + $L21:. (6-190) If one of the circuiti is given a virtual displacement under the condition of constant currents, there wodd be a change in Wm and E q (6-184) applies. Substirution of Eq. (6-190) in Eq. (6-184) yields 0 .
1 Similarly, we obtain from Eq. (6-185), Example 6-21 Determine the Torce between two coaxial circular coils of radii b, m d 6 , ~ c p i i r i ~ t ~ l hy iI ~iiisi~~nrn 11 which is m ~ c h Iiwgcr ~ I ~ : I I I tllc radii ( t i >> b,, h,). , I , IIC coils &:;)llsisl of N ill14 N 2 c10scIy woudd turns and carry currents I , and 1, rcspcctivcly, Solution: This problem is rather a difficult one if we try to solve it with Ampere's law of force, as exdressed in Eq. (6-1614. Therefore we will base our solution on Eq. (6-191). First, W e determine the mutual inductance between the two coils. In Fig. 6-28 Coaxial current-carrying circular loops (Example 6-21). - I -.
- Example 6-7 we found, in Eq. (6-43), thavector potential at a distant point, which was caused by a single-turn circular loop carrying a current I. Referring to Fig. 6-28 for this problem, at the point P on coil 2 we have A,, due to current I , in coil 1 with N , turns as follows: PoNiI1b: 4412 = a, ----- sin 8 4R2 In Eq. (6-193), z, instead of d, is used because we anticipate a virtual displacement, and : is to be kept as a variable for the time being. Using Eq. (6-193) in Eq. (6-24), we find the mutual flux. The mutual inductance is then, from Eq. (6-Ill), I . i' REVIEW QUESTIONS 257 ' i I: On coil 2, the fotc+dud to the magnetic field of b i l 1 can now be obtained directly by substituting Ed: 16-195) in Eq. (6-191). $ ' which can be written ad The ncgative sign in Eq, (6-196) indicates that I ; , , is a forcc of attraction Tor currents flowing in the same directioa. This force diminishes very rapidly as the inverse fourth power of the distance of separation. REVIEW QUESTIONS R6-1 What is ths.expression for the force on a test charge y that moves wrth vrloc~ty u in a magnetic field of flux density B? H.6-2 Vsrify that tesla (T), the unit for magnetrc flux denslty, is the same as volt-xcond per square meter (V.s/niZ). R.6-3 Wrrte Lorentz's force equation. R.6-4 Which postulate bf magnetostatics denies the existence of isolated magnetic charges? R.6-5 State the law of cthservation of magnetic flux. R.6-6 State Ampere's chcuital law. R.6-7 In applying Anlpeh's circuital law nlust the path of integration be circular? Explain. R.6-8 Why cannot the B-field.of ad infinitely long straight, current-wrrying conductor have n componolt in thc dircctlon or the currcnl? R.6-9 DO the foriulas for R. A s derr;h in Eqr (6-lp) and (6-11) for .
L round conductor, apply 10 co~~ductor Lavillg il UllliM crobh SIX~IOII of the same ilrea rild carryrng the ramr currcnt? ~ x ~ l n i 2 ' - - H.6-10 in what niasdcr does tllc U-lield of an rnlinitely long stnight filament urrying a drren current I vary with distance? R6-11 Can B-field exist b a p o d conductor? Elplain. R.6-12 Define in words udrtor magnetic potential A. What is its S1 unit? 258 STATIC MAGNETIC FIELDS / 6 \ R6-13 What is the relation between magnetic flux density B and vector magnetic potential A? Give an example of a situation where B is zero and A is not. R6-14 What is the relation between vector magnetic potential A and the magnetic flux through a given area? R.6-15 State Biot-Savart's law. R.6-16 Compare the usefulness of Arnpkre's circuital law and Biot-Savart's law in determining B of a current-carrying circuit. R6-17 What is a magnetic dipole? Define magnetic dipole moment. R6-18 Define scalar magnetic potential Vm. What is its SI unit? R6-19 Discuss the relative merits of using the vector and scalar magnetic potentials in mag- netostatics. r R6-20 Define magnetization vector. What is its SI unit? R.6-21 What is meant by "equivalent magnetization current densities"? What are the SI units for V x M and M x a,? -1 . R.6-22 Define magneticfield intensity vector. What is its SI unit? R6-23 Define magnetic susceptibility and relati& permeability. What are their SI units? R.6-24 Does the magnetic field intensity due to a current distribution depend on the properties of the medium? Does the magnetic flux density? R.6-25 Define magnetomotive force. What is its SI unit? R6-26 What is the reluctance of a piece of magnetic material of permeability p, length C, and a constant cross section S? What is its SI unit? R.6-27 An air gap is cut in a ferromagnetic toroidal core. The core is excited with an mmf of N1 ampere-turns. Is the magnetic field intensity in the air gap higher or lower than that in the core? R.6-28 Define diamagnetic, paramagnetic, and ferromaynetic materials. R6-29 What is a magnetic domain? R6-30 Define remanent j4ux density and coercive field intensity. R6-31 Discuss the difference between soft and hard ferromagnetic materials. . R6-32 What is curie temperature? R6-33 What are the characteristics of ferrites? R.6-34 What are the'boundary conditions for magnetostatic fields at an interface between two $jp , y i , . different magnetic media? R.6-35 Explain why magnetic flux lines leave perpendicularly the surface of a ferromagnetic medium. I I - R.. at th. R.I slr R.c tur R.I thl R.c R. I in - R.r St[ A. ( cu: CO' PROBLEV \ P.6 reg ch. P.6 fic~ Dl / 1 P. t. 4 the f 'OU - pic P .
C an to ' ,.,. . , . . , . , .
: . 1. , , ',':,.: : potential : :.;;j,jj::.;;1, ' , ; , .
, $ ; ' , :. . . . . . ', , , '" , . '; ' ,\ .. :. : 1 ; 4 j : : {,-)$ t ~ l ~ o u y h ,;..!; 2 : ' 6 <: L~ ! ; . , 'j . i i.? ! ; ; . 1 : : !! , . , ' , ' , ; . ,> . . . , , , .?;'',!. ' I . 8 , , , . : S1 units th P. and n mmf of nat in the magnetic PROBLEMS 259 ' \ , ; 4 ! 1, R6-36 What boundaryi condi!ion must the tangenVal components of magnetization satisfy , at an intcrfacc? If regiorl2 i6 n:onmaynctic, what is the relation betwcen thc surface current and thc tanycntial compbac~t of M I ? . R.6-37 Define (a)'the mhtual j?ductance betweka two circuits, and (b) the self-inductance or a single coil. , . . $ , % : R.6-38 Explain how the,.self-hd~lctance of a wire-wd~nd inductor depends on ~ t s number of turns. b, i , I R.6-39 In Example 6-f$, would the answer be ;he dame if the outer conductor is not "very thin"? Explain. 1 i 4 R.6-40 Give an expression oi magnetic energy In terms of B dndlor H. I R.6-41 Clve thc integral exprebslon for the force on a closed clrcult thd c m x s d current 1 In a magnetic field B. R.6-42 Discuss first the net force and then the net torque acting on a current-carrying cllcult situated in a uniform mlgnetq field. R6-43 What IS the reladon tztween the force and the stored magnetic energy In system of current-carrymg c~rcuits dnder the condition of constant flux linkages? Under the condmon of constant currents? 1 PROBLEMS P.6-1 A positive point charge q of mass m 1 s Injected with a velocity u, = a,u, into the J > 0 region where a uniform magnetic fleld B = a,B, exists. Obtarn the equat~on of motion of the charge, and describe the pdth that the chargc follows. I P.6-2 An electron is injected with a velocity u , = a,ilt, into a region where both an electrlc field E and a magnetic field B exist. Describe the motion of the electron if a) E = a,E, and B = a$,, b) E = -a,E, and B = -a,B,. Discuss the effect of the relative n~agnitudes of E, and B, on the electron paths of (a) and (b). I P.6-3 A current I Ilows it) (lie inncr I A ~ I W I O I - (,~:!II idillrlcly long co:i\i.~I Iinc atld roturnb wa lhc 0 \ 1 1 ~ ~ L X ~ I I ~ U ~ I ~ . , 'I'hc I&~IIIS oI' LIIC IIIIICT CL)~~LIC'IOT is 1 1 , and the Illlicr and outer radil of the O U ~ C I ' L ' O I ~ ~ I C ~ W L& 1) I I I \ ~ e ~ ~ c h ~ ~ c l ~ ~ c l y , I ; ~ I I ~ 111e t ~ u ~ g l i c k Ihu clcudy 11 Vor ,dl ucgionh ;mu plot I B I versus r. \ P.6-4 Determine the magnetic flux denslty at a point on the axis of a solenold with radius h and length L, and with a current 1 in its N turns of closely Wound coil. Show that the result reduces to that given in Eq. (6-13) when t approaches infinity. 260 STATIC MAGNETIC FIELDS 1 6 . . , , , P-6-5 Starting from the e ~ p r e s ~ i o n for vector magnetic potential A in Eq. (6-22), prove that Furthermore, prove that V . B = 0. J P.6-6 Two identical coaxial coils, each of N turns and radius b, are separated by a distance d, as depicted in Fig. 6-29. A current I flows in each coil in the same direction. a) Find the magnetic flux density B = axBx at a point midway between the coils. b) Show that dB,.dx vanishes at the midpoint. c) Find thc relation between h and (1 such that r12B,/rl.~2 also vanishes at the rnidnoint . r - --- Such a pair ufcuils arc used to obtain an appruxi~l~ately uniform magnetic field in the midpoint region. They are known as Heli~rholt~ coils. d (Problems P.6-6). dP.6-7 A thin conducting wire is bent into the shape of a regular polygon of N sides. A currcnt I flows in the wire. Show that the magnetic flux density at the center is !JON[ n . B = a , - tan -, 2nb N where b is the radius of the circle circumscribing the polygon and a, is a unit vector normal to the plane of the polygon. Show also that as N becomes very large this result reduces to that given in Eq. (6-38) with z = 0. P.6-8 Find the total magnetic flux through a circular toroid with a rectangular cross section of height h. The inner and outer radii of the toroid are u and b respectively. A current I flows in N turns of closely wound wire around thc toroid. Determine the percentage of error if the flux is found by multiplying the cross-sectional area by the flux density at the mean radius. P.6-9 in certain cxpcriments it is dcsirshle to havc a rcgion of constant magnetic [lux density. This can be created in an off-center cylindrical cavity that is cut in a very long cylindrical conductor I ' we that . . ' / , ! -1 (6-197) , , , : ' tmce (i, 1 current s to that il A density. I .onductor !,, carrying a uniform current density. Refer to the cross section in Fig 6-30. The uniform i~rial currcnt density is .I : - ;I,J: Find [hc in;~yilil~~dc xnd dirCcii011 of U ill ihc vyiindiicll cavity who axis is displaced from that of the conducting part by a distance d. (Hint: Use principle of super- position and consider B i n thecavity as that due to tdo long cylindrical conducton with radii b and a and current densities J and -J respectively.) - , P.6-10 Prove the following: t p.6-12 Startrng from the expressron of A In Eq, (6-34) for the vector magnet~c potent~al at a point in the brsecting ~lafle of 8 straight wlre of length 2L that carrles a current 1 a) Find A at pomt k x , y.0) in the blsectrng plane of two parallel r ~ r e s each of length 21, located at y = &!/2 and carrylng equal and obpos~te currents, as shorn in Fig. 6-31. b) Flnd A due to ecjbal and opposite currents m n very long two-w~re transm~ss~on line. c) Find B from A in part (b), and check your answer against the result obtarned by ~pplying Ampkre's circuital law. Fig. 6-31 parallel wires carrying equal and apposite currents (Problem P.6-12). 262 STATIC MAGNETIC FIELDS I 6 P.6-13 For the small rectangular loop with sides a and b that carries a current I, shown in Fig. 6-32: a) Find the vector magnetic potential A at a distance point, P(.u, y,d. Show that it can be put in the form of Eq. (6-45). b) Determine the magnetic flux density B from A, and show that it is the same as that given in Eq. (6-48). --- -- P.6-14 For a vector field F with continuous.first derivatives, prove that & ( v x ~ ) d u = - $ , F ~ L , where S is the surface enclosing the volume V. (Hint: Apply the divergence theorem to (A x C), where C is a constant vector.) P.6-15 A circular rod of magnetic material with permeability p is inserted coaxially in the long solenoid of Fig. 6-4. The radius of the rod, a, is less than the inner radius, b, of the solenoid. The solenoid's winding has n turns per unit length and carries a current I. a) Find the values of B, H, and M inside the solenoid for r < a and for a < r < b. b) What are the equivalent magnetization current densities J, and J,, for the magnetized rod'? P.6-16 The scalar magnetic potential, Vm, due to a current loop can be obtained by first dividing thc loop area into many small'loops and then summing up thc contribution of thcse small loops (magnetic dipoles); that is, where dm = a,I ds. (6-198b) Prove, by substituting Eq. (6-198b) in Eq. (6-198a), that where R is the solid angle subtended by the lbop surface at the field point P (see Fig. 6-33). ) (A x C), i the long no~cl. The dividing :;lit loops i i I \ PROBLEMS Fig. 6-33 Subdivided current loop for determination of scalaf magnetic potential (Problcm P.6-16). P.6-17 DO the following by ilsing Eq. (6-199): a) Determine the scalar magnetic potential at a point on the axis of a circular loop having radius h and carryinb' a current I. b) Obtain the maghetic flux density B from -Po OK,, and compare the rcsult with Eg. (6-38). P.6-18 A ferromagnetikpphere of radius b is magnetized uniformly wlth a magnetlation M = azM0. a) Determrne the equivalent magnetlzatlon current densltles J , m d J , b) Dctcrmlnc the nlagnclic llux densrty at the center of the qhere. P.6-19 A toroidal lron core of relative permeab~l~ty 3000 has a mean radlus R = 80 (mm) m d .i c~rcular cross section wlth rrd~ils b = 25 (mm). An .ur gJp fU = 3 (I,,) exlatr, J I I ~ .I cui rc11, i ilou \ in a 5W-turn wlndlng to poducea lnagnetlc flux of (Wb). (See Rg. 6-34.) Neglecting leakage Pig. 6-34 A toroidal iron core with air gap (Problem P.6-19). 264 STATIC MAGNETIC FIELDS 1 6 and using mean path length, find a) the reluctances of the air gap and of the iron core. b) B, and H , in the air gap, and B, and H, in the iron core. C) the required current I. P.6-20 Consider the magnetic circuit in Fig. 6-35. A current of 3 (A) flows through 200 turns of wire on the center leg. Assuming the core to have a constant cross-sectional area of (m2) and a relative permeability of 5000: a) Determ~ne thc magnctlc llux in cach leg. b) Dctcrrninc thc magnetic liclcl intensity ill c;~ch Icg of l l i c core i111tl ill the itlr gi~p. Fig, 6-35 A magnetic circuit with air gap (P~oblem P.6-XI), P.6-21 Consider an infinitely long solenoid with n turns per unit length around a ferromagnetic core of cross-sectional area S. When a current is sent through the coil to create a magnetic field, a voltage v , = - 1 1 dO/dt is inducod pcr unit length, which opposes the current change. Power P, = -v,I per unit length must be supplied to overcome this induced voltage in order to In- crease the current to I. a) Prove that the work per unit volume required to produce a final magnetic flux density Bj is W, = J O E ' H dB. (6-200) b) Assuming the current is changed in a periodic manner such that B is reduced from BJ to -B, and then is increased again to BJ, prove that the work done per unit volume for such a cycle of chayge in the ferromagnetic core is represented by the area of the hysteresis loop of the core material. P.6-22 Prove that the relation V x H = J leads to Eq. (6-99) at an interface between two media. P.6-23 What boundary conditions must the scalar magnetic potential Vm satisfy at an interface between two different magnetic media? P.6-24 Consider a plane boundary (y = 0) between air (region 1, p,, = 1) and iron (region 2, P a = 5000). a) Assuming B, = a,0.5 - aJ0 (mT), find B, and the angle that B, makes with the interface. b) Assuming B, = a,10 + a,OS (mT), find B, and the angle that B, makes with the normal to the interface. P.6-25 The method o f images can also be applied to certain magnetostatic problems. Consider a straight thin conductor in air parallel to and at a distance d above the plane interface of a magnetic material of relative permeability p,. A current I flows in the conductor. I . , 6 ~ 1 turns , / -' ~yllctlc IC licid. Power to 111- density ,6-200) rom L ?
, wlume I of the I media. nterface ! ~terfx. normal hsider :riace of and these cuhents ~irc equidistant from the interface and situated in alr, ii) the magnet~dfield below the boundary plane is calculated from I and -I,, both at the same lodtion! These currents ari sitbated in an infinite magneuc material of relative perx&bility p,. < b) For a long m d b c t o r carrying a current I ahd for p? > > 1 , determine the magnetic flux density B at thejoint P in Fig. 6-36. - ' P k Y ) I@ t I d C 0 Fig. 6-36 A current-carrying conductor Ferromagnetic rncd~pm ( h 2 - 1 ) , near a ferromagnetlc medium (Problem P 6-25). I'.6-26 Dctcrmme the 811-inductance of a toroidrl coil of N turns of wlrc wound on an ilr frame with mcan radius 'ro and a circular cross section of radius b Obtain an rpproxlmate ex- pression assuming b < < r,, P.6-27 Refer to ~ x a r n & 6-13. Dctcrminc lhc inductmcc pcr unit icngtii of tile sir coaxial transmission line anumibg that its outer conqu&r is dot very thin but is of a thickness d. V P.6-28 Calculate the int$tnal and external inductances per unit length ofa two-wire transmission line consisting of two lohg parallel conducting wires of radius n that carry currents in opposite directions. The wires are separated by an axis-tohis distance d, which is much larger than a. "P.6-29 Determine the inductance between a very long straight wire and a conducting equilateral triangular ~odb, as shown Fig, 6-37 A long straight- wire and a conducting equilateral triahgular loop (Pr'oblem P.6-29). 266 STATIC MAGNETIC FIELDS 1 6 Fig. 6-38 A long straight wire and a conducting circular loop (Problem P.6-30). ' P.6-30 Determine the mutual inductance between a very long straight wire and a conducting circulx loop, as shown in Fig. G -.is. . P.6-31 Find the mutual inductance between two coplanar rectangular-bo~with parallel sides, , -. as shown in Fig. 6-39. Assume that 11, > > / i , ( l ~ , > w2 > ti). Fig. 6-39 Two coplanar rectangular loops, h , > > h2 (Problem P.6-31). P.6-32 Consider two mupled circuits, having self-inductances L, and L,, that carry currents . I , and I , respectively. The mutual inductance between the circuits is M. -. a) Using Eq. (6-140), find the ratio 1,/12 that makes the stored magnetic energy W2 a mirimum. b) Show that M 5 a. P.6-33 Calculate the force per unit length on each of three equidistant, infinitely long parallel wires 0.15 (m) apart, each carrying a current of 25 (A) in the same direction. Specify the direction of the force. \i P.6-34 The cross section of a long thin metal strip and a parallel wire is shown in Fig. 6-40. Equal and opposite currents I flow in the conductors. Find the force per unit length on the conductors. currents :gy W, a n , pF' " J I ~ c ~ . _. ig. 6-40. .h on the .
.
I . .
I " ' 4 , < PROBLEMS 267 1 - - 1 palkillel strip and wire ' codduct& (Problem P.6-34). \/ P.6-35 Refer to robl lei 6-30 and Fig. 6-38. ~ i h d the force on the crrcular loop that IS exerted by the magnetic field du& to an upward current I, in the long straight wire. The circular loop carries a current I, in th~'counterc1ockwise direction. "6-36 Assuming the circulal loop in Problem P.6-35 a rotated about its horlaontal axis bj an angle a, find the torque exerted on the circular :oop. P.6-37 A small circular !urn of wire of radius i, that carrier a steady current I , is @aced at the center of a much larger turn:of wire of radius r, (r, s r,) that carries a steady current 1: in the same direction. The angle between the normals of the two circuits is 0 and the small circular wire is free to turn about its diameter. Determine the rni~gnitude anci the ~iirnc~ioli of ihc torquc ( 1 1 1 I ~ I C SIT~:III sirct~I:~r wire, . . P.6-38 A magnetized coinpass needle will line up with the earth's magnetis licld. A srnilil bar magnet (a magnetic dipole) with s magnetic moment I (A.m2) is placed st a disrmce 0.15 (m) from the center of a cornp& needle. Assuming the earth's magnetic flux density at the needle to be 0.1 (mT). find the maximum angle at which the bar magnetecan cause the needle to dcuiatc from the north-south direction How should the bar magnet be oriented? P.6-39 The total mean length of the llux path in iron for the electromagnet in Fig. 6-27 is 3 (m) and the yoke-bar contaci areas measure 0.01 (m2). Assuming the permeability of iron to be 40%0 and each of the air gaps to be 2 (mm), calculate the mmf needed to lift a total mass of 100 (kg). - x Fig. 6-41 A long solenoid with iron core pattially drawn (Problem P.6-4). p.6-40 A current I flowd in a long solenoid with 11 closeiy wound coll-turns per unlt length. The cross-sectional area of its iron core, which has permeability n S. Detemme the force acting on t h e W . i t is withdrawn to the pos~tion shown in Fig. 6-41. 7 / Time-Varying Fieids and . Maxwell's Equations 7-1 INTRODUCTION In constructing the electrostatic model, we defined an electric , . .- E. and an electric flux density (electric displacement) vector, C . . governing dilrcrcntial cclu;~t ions arc For linear and isotropic (not necessarily homogeneous) media, E and D are related by the constitutive relation D =EE. (3 -97) . For the magnetostatic model, we defined a magnetic flux density vector, B, and a magnetic field intensity vector, H. The fundamental governing differential equations are V . B = O (6-6) V x l l = J . (6 -68) The constitutive relation for B and H in linear and isotropic media is These fundamental relations are summarized in Table 7-1. We observe that, in the static (non-time-varying) case, electric field vectors E and D and magnetic field vectors B and H form separate and independent pairs. In other words, E and D in the electrostatic model are not ;elated to B and H in the magneto- static model. In a conducting medium. static electric and magnetic fields may both , , exist and form an electromagnetostaric$eld (see the statement following Example 5-3 on p. 187). A static electric field in a conducting medium causes a steady current to flow that, in turn, gives rise to a static magnetic field. However, the electric field can be completely determined from the static electric charges or potential distributions. Table 7-1 hndamental Relations for J$ectr~~tatic and Governing equations v ~ t ) = ~ V x H = J ~onstitudve Relations (linear a(d isotropic media) 1 c related (3 -97) r, B, and :quatiom x s w d I n , -r magnPt+ < nay L-.l , mple 5-3 ' , :urrent to : fieldxan ' ributions. The magnctic field is:a cmscqucnce; it docs not enter into the calculation of the electric field. In this chuptcr wc will skc that :I ch;inging mlignctic licld gives rise to an dcctric licld. and vice versa.',To explain clcctromagnctic phenomena ~ ~ n d c r time-varying conditions, it is necessary loconstruct an electromagnetic model in which the electric field vectors E :ind 6 arc properly related to the magnetic field vectors B and 'H. The two pairs cf the governing equations in Table 7-1 must thcreforc bc modificd to show a mutual depeqence between the electric and magnetic field vectors in the time-varying case. Wc will begin witti a r~~ntlan~cnlal postulate that modifics ~ h c V x E cquation in Table 7-1 and leads to Faraday's law of electromagnetic induction. The concepts c;f transformer emf and rflotional emf will be discussed. With the new postulate we will also need to modify the V x H equatiop in order to make the governing equations consistent with the eqitation of contindty (law uf conservation of charge). The two modified curl equatiohs together with the two divergence equations in Table 7-1 are known as Maxwell's equations and form the foundation of electromagnetic theory. The governing eq~~ations for electrostatics and magnetostatics are special forms of Maxwell's equations when all quantities are independent of time. Maxwell's equations can be conibined to yield wave equl-itions that predict the existence of electromagnetic waves propagating wit11 the velocity of light. The solutions of the wave equations, especikilly for time-harmonic fields, will be discussed in this chapter. '7-2 FARADAY'S LAW OF ELECTROMAGNETIC INDUCTION , -\ . , A major advance-in electromagnetic theory was made by Michael Faraday who, in 1831, discovered expetimeil.tally that a current was induced in a conducting loop when the magnetic flux linking the loop changed.+ The quantitative relationship ' There is evidence that Joseph Henry independently made similar discoveries about the same time. 270 TIME-VARYING FIELDS AND.MAXWELLgS EQUATIONS 1 7 between the induced emf and the rate of change of flux linkage, based on experimental observation, is known as Faraday's law. It is an experimental law and can be con- sidered as a postulate. However, we do not take the experimental relation concerning a finite loop as the starting point for dcvcloping the theory of clcctromagnctic induc- tion. Instead, we follow our approach in Chapter 3 for electrostatics and in Chaptcr 6 for magnetostatics by putting forth the following fundamental postulate and devel- oping from it the integral forms of Faraday's law. Fondnmmtal Postulate for Elrctramngnctic Induction Equation (7-2) is valid for any surface S it11 a bounding contour C, whclhcr or not a physical circuit exists around C. Of course, in a field with no time variation, dB/ir = 0 , Eqs. (7-1) and (7-2) reduce, respectively, to Eqs. (3-5) and (3-8) for electrostatics. In the following subsections we discuss separately the cases of a stationary circuit in a time-varying magnetic field, a moving conductor in a static magnetic field, and a moving circuit in a time-varying magnetic field. 7-2.1 A Stationary Circuit in a Tirne-Varying Magnetic Field . For a stationary circuit with a contour C and surface S, Eq. (7-2) can be written as If we define YT = $ c E . d l ' = emf induced in iircuit with contour C (V). (7-4) r .) (7-2) :r o; not i& = ostatics. y c~rcuit ield. and !
, . k ; , , i 1 $ 1 and 1 8 = ds = magnetic flux :rooding surface S (Wb); (7-5) then Eq. (7-3) becomg i: .
% ." I Equation (7-6) states +at r11e electromotiue.fwce induced in u , v o t ~ o ~ a r y c l o ~ d circuil is q i ~ d l o t h t1~q(i/i1w r ~ o (I/ ~IICIYYI,W of 111v I M I ~ ~ I V ~ ~ C /I11 K / I I I / , ~ ~ , ( , / / I ( , circ [ l i t . TIUS C , a \l;~tcnic~lt oI' I : w d t / y ' \ I w o/'cloc/ro~tioq~~er~c i t l d ~ r / ~ o ~ i . A tiolc-r.~lc of ch'inge ol m;ignetic llux i~iduces a11 electric licld according to Eq. (7-3), even in the abbence of r ' physical closed circuit.sThe negallve sign in Eq. (7-6) is an assertion that the induced emf will cause a turrent to flow in the closed loop in such a direction as to oppose the change in the linking magnetic flux. This assertion is known as Lm?s lair. The emf induced in a stationary loop caused by a time-varying magnetic field is a tronsjormer emf. Exrmplc 7-1 A circular loop of N turns of cuhduming wire lies in the xy-plane with its center at the origin ofa magnetic field specified by B = a,B, cos (nrl2bJ sin wt, where b is the radius ofthe loop and o is the angqlar frequency. Find the emf induced in the loop. Solution: The problem specifies a stationary loop in a time-varying magnetic field; hence Eq. (7-6) can be used directly to find the induced emf, Y The magnetic flux iinking each turn of the circular loop is 711' = Jfib [a,B, cos 26 sin w t . (1,271r dr) I ; = ?f (i -.-I)Bo sin or.: n Since there are N turns,:thc total flux linkage is NQ, and we obtain 1 . , B,o cos ot (V). 71 . . The induced emf is seen to be ninety degrees out of time phase with the magnetic flux. 272 TIME-VARYING FIELDS AND MAXWELL'S EQUATIONS 1 7 0 0 0 0 0 0 d f ? --+ u @ Fig. 7-1 A conducting bar moving 0 0 @ B @ in a magnetic field. 7-2.2 A Moving Conductor in a Static Magnetic field When a conductor moves with ' a velocity u in a static (non-time-varying) magnetic field B as shown in Fig. 7-1, a force F, = qu x B will cause the freely movable electrons in the conductor to drift toward one end of the conductor and leave the other end positively charged. This separation of the positive and negative charges creates a Coulombian force of attraction. The charge-sepa3tion process continues ~ ~ n l i l (IIC ~'Ieclric a11d n\;lgwlic I~)I.~,CS 1~1I;111cc C:ICII 01llcr ;IIIO ;I sl;\lc' oI'c(ll~ilil>rill~~l is rcachcd. A t ccluilib~.iu~n. wl~~cli is tcnclicd vcry ropiclly, the ncl rorcc on tllc kcc cllargcs in the moving conductor is zero. To an observer moving with the conductor, there is no apparent motion and the magnetic force per unit charge F,/q = u x B can be interpreted as an induced electric field acting along the conductor and producing a voltage V2, = J : (U x B). dt'. (7-7) If the moving conductor is a part of a closed circuit C, then the emf generated around the circuit is This is referred to as a flux-cutting emf, or a motional emf. Obviously only the part of the circuit that moves in a direction not parallel to (and hence, figuratively, "cutting") the magnetic flux will contribute to V' in Eq. (7-8). Example 7-2 A metal bar slides over a pair of conducting rails in a uniform magnetic field B = a,Bo with a constant velocity u, as shown in Fig. 7-2. (a) Determine the open-circuit voltage Vo that appears across terminals 1 and 2. (b) Assuming that a resistance R is connected between the terminals, find the electric power dissipated in R. (c) Show that this electric power is equal to the mechanical power required to move the sliding bar with a velocity u . Neglect the electric resistance of the metal bar and of the conducting rails. Neglect also the mqchanical friction at the contact points. on and the .n induced ted around ily. the part iguratively, m m n t i c ermine, the I ' a . dissipated required to c mctol bar tact points. F,ig. 7-2 A metal bar sliding over conducting rails (Example 7-2). a) The moving bar generates a flux-cutting emf. We use Eq. (7-8) to find the open- circuit voltage Vo: , V, = V, - v2 -; gC (U x Bj 're = J2! (a,u x a , ~ , ) . (a, ' i t ) = - ~rB,h (V). (7-9) b) Wlicn a rcsis~ancc R ~i conneclcd between terminals 1 and Z,3 current I = icB,li R will flow from terminal 2 to terminal I, so that the electric power. P,, dlsslpiited in R is c) The mechanical power, P,,,, required L o move the sliding bar is P , = F . u (w), (7-1 1 ) where F is the mechanical force required to counteract the magnetic force, F,, which the magnetic field exerts on the current-carrying metal bar. From Eq. (6-159) we have F,.= 1 S2t' dP x B = -a,IBoh (N). (7-12) The negative sign in Eq. (7-12) arises because current I flows in a direction opposite to that of dP. Hence,. F = - F , = axIBoh = a,u~;h'/~ (N). (7-13) -1 Substitut~on-of Eq. (7'13) in Eq. (7-1 1) proves P,,, = P,., which i~pholds tlic principle of conservation of cncrgy. Ehmple 7-3 The Faroilny disk gerwrntor consists of n circular nietnl disk rot;ltmp with :l constant ongulilr vclocily (u in a ull~l'urm and constant magnetic lield of flux density B = a,B, that is parallel to the axis of rotation. Brush contacts are provided 274 TIME-VARYING FIELDS AND MAXWELL'S EQUATIONS 1 7 1 a I' 2 Ld Fig. 7-3 Faraday disk generator (Example 7-3). at the axis and on the rim of thepdisk, as depicted in Fig. 7-3. Determine the open- circuit voltage of the generator if the radius of the disk is b. Solutiorz: Let us consider the circuit 122'341'1. Of the part 2'34 that moves with the disk, only the straight portion 34 "cuts" the magnetic flux. ~ & a v e , from Eq. (7-8). which is the emf of the Faraday disk generator. To measure V, we must ,lac n volt- meter or :I vcry high rcsisl;~nm so tI1i11 n o ;~pprcci:ihle current Ilowr ill llle circuit lo modify thc cxlcrn;~lly t~pplicd n1:tgnclic licltl. 7-2.3 A Moving Circuit in a Time-Varying Magnetic Field When a charge q mdves with a velocity u in a region where both an electric field E and a magnetic field L1 cxist, the clcctrornugnetic brcc 17 on y, us mcasured by a laboratory observer, is given by Lorentz's force equation, Eq. (6-5), which is repeated below: F = q(E + u x B). (7-15) To an observer moving with q, there is no apparent motion, and the force on q can be interpreted as caused by an electric field E', where :e open- e a volt- 2 circuit !c'iicld E :-ed by a repeated on q can t ' 4 4 il Hence, when a conducting circuit with contbur C and surface S moves with a velocity u in a field (E, B), weme Eq. (7-17) in Eq. [7$) to obtain sv: \ dB $ ~ ' ? d t ' - L d t . d s + (u x B).dt' (V). t (7-18) Equation (7-18) is tHe general form of Faraday's law for a moving circuit in a time- varying magnetic fidd. The line integral on tge left side is the emf induced in the moving frame of t'&fbence. Thc first term on thd right sidc represents the transformer emf due to the time inriation of B; and the second term represents the motional emf due to the motioli of the circuit in B. The division of the induced emf between the . transformer and the motimal parts depends on the chosen frame of reference. Let us consider P circuit with contour C that moves from C, at time t to C, at limo r + AI in a chilngi~ig m;ignolic lield U. Thc motion may include trunslaticn, rotation, and distortibn in an arbitrary manner. Figure 7-4 illustrates the situaticn. The time-rate of chatlge of magnetic flux through the contour is B(r + At) . ds, - J. ~ ( t ) ds, S 1 (7-19) B(r + At) in Eq. (7-19) can be expanded as a Taylor's series: a q t ) B(i + At) = B(t) + - At + H.O.T., at where the high-order tkrms (H.O.T.) coniain the second and higher powers of (Ai). Substitution of Eq. (7-20) in Eq. (7-19) yields Fig. 7-3 A moving circuit time-varying magnetic field. 276 TIME-VARYING FIELDS AND MAXWELL'S EQUATIONS I 7 where B has been written for B(t) for simplicity. In going from C, to C,, the circuit coven a region that is bounded by S,, S,, and S,. Side surface S3 is the area swept out by the contour in time At. An element of the side surface is d ~ , = dP x u At. (7-22) We now apply the divergence theorem for B at time r to the region sketched in Fig. 7-4: where a negative sign is included in the term involving ds, because outward normals must be used in the divergence theorem. Using Eq. (7-22) in Eq. (7-23) and noting that V B = 0, we have L, B. h, - k, B . ch, = -Ar$Ju x B)-dP. Combining Eqs. (7-21) and (7-24), we obtain t i c'R - C 0 'k = (' - . (1s - (f, (a x R) . l i t . tlr -.s ; / ., (. - - --- which can be identified as the negative of the right side of Eq. (7-18). If we designate 1 ' = $ c El dP = emf induced in circuit C measured in the moving frame Eq. (7-18) can be written simply as which is of the same form as Eq. (7-6). Of course, if a circuit is not in motion. f ' reduces to Y7 and Eqs. (7-2.7) and (7-6) are exactly the same. Hence. Faraday's law that the emf induce? in a closed circuit equals the negative time-rate of increase of the magnetic flux linking a circuit applies to a stationary circuit as well as a moving one. Either Eq. (7-18) or Eq. (7-27) can he used to evaluate the induced emf in the general case. IIil high-impedance .~oltmcter is insenal in a conducting circuit, it will read the open-circuit voltage due to electromagnetic induction whether the circuit is stationary or moving. We have mentioned that the division of the induced emf in Eq. (7-18) into trimsformcr and motioo;ll emf's is not stiiqac, but their sum is ;~lw;lys eqi~al to that computed by using Eq. (7-27). In Example 7-2 (Fig. 7-2). we determined tho open-circuit voltage V, by using Eq. (7-8). If we use Eq. (7-27), we have in Fig. , . 4 ' ' I orn~nls . ' ; i ' . - noting . ; . (7-27) ion. Y " ~ y ' s law rease of movmg I fin the t, it fl 1 : cir~- .& ! c1nf in 6 I i alwi - ! )y using I ' I I 7-2 / F : WDAY~S LAW 6~ ELECTROMAGNETIC INDUCTION 277 , 1: . . I 6 1 and which is the sake &hq. (7-9). Similarly, for thd ~ a r d d a ~ disk generator in Example 7-3, the magnetic Aux linking the circuit. 1?2'341'1 is that whish passes through the wedge-shaped area 2'342'. and Example 7-4 An h by i v rectangular conducting loop is sltuatcd iil ;I changing magnetic field B = a,B, sin o)r. The normal of the loop initially n~;ikes an angle 1 w ~ l h a,, as shown in f)g. 7-5. Iiind thc induced emf in thc loop: (a) when the loop is at rest, and (b) when the loop rotates with an angular velocity w about the x-axis. . . b (a) Pcrspectivc vicw. (b) Vicw from +x direction. I:ig. 7-5 A rcvlllll~ullr c o ~ l d a c h y loop rolnli~lg ill n cliu~lpiag ~ ~ l i i ~ ~ l c ~ i c licld (Example 7-4). 278 TIME-VARYING FIELDS AND MAXWELL'S EQUATIONS / 7 Solution a) When the loop is at rest, we use Eq. (7-6). a , = J ~ . d ~ = (a,,Bo sin wt) . (anhw) = Bohw sin wt cos a. - Therefore where S = hw is the area of the loop. The relative polarities of the terminals are ' as indicated. If the circuit is qompleted through an external load, Ku will produce a current that will opposc thc changc in (I). b) When the loop rotates about the x-axis, both terms in Eq. (7-18) contribute: the first term contributes the transformer emf Y; in Eq. (7-28). and the second term contributes a motional emf r,': where \ I - an - w x (a,Bo sin 031 . (a, dx) = l [ ( ; ) - a. - w x (a$, sin cot)] . (a, dx) +s:L( s ) Note that the sides 23 and 41 do not contribute to Y, and that the contributions of sides 12 and 34 are of equal magnitude and in the same direction. If a = 0 at t = 0 , then cr = wt, and we can write f/" . ' , = B,Sw sin wt sin wt. (7-29) The total emf induced or generated in the rotating loop is the sum of f'. in Eq. (7-28) and V:, in Eq. (7-29): \ --, which has an ;tngulnr frcqucncy 20). We can determine the total induced emf Y: by applying Eq. (7-27) directly. At any time t, the magnetic Run linking the loop is @(t) = B(t) . [a,(t)S] = B,S sin o t cds cr = BoS sin wt cos o t = iBoS sin 2ot. ;ids are xoduce ~trlbute: . second 7 butio ions x = O a t (7-29) of % < in ( 7 C ) 1 1 I Jir& , . 1 i r , ' 1 - , 3-3 I MAXWELL'S EQUh-tONS 279 t . e L 8 ! Hence r: . ( ' < d@ i : y : F --= dl -- ri ( ; BOs sin 2wt % I , , I . = - B,So cos 20r , as before. The fundamental dbstulate for electromagn~tic induction assures us that a time- varying magnetic figld gives rise to an electric field. This assurance has been amply verified by numerous npcrimcnts. The V x E = 0 equation in Table 7-1 must ' therefore be replacekl by Eq. (7-1) in the time-varying case. Following are the revised set of two curl and two divergence equations from Table 7-1. In addition, we know that the principle of conservation of charge must be satisded at all times. The mathematical csprcssion of charge conservation is the equation of co~~lirluity, ISq. (5 XI), which IS ~'cpc;~tctl Iwlow. The crucial question here is whether the set of four equations in (7-31a, b, c, and d) are now consistefit +,with the requirement specified by Eq. (7-32) in a time-varying situation. That the Answer is in the negative is immediately obvious by simply taking the divergence of Ed. (7-31 b), which follows from the null identity, Eq. (2-137). We are reminded that the divergence of lhc cusl-of-;in$ MI-hc11;wd vector field 13 zero. Sincc Eq. (7-32) :lsserts V . J ~ O C S 11ot v:111i~h ill L I ' I ~ I I I C - V ; I ~ ~ ~ I \ ~ S~~LI:I[~OII, 1 i 1 , ( 7 : ~ ) i,, s ~n ' gcllcr;~I, not true. How should Eqfi. (7-31a, b, c, and d) be modified so that they are consistent with Eq. (7-32)? First of all, a term ap/St must be added to the right side of Eq. (7-33): - Equation (7,361 indicates that a time-varying electric field will give rise to a magnetic field. even in the absence of a current flow. The additional term aD/& is necessary in order to make Eq. (7-36) consistent with the principle of conservation of charge. It is easy to verify that 2DBt ha's the dimension ofa current density (SI unit: Aim2). The term SD/dt is called displacement e~rrrent density, and its introductibn in the V x H equation was one of the major contributions of James Clerk M;i\vcll (1831-1579). In order to be cotisistent with tlie cqu;ltion of continuity in a time- varying situation. both of the curl equations in Table 7-1 muh'be generalized. The set of four consistent equations to replace the inconsistent equations, Eqs. (7-31a. b. c, and d), are They are known as Maxwell's equations. These four equations, together with the equation of continuity in Eq, (7-31) and Lorentr's force equation in Eq. (6-5), form the foundation .of electromagnetic theory. These equations can be used to explain and predict all macroscopic electromagnetic phenomena. Although the four Maxwell's equations in Eqs. (7-37a. b, c, and d) are consistent, they are not all independent. As a matter of fact, the two divergence equations, Eqs. (7-37c and d). can be derived from the two curl equations, Eqs. (7-37a and b), by making use of the equation of continuity. Eq. (7-32) (see Problem P.7-7). The four fundamental field vectors E, D, B, H (cach having three components) represent twelve unknowns. Twelve scalar equations are required for the determination of these twelve unknowns. The required equations are supplied by the two vector curl equations and the two vector constitutive relations D = aE and H = B/p, each vector equation being equivalent to three scalar equations. (7-36) lgnetic xssary large. 1/m2). in the dxwell L 11 ,--- d. The 7-3' 7-374 7-37b) 7-37c) 7 -37d) ~ t h the (6-5), !sed to sistent, . ; i t i ~ - P and u), 7). 7;' ?rese. .
. ~f these or curl u , each . ? j . , 3 . " "I.. . ., - . ' . I: .. ,+. . , . I ! . . ' . 3 . 2 ~, i . , f i , i , I ; . . . ,' . , - . , . . , ' I . , i . , I / i I ' 8 i ' . - , , 8 - 1 . 7-3.1 Integral Form o ~ , ~ ~ x w e l l ' s Equations; : . . i 7 ' i 1 ' The four Maxwell's bq~a'lions in (7-37a b, c,.and d) are differential equations that ,are valid at ever$ $oint;h space. In explain/ng electromagnetic phenomena in a physical environm t, we must deal with dhite objects of specified shapes and boundaries. It is bo ,venient to convert the differential forms into their integral-form equivalents. We ta f e the! sdrface integral of both sides of the curl equations in Eqs. (7-37a) and (7~37b) over an open surface'! with a contour C and apply Stokes's theorem to obtaiti I . . , , " . i) I I E . d P - .-• s at ds (7-38a) I . - and - Taking the volume ihtegdl of both sides'of the divergence equations in Eqs. (7-37c) and (7-37d) over a ~olumk v with a closed surface S and using divergence theorem, we have and The set of !
O L & equat~ons in (7-38a, b. c and d) are the integral form of Ma~well's t liq. (7-38a) is the same as Eq. (7-2), which is an expression ctrarnrignetic induction. Equation (7-38b) is a generalization law given in Eq. (6-701, the latter applying only to static magnetic urrent density J may consist of a convection current density of ;I irec-charge distribution, as well ;a ; ; canduction current t h presence of,an electric field in a conducting medium. The s the:c~rrent I flowing through the open surface S. a n b e recognized as Gauss's law, which we used extensively icli remains the same i n the time-varying case. The volume ihtegral ofAp eqvals the total charge Q that is enclosed in surface S. No particular h w is associated with EQ (7-38d); but, incdmparlng it with Eq. (7-38c), we conclude that there are no isdlatrd magnetic charges and that the total outward magnetic . .. _. " - - i ; > ,',""' > 2 : : . " . ?
' 282 TIME-VARYING FIELDS AND MAXWELCS EQUATIONS / 7 rh Table 7-2 Maxwell's Equations :.+ r..fl Differential Fonn Integral Form Significance dB V X E = - - at dt Faraday's law. dD dD f C H . de = I + 6 - ds Amfirs's circuital law V - D = p Gauss's law. V . B = o '$B.L=o No isolated magnetic charge. 6 flux through any closed surface is zero. Both the differential and the integral forms of Maxwell's equations are collected in Table 7-2 for easy reference. Example 7-5 An AC voltage source of amplitude V, and-angular frequency w, U . = V, sin wt, is connected across a parallel-plate capacitor C,. as shown in Fig. 7-6. (a) Verify that the displaccmcnt current in tl~c capacitor is the same us thc conduction current in thc wires. (b) Detcrminc the magnetic licld intensity at il distancc r from the wire. Solution a ) The conduction current in the connecting wire is i c = c l 3 = c v dt 1 0 a cos at (A). For a parallel-plate capacitor with an area A, plate sepmtion d, and a dielectric medium of permittivity e, the capacitance is 0 - 4 ( Fig. 7-6 A parallel-plate capacitor connected to an AC voltage source (Example 7 -5 ) . 1 forms -I acy o, ig. 7 luctioil r from electric I , .
I ( c < . I > . ; a % ; ' : ;i , , ' 7-4 / POTENTIAL FUNCTIONS 283 , ; 1 : , 1 , I 1 t !-I ,' With a vdltage 0, npidating betweenthe plates, the uniform electric field intensity E in the dielectfic is dqual to (neglecting fringing effects) E = vJd, whence ' The displacemeht cupent is then t I - = C,Vow cOs wt = i , Q.E.D. b) The magnetic fleld ntensity at a distanck r from the conductin, 0 wire can be found by applyi!lg the generalized Amp~re'scircuital law, Eq. (7--3Sb), to contocr C in Fig. 7-6 Two t/pical open surfaces Vith rim C may be chosen: (1) a planar diak surfacc S, i (2) .t curved surface S2 $assing through the diclectrlc medlum. Symmetry arouhd tl e wire ensures a constant Hd along the contour C The iine integral on the feft s de of Eq. (7-38b) is For the surface S,, c nly the first term on the right side of Eq. (7-38b) is nonzero because no chiUges ire deposited along the wire and. consequently, D = 0. j ' , J . dr = ic.= k1 vow cos wt. Since the surface S , )asses through the dielectric medium. no conduction current 8,. If the second surface idtegral were not there, the right side of ohld 11e zero. This would rdsult in a contradiction. The inclusion of the displgcerhent current term by Maxwell eliminates this contradiction. As we have showti in ]]art (a), iD = i,. Hence we obtain the same result whether surface S , or sdkfacc S, is chosen. Equating the two previous integrals, we find !
n Section 6-3 the concept of the vector magnetic potential A was introduced because of the solenoidal nature of B (V B = 0): . . 284 TIME-VARYING FIELDS AND MAXWELL'S EQUATIC$S 1 7 b c . s ' 1' If Eq. (7-39) is substituted in the differential form of faradiY9s law, Eq. (7-I), we get Since the sum of the two vector quantities in the parentheses of Eq. (7-40) is cur!-free, ' it can be expressed as the gradient of a scalar. To be consistent with the definition of the scalar electric potential Vin Eq. (3-38) for electrostatics, we write ( I from which we obtain In the static case, aA/dt = 0, and Eq; (7-41) reduces to E = - VV. Hence E can be determined from V alonc; and B, from A by Eq. (7-39). For time-varying fields, E depends on both V and A. Inasmuch as B also dcpcnds on A, E and B are coupled. The electric field in Eq. (7-41) can be viewed as composed of two parts: the first part, -VV, is due to charge distribution p; and the second part, -dA/dt, is due to time-varying current J. We are tempted to find V from p by Eq. (3-56) and to find A by Eq. (6-22) However, the preceding two equations were obtained under static conditions, and V and A as given were, in fact, solutions of Poisson's equations, Eqs. (4-6) and (6-20) respectively. These solutions may themselves be time-dependent because p and J may be functions of time, but they neglect the time-retardation effects associated with the finite velocity of propagation of time-varying electromagnetic fields. When p and J vary slowly with time (at a very low frequency) and the range of interest R is small compared with the wavelength, it is allowable to use Eqs. (7-42) and (7-43) in Eqs. '.- (7-39) and (7-41) to find quasi-static fields. We will discuss this again in subset- . L -( tion 7-7.2. Quasi-static fields are approximations. Their consideration leads from field theory to circuit theory. However, whcn'thc sourcc frequency is high and thc range be get ' (7-40) &free, ition of (7-fl . 1- . 1 Iicld\, C~uplcd. hc first tluc to (7-42) 4 (7-43) Ins, m d j (6-30) I and J I C d ( "P ' is smll in l ! subsec- I m field le range 7-4 1 "TENTIAL FUNCTIONS 285 : b, , I' : > , of interest is no longer small in comparison to the wavelength, quasi-static solutions ' will not suffice. Tirhe-rethrdation effects mtilt then be included, as in the case of electromagnetic radiation from antenn&.!~hese points will be discussed more fully ' when we study soldions to wave equations. i Let us substitute Eqs. (7-39) and (7-41) into Eq. (7-37b) and make use of the constitutive relations H = B/p and D = EE. We have where a homogeneous r-iediutn has been'assumed. Recalling the vector identity for V x V x A in Eq. (6-16a), we can write Eq. (3-44) as d2A V(V A) - V'A = pJ - V 0 r i12A V2.4 -pi----= -uJ + V (7-45) 3tZ Now, the definition of .L vcctor rcquircs the specification of both its curl and its tlivcrycncc. A I ~ I I O L I ~ ~ LIIL w1.1 o1.A is d~sig11;~~cd U in ECI. (7 -30), wu arc slill :I[ libcrly to chnose the divergence of A. We let which makes the se'konc~ term on the right side of Eq. (7-45) vanish, so we obtain I Equation (7-47) is the l~o~~/w~noyolcous. w v e eyrrutiorl for vector. pot~rlrid A. It IS called a wave equatim bxause its solutions represent waves traveling with a velocity equal to I/,&. This will become clear in Section 7-6 when the solution of wave equations is discusseti. The relation between A and V in Eq. (7-46) is called the Lorcnt: corditior~ (or Lora~tz galye) for potentiat. It reduces to the condition V . A = O in Eq. (6-l9)+x$atic fields. The Lorentz condition can be shown to be consistent with the equation of continuity (Problem P.7'8). A correspondini wave equation for the scalar potential V can be obtained by substituting Eq. (7-41) in Eq. (7-37c). We have which, for a constant e, leads to d P V 2 V + - ( V ' A) = --. (7-48) at E Using Eq. (7-46), we get I which is the nonhomogeneous wave equatiun for scalar potential V. The nonhomo- geneous wave equations in (7-47) and (7-49) reduce to Poisson's eqcations in static cases. Since the potential functions given in Eqs. (7-42) and (7-43) are solutions of Poisson's equations, they cannot:be expected to be the solutions of nonhomo, aeneous wave equations in time-varying iituations without modification. 7-5 ELECTROMAGNETIC BOUNDARY CONDITIONS -1 In order to solve electromagnetic problems involving contiguous regions of different constitutive parameters, it is necessary to know the boundary conditions that the field vectors E, D, B, and H must satisfy at the interfaces. Boundary conditions are derived by applying the integral form of Maxwell's equations (7-38a, b, c, and d) to a small region at an interface of two media in manners similar to those used in obtain- ing the boundary conditions for static electric and magnetic fields. The integral equations are assumed to hold for regions containing discontinuous media. The reader should review the procedures followed in Sections 3-9 and 6-10. In general, the application of the integral form of a curl equation to a flat closed path at a bound- ary with top and bottom sides in the two touching media yields the boundary con- dition for the tangential components; and the application of the integral form of a divergence equation to a shallow pillbox at an interface with top and bottom faces in the two contiguous media gives the boundary condition for the normal components. The boundary conditions for the tangential components of E and H are obtained from Eqs. (7-38a) and (7-38b) respectively: ) u C C )..L:l' .-< r - - We note that Eqs. (7-50a) and (7-50b) for the time-varying case are exactly the same ! as, respectively, Eq. (3-1 10) for static electric fields and Eq. (6-99) for static magnetic fields in spite of the existence of the time-Varying terms in Eqs. (7-38a) and (7-38b). (7 - 49) , I homo- . ; ' 1 static Ions of 3ncous i!ferc I u t the .ill\ are I(! dl to b ~ h ( s ~ i ~ l - ntcgr:il 13. The ~cneral, bound- - ry con- rm of a m faces )orients. btained he same riagnetic (7-38b). fi I'i ' h The reason is that: intlettiq the height oflt6e flQt closed path (obcda in Figs. 3-22 and 6-17) approach zerd, the:area bounded by the path approaches zero, causing the surface integrals of akpt i n d a ~ / a t to vanish' Similarly, the b&ndary conditions fot the normal components of D and B are obtained from EqS. P-385) and (7-38d); , " I . . , These are the same a$, re: p6ctively, Eq. (3-l'l3qj for static electric fields and Eq. (6-95) , for static magnetic %Ads Gecause we start from the same divergence equations. We can make thk follawing general (Gtements about electromagnetic boundary conditions: (1) Tltc iarig~~&i conlporlent of an E jeld is continuoris across nn inter- .race; ( 2 ) Thc ~ungcnliul cunzponcnl c f an H Jielzl is di.sc~ontit~rlo~i.s ucross t r n intc f 'I' Lice where u surface current e&s, the a~nount' of disconti~~uity b e i y detcrtnineti by Ey. (7-506); (3) The noimal component of a D jield is discontinuous across an inrcrjiuce . where u surfucc dlurqe c.xisi,s., the umount oj' tliscontinuity hcing dctotn~tled hj. Eq. (7-50~); and (4) The normal component of a B jieltl i s continuous across nn inrefice. As we have noted previoikly, the two divcrgcllcc equ;!lions can bc dcrivcd from the Lwo curl cqualions find Lllc crl\li~~ioti oScotili!l~rily: IICIICC, tllc hot~~i~I;~ry conditions i n Eqs. (7 504 ; I I ~ (7 50d), wl~icl~ ;tr( ~ ? l w i ~ ~ c d [IX)IN tl~c dive~~gc~~cc cqu;~tio~~s, cant101 be iti~1cpc11~1~ti1 I'rotn ~IIOSC in Lqs. (7-50a) : m 1 (7 --job), which arc obtained from the curl equatibns. As a matter of fact, in the time-varying case the boundary condition for the tangential component o f l ~ in Eq. (7-50a) is equivalent to that for the normal compodent of B in Eq. (7-50d), and the boundary condition for the tangential component ot'H in Eq. (7-50b) is ekluivalent to that of D in Eq. (7-51)~). The simultaneous s$ecific'ation of the tangential component of E and the normal component of B at.. a boundary surface in a ,timevarying situation, for example, would be redundant,and, if we are not careful, could result in contradictions. We now examine the.important special cases of (1) a boundary between two lossless line~r medk, and (2) a boundary between a good dielectric and a good ' C . conductor. - . 7-5.1 Interface'twtween two ~Qss~ess . Linear Media A lossiess linear medium can be specified by ti permittivity E and a permeability p, with a = 0. There are usually no free charges and no surface currents at the interface between two lossless media, v e set p, = 0 and J, = 0 in Eqs. (7-50% b, c, and d) and obtain the boundary conditions listed in 'Table 7-3. 288 TIME-VARYING FIELDS AND MAXWELL'S EQUATIONS 1 7 Table 7-3 Boundary Conditions between Two Lossless Media 1 (7-51a) I i 7-5.2 Interface between a Dielectric and a Perfect Conductor A pcrl'cct C O I I I ~ I I C ~ O S is o t i c with a11 i ~ l l i l i i l c L'OII~IIC~~V~L~. 111 llle physical world we only have "good" conductors such as silver, copper, gold, an'szluminium. In order i to simplify the analytical solution of field problems, good conductors are often con- 1 I sidered perfect conductors in regard to boundary conditions. In the interior of a perfect conductor, the electric field is zero (otherwise it would produce an infinite 1 current density), and any charges the conductor will have will reside on the surface only. The interrelationship between (E, D) and (B, H) through Maxwell's equations ! ensures that B and H are also zero in the interior of a conductor in a time-varying i i sit~ation.~ Consider an interface between a lossless dielectric (medium I) and a per- fect conductor (medium 2). In medium 2, E2 = 0, M 2 = 0, D2 = 0, and B, = 0. The I I f 1 I Tablc 7-4 Boundary Conditions between a Dielectric (Medium I) ; l and a Perfect Conductor (Medium 2) (Time-Varying Case) i On the Side of Medium 1 On the Side of iMedlurn 2 El, = 0 E2, = 0 (7-52a) (7-52~) -\ a n 2 ' DI = P s D,, = 0 B,, = 0 R,, = 0 (7-52d) t - ' In the static case, a steady current In a conductor produces a static magnetic field that does not affect the electric field. Hence, E and D within a good condudtor may be zero, but B and H may not be zero. orW-3 .n oruer cn iOr 0 1 ; i infinite \urface jumons - L aryirtg .d a per- = 0. The (7-52a) (7-52b) (7P.c) (7-, -4) I - 2s not affect L be zero. 1 C I general boundary ctmditibns in Eqs. (7-50, b, c , and d) reduce to those listed in Table 7-4. When \uj apply Eqs. (7-52b),and ('I-52c), it is important to note that the reference unit nokmdl is ail outward normal /ram m d i u m 2 in order to avoid an error in sign. As mentionM in Section 6-10, currents in media with finite conductivities are expressed in tenhs orvolume current denjities, and surface current densities de- fined for currents flohing through an infinitesimal thickness is zero. In this case, Eq. (7-52b) leads to thq condition that the tAngCntia1 component of H is continuous across an interface $th a conductor having a finite conductivity. . , 1 Exan~ple 7-6 The k arid H field of a certain propagating mode (TE,,) in a cross section of an a by b riectallgular waveguide are k = a,E, and H = a,H, + a,H., where a IIX E, = - j o p - H, sin - II a (7-534 where H,, w, p,, and / 3 are constants. Assumiqg the inner walls of the waveguide are perfectly conductin& determine for the four inner walls of the waveguide (a) the surface charge densities m d (b) the surface current densities. I Solution: Figure 777 sliows a cross section of the waveguide. The four inner walls are specified by x = 0, x = a, y = 0, and y = b, The outward normals to these walls (medium 2) are, respkcti~ ely, a,, -a,, a,,and -a,. Fig.. 7-7 Cross section of a rectangular x waveguide (Example 7-6). 290 TIME-VARYING FIELDS AND MAXWELYA'EQUATIONS 1 7 . a) Surface charge densities-Use Eq. (7-52c): ps(x = 0) = a, . E,E = 0 a nx PAY = 0) = a), . r,E = E , E,, = - j o p r , - H , sin - 71 a a nx PAY = b) = - aye = -roEy = j o p o - H, sin - n a = -ps(y = 0). b) Surface current densities - Use.Eq. (7-52b): nx n 7 t .
Y = a,H, cos - - a, j p - H , sin - a n a nx a nx = -a,H, cos - + azjp - H , sin - a n ( I In this section we have discussed the relations that field vcctors must satisfy at an interface between different media. Boundary conditions are of basic importance in the solution of electromagnetic problems because general solutions of Maxwell's equations carry little meaning until they are adapted to physical problems each with a given region and associated boundary conditions. Maxwell's equations are partial differential equations. Their solutions will contain integration constants that are determined from the additional information supplic$ by boundary conditions so that each solution will be unique for each given problem. 7-6 WAVE EQUATIONS AND THEIR SOLUTIONS At this point we are in possession of the essentials of the fundamental structure of electromagnetic theory. Maxwell's equations give a complete description of the @ t t l " ' relation betwcen elcctromagnctic fields and chsrgs and curmnt distributions. Thcir . solutions provide the answers to all electromagnetic problems, albeit in some cases the solutions are difficult to obtain. Special 'analytical and numerical techniques mr thc i I 7-6.1 Sol I ! for Potenti I L\'< for d c an( Po of 1 ' Eq by of wi' soi Eq - 11 / -' satisfy at portance laxwell's :ach with : e partial that are ltions so A uctur. f )n of the Ins. Their m e cases :cliniques ! 7- -) WAVE EQUAT~ONS AND THEIR SOLUTIONS 291 I 1 . t may be devised to aid in the solution proceduie; but they do not add to or refine the fundamental structure. Such is the importanbe of Maxwell's equations. , For given charge $nd current and J, we finf solve the nonhomo- geneous wave equations, Eqs. for potentials A and V. With A and V determined, E dild Bacan be found ffbml.kespectively, Eqs. (7-41) and (7-39) by differentiation. 4 t . L ' 7-6.1 Solution of Wave ~ ~ d t i o n s for Potentials I: I . I I : . .. . Wc now 'cokidci the'kol~hion of the nonbom&encous wave cqu:ltion. Eq. (7-49,, for ~cilliir electric potenti.~l V. Wc can do thisby first finding the solution for an , clcmental point oliarpc ili timc i , p ( / ) Ad, locticd at tbc origin ol' tile coordinates and lhen by summing the elkcts of all 1 1 1 e charge elements in a given region. For a point charge at tfie origin, it is most convenient tb use spherical coordinates. Because of spherical symmetry, V depends only on R and t (not on 6 ' or 9). Except at the origin. V satisfies the following homogeneous equation: 1 d --(,2?\ - d2V , R2 iiR dR) ,LlE ---7 - = 0. dt- We introduce a new variaole I V(R, t ) = - U(R, t), R which convcrts Eq. (7-54) to a2u z2u -- dR2 / l E ---- = 0. 2 z Equation (7-56) is a one-d~mensional homogeneous wave equation. It can be verified by direct substitution (see Problem P.7-15) that any twice-differentiable function of (1 - ~fi) or of (t + R \,z) is a solution of~Eq. (7-56). Later in this section we will see that 3 f ~ n ~ t i d n o i l t + R&) does not correspond to a physically useful solution. Hence w : have , Equation (7-57) represents a wavestraveling m the positive R direction wth a veloclty 1/42. As we see, the Function at R + AR at a later time i + At is . 1- U(R + AR, i + Arj = f [r + At - (R + AR)&] = jlr - ~ ~ 1 % ) . Thus, the hnction retains its form if At = AR& = AR/u, where u = 1/& is the velocity o f propagation, a characteristic of the medium. From Eq. (7-55), we get To determine what the specific function f(t - R/u) must be, we note that for a I static point charge p(t) Av' at the origin, Comparison of Eqs. (7-58) and (7-59) enables us to identify !
. p(t - Rlu) Au' i, A f (t - R/u) = I 6 4nc f The potential due to a charge distribution over a volume V' is then f I I Equation (7-60) indicates that the scalar potential at a distance R from the source at time 1 dcpe~ids on tlle vdue of the c l l q e dcllsity at an scs./icr tilne (i - Rle). It , takes lime R ~ I for the ellect of p to be fclt at dist~~nce R:.Fw. this rcnson I,'(R, r) in Ell. (7 - 0 ) i d c i l l / I I . I t is IIOLV clcal. Illat a function of (t + R/u) cannot be a physically useful solution. since it would lead to the impossible 1 situation that the effect of p would be felt at a distant point before it occurs at the j source. I The solution of the nonhomogeneous wave equation, Eq. (7-47), for vector 1 magnetic potential A can proceed in exactly the same way as that for V. The vector ; equation. Eq. (7-47), can be decomposed into three scalar equations, each similar to Eq. (7-49) for V. The retarded vector potential.is, thus, given by I The electric and magnetic fields derived from A and V by differentiation will : obviously also be functionspf (t - Rlu) and, therefore, retarded in time. It takes time for electromagnetic waves to travel and for the effects of time-varying charges and / currents to be felt .at distant points. In the quasi-static approximation, we ignore i this time-retardation effect and assume instant response. This assumption is implicit I in dealing with circuit problcms. i n i I 7-6.2 Source-Free Wave Equations i 717 -In problems of wave propagation we are concerned with the behavior of an electro- magnetic wave in a source-free region where p and J are both zero. In other words. we are often interested not so much in how an electromagnetic wave is originated. l ~ t in how il propag;;ltcs. Ifllic wave is iu a sinlpi,c(li~lcar, iwtropic, il~ld homogeneous) t (7-59) ' ( 7 - 60) : ',i)llrLC li,f-Y (it. r j in ctlc' f ipo\slole 1 5 ,it the )r v c C t O r IK ~ector h slmllar (7-61) .ltion wdl .aka time arges and we Ignore is i m R t I :in clcctro- Ilcr words. or!gmated, logeneous) , r . , nonconducting rnedidL cihracterized by and p (0 = 0 1 , Maxwell's equations (7-37a, b, c, and d) reduce to: Equations (7-62a, b, c, and d) are first-order diffetcntlal equations in the two vnr~ables E and H. They can bd combined to give 3 second-order equation in E alone. To do this, we take the curl of Eq. (7-62a) and use Eq. (7-62b): Now V x V x E = V ( V - E) - V'E = -V2E because of Eq. (7-62c). Hence, iic hi^^ or, since u = I/& In an entirely similar way we also obtain an equation in H: I Equations (7-64) and (7-54) \ire homogeneous vector w a w equatrons. We C X ~ see t h a ~ ih Cartesian coordinatcs Eqs. (7-64) and (7-65) can each be decomposed into three on;-dimensional, homogeneous, scalar wave equatlons. Each component of E arid of H wdl satisfy an equation exactly like Eq. (7-56), whose so1ut1p.s represent whves. We will extensively discuss wave behavior in varlous envirpnmerm in,the next two chapters. 7-7 TIME-HARMONIC FIELDS Maxwell's equations and .III the eqo;~tionodcsived from thum so far in this chapter hold h r clwlso~aignclic qu;mlilics with .lo ;trbi[r:~ry ~~nru-dape~~dc~lc~.. '1.11~ aclu:d type of time functions that the field quantities assume depends on the source func- tions p and J. In engineering, sinusoidal time functions occupy a unique position. 294 TIME-VARYING FIELDS AND MAXWELL'S EQUATIONS 1 7 They are easy to generate; arbitrary periodic time functions can be expanded into Fourier series of harmonic sinusoidal components; and transient nonperiodic func- tions can be expressed as Fourier integralst Since Maxwell's equations are linear differential equatibns, sinusoidal time variations of source functions of a given fre- quency will produce sinusoidal variations of E and H with the same frequency in the steady state. For source functions with an arbitrary time dependence, electrodynamic fields can be determined in terms of those caused by the various Trequency cuniponcnts of the source functions. The application of the principle of superposition will give us the total fields. In this section wc examine time-hurrnor~ic (stcady-state sinusoidal) field relationships. The Use of Phasors - A Review For time-harmonic fields it is condenient to use a phasor notation. At this timc we digress briefly to review the use of phasors. Conceptually it is simpler to discuss a scalar phasor. The instantaneous (time-dependent) expression of a sinusoidal scalar quantity, such as a current i, can be written as either a cosine or a sine function. If we choose a cosine function as the rrfel+rizce (which is usua1l:v'dictated by the func- tional form of the cxcitntion), thcn all derived results will rcfer to the cosine function. The specification of a sinusoidal quantity requires the knowledge of three parameters: amplitude, frequency, and phase. For example, i(t) = I cos (wt + 4), (7-66) where I is the amplitude; w is the angular frequency (rad/s)--c:, is always equal to 2nf, f being the frequency in hertz; and cb is the phase referred to the cosine function. We could write i(t) in Eq. (7-66) as a sine function if we wish: i(t) = I sin (wt + 47, with 4' = 4 + 7112. Thus it is important to decide at the outset whether our rcfcrence is a cosine or a sine function, then to stick to that decision throughout a problem. To work directly with an instantaneous expression such as the cosine function is inconvenient when differentiations or integrations:of i(t) are involved because they lead to both sine (first-order differentiation or integsktion) and cosine (second-order differentiation or integration) functions and bccausc it is tedious to combine sine and cosine functions. For instknce, the loop equati6n for a series RLC circuit with an applied voltage e(t) = E cos wt is: If we write i(t) as in Eq. (7-66), Eq. (7-67) yields . (cot + 4) + R cos (wt + 4) + l sin (wt + 4) = E cos wt . (7-68) w c 1 ' D. K. Cheng, Analysis o f Linear Systems; Addison-Wesley Publishing Company, Chapter 5, 1959 ar:. of Exa Az : odic func- are lineur given fre- 11cy in the odynamic mponcnts will give inusoidal) , llllle we dlsciuss a da1 scaldr l l L l l r p If [!LC IL,IC- run!. 7. Kln.-..As: 17 -66) q u 2 I to r i l ~ l ~ l l o l l . ( , I / + (I,'), rcfcrence problem. functlon l u x thcy nd-order bine sine cuit with '767) (7-63) 1959. i f " 4 1. , . ndcd into : ' , - I? . ., 7-7 I TIME-HARMONIC FIELDS r ~ i , t 7 4 ,?' ' i . 4 Complicated mathe~dticalimanipulation~ ,$re iequired in order to determine the unknown I and 4. It 1' . +I 1 ' C , It is much sim$ef,ho u& exponential fGnctidns by writing the applied voltage as ' ii i ' e(t) = E cos wt = &[(EejO)yjw] \ = %(E,eJWL) , (7-69) and i(t) in Eq. (7-66) & . I C . 1 J; i(t) = 9 & [(IeJ@)eju'] I f: . :i = We (I,ejO'), (7-70) where .i rncnns ' U c ;cal &rt of." In Eqs.'(7-69) and (7-701, , I are (scalar) phasors that contain amplitude and phase information but am independent of r. The phasor E, in Bq. (7-71a) with zero phase angle is the reference phasor. Now, I Substitution of Eqs. (7-69: through (7 7%) in Bq. (7-67) yields , R f j oL-- I,=E,, [ (' J l from which the current ph.rsor I, can bc solvcd clslly. Note that thc time-dcpendcnt factor eJ'"' disappears from Eq, (7'-T! because it is present in every term in Eq. (7-67) after the substitution dnd ,s i i i c r t k . ~ i h i ~ ~ l e d . ?his is the essence of the usefulness of phasors ir. the analysis Oi linear ~ysicn~s ui!h time-harmonic excitations. After I, has beer, determined, the instantaneous current response i(t) can be found from Eq. (7770) by (1) multiplying I, by eJ", and (2) taking the real part of the product. If' the applied voltage had been given as a sine ]u~~rtion such as e(t) = E sln ojr. the series RLC-circuit broblem would be solved in terms of phasors in exactly the same way; . q y the idstantaneous expressions would be obtained by taking the iiriqinory part ofthe product of the phasors with i>j("'. The complex phasors represent the magnitudes and the phase shifts of the quantities in the solution of time-harmonic problems. Example 7-7 Expres9 3 cos wt - 4 sin wr us first (a) A , cos (wr + O,), and then (b) A, sin (or + 0,). Detenhina A,, dl, A,, and 0,. 296 TIME-VARYING FIELDS AND MAXWELL'S EQUATIOW?%f 7 Solution: We can conveniently use phasors to solve this problem. a) To express 3 cos o t - 4 sin o t as A , cos (ot + 01), we use cos o t as the reference and consider the sum of the two phasors 3 and -4e-j"I2(=j4), since sin wt = cos (ot - n/2) lags behind cos cot by n/2 rad. Taking the real part of the product of this phasx and ej"', we have , - 3 cos o t - 4 sin o t = 9s[(5ej53.1')ejw'] = 5 cos iwt + 53.1"). (7-74a) So, A, = 5, and 0, = 53.1" = 0.927 (rad). b) To express 3 cos wt - 4 sin ot'as A, sin (ot + I),), we use sin wt as the reference and consider the sum of the two phasors 3ejnI2 ! =j3) and - 4. j3 - 4 = ~~j1an-l 3/(-4) = 5ej143.1' (The reader should note that the angle above is 143.1°;-no~36.9".) Now we take the imaginary part of the product of the phasor above and ejw' to obtain the desired answer: 3 cos o t - 4 sin wt = ~m[(5e~'~~.'')e'"'] = 5 sin (ot + 143.1"). (7-74b) Hence, A, = 5 and 0, = 143.1" = 2.50 (rad). The reader should recognize that the results in Eqs. (7-74a) and (7-74b) are identical. Time-Harmonic Electromagnetics Field vectors that vary with space coordinates and are sinusoidal functions of time can similarly be represented by vector phasors that depend on space coordinates but not on time. As an example, we can write a time-harmonic E field referring to COS C0tt US E(x, y, Z, t) = .
% k [E(x, y, z) ejw'] , (7-75) where E(x, y, z) is a vector phasor that contains information on direction, magnitude, and phase. Phasors are, in general, complex quantities. From Eqs. (7-75), (7-70), (7-72a), and (7-72b), we see that, if E(x, y, z, t) is to bc rcprcscntcd by thc vcctor phasor E(x, y, z), then dE(x, y, z, t)/& and 1 E(x, y, z, t) dt would be represented by, respectively, vector phasors joE(x, y, z) and E(x, y, z)/jw. Higher-order differentia- tions and integrations with respect to t would be represented, respectively, by multi- plications and divisions of the phasor E(x, y, z) by higher powers of jo. ' If the time reference is not explicitly specified, it is custom;arily taken as cos cot. i .efermce ;in o t = (7-74a) ,eJerence 'io.Tue 3 0btc11n (7 -74b) :.lib) dlIC s of tlme ~rdinates crrrny to (7-75) sgmtude, L i-f-x le vector cnta' ' ' Ikren,.~- by multi- 4 :I , .
a ' \ ?!-7 1 TIME-HARMONIC FIELDS 297 1 WC now,t&td timu-hsnnonic Maxwell's ciuiitions (7-37a, b, c, ;,nd d) in terms of vector field3hasors (E, H) and source phasors (p, J) in a simple (linear, isotropic, and homogenkoud) medium as follows. I' Thc sp;~ce-codrdin:~td ilrg iments llavc been omitted for simplicity. Tlir f ~ t [hilt the same notations are used b r thc phasors as arc used for their corresponding time- dependent qunntities shc lild create lit tle conft~sion. bec;lose ivc will deal almost exclusively with timeh-~nooic liclds (md tllercfore with phasors) in the rest of this book. When there is a na.d to distinguish an instantaneous quantity from a phasor, the time dependcnp of th:: instimtaneous quantity will be indicated erpiicitly by the inclusion of a f in its arqrment. Pha~or qw~tities are not functions of i . It is useful to note that any quantity xntaining j most nccess;~rily be a ph:rsor. The time-harmonic \rave equations for scalar potential V and vector potential A-Eqs. (7-49) and (7-47)- become. respectively, and V ' A t- k2 A = - p,J , whcrc (7-78) is called the waven~mber. Equations (7-77) and (7-78) are referred to as sm~honio- geneous Helmholtz's e q l d x s . The Lorentz condition for potentials, Eq. (7-46), is now The phasor soludbns oi Eqs. (7-77) and (7-78) are obtained from Eqs. (7-60) and (7-61) respectiydy : Je-'tR =.& J " - ' R . dv' (Wb/m). I 298 TIME-'.'2RYING FIELDS AND MAXWELL'S EQUATIONS / 7 These are the expressions for the retarded scalar and vector potentials due to time- harmonic sources. Now the Taylor-series expansion for the exponential factor e-jkR is where k, defined in Eq. (7-79), can be expressed in terms of the wavelength 1 = u/f in the medium. We have Thus, if or if the distance R is small compared to the wavelength I., e-jkR can be aiproximated by 1. Equations (7-81) and (7-82) then simplify to the static expressions in Eqs. (7-42) and (7-43), which are used in Eqs. (7-39) and (7-41) to find quasi-static fields. The formal procedure for determining the electric anhiiagnetic fields due to time-harmonic charge and current distributions is as follows: 1. Find phasors V(R) and A(R) from Eqs. (7-81) and (7-82). 2. Find phasors E(R) = - V V - jwA and B(R) = V x A. 3. Find instantancow E(R. t ) = 9, [ E ( R ) cj'"'] and B(R. t) = 9?, [B(R)~~"'] for a cosine reference. The degree of difficulty of a problem depends on how difficult it is to perform the integrations in Step 1. j Source-Free Fields in Simple Media In a simple, nonconducting source-free medium iharacterized by p = 0, J = 0, a = 0, the time-harmonic Maxwell's equations (7-76a, b, c, and d) become V x E = -jwpH (7-85a) V x 11 = ~ o ) E E (7-85b) V . E = O (7-85~) V . H = O . , (7-85d) Equations (7-85% b, c, and d) can be combined to yield second-order partial differ- ential equations in E and H. From Eqs. (7--64) and (7-65), we obtain V2E + k2E = 0 (7-86) and V211 + li21,1 = 0, (7--87) ninlated \ in Eqs. tlC fif-$. !s LibL .o "'1 for a -form the t . - ?- ' TIME-HARMONIC FIELDS 299 1 . . 3 , ; , ,- . , f: which arc hornopnmu.~ pctor Helmholizk eguutiunr. Solutions of homogeneous Helmholtz's equations with various bouqdae conditions is the main concern of Chapters 8 and 10. Example 7-8 Show that if (E, H) are sol&iohs of source-free Maxwell's equations in a simple medium characterized by 6 and a, then so also are (E', B'), where In Eqs. (7-88a) and (7-~8b), o is an arbitrary angle, and q = is called rhe intrinsic impedulzce of the medium. Solutios: We prove the st'atement by taking the curl and the divergence of E' aod H' and using Eqs. (7-,85a, b, 6 , and d): V x E'= ( V % E) cos 2 -t )1(V x H)sin u = ( -jw,uH) cos u i , q(JoeE) sin u I 1 V x h ' = -- (V x E) sin u + LV x H) cos u YI 1 = -- (-jopH) sin u + (jweE) cos a 1 ; YI ' = jtuc(qH sin a + E cob a) = joeEf; (7-89b) I V d k' = ( V . E) cos u + q(V. H) sin a = 0 ; (7-89c) ' 1 V . k t = --(V ~ : ~ i n u ( ~ . ~ ) c o s o = o . ' 1 (7-89dj Equations (7-S9a, b. c, anJ ?J are source-free ~ v i ~ ~ r e l l ' s equations in El and HI, This example shbws that ~~urce-free M^xwell9s equations for free space are invariant under the linear ,trogrf?rmation spedified by Eqs. (7-88a) and (7-8Sbl. An interesting special case is for ii = 2712. Equations (7-88a) and (7-88b) become H ' = -r: 7 , (7 -90b) Equations (7-90a) and (7-90b) show that $(E, 8) are soiutioss of source-jke Max- well's equations then so also are (E' = qH,.Hf = -Eh). This is a statement of the principle of duulity. his principle is a consequence of the symmetry of source-free Maxwell's equations. 300 TIME-VARYING FIELDS AND MAXWELL'S EQUATIONS / 7 . 8 3 If the simple medium is conducting (a # 0), a current J = aE will flow, and Eq. (7-85b) should be changed to = jw€,E with where E' = E and E" = a/w. The other three equations, Eqs. (7-85a, c, and d), are unchanged. Hence, all the previous equations for nonconducting media will apply to conducting media if E is rcplaccd by the cor~plcs p~wlirtivity E,. The rcal wuvc- number k in the Helmholtz's equations, Eqs. (7-56) and (7-87), will have to be changed to a complex wavenumber kc = o a . The ratio E"/E' measures the magnitude of the conduction current relative to that of the displ;~~cn~cnt CUI-I-~111. It is c ; I I I c ~ : I loss t l ~ i ~ ~ q c n t ~CC;IUSC it is a measure of thc ohmic loss in the medium: --. --. E" a tan 6, = -= -. E' .
.
W E The quantity 6, in Eq. (7-93) may be called the loss angle. A medium is said to be a good conductor if a > > OK, and a good insulator if WE > > a. Thus, a material may be a good conductor at low frequencies, but may have the properties of a lossy dielectric at very high frequencies. For example, a moist ground has a dielectric constant 6, and a conductivity o that are, respectively, in the neighborhood of 10 and lo-' (S/m). The loss tangent ajoe of the moist ground then cquals 1.8 x lo4 at 1 (kHz), making it a relatively good conductor. At 10 (GHz), a/oc becomes 1.8 x and the moist ground behaves more like an insulator.? Example 7-9 A sinusoidal electric intensity of amplitude 50 (Vjm) and frequency 1 (GHz) exists in a lossy dielectric medium that has a relative permittivity of 2.5 and a loss tangent of 0.001. Find the average power dissipated in the medium per cubic meter. Solution: First we must find the effective conductivity of the lossy medium: CJ tan 6, = 0.001 = - OEoEr ' Actually the loss mechanism of a dielectric material is a very complicated process, and the assumption oTa constant conductivity is only a rough approximation. T REVIEW C R. Cl' R. :c. R.' u.- I<. K .
' -n I R.7 R.7 R.7 R.7 Pro R.7 R.7 R.7 R.7 110. R.7 R.7 tlJ1 7 R.7 , - -7rr < , - R.7 OOlT R.7 t l r n ~ H.7 Ind Eq. (7-91) (7-92) d), are 1 apply i wave- : to be l u Illat 3 0 1 f i r 1 0 he a ~y ?x a C I C S ~ ~ I C 3tdllt E, ' tS/m). naking moist - quency 2.5 and r cubic /? 1 - unlption REVIEW QUESTIONS 301 , i d , i + Z ., at; - t The average power di&ipadd per unit volb$e id I p i +JE& $aEZ :I? =!+x (1.389 x loe4) x 50' =0.174(W/m3). ' REVIEW QUESTIONS ', 5 R.7-1 What constitute$ an'ekctromagnetostatic field? In what ways are E and B related in 3 , conducting medium unddjr static conditions? R.7-2 Write the fundatnental postulate for electromagnetic induction, and explain how it leads to Faraday's law. R.7-3 State Lenz's law. R.7-4 Writc thc expression for trxnsforrncr ern(. R.7-5 Write the cxprestlion f i r llux-cuttlng emf. R.7-6 Write the expression :or the induced emf in a closed circuit that moves in a changing rnagnctic field. K.7-7 What is a Faraday d ~ s k generator'? R.7-8 Wrlte the differential folm of Maxwell's quatlons. 7 Arc all four Maxwell's equations ~ndcpcndcnt'l Explain. R.7-I0 Write the integral form of Maxwell's equatiohs, and identify each equation with the proper cxpcrimcntal law. 1 R.7-11 Explain the significance of displacement current. R.7-12 Why are potentid functions used in electromaghetics? R.7-13 Express E and B jn terms of po,ential funct~ons V and A. R.7-14 What do we mean by ,l:/:rsl-\iuir~ f i ~ . r t s ? Are they exact solut~ons of Mauwell's q u ~ - tions'? Explain. R.7-15 What is the Lorentz ciind~tion for pot.ent~als? what is ~ t s physical s~gn~ficance? R.7-16 Write the n~nhoino~encous w,byc oq.~ation forlcalar potcnti:ll I / ,und for vector porcn- Linl A. d R.7-17 State%e-boundary ccmditions for the tangent\al component of E and for the normal component of B. -. R.7-18 Write the boundary conditions for the tangen,tjal component of H and for the normal coniponent of D. R.7-19 Can a static magnetic field exist in the Interior of a perfect conductor? Explain. Can a time-varying magnetic field? Explain. K.7-20 What do we mead by a retarded potential? 302 TIME-VARYING FIELDS AND MAXWELL'S EQUATIONS 1 7 w .
; % 4 & 3 < ~.7-21 In what ways do the retardation time and the velocity of wave propagation depend on the constitutive parameters of the medium? R.7-22 Write the source-free wave equation for E and H in free space. R.7-23 What is a phasor? Is a phasor a function oft? A function of w? R.7-24 What is the diAerence between a phasor and a vector? R.7-25 Discuss the advantages of using p h u o n in electromagnetics. R.7-26 Write in terms of phasors the time-harmonic Maxwell, equations for a simple medium. ,' R.7-27 Define wavenumber. . . R.7-31 In a time-varying situation how do we define a good c o n d & i g - ~ lossi dielectric? R.7-32 Are conduction and displacement currents in phase for time-harmonic fields? Exphin. PROBLEMS P.7-1 Express the transformer emf induced in a stationary loop in terms of time-varying vector potential A. P.7-2 The circuit in Fig. 7-8 is situated in a magnetic field B = a= 3 cos (5xi07t - :rrx) (pT). Assuming R = 15 (Q), f nd the currcnt i. Fig. 7-8 A circuit in a time-varying magnetic field (Problem P.7-2). P.7-3 A conducting equilateral triangular loop is placed near a very long straight wire, in Fig. 6-37, with d = b/2. A current i(t) = I sin o r flows in the straight wire. a) Determine the voltage registered by a high-impedance rms voltmeter inserted in the loop. b) Determins the voltmeter reading when the triangular loop is rotated by 60' about a perpendicular axis through its center. ipls medium. , tiais in terms nconducting, it wire, snown ud in the loop. 60 about a , . r I ; I I : , i PROBLEMS 303 . i P.7-4 A conducthg Oircudr loop of s rudiuafl.1 (h) is rituatcd in the neighborhood of a very long power line carrying a 60-(Hz) current. as shown in Fig. 6-38, with d = 0.15 (m). An AC milliammeter inserted the luop reads 0.3 (mA). Ansume the total impedance of the loop including the milliammeter to bqb.01 (0). , . a) Find the mngdhde gf the current m ihe po&er line. b) To what angle about the horizontal axrs should the c~rcular loop be rotated in order to reduce the milhammeter teading to 0.2 (mA)? , I P.7-5 A conducting 01iding:bar oscillates oCer twd parallel conducting ra~ls in a s~nuso~dally varying magnetic field 1 / , B = a, 5 css O J ~ (rnT), as shown in Fig. 7-9. The posltion of the sliding bar is glven by x = O.ii(l - cos or) (m), and the rails are termmated in a resistance R = 0.2 (R). Find i. R - Fig. 7-9 A conducting bar slldlng over parallel ralls In a tlme.varylng magnetlc field (~rbblem P.7-5). P.7-6 Assuming that a rcslstsnce R n connected across the slip rings of the rectangular con- ductlng loop that rotates in a constant magnetic field B = ap,, shown in Fig. 7-5. prove that the power dissipated ih R is equal to the power required to rotate the loop at an angular fre- quency o. i P.7-7 Derive the two divergence equations, Eqs. (7-37c) and (7-376). from. the two curl equations, Eqs. (7-3711 and 17-37b), and the equation of continuity, Eq. (7-32). .
I P.7-8 Prove that the Lor:ntz condition for potentials as expressed in Eq. (7-46) is consistent with the equation of cdfitinxty. P.7-9 Substitute ~ ~ 3 . (7-30) and (7-41) in Maxwell's equations to obtain wave equations for scalar potentiA V and vector potential A for a linear, isotropic but inhomogeneous medium. P.7-10 Write the set of four Maxwell's equations, Eqs. (7-37a, b, c, and d), as elght scalar equations -'\ - a ' in Cartesian cdordi,l~tes, b) in cylindrical ~bordinates, c) in spherical coordinates. P.7-11 Supply the detailed steps for the derivGion of the electromagnetic boundary conditions, Eqs. (7-50a, b, c, and d). 304 TIME-VARYING FIELDS AND MAXWELL'S EQUATIONS 1 7 P.7-12 Discuss the relations a) between the boundary conditions for the tangential components of E and those for the normal components of B, b) between the boundary conditions for the normal components of D and those for the tangential components of H. P .
7 -1 3 Write the boundary conditions that exist at the interface of free space and a magnetic material of infinite (an approximation) permeability. P.7-14 The electric field of an electromagnetic wave is the sum of E, = ex 0.03 sin 10% and El = ex 0.04 cos [L08x(t - j ) - 3 -'. Find Eo and 8. P.7-15 Prove by direct substitution that any iwice differentiable function of (t - R&E) or of (t + R f i ) is a solution of the homogeneous wave equation, Eq. (7-56). P.7-16 Prove that the retarded potential in Eq. (7-60) satisfies the nonhomogeneous wave equation, Eq. (7-49). P.7-17 Write the general wave equations for E and H in a nonconducting simple medium where a ,charge distribution p and a current distribution J exist. Convert the wave equations to Hclrnholtz's cqu;ltions for sinusoidnl time dcpcndence. P.7-18 Given thi1t E = a, 0.1 sin (IOnx) cos (6nlO"f - p i ?
) (V/ln) in air. find M and /L P.7-19 Given that H = a, 2 cos ( 1 5 ~ ~ ) sin (6x109t - pz) (Aim) in air, find E and P. P.7-20 It is known that the electric field intensity of a spherical wave in free space is Eo . E = a, - sin 0 cos (wr- kR). R Determine the magnetic field intensity H. P.7-21 In Section 7-4 we indicated that E and B can be determined from the potentials V and A, which are related by the Lorentz condition, Eq. (7-80). in the time-harmonic case. The vector potential A was introduced through the relation,B = V x A because of the solenoidal nature of B. In a source-free region, V . E = 0, we can definc another type of vector potential A,, such PROBLEMS 305 those for the P.7-22 For a source-ffke me'dium where p ='O, J = 0, p = po, but where there is a volume density of polarization P; a single vector potentlal n, d a y be defined such that I H = jm$, x 'n,. (7-94) a) Express electricjfield iptendty E in temp of n and P. b) Show that x, satisfies the h~nhomo~enkous elmholtz equatlon .
I .
; I-/ P V2n, + k2z a e ' - - --' (7-95) € 0 id a magnetic ' I ;. i : I : ! , ! P.7-23 Calculations cdhccrning thc clcctrornjlgnctit! clTect of currents in a good conductor usu:~lly ncglcct llic tlisplllcc~i~x~ll cu~.rcnl cvw ;iL ~iii~rnw:~vc rrcqucncics. (a) Assunling t, = I and u = 5.70 x lo7 (S/m) for copper, compare thc magnitude of the displaccment current density with that of the conduction current density at 100 (GHz). (b) Write the governing diRerentia1 equation for magnetic field intensity H in a source-free good conductor. ;t'cnlials V and ase. The vector ' rnoidal nature ential A,, such i 8 i Plane Electromagnetic Wqyes 8-1 INTRODUCTION In Chapter 7 we showed that in a sburce-free simple medium Maxwell's equations, Eqs. (7-62a, b. c, and d) can be combincd to yield homogcncous vcctor wavi: equa- tions in E and in H. These two equations, Eqs. (7-64) and (7-65). huvc exactly the same form. In free space. the sourcc-free wave equation for E is -.
. 1 . where is the velocity of wave propagation (the speed of light) in free space. The solutions of Eq. (8-1) represent waves. The study of the behavior of waves which have a one- dimensional spatial dependence (plane waves) is the main concern of this chapter. We begin the chapter with a study of thc propaption of time-harmonic plane- wave ficlds in an unbounded homogcncous nicdiurrl. Medium paramclcrs sirch xs intrinsic irnpcdance, attenuation constant, and phase constant will be introduced. The mcaning c~T.\kiri dcptll, tlic clcpth of wave pcncttxtion into : I good concluctor, will he explained. Electromagnetic waves carry with them clectromagnetic power. The concept of Poynting vector, a power flux density, will be discussed. We will examine the behavior of a plane wave incident normally on a plane boundary. The laws governing the reflection and refraction of plane waves incident obliquely on a plane boundary will then be discussed, and the conditions for no reflection and for total reflection will be examined. A uniform plane wave is a particular solution of Maxwell's equations with E (and also H), assuming the same direction, same magnitude, and same phase in infinite planes perpendicular to the direction of propagation. Strictly speaking, a uniform plane wave does not exist in practice, because a source infinite in extent would be required to create it, and practical wave sources are always finTte in extent. But, if we are far enough away from a source, the .wavefi.ont (surface of constant phase) 8-2 PLANE WAVES IN L~SSLESS MEDIA In this and future focus our attention on wave behavior in the sinusoidal , , ,. great advantagd. The source-free, wave equation, Eq. ' . a h ~ m o g e ~ e o u s ~ ~ e c t o r Helmholtz's equatioli (see Eq. . , : 7-86): I C -P P; (8-3) . where k, is the free-bpace w~ver~umher :quatlons, b e a one- . In Cartesian cobrdiriates, Eq. (8-3) is equivalent to three scalar Helmholtz's -; equations, one each ih thc components E,, E,, and E,. Writing it for the component Ex, we have Consider a uniform plane wave characterized by a uniform E, (uniform magnitude and constant phase) over plane surfaces perpenducular to z; that is, aiL',/as2 = o and S2E,/ay2 = 0. Equation (8-5) simplifies to which is an ordinary differential equation becake Ex, a phasor, depends only on r. The solution of Eq. (8-6) is readily seen to be E,(r) E; (z) + El(:) , = Eg+e-~bo~ + ~;&koz, ' -' .
- (8-7) where ki and E; aie arbitrary (and,'in general, complex) constants that must be determined by boundaj conditions. Note that since Eq. (8-6) is a second-order equation, its general sblut~m in Eq. (8-7) contains two integration constants. Now let us examine what the first phesor term on the right side of Eq. (8-7) represents in real time. Using.cos wt as the reference and assuming E l to be a real . . - 308 P~ANE ELECTROMAGNETIC WAVES 1 8 , : , , I , . 0 z Fig. 8-1 Wave traveling in positive z direction E:(z, t) = E,f cos (wt - k,~), for several values oft. constant (zero reference phase af z = 0), we have = E; cos (wt - koz) (V/&P (8-8) Equation (8-8) has been plotted in Fig. 8+ for several values oft. At t = 0, E:(r, 0) = , E; cos koz is a cosine curve with an amplitude E;. At successive times, the curve ellcctively travels in the positive i direction. We have, then, a truuelitly wove. If we fix our attention on a particular point (a point of a particular phase) on the wave, we set cos (tot - k,:) = a constant or wt - koz = A constant phase, from which we obtain dz w - - --- - c. dr k, (8-9) Equation (8-9) assures us that the velocity of propagation of an cquiphase front (the phase velocity) in free space is equal to the velocity of light, which is approximately 3 x lo8 (m/s) in free space. The quantity k, bears i definite relation to the wavelength. From Eq. (8-4), k, = 2nflc or which measures the number of wavelengths in a domplete cycle, hence its name. An inverse relation of Eq. (8-10) is I ( 2 . c 1~ curve c. If we e wave, mt (the mately I ' , . 8-2 1 P~ANL WAVES IN LOSSLESS MEDIA 309 . . I I ' I .
A 1 t I i . ' < ! ! i f" . ~ ~ u a t i o n s (8-10) adb (8711) are valid withoet the subscript 0 if the medium is a lossless material sucd as a.perfect dielectric. : It is obvious hitbout replottins that thk sehond phasor term on the right side of 'Eq. (8-7), E;#O, rdbresents a cosinusoidal wive traveling in the - .- direction with the same velocity e. Id an anbounded regiori w C are concerned only with the outgolng wave; hcncc, if tllc S(IIII.CC is on IIic ldl. Ibc:ncg~livcly goi~lg WiiVI. does not exist, and Eg = 0. However, if there are dmontinuities injhe medium, reflected waves traveling in the opposite direction must also be coniidered, as we will see later in this chapter. The associated rbagne'tic field H can 6 5 folind from Eq. (7-85a) - - , ! : I , 1 ' ' I 0 0 1 which leads to I aE+(z) H,+ = -A - j y ~ ~ dz , H+ = O . Thus, H,? is the only hon cero component of EI: and since . - We have introduced a nej i quantity, q,, in Eq. (8-1)): which is called the intrinsic in~ped~?dnnee of theflee qoce. Because ,lo is 3. real number H;(z) is in phase with ~ : ( r ) ; and we can write tlie instantaneous expression for H as ' Q(z, .) ;,H; (z, i) = i,, 9a[HT(r)ej"] Hence, for a uniform plhne wave, the ratio o;the magnitudes of E and H is the intrinsic impedance of the medium. We al~o~note that H is perpendicular to E and that both are normal to the direction of propagation: The fact that we specified E = axEx 310 PLANE ELECTROMAGNETIC WAVES 1 8 is not as restrictive as it appears, inasmuch as we are free to designate the direction of E as the +x direction, which is normal to the direction of propagation a,. Example 8-1 A uniform plane wave with E = axE, propagates in a lossless simple . medium ( E , = 4, p, = 1, c r = 0) in the + z direction. Assume that Ex is sinusoidal with a frequency 100 (MHz) and has a maximum value of + low4 (V,/m) at t = 0 and 1 = (m). a) Write the instantaneous expression for E for any t and z. b) Write the instantaneous expression for H. c) Determine the locations where E, is a positive maximum when t = (s). Solution: First we find k. a) Using cos ot as the reference, we find the instantaneous expression for E to be E(z, t) = a,E, = a,10-4 cos (277 1O8t - kz + $). Since E , equals + when the argument of the cosine function equals zero- that is, when 2rr 108t - k: + $ = 0, we have, at t = 0 and z = $, Thus, E(Z, t) = ax10-4 c ~ s 2~ io8t - -z + - -. ( 3 6 ! = axlo-' cos [Zn lost - (i - : ) I (~/rn). This expression shows a shift of a mere $ in the +z direction and could have been written down directly from the statement of the problem. b) The instantaneous expression for H is where = 0 and ! (s). / 7 E e s mo- ~uld have n t I , 8-2 1 PLAhE WAVES IN LOSSLESS MEDIA 311 I - 1 Hence, 3 . r 1 , H~Z, t ) = a , , cos [b IO'L - 7 (z - i)] (/mi. ' c) At t = lo-', witequate the argument of tke cosine function to +2nn in order to make E, a maximumi Examining this result more closely, we note that the wavelength in the given medium is Hence, the positive naximum value of Ex occurs at 13 Z,,, = , - L hL (m). 8 The E and H fleldr are shown in Fig. 8-2 as functions of z for the reference time t = 0. Fig. 8-2 E and H fields of a uniform plane wave at f = 0 (Example 8-1). 312 PLANE ELECTROMAGNETIC WAVES 1 8 8-2.1 Transverse Electromagnetic Waves We have seen that a uniform plane wave.characterized by E = a,E, propagating in the + z direction has associated with it a magnetic .field H = ayHy. Thus E and H are perpendicular to each other, and both are transverse to the direction of propaga- tion. It is a particular case of a transverse electromagnetic (TEM) wave. The ph;l:or field quantities are functions of only the distance z along a single coordinate <.-,is. We now consider the propagation of a uniform plane wave along an arbitl~iry , direction that does not necessarily coincide with a coordinate axis. The phasor electric field intensity for a uniform plane wave propagating in the + z direction is E(z) = Eoe-jkz, (5-16) where E, is a constant vector. A morc gcncrnl form o T Eq. (8-1 6) is , E(-Y, yr =) = Eoe-lk+-lkd"~ki' (8-1'7) It can be easily proved by direct substitution that this expression satisfies the homo- geneous Helmholtz's equation, provided that --. 1 - /if + k; + k : = W ~ , D E . (8-15) If we define a wavenumber vector as and a radius vector from the origin then Eq. (8-17) can be written compactly as E(R) = Eie-~k ' R - - E e-jk%. R 0 (v/m), (8-21) where a, is a unit vector in the direction of propagation. From Eq. (8-19) it is clear that k, = k a, = ka, . a, (8-22a) ky = k a, = ka, a, (8 -22b) k, = k . a, = ka; a,, (8-22c) and that a,. a,, a, a, and a, . aZ are direction cosines of a,. Thc gcomctrical relations or a,, and I< arc illustrated in Fig. 8-3, from which we see that a, R = Length OP (a constant) is the equation of a plane normal to a,, the direction of propagation. Just as z = Constant denotes a plane of constant phase and,uniform amplitude for thc wave in , r . a I ' " i t , . . ?
, , !
I , agating in ' " ' ! 3 ; E a n d H # . !! f propaga- I 1 'he phacor I 1 inate :i?;k. , 1 . ting i3 the . , L; 1 i (8-16) the homo- P I 8) (8-19) (5 -20) (8-21) 3) it is clear ' (8 -22a) (8-22b) (8 -224 f r o ?
?
i c h (\ Just rfs = ihc wave in Fig. 8-3 Radius vector and wave normal Y Plane of constant to a phdk front phase (phase front) of a uniform plane wave. Eq. (8-16), a,, . R = Constant is a plane of constant phase and uniform amplitude for the wavc in Eq. (8-21). In a charge-free region, Y - & = 0. As a result, Hence Eq. (8-23a) can be written as a,, -E, = 0. (8-23b) Thus the plane-wave solution in Eq. (8-17) implies that E, is transverse to thc Jirec- tion of propagatioli. The magnetic held associated with E(R) in Eq. (8-21) may be obtained from Eq. (7-85a) as --------, where This is a consequence Of the iact that 7 E , = 0, where E , is a constant vector (see problem P.2-18). 314 PLANE ELECTROMAGNETIC WAVES / 8 is the intrinsic impedance of the medium. Substitution of Eq. (8-21) in Eq. (8-24) yields It is now clear that a uniform plane wavc propagting in an :irbitmry direction. a,,, is a TEM wave with E I H and that both E and H arc t~orn~al to n,. Polarization of Plane Waves The polarization of a uniform planewave describes the time-varying behavior of the electric field intensity vector at a given point in space. Since the E vector of the plane wave in Example 8-1 is fixed in the x direction (E = a,Ex, where Ex may be positive or negative), the wave is said to be linearly polurized in the x direction. A separate description of magnetic-field behavior is not necessary. inasmuch as the direction of H is definitely related to that of E. In some cases the direction of E of a plane wave at a given point may change with time. Consider the superposition of two linearly waves: one polarized in the x direction; the other polarized in the y direction and lagging 90" (or n/2 rad) in time phase. In phasor notation we have where El, and E,, are real numbers denoting the amplitudes of the two linearly polarized waves. The instantaneous expression for E is = axElo cos ( o t - kz) + a,Ezo cos In examining the direction change of E at a givcn point as t changes. it is convenient to set z = 0. We have E(0, t) = axEl(O, t) + a,Ez(O, t) = axElo cos o t + h,Ezo sin o t . (8 -28) As wt increases from 0 through 42, n, and 3x12 - the cycle at 2n - the tip of the vector E(0, t) will traverse an elliptical direc- tion. Analytically, we have El(O9 t) COS W t = --- El, 4. (8-24) ,$--26) :ction, a,, 3 !or of the the plane e posit~ve wparate :ec:~on of ,P L c; ii " . . ' . I ' . L .I E2(0,t) $ 1 7 sih o t = - E2 0 which leads to the foilowing equation for an ellipse: (8 -29) Hence E, which is tqe sum of two linearly polarized waves in both space and time quadrature, is ellipticully pobrired if E20 # Ela. and in circelurly polilr~zed if Ezb = . El,. A typical polarization clrcle is shown in Fig. 8-4(a), When E2, = ElQ1 thc instantaneous angle ct which E makes with the r-axis at -. = 0 is I which indicates that E lotates at a uniform rate with an angular velocity io in a countcrclockwisc directicn. When the fingers of the right hand follow the direction Fig. 8-4 Polarization diagrams for sum of two linearly polarized waves in space quadrature at z = 0: (a) circular polarization, E(0, t) = E,,(a, cos o r f a, sin at); (b) linear polarizhtion, E(0, t) = (a,E,, + ayE2,) cos wr 316 PLANE ELECTROMAGNETIC WAVES 1 8 ' of the rotation of E, the thumb points to the direction of propagation of the wave. This is a right-hand or positive circularly polarized wave. If we start with an E2(z), which leads El(z) by 90" (71.12 rad) in time phase, Eqs. (8-27) and (8-28) will be, respectively, E(z) = axEl ,e-jkz + a YJ 'E 20 e-jk? (8-31) and E(0, t) = a,Elo cos ot - a,,E2, sin ot. (8-32) Comparing Eq. (8-32) with Eq. (8-28), we see that E will stiJJ. be elliptically polarized. If Ez0 = El,, E will be circularly polarized and its angle measured from the x-axis at z = 0 will now be -ot, indicating that E will rotate with an angular velocity w in a clockwise direction; this is a left-hand or negative circularly polarized wave. If E,(z) and El(z) are in space quadrature but in time phase, their sum E will be linearly polarized along a line that makes an angle tan-' (E20/E,o) with ,the x-axis. as depicted in Fig. 8-4(b). The instantaneous expression for E at : = 0 is E(0, t) = (n,Elo + a,E,o) cos tot. (8-33) -----. The tip of the E(0, t) will be at the point P , when wt = 0. Its magnitude will decrease toward zero as wt increases toward n/2. After that, E(0, t) starts to increase again, in the opposite direction, toward the poiit P, where wt'= n. In the general case, E2(z) and El(:), which are in space quadrature. can have unequal amplitudes (E2, # El,) and can differ in phase by an arbitrary amount (not zero or an integral multiple of 7~12). Their sum E will be elliptically polarized and the principal axes of the polarization ellipse will not coincide with the axes of the co- ordinates (see Problem P.8-4). Example 8-2 Prove that a linearly polarized plane wave can be resolved into a right-hand circularly polarized wave and a left-hand circularly polarized wave of equal amplitude. Solution: Consider a linearly polarized plane wave propagating in t h ~ + z direction. We can assume, with no loss of generality, that E is polarized in the x direction. In phasor notation we have E(z) = axl&c-jk', But this can be written as E(z) = E,,(z) + E d 4 where Eo E,&) = - (a, - ja,)e- jkz , , . 2 (8 -34a) and " . i ' Eo Ek(~) = 7j- (a,., + jay)e-~kz. (8 -34b) 8 -3 COh of the wave. phase, Eqs. (8-31) (8 -32) y polarized. n the s-axis r velocity 0 1 ' M'NL'L'. m E will be 1 rhs x-axis, vtd into a xi wave of : direction. iection. In , ' I 8-3 / PLANE WAVES IN CONDUCTING MEDIA 317 3 C '. .r, i C r From previous dig$ussions we recognize thdt E & ) in Eq. (8-34a) and E,,(r) in ; Eq. (8-34b) represent, iespectively, right-h&d and left-hand circularly . waves, each havingian amplitude E,/2. ~he;btatement of this problem is therefore . , proved. The conte:se statement that the sum of two oppositely rotating circuiariy polilrimd waves of equal amplitude is a linearly polarized wave i s of course, also true. ~. 8-3 PLANE WAVES lhj , ; CONDUCTING MEDIA ' T . ' ' I In a source-free codducting medium, the homogeneous vector Helmholtz's equation to be solve is where the wavenumber kc = w& is a complex number because s , = 6 - jd' is complcx. 2 s dcfincd in Eq, (7792). The dcrlvations and discussions pcruin~ng to plane waves in a lossless medium in Section 8-2 can be modified to apply towave propagation in a cpndo:ting medium by simply replacing k with kt. However, in an effort to conform with tle conventional notation used in transmission-line rheory. it is cusmn~:~ry to dcflii~. :I psirp:~g;\~io~~ const;ld/, 11, such th:\r I -- I S i n c ~ 7 is c~mplex, wc i:ritc, with the help of.& (7-92) where u and /I are, resjactively, the real and imaginary parts of 7. Their physical significance will be explained presently. For a lossless medium, 0 = 0, 1 = 0, and / I = k = t o & The Helmhoitz's cqwtion, Eq. (8-X), bcco~nes The solution of E ~ : (8-38), whi& corresponds to a uniform plane wave propagating in the + s direction, is .- 1- E = a,E, = a,E,e-Yz, (5-39) where we have assbmed that the wave ii linearly polarized in the .x direction. The propagation factor e-"' can be written as a product of two factors: E, = ~ , i ~ - " ~ ~ - j L ' ' , As we shall see, both x and j 3 are positive quantities. The first factor, e-", decreases 318 PLANE ELECTROMAGNETIC WAVES 1 8 1 .
.
. as z increases and, thus, is an attenuation factor, and a is called an attenuation con- I stant. The SI unit of the attenuation constant is neper per meter ( N P / ~ ) . ~ The second factor, e-jt', is a phase factor; P is called a phase constant and is expressed in radians per meter (rad/m). The phase constant expresses the amount of phase shift that occurs as the wave travels one meter. General expressions of a and in terms of o and the constitutive parameters-E, p, and a-of the medium are rather involved (see Problem P.8-6). In the following paragraphs we examine the approximate expressions for a low-loss dielectric and a good conductor. 8-3.1 Low-Loss Dielectric A low-loss dielectric is a good but'imperfect insulator with a nonzero conductivity, such that E" << E' or a / w ~ << 1. Under this condition y in Eq. (8-37) can besapproxi- mated by using the binomial expansion. from which we obtain the attenuation conitant - and the: phasc constant It is seen from Eq. (8-40) that thc attcnualion constant of a low-loss dielectric is a positive constant and is approximately directly proportional to the conductivity 0. The phase constant in Eq. (8-41) deviates only very slightly from the value w f i for a perfect (lossless) dielectric. The intrinsic impedance of a low-loss dielectric is a complex quantity. Since the intrinsic impedance is the ratio of Ex and H, for a uniform plane wave, the electric and magnetic field intensities in a lossy dielectric are, thus, not in time phasc, as they would be in a lossless medium. . - - -- - -. . . -. - - Ncper is a dimensionless quantity. If s ! = 1 (Np/mJ, thcn a unit wave amplitude dccreascs lo :I magnitude t . - I (=0.368) as it travels a distance of I (m). In terms of'field intensities 1 (Np/m) equals 20 log,,t = 8.69 (dB/m). ductivity, : ~pproxi- lectdic is a uctivity a. due a & € - 8-3 1 PLANE ~ A V E S IN CONDUCTING MEDIA 319 ! ;- 1 L The phase vhocky up is obtained from the ratio o/b in a manner similar to that - 1 in Eq. (8-9). (8t41), we have " , I - I (mls). 8-3.2 Good Conductor .. 1 , A good conducto~ih a medium for which C . c' or o / w > > 1. Under this condition we can neglect 1 in tomparison with the terd o/jw in Eq. (8-37) and write i I + j '7 i j u & J g = ' ~ ~ = -= Jz or ~ = + j j l j z ( l +j)&, where we have used thc relations and o = 2nf. Ecjudtion (8-44) indicates that i l and /j for a sood conductor arc approximately equal an.1 both increase us $ and &. For a good conductor, L I The intrinsic impedi-nce of a good conductor is which has a phase angh: of 1 5 ' . Hence the msgnctic field intensity lags behind the electric field intensity by 45". The phase velocity in a good conductor is which is proportiunul to ,/r and I/&. Consider copper as an example: o =i.80 x lo7 (S/m), % .
-% . p = 4 7 c x 10-'(H/m), / I , , 770 (~nis) :It 3 (MI 17). \vhicl~ is :lboul laicc 1I1c vcl~cily of suuiid in ~ l i r il~ld is many ordors of magnitude ;lower than the velocity of light in air. The Wavelength of a plane wave in a good conductor is , 320 PLANE ELECTROMAGNETIC WAVES 1 8 For copier iti3 (MHz), i = 0.24 ( m d . As a comparison, a 3-(MHz) electromagnetic wave in air has a wavelength of 100 (m). At very high frequencies the attenuation constant a for a good conductor, as given by Eq. (8:45), tends to be very large. For copper at 3 (MHz), cc = Jn(3 x 106)(4x x 10-7)(5.80 x lo7) = 2.62 x lo4 (Nplm). Since the attenuation factor is e-"', the amplitude of a wave will be attenuated by a factor of e-' = 0.368 when it travels a distance 6 = l/a. For copper at 3 (MHz), this distance is (112.62) x (m), or 0.038 (mm). At 10 (GHz) it is only 0.66 (pm)-a very small distance indeed. Thus, a high-frequency electromagnetic wave is attenuated very rapidly as it propagates in a good conductor. The distance d through which the amplitude of a traveling plane wave decreases by a factor of e- ' or 0.368 is called the s k i ~ l depth or the depth ?f'yrnetrcdion of a conductor: Since cc = p for a good conductor, 6 can also be written as At microwave frequencies, the skin depth or depth of penetration of a good conductor is so small that fields and currents can be considered as. for all practical purposes, confined in a very thin layer (that is, in the skin) of the conductor surface. Example 8-3 Thc clcotric Gcld inlcnsity ol'a lincariy polarized uniform plane wave propagating in the + z direction in sea water is E = aJ00 cos (107nt) (V/m) at z = 0. The constitutive parameters of sea water are 6, = 80, pr = 1, and a = 4 (S/m). (a) Determine the attenuation constant, phase constant, intrinsic impedance, phase velocity. wavelength, and skin depth. (b) Find ihc distance a: which ihc amplitude of E is 1% of its value-at z = 0. (c) Write the expressions for E(z, t) and H(z, t) at z = 0.8 (m) as functions oft. Solution o f = - = 5 x lo6 (Hz), 2 1 1 . I' 8-3 1 PL&>E WAVES IN CONDUCTING MEDIA 321 f ? . h I uated by a MHz), this - I I 6 (pm) -a attenuated I ! which the I calicd the ;onductor 1 purposes, )lane wave m) at r = = 4 (S/m'). nce. phase ~plitude of r, 1) at z = f Y 4 I f ' ; 5 Hence we cart. b&! theLformulas for good c'ondiktors: a) Attenuation consfant, a = - = ,/5nlo6(4n10-!)4 = 8.89 (Np/m). Phase constant, P = = 8.89 (rad/m). Intrinsic impedatice, . ! + j -% = ( I + j) $ T x 106)(4n x lo-') 4 - - ne'"'" (R). Phase velocity, Skin depth, 1 1 S=-=-=Q112( cc 8.89 . m). -.. , b) Distance z, at which the amplitude of wavd decreases to 1 9 o f I ~ S ' value at 2 = 0: 1 4.605 z , =-In 100 = -- ci 8.89 - 0.518 (m). c) In phasor notation, E(=) = 3x10()e-eze-~/'=. The instantaneous expression for E is E(z, t) = Wt [E(z)eTa'] 1 % B d [axlOOe-"zeJ('-~q] = axlOOe-az cos (or - 8;). , At r7 = 0.8 (m), we have E(O.8, r ) = aJOOe-0.8a cos (107nt - 0.81~') = a,0.082 cos (107nt - 7.11) (V/m). We know that a uniform plant wave is a TEM Lave with E I H and that both are normal to the direct~on of wave propagadon a,. Thus H = a,,H,. To find I 322 PLANE ELECTROMAGNETIC WAVES I 8 H(z, t), the instantaneous expression of H as a function of t, we must not make the mistake of writing Hy(z, t) = E,(z, t)/v,, because this would be mixing real time functions E,(z, t) and HZ(z, t) with a complex quantity q , . Phasor quantities E,(z) and Hy(z) must be used. That is, from which we obtain the relation between instantaneous quantities For the present problem we have, in phasors. , Note that both angles must be in radians before combiniug. The instantaneous -- expression for H at z = 0.8 (m) is then We can see that a 5-(MHz) plane wave attenuates very rapidly in sea water and becomes negligibly weak a very short distance from the source. This phe- nomenon is accentuated at higher frequencies. Even at very low frequencies, long-distance radio communication with a submerged submarine is extremely .difficult. 8-3.3 Group Velocity In Section 8-2 we defined the phase velocity, up, of a single-frequency planc wavc as the velocity of propagation of an equiphase front. The relation between up and the phase constant, /3, is I For plane waves in a lossless medium, /3 = w@ is a linear function of w. As a conscqucncc, the phase velocity u , = 1 1 4 ; is a constant that is indcpcndcnt of frequency. However, in some cases (such as wave propagation in a lossy dielectric, as discussed previously, or along a transmission line, or in a waveguide to be dis- cussed in later chapters) the phase constant is not a linear function of o; waves of different frcquencics will propagate with dircrcnt phxx vclocitics. Inasmuch as all information-bearing signals consist of a band'of frequencies, waves of the component not make . lixing real quantities n water : . This phe- Srequencies, IS extremely plane wave ..em r r , and P ~q 4s :I icpcnc- >t of sy dielectric, Je to be dis- w : waves of d much as all e component 1 . frequencies travel with different phase velocities, causing a distortion in the signal wave shape. The signitl "disperses." The phenomenon of signal distortion caused by r dependence of the pHase ielocity on frequency 1s called dispersion. Given Eq. (8-43). we conclude that a lossy dielectric is obviously & dispersive medium. An information-bearing signal normally has a small spread of frequencies (sidebands) around a high carrier frequency. Such a signal comprises a "group" of frequencies and forms,a wave packet. A group u~locity is the velocity of propagation of the wave-packet erlielope. C~nsider the simblest case of a wave packet that consists of two traveling waves having equal amplidde and s!ightly different angular frequencies wo + Aw and wo - A o (Aw < < w,). The phase constants. being functions of frequency. will also be slightly different. Let the phase constants corresponding to the two frequencies be Do + AP and Po - AD. We have E(:, t) == E~ cos [(coo - Aw)t - (Po + Ail):] + Eg cos [((oO - Ao))t - (/j0 - A/]):] == LEO cos (1 AOJ - : AS) cos (oot - Po:). (8-51) Since Aw << wO, the exprwion in Eq. (8-51) represents a rapidly oscillating wave having :ln i~ngi~lar lrcquc~.cy in,, and a n i~mplitildc that varies slowly will] ;lo ;~npul;ir li-cquc~~cy All). 'I'his is depicted in Fix. 8- 5. The wave inside theenvelope propagates with a phase velocity found by setting wot - p0z = Constant: The velocity of the envelope (the group velocity 11,) can be determined by setting the argument of the first cosine factor in Eq. (8-51) equal to a constant: t Aw - z AD = Constant, I Fig. 8-5 Sum of two tihe-harmonic traveling waves of equal amplitude and slightly different frequenb~es at a given t. 324 PLANE ELECTROMAGNETIC WAVES / 8 from which we obtain 1 dz Aw u = - = - dt Ap -ma In the limit that A o - 0, we have the formula for computing the group velocity in a dispersive medium. This is the velocity of a point on the envelope of the wave packet. as shown in Fig. 8-5. and is identified as the velocity of the narrow-band signal. .
, A relation between the group and phase velocities may be obtained by combining Eqs. (8-50) and (8-52). From Eq. (8-50), we have ---. Substitution of the above in Eq. (8-52) yields lip U g = w du 1 --P up nw From Eq. (8-53) we see three possible cases: a) No dispersion : du P = 0 (up independent of w, /3 linear function of w), dw U g = U p . b) Normal dispersion: duP - -? 0 (up decreasing with w), . dw ll,, < 14,). c) Anomalous dispersion: dup - > 0 (up increasing with w), dw Example 8-4 A narrow-band signal propagates in a lossy dielectric medium which has a loss tangent 0.2 at 550 (kHz), the carrier frequency of the signal. The dielectric (8-52) I I Fig. 8-5, . . ombining lium which IC dielectric I ' ent o/oc = 0.2 and 02/8(&c)' < < 1, Eqs. (8-40) and (8-41) can a and , O respectively. But first we find o from the loss tangent: a = 1.53 x (S/m). Thus, .- d = 0.0182 x 1 .005 = 0.01 $3 (radlm). b) I'hasc velocity: From Eq. (8-41) we h ~ v e c) Group velocity: Since ug # 5, the medium is dispersive As we can see, the computed values of u, and up do not differ much because of the Small value of the loss tangent. . % 8-4 FLOW OF ELECTROMAGNETIC i I - d POWER AND THE POYNTING VECTOR i ' I I Electromagnetic waves carry with them electromagnetic power. Energy is transported : through space to distant receiving points by electromagnetic waves. We will now 1 derive a relation between the rate ofsuch energy transfer and the electric and magnetic field intensities associated with a traveling electromagnetic wave. We begin with the curl equations I The verification of the following identity of vector operations (see Problem P. 2-23) [ is straightforward: I I 1 V . (I3 x 1-1) = 11. (V x E) - E . ( V x 1-11. (8 -56) .
? Substitution of Eqs. (8-54) and (5-55) in Eq. (5-56) yields -'-. In a simple medium, whose constitutive parameters r, 11, and D do not chan, oe with time, we have 1 dB H.- - -H .
= - d(pH) l i ( p H s H ) - < ( l - ) dr at 2 ijt st p H 2 GO I I I nccrt: - 1s) (7 15 .---= E .---- - - (7 t (7t 2 (11 E - J = E - ( c E ) = a E 2 . I Equation (8-57) can then be written as i, V(E x H ) = -- (8-58) e .
( - which is a point-function relationship. An integral form of Eq. (8-58) is obtiined by integrating both sides over the volume of concern. 6 (E x H) ds = oE2 do, (8-59) where the divergence theorem has been applied to convert the volume integral of V - (E x H) to the closed surface integral of (L x H). i magnetic 9) (8-54) h) (8-55) 3 1 I'. 2-23) (8-56) t p 57) hmge with (8 -58) 1 s obtained 0 5')) --. integral of -ansported : + :'will now + ! the first and second t&ms on the right side of Eq. (8-59) of ,change of the energy stored, respectively, in the electric with Eqs. (13-i46b) and (6-151c).] The last term is volume as a rk?sult of the flow ofconduction current electric ficld E. Hence we may interpret the right side of Eq. (8-59) as the rate o f decrease of the electric and magneuc energies stored, subtracted by the ohmic power dissipated as Heat in the volumc V. Tn order to be consismt kith thc ltlw of conscrva~ion ol' cllcrgy, this must equal the power (rate of energy) leaviilg the volume through its surface. Thus the quantity (E x H) is a vector representing the power flow per unit are'd. Define Quantity : P is knoivn as rhc Poyillitlq ~icpr.Lo/., which is a power dcnsity vector asso- ciatcd with an' elec~roma~nctic ficld. The assertion that the surhce integral of .
P over a closed surface, as given by the left side of Eq. (8-59), equals the power leaving the enclosed volume is referred to as P O > . ) I ~ ~ J I ~ ' S theorem. Equation (8-59) may be writren in another form where we = ~ E E ' = Electric energy density, w,, = $pH2 = Magnetic energy density, p, = O E ' = J ~ / O 7 Ohmic power density. In words, Eq. (8-61) states that the total power flowing ittto a closed surface at any instant equals the sum of the rates of increase of the stored electric and magnetic energies and the ohmic power dissipated within the enclosed volume. Two points concerning the Poynting vector &re worthy of note. First, the power relations given in Eqs. (8-59) and (8-61) pcrtain to the total power flow across a closcd surface ohtaincd by Lhc surhcc integral of (E x 1-1). The definition of the Poynting vector in Eq. (8-60) as the power density vector at every point on the surface is an arbitrary, albeit useful, concept. Second, the Poynting vector :P is in a direction normal to betkEE and H. Ifthc region ofconccrn is losslcss (cr = 0). rhcn the I:N tcrm in Eq. 18-61) vanishes. ~ind the total power flowing into a closed surface is equal to the rate of increase of the.stored electric and magnetic energies in the enclosed volume. In a static situation, the first two terms on the right side of Eq. &61) vanish, and the total power Howing into a closed surface is equal to the ohmic power dissipated in the enclosed volume. 328 PLANE ELECTROMAGNETIC WAVES 1 8 . , Example 8-5 ' ~ i n d the Poynting vector on the surface of a long, straight conducting wire (of radius b and conductivity v) that carries a direct current I. Verify Poynting's theorem. Solution: Since we have a DC situation, the current in the wire is uniformly dis; tributed over its cross-sectional area. Let us assume that the axis of the wire coincides with the z axis. Figure 8-6 shows a segment of length t of the long wire. We have and J I E = - = a - - ' a - anb' On the surface of the wire, Thus the Poynting vector on the surface of the wire is -. I" P = E x H = ( a Z x a,)----- 2an2b3 - 1 ' - -ar---- 2 4 h 3 ' which is directed everywhere into the wire surface. Fig. 8-6 Illustrating ~oynting's 1 theorem (Example 8-5). , 8-4.1 Ir Power DE onducting 'oynting's . ~rmly dis- coincides We have 8-4 I FLOW OF EL~CTROMAGNETIC P ~ W E R ~ N D THE POYNTING VECTOR 329 , . \ 1 i : ' I . , ' , I In order to verify ,Poy$ting3s theorem, y e &negate P over the wall of the wire Segment in Fig. 8-6. 'i - i' , . a P . a , d ~ . = ' (,Sb3) - 2 n b ' where the formulz for the iesistance of a straight wire in Eq. (5-13), R = //US, has bcen used. The aboved-esult afFjnnr that the negdtive surface integral of the Poynting , vector is exactly equal to the 1'R ohmic power loss in the conducting wire. Hence Poynting's theorem is verified. I 8-4.1 Instantaneous and AVer~ge Power Densities In dealing w~th time-harrronlc electromagnetic waves. we have found it convenlcnt to use phasor notations. The instantaneous value of a quantity is then the real part of the product of the phdsor quantity and eJu' wheh cos o t is used as the reference. For example, for the phasor E(:) = axEx(:) = a , ~ ~ e - ( " + ~ ~ ! ' , the instantaneous expressim is For a uniform plane wave propagating in a lossy medium in the +: direction, the associated magnctic fleld illtensity phasor is where 0, is the phase angle of the intrinsic impedance q = Iq(rjopi of the medium. The corresponding instantaneous expression for H(z) is h 0 H(z, t) = %[H(z)ej?'] = a, - e -' cos (or - /?z - 0,). (8 -64b) I4 This proccdurc is pcrmissihle ils long as thc operations andlor thc equations involving the quantities with sinbsoidal time dependence are linrur. Erroneous results will be obtained if this procedure is applied to such nonlinear operations as a product of two sinusoidal quantities. (A Poynting vector, being the cross product of E and H, falls in this category.) The reason is that , . , . . . -. a . I I . 330 PLANE ELECTROMAGNETIC WAVES 1 8 1 . - The instantaneous expression for th;'poynting vectoi,or p&w density vector is on the one hand, from Eqs. (8-63a) and (8-64a), E: = a= - e-2az cos (wt - /jz) cos ( ~ t - /3z - O , , ) I 1 4 ,5Z = a, 4 e-'" [COS U,, + cos (20t - 2pz - O,)]. ! 2141 (8-65)' 1 On the other hand, ! E; 9e[E(z) x ~(z)ej@'] = a= - e-'"' cos (wt - 2/32! - O,), lvl which is obviously not the same i s the expression in Eq. (8-65). As far as the power transmitted by an electromagnetic wave is con'cerned, its average value is a more significant quantity tli;~n its inst;~iit;~nco~is v:duc.. Fro111 Eq. (8 -65) wc obtain l h c tinic-;~\wxgc I'oynlilig \ cctor, .
Y ' , , , , !
: ) , 1 . where T = 2n/w is the time period of the wave. The second term on the right side of ' Conslder two general complex vectors A and B. We know that Wc(A) = +(A + A) and &(B) = $(B + B), where the asterisk denotes "the complcx conjugate of." Thus, 9 ( ( A ) x :'Ac(B) = +(A + A) x +(B + B) = J[(A x B + A x 4%) + (A x B + A x B)] = :%<(A x B + A x U). (8-66) This relation holds also for dot products of vcctor functions and for products of two conlplex scalar functions. It is a strniglitforward ekercise to obtain the restlit in Eq. (8-65) by identifying the ccctors A and B in Eq. (8-66) with E(:)eJm' and H(z)eJw' respectively. Equation (8 -67) is quite similar to the formula for computing the power dissip;~tcd in an impcdnncc % = I%lriO. w l l c ~ ~ ;I si~~usoitl:~l volti~gc I ) ( / ) = 6, cos rt11 appears ucross its tcrn1in;tls. Tllc inht:~~~t;tt~co~~s expression for the current i(/) through thc impcdancc is vo i(t) = - cos (ot - 0,). 1 -4 From the theory of AC circuits, we know that the average power dissipated in Z is where cos 0; is the power factor of the load impedance. The cos 0, factor in Eq. (8-67) can be considered the power factor of the intrinsic impedance of the medium. ncerncd, its due From i -67)' J right side of (8-66) :ornplex scalar ng the vectors d n ~mpedance instdntaneous A { \ - - i he considere? . \ 8 , EL,. (8-65) is a cosind Sunuiun of a doub~c frJquency whose average is zero over a 1 fundamental periad. : , ; , ; I Using Eq. (8-66),\we q n express the instan)hneous Poynting vector in Eq. (8-65) as the real part of ~ H L sum-of two terms, instelld of the product of the real parts of !wo complex vectors.' ': I = !;RI[E(i) x H(r) + E(;) x H(.z)e~"~']. (8 -65) The average power 8ensity, Pav(z), can be obtained by integrating P ( z , t ) over a fundamental period;;?. Since the average of the last (second-harmonic) term in Eq. (8-68) vanishes, W e have I , 3 .YIV(:) = i ~ 9 ~ [E(z) X f-I(:)]. In the general case, wk may not be dealing with a wave propagating in the z direction. We write , = . E x 1 I ) (W/mZ), (S -69) which is 3 general forhu1.l for computing the average power density in a propagating . wave. Example 8-6 The far firld of a short vertical currem element I dl located at the origin of a spherical coordinate system in free space is E(R, 0) = a,,E,(R, 0) = a, (b";Ld' - sin 0) e-~p' ( ~ / m ) and &(R, 0) H(R, 0) = a,,, ----- - -( s O ) e J (Aim), '70 21.R where i = 271/lj is the wa.elength. a) Write the expression for instantaneous Poynting vector. b) Find thc total avcragc power :diiiice by the current element. Solution a) We note that ~,/k, = : '7, = 12On (Q). The instantaneous Poynting vector is 9(Za: 4 = &xE[E(R, O ) P J ~ I x ~C[H(R, e)ejmfl I 332 PLANE ELECTROMAGNETIC WAVES 1 8 - b) The average power density vector is, from Eq. (8-69), which is seen to equal the time-average value of B(R, 8; r) given in the first equation of this solution. The total average power radiated is obtained by in- tegrating .Ya,(R. 0) ovcr thc surface of thc sphere of radius R. I dd , , Total Pa. = $ s Pa"@, 0) . ds = SozR S : [ISn (x) sin2 0 ] R2 sin 0 dO1dd 2 = 40a2 ( : ) l2 (W), where 1 is the amplitude (3 timcs thc cn'cctivc valuc) of the sinusoidal currcnt in dd. 8-5 NORMAL INCIDENCE AT A PLANE CONDUCTING BOUNDARY Up to this point wc have discussed the' propagation of uniform plane waves in an unbounded homogcneous medium. In practice, wavcs oftcn propagate in bounded regions where several media with different constitutive parameters are present. Whcn an clcctrotnagnctic wave traveling in onc medium impinges on another medium with a dilkrent intrinsic impedance, it experiences a rellcction. In Scct~ons 8-5 and 8 -6 wc examine the bchavior of a planc wave when it is incidcnt upon a plane conducting boundary. Wave behavior at an intcrfacc betwecn two cliclcctric media will be discussed in Sections 8-7 and 8-5. For simplicity we shall assume that the incident wave (Ei, Hi) travels in a lossless medium (medium l:o, = 0) and that the boundary is an interface with a perfect conductor (medium 233, = co). Two cases will be considered: normal incidence and oblique incidence. In this section we study the field behavior of a uniform plane ,'- wave incident normally on a plane conducting boundary. Consider the.situation in Fig. 8-7 where the incident wave travels in the + z direction, and the boundary surfacc is thc planc z = 0. The incidcnt electric and magnetic field intensity phasors are: l<,(z) = J'llz (8 -70a) where Eto is the magnitude of El at z = 0, and 8, and 11, are, respectively, the phase constant and the intrinsic impedan3 of medium 1. It is noted that the Poynting vector of incident waves, Pi(z) = E,(z) x Hi(z), is in the a, direction, which is the direction of energy propagation. The variable z is negative in medium 1. )idal current waves in an in bounded .ire present. on mother I . In Sections idcnt upon a wo dielectric s inJF? lossless ith a perfect ncidcnce and nirorm plane :Is in the + z t electric and ely, the phase the Poynting , which is the 1. Insiclc medium 2 [a perfect conductor) both electric and magnetic fields vanish. E, = 0. 11, = 0; hcncb no wave is iransn~itted across the boundary into the z > 0 region. The incident wave is reflected, giving rise to a reflected wave (E,, H,). The reflected electric field intensity can be written as E,(I) = a,Eroe+jl'l=, (S-71) where the positive sign in the exponent signifies that the reflected wave travels in the - z direction, as tliscusscd in Section 8-2. The total electric field intensity in medium 1 is the sum of E and E,. E,(z) = E,(z) + E,(z) = a,(Eloe-JP1' + EroefJP1z). (8-72) Continuity of the tangenti 11 component of the E-field at the boundary r = 0 demands that I;,(O) = a,(Elo + E,,) = E,(O) = 0, which yields Ero = -Aio. Thus, Eq. (8-72) becomes El(z) f a,~,~(~-jPl' - e+jP1z) = -a, j2Ei0 sin / I , : . (8 -73,) The msirctic field il.tensity H, of the reflected wave is related to Er by Eq. (8 -24). .. I Combining H,(z) with Hi(z) in Eq. (8-70b), we obtain the total magnetic field inten- t > I . . _ . , . . , . '' sity in medium 1 : . " k - Hl(z) = Hi(z) + H,(z) = a,2 - Eio cos P1z. (8 -73 b) ' I 1 It is clear from Eqs. (8-73a), (8-73b). and (8-69) that no average power is associated with the total electromagnetic wave in medium I, since El(z) and Hl(z) are in phase quadrature. In order to examine the space-time behavior of the total field in medium 1 . we first write the instantaneous expressions corresponding to the electric and magnetic field intensity phasors obtained in Eqs. (8-73a) and (8-73b): E,(z, t) = .%'o[~,(z)ej"'] = a,2Eio sin PI? sin r ~ . (8-74a) Eio H ~ ( z , t) = .%e[H,(:)eJm'] = a,2 - cos Plz cos wt. (8 -74b) '11 Both El(i, t ) and Hl(z, t) possess zeros and maxima at fixed distances from the conducting boundary for all t, as foilows: 1 -' . Zeros of El(,-, t) i i . occur at /Il: A -nx, O r = = -"z' Maxima of H ,(z, t) n = 0, 1.2,. . . Maxima of El(;, i) ) 7i ; . occuratB,z= -(2n+ I)-, or:= -(2n+ I)-, 2 4 Zeros of Hl(z, t) n = 0 , 1 , 2 . . . . The total wave in medium 1 is not a traveling wave. It is a standing wave, resulting from the superposition of two waves traveling in opposite directions. For a given t. both El and H1 vary sinusoidally with the distance measured from the boundar) plane. The standing waves of El = a,E, and H, = a,H, are shown in Fig. 8-8 for several values of wt. Note the following three points: (1) El vanishes on the con- ducting boundary (E,., = - Eio); (2) H, is a maximum on the conducting boundary (H., = Hio = Eio/vl); (3) the standing waves of El and H, are in time quadrature (90 phase dillercncc) and are shifted in spiicc by a qu;iricr wavelength. Example 8-7 A ):-polarized uniform plane wave.(Et, Hi) with a frequency 100 (MHz) propagates in air and impinges normally on a perfectly conducting plane at x = 0. Assuming the amplitude of Ei to be 6 (mV/m), write the phasor and instan- . .- taneous expressions for: (a) E, and Hi of the incident wave; (b) Er and H, of the reflected wave; and (c) El and H, of the total wave in air. (d) Dcierminc tile location nearest to the conducting plane where El is'zero. 1 1 icld inten- , , , 1 + I .
; { , I - . I t (8-73h) ' . 1 issociated i e in phase ium 1, we magnetic I I (8 -744 i Y -74b) Irom the ,-- I I ) :. 4 7 -, . . . . resulting a given t, boundary 2. 5-8 for ! the con- boundary uadrature 1 e n c n 0 : I . .I ~d Instan- If, of the : I~~c'l~lclll Fig. 8-8 Standing waves of ;i, = r,E, md H I = r,H, for several values o f o ~ . Solution: At the giveh frequency 100 (MHz), ' a) For the incident wive ( ( 1 travel~ng wave): i) Phasor expressions E,(x) = ily6 x 10-3e-J2nxi3 (v/m), 1 H,(x) = - ax X El(.;) = a, - e-J2"x'3 Y 1 2n (A/m). 336 PLANE ELECTROMAGN"ETIC WAVES 1 8 b) For the rejected wave (a traveling wave): . , . i) Phasor expressions . . Er(x) = ,a,6 x 1@3ej2n"/3 (v/m), ii) Instantaneous expressions Er(x, r) = Wc[E,(x)eja'] = - 5 6 x cos 2n x 108t + - s 3 Hr(x, t) = a= - cos 2 7 1 x 108t + lo-' ( , . 2n c) For the total wave (a standing wave): i) Phasor expressions ii) Instantaneous expressions E,(x, t) = ~%e[E,(x)e~"'] = a 9 2 x sin -x sin(2n x 10") (? ) H (x, t) = a, - cos ( 2 7 7 x 108t) (A/m). d) The electric field vanishes at the surface of thc conducting plane at x = 0. In medium 'I, the first null occurs at 8-6 OBLIQUE INCIDENCE AT A PLANE CONDUCTING BOUNDARY Whcn a uniform plane wave is incidcnt on a plane conducting surfacc obliquely, the behavior of the reflected wave depends on the polarization of the incident wave. In order to be specific about the direction of Ei, we define a plane of incidence as the plane containing the vector indicating the direction of propagation of the incident wave and the normal to the boundary surface. Since an Ei polarized in an arbitrary direction can always be decomposed into two components-one perpendicular and Medium I , I - ' Fig. 8-9 Planc wave inc~dcnt (01 = 0) obliqacly 011 a plane conducting 2 4 0 boundary (perpendicular polarization) the other parallel to the pime of incidence--we consider these two cases sepnmtelj. The general case is ostilined by superposing the results of the two component cas~s. 8-6.1 Perpendicular ~ o l a r i r a t i o n ~ In the wsc of perpendicuiilr pduri~ution, Ei IS p&+endicular to the plane of incidence. as illustrated in Fig. 8-9. Noting that ani = a, sin Oi + a, cos Oi, (8-753 where Oi is the ungle o f inci~lence measured from the normal to the boundary surface, we obtain, using Eqs. (8-17) and (8-23), - ' Also referred to as horizontal poiari:ation or E-polcrrlzatioh. i -, I . 338 PLANE ELECTROMAGNETIC WAVES / 8 . - At the boundary surface, z = 0, the total electric field intensity must vanish. Thus, In order for this relation to hold for all values of x, we must have Ero = - Eio and I Or = Oi. The latter relation, asserting that the angle of reflection eqtlals the artgle of 3 irtcideilce, is referred to as Snell's law oj' reJectiort. Thus, Eq. (8-78) becomes The corresponding Hr(x, z) is The total field is obtained by adding the incident and reflected fields. From Eqs. (8-76a) and (8-79a) we have El(x, z) = E,(x, z) + E,(.Y, Z ) = a E ( e - ~ 8 ~ Z C ~ d e , - &PI: cos8, J b ~ x s r n O , Y 1 0 )e - = - a,j2Eto sin (P,z cos Ol)e-J"lX e l . (8-80a) Adding the results in Eqs. (8-76b) and (8-79b), we get E;0 I M,(.K, z) = - 2 - [ a , cos Ui cos (/jl; cos Ui)e-JI1~X"nol ' 1 1 + a,j sin 0, sin (plz cos Oi)e-j51xsin o1 1. 18-80b) Equations (8-8Oa) and (8;80b) are rather complicated expressions, but we can make the following observations about the oblique incidence of a uniform plane wave with polarization on a plane conducting boundary: 1. In the direction ( z direction) normal to the boundary; El, and H I , maintain standing-wave patterns according to sin Plzz and cos filzz, respectively, where fil= = PI cos Oi. No average power is propagated in this direction since El, and HI, are 90" out of time phasc. 2. In the dircction (.K direction) parnllcl to thc boundary, E , , and If,, arc in both time and spacc phasc and propagate with a phasc vclocity (I) V ) u1 U l x =-=----=-. PIX /3, sin Oi sin 8, 1 ... : . . ' I . / ': ' ,..':;:, : i ' st vanish. ..: . , , I : . 1 r , .
. b ., ,, ! 1 ... , , : , 1.. . , ,.' ' I , , :. - E,, and unyle o f : s (8-79a) (8-79b) :on1 f-". ( x -- aoa) is-sob) t \VC call t 1 1 plane 7x1 infain >. \~llcl.o L,. apq in . ' 3. The propagating wave in the r direction is a nonunijom plane wave because its amplitude varies witIi'z. I. Since E, = 0 for all x when sin (P,z cos OiJ = 0 or when ^ 2n fllz cas oi = - A - cos 0, = - 1117t, )' 1 nz=1,2,3 ,..., a conducting plilte can be insertdd at without changinh tht: field pattern that exists between the conducting plate and the conducting ~~~~~~~~~y at I = 0. A tiitn.queiac elearic (TE) IIYIC~ ( E l , = 0) will bounce back and forth between the conducting pinnes ilnd propagate in tne r direction. We hitVe, i I effect, a parallel-plate waveguide. Example 8-8 A uniforrrl plilhe wavc (El, H,) of an angular Srequcncy w ir lncldent from air on a very ILrge. perfectly conducting wail ar an angle of incidence 0, with perpendicular polarihllun. Find (a) thc current induced on the wall surface, and (b) the time-average Poynting vector in medium 1. a) The conditions df this problem are exactly those we have just discussed: hence we could use the kormulas directly. Let r = 0 be the plane representing the rurfxe of the perfectly cbnducting wall. and let Ei be polarized in the y direction. as was shown in Fig. 8-9. At z = 0, El(x, 0) = 0, and H,(r. 0) can be obtained from Lq. (8-80b): Inside the perfectly conducting wall, both E2 and H, must vanish. There is then a discontinuity ih iho milgnktic field. Thc amount of discontinuity is equal to ilic surfacu currchl. Iqroin Lcl. (7-52b), wc have .
- - -The instantaneous expression for the surface current is Ei0 J,(x, t) = a, - cos 4 cos o 6 0 ~ (8-82) It is this induced current on the wall surface that gives rise to the reflected wave in medium 1 and cancels the incident wave in the conducting wall. b) The time-average Poynting vector in medium 1 is found by using Eqs. (8-80a) and (8-80b) in Eq. (8-69). Since El, and HI, are in time quadrature. Pa, will have a nonvanishing x component. E 2 - - a,2-C sin Oi sin' /l,,r, '/I where /IlZ = /3, cos 0,. The time-average Poynting vector in medium 2 (aprfcct conductor) is, of course, zero. 8-6.2 Parallel polarizationt -1 We now consider the case of E, lying in the plane of incidence while a uniform plane wave impinges obliquely on a perfectly conducting plane boundary, as depicted in Fig. 8-10. The unit vectors a,, and a,,, representing, respectively, the directions of propagation of the incident and reflected waves. remain the same as those given in Reflected wave Incident wave Fig. 8-10 Plane wave incident obliquely on a plane conducting boundary (parallel polarization). ' Also referred to as tiertical polarizatiotl or H-polarizatidt~. (8-82) :tcd wave ; . (8-80a) . .Pav will (8 -83) (a perfect /? )I n- 1 1 e Cplc'l,., 117 cctlolls of L' given in m - 8-6 1 OBLlOUE !NCIDENCE AT A'PLANE CONDUCTING BOUNDARY 341 , . f"', , I 1 i ? Eqs. (8-75) and ($-I!). Boih Ei and Er now have components in x and z directions, whereas Hi and H, hdve only a y component. w e have, for the incident wave, t 1, i 1 Er(x, 2) = EIO(ax cos 0, -' a: sin Oi)e-jbl(x 'ln ' 1 + z cosUf), (8 -84a) H,(x, = ay 5 e - ~ \ l ~ ( x s t n oi+z tb\ 0 1 ) . "t 1 (8 -83b) The reflected wave (E,, H:) have the following phasor expressions: E,(x, k) = Ero(a, cos 0, +'a, sin Or)e-jP1(xsln cos Or), (8-85a) ' : H , ( ~ , i) = - B I E r o e - ~ ~ ~ ( x s l n 8 r - z c o s ~ r ~ . (8-8jb) YI 1 A 1 thc surklcc ol'~lic pclfcct conductur, : = O, ~ h c tanyentiti1 compolicn~ ( t l x .
Y component) of the total electric field intcnsitymust vanish for a 1 1 s, or E,,(,Y. 0) - 6,,(.u, 0) .
= 0. From Eqs. (8.- 841) ant1 (8-.X~:I), wc 1i;ivc whlch requires Ero = - f;,, and 0, = 0,. The total electric field intensity in medlum 1 is the sum of Eqs. (8+84id and (8-85a): Adding Eqs. (8-84b) and (8-85b). we obtain the total magnetic field intensity in medium 1. Hl(s. z) = H,(s, z) + Hr(x, z) El 0 = ay2 - cos (/jlz cos O , ) C - J D ~ ~ ' ~ ~ " 4 , (8-86b) YI 1 I The interpretation oi' Eqs. (8-86n) and (8-86b) is similar to that of Eqs. (8-80a) and (8-SOW b r the erpendicular-pol:~rizi~tio~~ cclsc, cxcept that E,(-. z), ~nbtc;id of fl,(s. : ) , now 11;)s 1 x 1 P 11 ; I I I Y :rnti :I : componc~it. Wc C O I I C ~ L I ~ C . I~ICI.L'~OI.C: ,1. In the direction (2 direction) normal to the boundary, E l , and H,, maintain standing-wave patte-ns according to,sin P1,z and cos Dl,z, respectively, where j , , = Dl cos Oi. No average power is-propagated in this direction, since El, and H,, are 90" out of tipe phase. . . , ..r, l I . < . 342 PLANE ELECTROMAGNETIC WAVES 1 8 2. In the x direction parallel to the boundary, El: and H,, are in both time and space phase and propagate with a phase velocity u,, = u,/sin O,, which is the same as that in the perpendicular polarization. 3. The propagating wave in the x direction is a nonuniform plane wave. 4. The insertion of a conducting plate at z = -nzi,/2 cos Oi (nt = 1 , 2, 3, . . .) where E ,, = 0 for all x will not affect the field pattern that exists between the conducting plate and the conducting boundary at z = 0, which form a parallel-plate wave- guide. A transverse magnetic (TM) wave ( H , , = 0) will propagate in the x direction. 8-7 NORMAL INCIDENCE AT A PLANE DIELECTRIC BOUNDARY When an electromagnetic wave;s incident on the surface of a dielectric medium that has an intrinsic impedance different from that of the medium in which'the wave is originated, part of thc incidcnt powcr i s rcflcctcd and part is transmitted. We may think of the situation as being like an impedance mismatch in circuits. The case of wave incidence on a perfectly conducting boundary disc&ed- in the two previous scctions is like terminating a generator that has a certain internal impedance with a short circuit: no powcr is transn~ittcJ i~ito thc conducting rcgion. As before. we will consider separately, the two cases of the normal incidence and the oblique inciclencc of a unili,rrn planc wavc on a planc diclcctrii: medium. 130th media are assumed to be dissipationless (a, = o2 = 0). We will discuss thc wave behavior for normal incidence in this section. The case of oblique incidence will be taken up in Section 8-8. Consider the situation in Fig. 8-11 where the incident wave travels in the + z direction and the boundary surface is the plane z = 0. The incident electric and Transmitted Reflected +- wave H, b":+ wave H t an1 Incident wave H, Medium 1 Medium 2 (€2, P2) z = o Fig. 8-11 Plane wave incident normally on a plane dielectric boundary. . . .) where onducting . late wave- . direction. :n the +: ctric and ' Y 1 magnetic field intensity phasors are [ . . " 171 .
< 2 I. .
.
, . ; " .
; k . . ,+. .
; 4 1 ; , 3 .
. - J , Ei(z) = a , ~ ~ ~ e - ~ p ~ , i : (8-873) Ei, H,(z) = a - e-JPlz. I I (8-87b) 1 : t y ' I 1 3 These are the same expressions as those given in Eqs. (8-70a) and (S70b). Note that z is negative in medltlm 1. Because ofthe medium discontinuity at z = 0, the incident wave is partly reflected back into m e d i u ~ and partly transmitted into medium 2. We have a) For the reJecte3 wave (E,, H,): where El, is the magnitude of E, at r = 0, and p2'and ' 1 , are the phase constant and the intrinsid impedance of medium 2 respectively. Note that the directions of the arrows for E, and E, in Fig. 8-1 1 are arbitrarily drawn, because ErO and E,d may, themseives, be positive or negative. depending on the relative magnitudes pf the constitutive parameters of the two media. Two equations $re needed for determinidg the two unknown magn~tudes Er, and El,. These equations are supplied by the boundary conditions that must be satisfied by the electric and magnetic fields. At the dielectric interface r = 0. :he tangential componed~s (the x components) of the electric and magnetic field intensltles must be continuous. b e have ~ ~ ( b ) + E.(O) = E,(O) or El, + Ero = El, (8-90a) and 1 El0 H.(o) + H.(o) = &o) or - ( E , ~ - E ~ ~ ) = -. ' I I (8 -9Ob) -- 'I2 Solving ~ q s . &90n) nnd (8-90b), wo ihl;lio . 3 344 PLANE ELECTROMAGNETIC WAVES / 8 . .
. I t i - The ratios Ero/Eio and Eto/Eio are called, respectively, rejection coefficient and i i transmission coefficient. In terms of the intrinsic impedances, they are and L El(z) = a,Eio[~e-jblz + T(j2 sin /I,z)]. (8 - 96) We see in Eq. (8-96) that E1(z) is composed of two parts: a traveling wave with an amplitude rEio, and a standing wave with an amplitude 21.Ei,. Because of the exis- tence of the traveling wave, E,(z) does not go to zero at fixed distances from the interface; it merely has locations of maximum and minimum values. The locations of maximum and minimum jE,(z)l are conveniently found by rewriting E,(z) as El(z) = a , ~ ~ ~ e - j " ~ ' ( l + TeJ2P1z). (8 -97) t=-=- EtO " (Dimensionless). EiO q2+q1 Note that the reflection coefficient I- in Eq. (8-93) can be positive or negative, de- pending on whether q, is greater or less than q,. The transmission coeflicient t, however, is always positive. The definitions for I. and t in Eqs. (8-93) and (8-94) apply even wh'en the media are dissipative: that is. even when 1 7 , and/or r 1 2 are corn- plcs. Thus I - ;rnd T may Ilic~nscl~cs hc co~nplcs in \lie y m x i 1 . c;lsc, A complcs (or r ) simply nicans tI1:11 : I pl~:~sc sllifl is i~i\roduccJ :I[ tile i~~tcrhcc ripuli 1.cllcctio11 (or tr:~nsrnission). RcI1c~li01i a~id ~ ~ ~ ; i ~ i s n i i s s i ~ ~ ~ i c ~ ~ l l i ~ i c n t s ;IIX rclittcd by the following equation: b i (8-94) 1 1 + T = T (Dimensionless). (8 -95) If medium 2 is a perfect conductor, = 0, Eqs. (5-93) and (5-94) yield r = - 1 and t = 0 . Consequently, Ero = - Eio, and E,, = O.'The incident wave will be totally reflected, and a standing wave will be produced in medium 1. The standing wave will have zcro and maximum points, as discussed in Scction 8-5. If medium 2 is not a perfect conductor, partial reflection will result. The total electric field in medium 1 can be written as El(,-) = Ei(:) + Er(:) = a , ~ ~ ~ ( e - j ~ ~ ' + rejal' 1 = axEiO[(l + r)e-Jplz + r ( e j p l i - ,-jPl: 1 1 . = a,Eio[(l + T)e-jpl' + T(j2 sin biz)] or, in view of Eq. (8-95), 8-7 I NORMAL INCIDENCE AT A PLANE DIELECTRIC BOUNDARY 345 I ' I a i c i m and , : : i ! '1 For dissipationless media, 4, and 4, are real, making both r and r also real. However, I :, I ' can be positive or hegqtive. Consider the following two cases. , , The maximum value of IE,(z)1 is Eio(l + n, which occurs when 2P,1,., = -2nn(n =0, 1,2,. . .),orat The minimum 'value of lE,(z)l is Eio(l - r), which occurs when 2/I,zm,, = -(2n + l)n, or i t The maximum vdue of /K,(z)/ is L..,(l - I -) , wliicll occurs at .-,,,,, given in Eq. (8-99); and the minimum valus of /E,(z) is Eio(l + r), which occurs at z,,, given in Eq,(S-OS). In other words. the locations for E,(;)I,,, and i i , ( ~ j l , , , ~ ~ when I- > 0 and when r < 0 are interchaflged. The ratio of the m;iimum value to the minimum vdue of the ~ I C C ~ ~ I C fidd intensity of a standing wave is called the standing-wuae rutio, S. I An inverse relation of Eq. (8-100) is S - 1 irl = s+r (Dimensionless). (8-101) WIiib the v:ilua o f r rangx from - I :o i- i, the value oss riinges fro~n I to z. 1t IS customary to express S on a logaritl 7;~. .i'~e. The standing-wave ratio in decibels 1 s 20 log,, S. Thus, S = 2 corresponds i o i ) rt.ioalng-wave ratio of 20 log,, 2 = 6.02 dB and 1 l -1 = (2 - 1)/(2 + 1) = ; . A. stnndiq-inve ratio of 2 dB is equivalent to S = 1.26 and ( T I = 0.1 15. 4 , The rnapetic field intensity in mcdium 1 is obtained by combining Hi(:) and H.(z) in Eqs. (8'-87b) and (8-88b), respectively: 346 PLANE ELECTROMAGNETIC WAVES 1 8 This should be compared with E,(z) in Eq. (8-97). In a dissipationless medium, r is real; and IH,(z)J will be a minimum at locations where /E,(z))I is a maximum, and vice versa. In medium 2, (E,, H,) constitute the transmitted wave propagating in + z direc- tion. From Eqs. (8-89a) and (8-94), we have Et(z) = a , ~ E ~ ~ e - j P ~ ~ . (8-103a) And from Eqs. (8-89b) and (8-94), t Z H,(z) = a, - Eioe-jP2'. (8-103b) 72 Example 8-9 A uniform plane wave in a lossless medium with intrinsic impedance vl is incident normally onto another lossless medium with intrinsic impedance q, through a plane boundary. Obtain the expressions for the time-avcragc power dcnsirics in both ~nctlia. .Pa, = $.%(E x H) . In medium 1. we use Eqs. (8-97) and (8-102). where r is a real number because both media are lossless. In medium 2, we use Eqs. (8-103a) and (8-103b) to obtain Since we are dealing with lossless media, the power flow in medium 1 must equal that in medium 2; that is, h m , r- is num, and f z direc- (8-103a) (8 - 103b) npedance euance q, LAC power 12-~vcragc f i (8 - 104) csy-as, 4 m 1 must J That Eq. (8-106) is true can be readily verified by using Eqs. (8-93) and (8-94). 3 ,a 8-8 NORMAL INCIDENCE AT MULTIPLE DIELECT R l ~ , l d f ERFACES ' f In certain pi-@tical situations a wave m q be incident on severai layers of dielectric. media with difierznt constitutive parameters. One such situlition is the ~1st of a dielectric cbhting dn g!ass in order to reduce glare from sunlight. Another is n raclomc, which is il clomc-shapcd crlclosurc clcsigncd no1 only to protect radar installations from inclement weather but to permit the propagation of electromag- netic waves through the enclosure with as little refection as possible. In both siltla- tions. determining the propcl- dielcctric material and its thickness is an important design problem. We now consider the three-region situation depicted in Fig. 8 - 12. A uniform plane wave traveling in the +: direction in medium.1 ( E , , p , ) impinges normally :it a plane boundary with medium 2 ( E ? , p2), a t : = 0. Medium 3 has a finite thickness and interfaces with medium 3 (c3, p 3 ) at z = d. Reflection occurs at both : = 0 and z = d. Assuming ah .u-polarized incident field, the total electric field intensity in medium 1 can always be written as the sum ofthe incident component a,EiOe-JP'' and wave +- H, Medium 1 kll PI) Hl Transmitted h,lediunl 3 ( ~ h 13) - 7 Fig. 8-12 Normal incidence 7 - 2 = 0 . . i = d at multiple dielectric interhces. ' , .,.. . . . . . , k 1 ,, . , . , . . , i . : ' i , , , , . .,:: .
, . . . . . . . % .
. 5 I , ~, . . . , . , a reflected component 'aX~,,ejPiz: , . i : . ! : . , , -:-:.,:: ?,k.- > . , , . . , I . . .
. ,... . . . .
. !p:>: 1 However, owing to the existence of a second discontinuity at z'= d, Ero is no longer I related to Eio by Eq. (8-91) or Eq. (8-93). Within medium 2 parts ofhaves bounce i back and forth between the two bounding surfaces, some penetrating into media 1 and 3. The reflected field in medium 1 is the sum of (a) the field reflected from the interface at z = 0 as the incident wave impinges on it; (b) the field transmitted back ) into medium 1 from medium 2 after a first reflection from the interface at z = d; L i (c) the field transmitted back into medium 1 from medium 2 after a second reflection ! at z = d; and so on. The total reflected wave is, in fact, the resultant of the initial 8 -8 .
1 reflected component and an infinite sequence of multiply reflected contributions within medium 2 that are transmitfed back into medium 1. Since all of the contribu- tions propagate in the - z direction in medium 1 and contain the propagation factor ejhZ. they can be combined into a single term with a coefficient Ero. But hdw do we determine the rclation bctwocn I:,,, ;ind E,,, n o w ? One way to find E,,, is to wrik don11 tilc clcctric ;tnd ~n:!gtictic licld illrct~sity vectors in all three regions and apply the boundary conditions. The 11, in region 1 that corresponds to the El in Eq. (8-107a) is, from Eqs. (8-S7b) and (8-SSb), The electric and magnetic fields in region 2 can also be represented by combinations of forward and backward waves: In region 3 only a forward wave traveling in + s direction exists. Thus. 1 On the right side of Eqs. (8-107a) through (8-109b), there are a total of four unknown amplitudes: E,,, E l , E;, and E l . They can be determined by solving the four boundary-condition equations rccluirctl hy t h i continuity of ~ h c t:~ngcrlti:tl I components of the electric and magnetic fields. no longer I es bounce I 3 media 1 I 1 from the itted back : at z = d ; l reflection &the initial ~trihutions : contribu- ;Ion factor low do we ti intensity In region 1 4b). - m binations 31aI of four solving the : t a f i t i a l . . (8 - 1 10a) . (8-110b) 8-8 1 ~ J O ~ M A L ~NCIDENCE AT' M U L ~ P L E DIELECTRIC INTERFACES 349 ! 4 ; L (8-1 10d) 1 I The procedure is stfaigl~tforward and is purely algcbraic (scc Problem P.8-23). In the following subsections we introduce the Concept of wave impedance and use it in an alternative approach for studying the prdblem of multiple reflections at normal incidence. 8 : I /. , / . > 8-8.1 Wave Impedance 6 1 Total Field We definc the wuur iiupedat~ce u j the ford jiuid at any piiine parallel to the plane. boundary as the ratio of the total electric field intensity to the total magnetic field intensity. With a I-dependent uniform plane wave as was shown in Fig. 8-12, we write, in general, TotalEJz) ' Z(:) = (Q) . is-111, Total H,.(z) For a single wave propapatinq in the + r direction in an unbounded mealum, the wave impedance eqilals the intrinsic impedance, q, of the medium; for a single wave traveling in the - z direction, it is - q for all z. In the case of a uniform plane wave inddent fiom medium 1 normally on a plane boundary with an infinite medium 2, shch as that illustrated in Fig. 8-1 1 and discussed in Section 8-7. the magnitudes of the total electric and magnetic field intensities in medium 1 are, from Eqs. (8-97) and (8-102), Their ratio defines the wave impedance of the total field in medium 1 at a distance : from the boundary plane which is obviously d functiorl of.:. At a distance 2 = - L to the left of the boundary plane, . 1 Using the definition of r = (s, - ql)/(ilz + q , ) in Eq. (8-1 13), we obtain 'I2 cis plt + jq, sin P,L Z1(-.4 = 111 ql cos Pld + jq2 ~ i n . / 3 ~ t ' 350 PLANE ELECTROMAGNETIC WAVES 1 8 .' which correctly reduces to q, when q, = 7,. In that case, there is no discontinuity at z = 0; hence there is no reflected 'wave and the total-field wave impcdance is the same as the intrinsic impedance of the medium. When we study transmission lines in the next chapter, we will find that Eqs. (8-1 13) and (8-1 14) are similar to the formulas for the input impedance of a trans- 1 mission line of length l that has a characteristic impedance q1 and terminates in an L impedance q 2 There is a close similarity between the behavior of the propagation t . of uniform plane waves at normal incidence and the behavior of transmission lines. I If the plane boundary is perfectly conducting, q2 = 0 and l- = -1. and Eq. f I (8 -1 14) becomes ' i Zl(-P) = jql tan P1d. (8-115) which is the same as the input impedance of a transmission line of length C that has a characteristic impedance 11, and ferminates in a short circuit. 8-8.2 Impedance Transformation with Multiple Dielectrics -1 The conccpl of lotal-liclci wavc impcdancc is \tc~.y useful in solving p r ~ b l e ~ ~ ~ s w t l l multiple dielectric interfaces such as the shuation shown in Fig. 8-12. The total field in medium 2 is the result of multiple reflections of the two boundary planes 2 = 0 and z = d; but it can be grouped into a wave traveling in the + z direction and another traveling in the - 2 direction. The wave impedance of the total field in medium 2 at the left-hand interface z = 0 can be found from the right side of Eq. (5-1 14) by replacing y, by q , by q,, fi, by P,, and C by d. Thus, q3 cos P,d + jq, sin P,d Z2(o' = " q2 cos /12d + jq3 sin BJ' As far as thc wavc in mcdium 1 is conccrncd, it cncountcrs o discontinuity at z = 0 and the discontinuity can bc charactcrizcd by an infinite medium with an intrinsic impedance Z2(0) as given in Eq. (8-1 16). The effective reflection coeficient at z = 0 for the incident wave in medium 1 is . We note that I-6 differs from r only in that g2 has been replaced by Z2(0). Hence 1 the insertion of a dielectric layer of thickness d and intrinsic impedance q, in front - of medium 3, which has intrinsic impedancc y3. has the effect of transforming q , to Z2(0). Given q1 and q3, ro can be adjusted by suitable choices of q 2 and d. -- I I . : - , Once I-, has been found from Eq. (8-117), Ero of the reflected wave in medium c 1 can be calculated: E,, = TOEi,. In many applications T o and E,, are the only 1 quantities of interest; hence this impedance-transformation approach is conceptudly simple and yields the dcsired answers in :i dircct manner. It' the ficlds E j , L:; and < b f . + I e - :hat Eqs. f a trans- ' 3 . tcs in an ' paga4ion sn llnes. and Eq. that has ,mt, 11 the only ceptually j , E; and $ _ I " I E, in media 2 and are also desired, t h y &n be determined from the bqundary conditions at z = 0 and z,= d (see Problem P,8-23). 71 i Example 8-10 A dielectric layer of thickness d and intrinsic impedance q, is belwccn media 1 a n t 3 having intrinsic impedances q , and q3 respectively. Determine d and q2 su& that no reflection occurs wheh a uniform plane wave in medium 1 impinges normally on the interface with medium 2. Solution: with the dielectric layer interposeti between media 1 and 3 as shown in . Fig. 8-12, thC condition of no I'eflection at intprface z = 0 requires To = 0, or Z2(0) = q From Eq. (8:11~) wc havc i - Equating the real and imaginary parts separately, we require and Equation (8-119) is satisfied if either cos /?,d q 0 , which implies that ( )(I llld trlli' Ilitli~I, 11 ~oll(lllioll ( X 121) holdh, I:q. (8 120) C ; I I ~ bc s111sIicd when cilhcr (a) r l Z = t13 7 tll, ~ l ~ i c l i ib L ~ C Lrivi;11 case of 110 dis~~ntlnulties 21 ;ill, or (b) sin P,d = 0, or d = hA,/-. On the other hdhd, il'relation (8-122) or (8-122a) holds, sin P,d does not vanish, and Eq. (8-120) cart be satisficd when 17, = 6. We have then two possibilities for thc condition of no rcllcction. 1. When q, = qi, we require 1 - --. 2 2 d = i ~ - - ? 11 = 0, 1, 2.. . . 2. d ' that is, the tqicknesr of the dielectric layer be a multiple of a half wavelength 6 ' in the dielectric at (he operating frequency. Such a dielectric layer is referred to as a half-wave dielectric window. Since 1 , = u,Jf = l / f a z , where / is the operating fre~ut!ncy. a half-wave diefectric window is a narrow-band device. and 1 1 2 d = ( 2 n + I)-, n=0,1,2, ... 4 When media 1 and 3 are different, q, should be the geometric mean of q , and 1 1 3 , and d should be an odd multiple of a quarter wavelength in the dielectric layer at the operating frequency in order to eliminate reflection 'Under these conditions the dielectric layer (medium 2) acts like a quarter-wave impedance tmnsformer. We will refer to this term again when we study analogous trans- mission-line problems in Chapter 9. 8-9 OBLIQUE INCIDENCE AT A PLANE n'K!.YCTR1C BOUNDARY 7 % i b 2 now consider the case of a plane wavc that is incident obliquely at 3 1 1 arbitrary angle of Incidence Oi on a plane interface between two die1eC'ti-i.~-media. The medla are assumed to be lossless and to have different constitutive parameters (el, p,) and ( E ~ , p2), as indicated In Fig. 8-13. Bccausc of the medium's discontinuity at the interface, a part of the incident wave is reflected and a part is transmitted. Lines AO, O'A', and O'B are, respectively, the interscctions of the wavefronts (surfaces of constant phase) of the incident, retqected, and transmitted waves with the plane of incidence. Slnce both the incident and the reflected waves propagate In medium 1 with the same phase velocity up,, the distances m' and m' must be equal. Thus, Reflected Incident wave Medium 1 ( € 1 . PI) - 00' sin 0, = 0 0 ' sin Bi Medium 2 Fig. 8-13 Uniform plane wave (€2, PZ) incident obliquely on a plane z = 0 dielectric bo;ndary. of 1 1 , and diekctric ~der these, n~pedance )us trans- ardltrary h? Iy-illa , . I:, , .Ad :ty C ~:d. I ,lncj ~ d x e s of : plme of ucd~um 1 . 'Thus, f-' I - , I I 8-9 1 OBLIQUE INCIDENCE AT /# PPUNE DIELECTRIC BOUNOARY 353 1 . '. i f : 4 1 f , t or . - . ,.:I r I > ! ., . . ' 5 ' , 5 1 , 1 , 1 -1 0, = [Ii. (8-123) -. i, Eqaation (8 -123) asrllrcr or that tllc ;~nglc ok rcllcctioli is ~ ~ l ~ i i l to the .inglc of inc~dencc, which is Snell'~ luw oJreJecrion. In medium 2, the timc it takes for the transmitted wave to travel from 0 to B equals the time for the incident wave to travel &om A to 0 ' . We have .: , i aB A T . -- -- UP2 ~ P I ( S i i 0 3 ' ~ i n i l , I , , ~ ~ . - - -- X I ~ ~ , - s ~ O , - 1% l from which we obtain sin 0 (8 - i Xi) Furthermore, if medium reduces to 1 is free space such that 5, = 1 and n, = 1, Eq. (8-l?ib) sin 0, 1 1 8-9.1 Total Reflection Let us now examine !hell's law in Eq. (8-124b) for c, > e2-that is, when the wave in medium 1 is idcident on a less dense medium 2. In that case, 8, > Bi. Since 8, increases with Oi, an interesting 'situation arises when 0, = 7~12, at which angle 354 PLANE ELECTROMAGNETIC WAVES 1 8 the refracted wave will glaze along the interface; further increase in Oi would result in no refracted wave, and the incident wave is then said to be totally reflected. The a , I angle of incidence 0, (which corresponds to the threshold of total rejection 0, = n/2) , . - is called the critical angle. We have, by setting 9, = n/2 in Eq. (8-124b), This situation is illustrated in Fig. 8-14 whcrc a,,,, a,,,, and a , , , are unit vcctors dc- ,noting the directions of propagation of the incident. reflected. and transmitted waves respectively. What happens mathematically if Oi is larger than the-critical angle 0, (sin 0, > sin 8, = \iIEZ/E1)? From Eq. (8-124b) we have 1 . 7 sin 8, = - sin 8, > 1, J : : (8-126) which does not yield a real solution for 0,. Although sin 9, in Eq. (8-126) is still real. cos 8, becomes imaginary when sin 0, > 1. Reflected wave a , , Surface . " z ... . Incident , wave Medium 1 ('I 1 KO) , , ~ e d i u m 2 Fig. 8 ; 1 4 Plane wave incident at J 5 2 : elqs ,-& z = 0 critical angle, el > e2. ected. The n 0, = 74) (8-125a) (8-125b) ktmors de- itted waves 0 ' (411 0, > r (b--126) I is 51111 rcal, (8 - 127) /" ' 1 J ! A . , - In medium 2, the unit vedtor n, in $he d k t i o d of propagation of a typical transmitted (refracted) waGe, as shown in Fig. 8-13, is i - 1 4 r 1 a,, = a, sin 0, + a, cos 0,. (8-128) J Both E, and H, vary .$atialIy in accordance with the following factor: Tllc upper sign in h l . i H - 1.27) l l x bccn i~hmduned bi.c;;losc it wo~ilcl ic;id to the impossihlc result of all inci-c;ising licld as 2 ii~crcasi.~. We can conclude frorn (8-119) that for Oi > 0, a wave exists olmg the interface (in .
Y direction), which is attenuated exponentially (rapidlyJ in nedium 2 in the,norm&l direction (i direction). This wave is tightly bound to the interface and is called a surface wove It is illustrated in Fig. 8-14. Obviously, it is li ncnuniform plane wave. Example 8-11 A didlcctric rod or fibcr ofa .transparent material can be used to guide light or an electromagnetic wave under the conditions of total internal reflection. Determine the minimtim dielectric constant of the guiding medium so that a wave inc~dent on one end at any iing10 will bc confined wlthin the rod unt~l it emerges from the other end. I! J ' Solution: Refer to Fi . 8 -15. For total infernal reflection, 0, must be greater than or equal to 8, for the uiding dielectric medium; that is, cos 0, 2 sin 8,. Fig. 8-15 Dielectric rod or fibcr guiding elcctrornagnetic wave by total internal reflection. 356 PLANE ELECTROMAGNETIC WAVES 1 8 . . . A,., : r , . . . . . 1 , 1 !
. t - ! From Snell's law of refraction, Eq. (8-124c), we have , .\ , I , . J I , ? , J c r ~ t ' , It is im~ortant to note here that the dielectric medium has been designated as medium 1 (the denser medium) in order to be consistent with the notation of this subscction. Combining Eqs. (8-130), (8-131), and (8-125a), we obtain Since the largest value of thk right side of (8-132) is reached when 0, = ni2, we require the dielectric constant of the guiding medium to be at least 2, which corre- i F t \ , . (8-131) ts medium ubsection. intensity -79a) and i l ,. +; ' Therevare fokr anknown quantities in Eqs. (8-133a) through (8-135b), namely, E,,, E,,, Or, ,hnd 8,. Their determlnarion follows from the requirements that the tangential compondnts of E and H be continuous at the boundary r = 0. From Bi,(x, 0) + E&, 0) .= EJx, O ) , we have Because Eqs. (8-136a) and (8-li6b) are to he satisfied for ull x, all three exponential bctors that are function5 of x must be equal. Thus, Plx sin Oi = /I,x sin 0, = /3,x sin Or, which leads to Snell's law of reflection (8,. = 0;) and Snell's law of refraction (sin 0.1 sin @i = /31/P2 = n1/h2!. Equations (8-136a) and (8-l36b) can now be written simply as Eio + Ero = E,o and (8-137a) 1 ' 40 - (Eio - E,,) cos 0, = - cos Or, (8-137b) 'I1 '7 2 from which Ero and Et0 can be found in terms of Eio. We have and El0 TI=-= 2t12 cos Oi EiO '72 cos Oi + q cos Or ~ o r n ~ a r i n & ~ e expressions with the formulas for thc rellcction ;lnd transmission coefficients at normal incidsnce, Eqs. (8-93) and (8-94). we see th:~t the same formulas apply if ill and 17, are changed to (il,/cos Oi) and (rli/cos 0,) rcspcctively. When 0 , = 0, making 0. = 0 , = 0, these cxprcssions reduce to those for normal incidence, as they 358 PLANE ELECTROMAGNETIC.WAVES 1 8 -. i should. Furthermore, T , and T, are related in the following way: (8-140) which is similar to Eq. (8-95) for normal incidence. If medium 2 is a perfect conductor. q, = 0. We have T , = - l(E,, = - Eio) and T, = O(E,, = 0). The tangential E field on thc surface of the conductor vanishes, and no energy is transmitted across a perfectly conducting boundary, as wc have noted in Sections 8-5 and 8-6. 4 Noting that the numerator for the reflection coefficient in Eq. (8-138) is in the form of a difference of two terms, we inquire whether there is a combination of q,, q2, and €Ii, which makes T , = 0 for no reflection. Denoting this particular Oi by O,,, we require , q, cos OB1 = q1 cos 0,. (8-141) Using Snell's law of refraction, we have , cos 0, = J - = 1 - 5 siii- J (8-142) I7 5 and obtain from Eq. (8-141) The angle 8 , , is called the Brewster angle of no reflection for the case of perpendicular polarization. For nonmagnetic media, p, = p, = po, the right sidc of Eq. (8-143) becomes infinite, and 0 , $ , docs /lot exist. In thc casc of E, = c2 and p1 # p2, Eq. (8 - 1 43) reduces to sin 0,, = 1 J = - G ~ ' which does have a solutioli whether p,/p, is greater or less than unity. However, it is a very rare situation in electromagnetics that two contiguous media have the same permittivity but different permeabilities. Parallel Polarization When a uniform plane wave with parallel polaiization is incident obliquely on a plane boundary, as illustrated in Fig. 8-1 6, thc incident and rcflectcd elcctric and magnetic field intensity phasors in medium 1 are, from Eqs. (8-84a) through (8-85b): E~(x, 2) = ~ ~ ~ ( a ~ cos Oi - a= sin Oi)e-jpl(xsinOl + zcos '3) (8-145a) (8-140) .
O = - E d r vanishes, IS we have 5) is ill the .tion of Ill, 4 by 8,,? (8-141) (8-142) / ' - (8-143) pendicul~r q 18-143) $. 18- 143) (8- 144) Ho+cver, i have the juely on a ectrir(3 h ( 5 ): (8-145a) (8-135b) I E,(x, Z) = C,,(a, cos 0, + 3, sin ~ , ) L ' - j b ~ ( ~ s i n O r - = 0 s or) (s-1463) Er0 H,(x, 1) = - 3y -. - j D l ( >;.I 0, - 2 a s 8,) 'I 1 (8-146b) The transmitted electric and magnetic field intensity phasors in medium 2 are Et(x, Z) = Eto(as cos 6, - a_ ,sin d,)e-jb:(xsinO~ + = m d t ) (8-1472) z) ay 5 e - j , , ~ ~ x ~ i c 0, + z c w o t , 'I2 (8-147b) Continuity requirements for the tangential components of E and H at ; = 0 lead again to Snell's laws of reflection and refraction, as well as to the following two equations: 1 1 10 ' (8-148b) - (El, - E,,) = - E ' I I 11 2 Solving for Ero and E,, in lerms o[ELO, we obtain Ero '12 cos 6, - cos Oi --= -1 - I I - Eio q2 cos 8, + 'I, cos Oi and 4 0 T I ! = - = 2112 cos 13, Eio Y / ~ C O , S Q ~ + ~ ~ C O S O ~ ' ' These are also referred to as Fresnel's e&ariorls. 360 PLANE EL~CTROMAGNETIC WAVES / 8 It is easy to verify that Equation (8-151) is seen to be different from Eq. (8-140) forperpendicular polariza- tion except when. Oi = 9, = 0, which is the- case-for normal incidence. At normal incidence rll and rll reduce to r G e n in Eqs. (8-93) and (8-94) respectively, as did I?, and 7,. If medium 2 is'a perfect conductor (qz = 0), Eqs. (8-149) and (8-150) simplify to rll = - 1 and r l l = 0 respectively, making the tangential component of the total E field on the surface of the conductor vanish. as expected. From Eq. (8-149) we find thai TI, goes to zero when the angle of incidence Bi ,, - equals €IBl1, such that q2 cos 9, = rl, cos OBtl which, together with Eq. (5-142), requires 1 - ~ ' E ~ , / ' / L , € ~ sin' OBI1 = . 1 - . The angle OBI1 is known as the Brewster angle of no reflection for the case of parallel polarization. A solution for Eq. (8-153) always exists for two contiguous nonmag- netic media. Thus, if p , = p2 = pO, a reflection-free condition is obtained when the angIe of incidence in medium 1 equals the Brewster angle OBI,, such that Because of the difference in the formulas for Brewster angles for perpendicular and parallel polarizations, it is possible to separate these two types of polarization in an unpolarized wave. When an unpolarized wave such as random light is incident upon a boundary at the Brewster angle O,,,, givcn by Eq. (8-IS), only the component with pcrpcnclici~lar polarization will bc rcllcctctl. 'I'lii~s, a Iircwstcr anglc is also referred to as a polarizing utzyle. Based on this principle, quartz windows set at the ' Brewster angle at the ends of a laser tube are used to control the polarization of an emitted light beam. Example 8-12 The dielectric constant of pure water is 80. (a) Determine the . , Brewster angle for parallel polarization, O , , , , , and the corresponding angle of trans- mission. (b) A plane wave with perpendicular polarization is incident from air on water suiface at Bi = OBI1. Find the reflection and transmission coefficients. REVIEW QUESTIONS 361 poldriza- .t normal pectivel y, ' I ) simplify ' the total :idence 0, '8-152) r\ I ' J) I( p~ralid connag- when the (8-154) )endicular larization s incident Imponent .le is also het at the tion of an /7 i -n~i~ie--rl~c : of trans- ) m air on Solution i a ) The Brewster angle of no reflection for darallel polarization can be obtained directly from Eq. (8-154): I - Osll = sin- ' ' 1 Jl+(ilE,2j 1 =sin-'d . = 8 l . P . 1 + (1/80) The corresponding angle of transmission is, from Eq. (8-124c), 1 ' = sin-' ( ; G ) = 6.38". b) For an incident Wave with perpendicu!ar polarization, we use Eqs. (8-138) and (8-139) to find TI m a r , at 0, = 81.0" and 0, = 6.38': q1 = 377 (R), ql/cos 0, = 2410 (Rj 377 q z = - - - 0 . 1 ( ~ J C O S 0. = 40.4 (R). Thus We note that the hlat~on between TL and r, given in Eq. (8-140) is satisfied. REVIEW QUESTIONS R.8-1 Define uiliform plane Imve. R.8-2 What is a wnvefionr? R.8-3 Write'th4~ornogeneo.1~ vector Hclml~oiiz's equation for E in free spce. N.8-4 Dclinc n r ~ n w ~ ~ n d r ~ l : t.hh is \v(.i~\cllt~t~lb~r icliiicd 10 ~ v a v c l c ~ ~ g ~ h ? R.8-5 Define phase ve[dcity. R.8-6 Define intrinsic itnpedance of a mediurn.'What is the value of the intrinsic impedance of free space? 362 PLANE ELECTROMAGNETIC WAVES 1 8 R.8-7 What is a TEM wave? - 1 R.8-8 Write the phasor expressions for the electric and magnetic field intensity vectors of an x-polarized uniform plane wave propagating in the z direction. R.8-9 What is meant by the polarization of a wave? When is a wave linearly polarized? Cir- 1 cularly polarized? 1 I R.8-10 Two orthogonal linearly polarized waves are combined. State the conditions under which the resultant will be (a) another lincarly polarized wave, (b) a circularly polari~ed wave, and (c) an elliptically polarized wave. R.8-11 Define (a) propagation constant, (b) attenuation constant, and (c) phase constant. i R.8-12 What is meant by the skin depth of a conductor? How is it related to the attenuation ? constant? How does it depend on a'? p n J ? I ! R.8-13 What is meant by the dispersion of a signal? Give an example of a dispersive medium. I t R.8-14 Define group uelocit~. In what ways is group velocity different from phase velocity? , PROE , - R.8-15 Define Poynting vector. What is the SI unit for this vector?-\ R.8-16 State Poynting's theorem. R.8-17 For a time-harmonic electromagnetic 'field, write the expressions in terms of electric and magnetic field intensity vectors for (a) instantaneous Poynting vector and (b) time-average Poynting vector. R.8-18 What is a standing wace? R.8-19 What do we know about the magnitude of the tangential components of E and H at 1 the intcrfacc whcn a wavc impingcs normally on a pcrfcctly conducting planc boundary? i I R.8-20 Define plane o f incidence. R.8-21 What do we mean when we say an incident wave has (a) perpendicular polarization and L : (b) parallel polarization? i I R.8-22 Define refiectiorl coejficient and rransritission cotfl'cier~t. What is the relationship between I them? R.8-23 Under what cohditions will reflection and transmission coefficients be real? R.8-24 What are the values of the reflection and transmission coeficients at an interface with a perfectly conducting boundary? li.3-25 A pl:~nc w:~vc origin:~li~)g it) 111cdiu111 I (r I, 1 1 , - / L , , . m l = 0) ix ~ I I C ~ C ~ C I I I ~lot.~~li~lfy OII i t plane intcrhcc ~ i l h medium 2 (6, # E,, p, = p,, a, = 0). under what condition will ~ h c electric field at the interface be a maximum? A minimum? R.8-26 Define standing-wave ratio. What is its relationship with reflection coefficient? R.8-27 What is meant by the wave impedance of the total field. When is this impedance equal to the intrinsic impedance of the medium? ' ectors of an Iar~/zd? Cir- Lt~ons under 'inzed wave, 'ant. : attenuation w e medium. elocity? f -' T Jric t~,~lc-~\erage 11 T . ,mcl 14 at I[! 11) ! aru.tlon and 1s111p between 7 merface with ormally on a 111 ~ h e n t r i c Y cnt? xdance equal PROBLEMS 363 R.8-28 Thin dielectric cbating is sprayed on optical hstruments to reduce glare. What factors determine the thickness df the coating? R.8-29 How should the thickness of the radome in a radar installation be chosen'? lt,N-30 Si;~tc Stlcll's ltrw IJ/' w/lvc/iorl. R.8-31 State Snell's law o f rejklcriorl. R.8-32 Define cr~rrcul dhyle. When does 11 ex~st at an interl'ice of two nonmagnetic rned~a" R.8-33 Define Brewstet angle. Whcn does ~t exist at ah interface of two nonmagnetic media? R.8-34 Why is a Brewster angle also called a polarizing angle? R.8-35 Under what conditions will the reflection and transmission coefficients for perpendicular polarization be the same as those for parallel polarization? PROBLEMS P.8-1 Prove that thc electric field intensity in Eq. (5-17) satisfies the homogeneous Helmho1:z's equation provided that the condition in Eq. (8-18) is satisfied. P.8-2 For a harmonic uniform pl:m wave propasating in a simple medium. both E and H ' \Iilry in accordance with thi hctor crp (-,jk R) as indicated in Eq, ( Y -?I 1 . Show :ha: the forr S1:lxwcll's cquaiioi~s Ibl. unilorm pla~lc W;L\C in :I source-frcc rcgiun L . O ~ I I C O LO L I 1 c k~llowiil~: k x I<'= (I)/L!~I k x H = --WEE k . F ; = 0 k.11 = O . P.8-3 The instantoncaus ixprcssioo for the magnetic field intensity of a uniform plane w:ivc propagating in the +y direction in air is givcn by a) Determine k , alld tt-.e location whzre H z vanishes at r = 3 (ms). b) Write the instantaneous expression for E. , , -- 1'3-4 Show that a plane u.avc with an instant;lncous expression for the electric field E ( , 0 = a,EI0 sin (cot - k:) + askzo sin (rot - l i ~ + $) is ellipticalIn~larized. Find the polarization ellipse. PA-5 I ' ~ O V ~ tlic rollo\~i~ig :I) An elliptically polarikd plane wave can be resolved into right-hand and left-hand circularly polatfzed waves. b) A circularly polarized plane wave can bg obtained from a superposition of two oppositely directed elliptically polarized waves. . 364 PLANE ELECTROMAGNETIC WAVES 1 8 I P.8-6 Derive the following general expressions of the attenuation and phase constants for conducting media: a = OJ &[/a - I]"' (Np/m) P.8-7 Determine and compare the intrinsic impedance, attentuation constant (in both Np/m and dB/m), and skin depth of copper [a,, = 5.80 x lo7 (S/m)], silver [c,, = 6.15 x lo7 (S/m)]. and brass [c,, = 1.59 x lo7 (S/m)] at the following frequencies: (a) 60 (Hz), (b) 1 (MHz), and (c) 1 (GHz). P.8-8 A 3 (GHz), y-polarized uniform plane wave propagates in the + x direction in a non- magnetic medium having a dielectric constant 2.5 and a loss tangent lo-'. a) Determine the distance over which the amplitude of the propagating wave will be cut ' in half. b) Dcterlninc the intrinsic impcd;lncc. thc w;~vclcngth, the plme vclocity. and thc group vclocity of the wave in the medium. ' \ .
. c) Assuming E = a,50 sin (671 109t + n/3) at r = 0, write the instantaneous expression for H for all r and x. P.8-9 The magnetic field intensity of a linearly polarized uniform plane wave propagating in the + y directi~ii in sea water [E, = 80. p, = 1. c.= 4 (S/m)] is 'a) Determine the attenuation constant, the phase constant, the intrinsic impedance, the phase velocity, the wavelength, and the skin depth, b) Find the location at. which the amplitude of H is 0.01 (A/m). c) Write the expressions for E(y, t) and H(p, r) at y = 0.5 (m! as functions oft. P.8-10 Given that the skin depth for graphite at 100 (MHz) is 0.16 (mm), determine (a) the 'conductivity of graphite, and (b) the distance that a 1 (GHz) wave travels in graphite such that its field intensity is reduced by 30 (dB). P.8-11 Prove the following relations between group velocity u, and phase velocity up in a dis- persive medium: ' P.8-12 There is a continuing discussion on radiation hazards to human health. The following calculations will provide a rough comparison. a) The US, standard for personal safety in a microwave environment is that the power dcnsity be less than I0 (mW/cm2). Calculate thc corresponding stmdard in terms of electric field intensity. In terms of magnetic field intensity. b) It is estimated that the earth receives radiant energy from the sun at a rate of about 1.3 (kW/m2) on a sunny day. Assi~ming a monochromatic planc wavc, calculate thc amplitudes of the electric and magnetic field intensity vectors in sunlight. istants for . 0th Np/m 0' (Sin)], AHz), and I In a non- dance, the .ne (a) the such that following n thr;' - 7 \ ltn~,.- .d : of about culate the PROBLEMS 365 P.8-13 Show that the instantaneous Poynting vector of a propagating cikularly polarized plane wave is a constant that is indcpendent of time add distance. P.8-14 Assuming that the radiatioh electric ficld interlsity of an antenna system 1 s E = a,& + a&,,, - find the expression for the average outward power flow per unit area. P.8-15 From the point of view of electromagnetics, the power transmitted by a IOSSIC:~ c~axial cable can be considered in terms of the Poynting vector inside the dielectric medium between the inner conductor and the otiter sheath. Assuming that a DC voltage V, applied between the inner conductor (of radius a) and the outer sheath (of inner radius b) causes a current I to flow to a load resistance, verif) that the integration of the Poynting vector over the cross-sectional area of the dielectric medium equals the power VJ that is transmitted to the load. P.8-16 4 right-hand c~rcularly polarized plane wave represented by the phasor impinges normally on a perfectly conducting wall at : = 0. a) Determine the polarization of the rcllec;ted wave. b) Find the induced current on the conducting wall. c) Obtain the instdhtaneous expression of the total electric intensity based on a cosine time reference. P.8-17 A uniform sinusoidal planc wavc in air with the following phasor expression for electric intensity is inciclcnt 011 a pcrkctly cond~~cting pl;~nc at : = 0. :I) k'iritl thc I'rcclwncy :ml wavclcngll1 of llic wave. b) Write the instantaneous expressions for E,(x, z; t ) and Hi(x, z ; t), using a cosine reference. C) Determine the angle of incidence. d) Find E,(x, z ) and H,(?c, z) of the reflected wave. e) Find E,(.u, : j and H ,(x, z) of the total field. P.8-19 For the case of oblique incidence of a uniform plane wave with perpendicular polnriza- tion on a perfectly conducting plane bounhary as shown in Fig. 8-9, write (a) the instantaneous expressions for the total fieldjn?edium 1, using a cosine reference; and (b) the time-average Poyctjng vector. P.8-20 For the case of oblique incidence of a uniform plnne wave with p:wallel polarization on p~rkctl!~ ond doc tiny pl;~nc Soul~clary as s l ~ u w ~ ~ in 1"ig. S-10. wilt: (a) lhc instantaneous csprcssions E,(x, z; r ) and ' H,(s, z ; r) for the total field in medium 1, tising a sine reference; and (b) the time-average Poynting vector. 366 PLANE ELECTROMAGNETIC WAVES 1 8 I I : P.8-21 Determine the condition under which the magnitude of the reflection coefficient equals that of the transmission coefficient for a uniform plane wave at normal incidence on an interface between two lossless dielectric media. What is the standing-wave ratio in dB under this condition? P.8-22 A unifomi plane wave in air with E,(z) = a,10e-j6' is incident normally on an interface \ at : = 0 with a lossy medium having a dielectric constant 2.5 and a loss tangent 0.5. Flnd the following: t a) The instantaneous expressions for E,(:, t), H,(z, t), E,(z, t), and H,(z, t), using a cosine reference. b) The expressions for time-average Poynting vectors in air and in the lossy medium. P.8-23 Consider the situation of norn~al incidence at a lossless dielectric slab of thickness tl in air. as shown in Fig. 8-12 with e l = E , 7 E , and p, = p, = 11,. i a) Find E,,, E,', E;, and E,, in terms of E,,, d, 6,. and 11,. b) Will thcrc l x rcllcction nt intcrfircc : = 0 if tl = iL,4'? Espl:lin. I P.8-24 A transparent dielectric coating is applied to glass ( E , = 4 , p, = 1) to eliminate the reflection of red light [ I = 0.75 (pm)]. 1 - a) Determine the required dielectric constant and thickness of the coating. b) If violet light [I = 0.42 (pm)] i s shone ormal mall^ on the coated glass, what percentage / of the incident powcr will be reflected? I P.8-25 Refer to Fig. 8-12, which depicts three different dielectric media with two parallel t i interfaces. A uniform plane wave in medium 1 propagates in the + z direction. Let TI, and T13 t denote, respectively, the reflection coefficients between media 1 and 2 and between media 2 and ! 3. Express the effective reflection coefficient, I -, , at 2 = 0 for the incident wave in terms of T I : . r,,, and P,d. i ! 1 Medium 1 ( € 1 9 PI) Fig. 8-17 Plane wave incident normally onto a dielectric slab backed by a perfectly conducting z = O z = d plane (Problem P.8-26). I lent equals In interface I t condition? In interfiicc 5 . Find the ~g a cosine adium. , thickness d ;no parallel i-, : and rZ3 media 2 and srms of r,,, f-? . I ' . PROBLEMS 367 . . ( I !'..'. , ' 1 , PA-26 A uniform wave with - E,(i, t) = a,E,, cos CCI I . in medium 1 (el, p i ) is incident normally onto a lossless dielectric slab ( E , , p,) of a thickness d backed by a perfectly coriducting plane, as shown in Fig. 5-17. Find c . a) E,(& t) b Elk, 4 Ez(z, 0 4 PaVh 4 (9'A - f) Determine the thickness d that makes E1(3, t) the same as if the dielectric slab were absent. ; P.8-27 A uniform plane wave with E,(z) = a,~~,,e-jfl~' in air propagates normally through a , thin copper sheet of thickness (1, as shown in Fig, 8-18. Neglecting multiple reflections within the coppcr shccts, lind ;I) Ef. 111 1)) 1;: , l l j C) E 3 0 , H30 4 v'JA'(,pJL Calcuiate (9aP,,)3/(9'av)i for a thickness d that equals one skin depth at 10 (MHz). (Note that this p~rtains to the shieldin$ elrcctivcncss of thc thin coppcr shcct.) Copper sheer t Z (€0. POI -,d+ A. r Fig. 8-18 Plane wave prop::gating through a t,liin copper sheet (Problem P.S-27). P.8-28 A lCL(k-Hz) parullcll) polarized elcctromagnctic wave in air is incident obliquely on :In ocean surface at a near-grazlrlg angle Oi = SS'. Using E, = 81. jl, = I , :~nd a = 4 (SJln) for sea- wntcr, find (a) thc anglc of rcfi.nctian 0,. (b) thc transmission cocificient T ~ ~ . (c) (.Pd,,),:(.Pd,j,, and (d) the distance below tllc ocean surface where the field intensity has been diminished by 30 (dB). P.8-29 A light ray is idcitienr from air obliquely on a transparent sheet of thickness d with an index of refraction n, as shown in Fig. 8-19, The angle of incidence is Oi. Find (a) 0,. (b) the distance L, at the point of exit, and (c) the amount of the lateral displacement l2 of the emerging ray. Fig. 8-19 Light-ray impinging. , f .
. -+ , a + A obliquely on a transparent sheet of . . v ". - . .t' . refraction index n (Problem P.8-29). P.8-30 A uniform plane wave with perpendicular polarization represented by Eqs. (8-133a) and (8-133b) is incident on a plmdinterface at .- = 0, as shown in Fig. 8-13. Assuming t r < r , and 0, > Oc, (a) obtain the phasor expressions for the transmitted field (E,.?,), pnd (b) verify that the average power transmitted into medium 2 vanishes. I\ p.8-31 Elcctromaglletic w;,vc from il~ldenv;~tcr source with perpcndlci ir p d ~ r i ~ l t i c ~ l l i.: P incident on a wrter-air interf~lcc at tii = 20, , Using cr = 81 rlidp,= 1 fresh uatcr. lind (a) critical angle 8.. (b) reflection coefficient r,, (c) ti;msmission coefficient 7,. and id) attenuation indB for each wavelength into the air. -. P.8-32 Glass isosceles triangular prisms shown in Fig. 8-20 are used in optical instruments. Assuming t , = 4 for glass. calculate the percentage of the incident light power reflected back by the prism. 7 . . Incident 1 - ' - . . Fig. 8-20 Light reflection by a right isosceles triangular prism (Problem P.8-32). -, % . . - r P.8-33 Prove that, under the condition of no reRection a an interface. the sum of the B r e w e r 1 : angle and the angle of refraction is n/2 for: k a) perpendicular polarization ( p i # pz), . , i f . b) parallel polarization ( E ~ # ct). , P.8-34 For an incident wave with parallel polarization: t . .' . ~ . a) Find the relation between the critical angle 0, and the Brewster angle OBI, for nonmagnetic i media. I b) Plot 0, and esll versus the ratio E , / E ~ . 1 L I P.8-35 By using sncll's jaw of refrxtion, (a) express r and r in tcrnms of r.,, f 2 , and 0,; and . t? (b) plot I - and T versus 0 , for r,,/e,, = 2.25. P.8-36 In some books the rcflection and transnmsion coerXcicnts for parallel polar~zatlon are defined as the tatios of the amplitude of the tangentla1 components of, respectively, the retlected and transmitted E fields to the amplitude of the tangential component of the incident E field. Let the coeficients defined in this manner be destgnated, respect~vely, Ti, and ril. a) Find ril and Til in terms of j l l , t 7 : , O,, and 8,; and compare them wlth TI, and r,, in Eqs. (8- 149) alld (8-150). b) Find the rslatioi between Ti, and ri,, and compare if with Eq. (8-151). 9-1 INTRODUCTION We have now developed an e~ectrdma~netic model with which we can analyze electro- magnetic actions that occur at a distance and are caused by time-varying charges and currcnts. These actions are explained in terms of clcctrorna~netic fields and waves. An isotropic or omnidirectional electromagnetic source radiates waves equally in all directions. Even when the source radiates through n hi$-ty directive antenna. its encrgy spreads over n widc.arca at Iargc dishnccs. This radiatctl cncrgy is not guided, and the transmission ol' power andinformation from thc source to a receivcr is incficient. This is cspccinlly true at lowcr frcqucncics for which dircctivc antcnnas would have huge dimensions and, therefore, would be excessively expensive. For instance, at AM broadcast frequencies, a single half-wavelength antenna (which is only mildly directivet) would be over a hundred meters long. At the 60-Hz power frcqucncy a wavelength is 5 million meters or 5 (Mm)! For cllicicnt point-to-point transmission of power and information, thc source energy must be directed or guided. In this chapter we study transverse electromagnetic (TEM) waves guided by transmission lines. The TEiM mode of guided waves is one in which E and H are perpendicular to each other and both are transverse to the direction of propagation along the guiding line. We have discussed thc propagation .
. .
.
+ of unguided TEM plane waves in the last chapter. We will now show in this chapter that many of the characteristics of TEM waves guided by transmission lines are the same as those for asuniform plane wave propagating in an unbounded dielectric medium. The three most common types of guiding structures that support TEM waves arc: . a) Parallel-plate transmission line. This.type of transmission line consists of two parallel conducting plates separated by a dielectric slab of a uniform thickness. See Fig. 9-l(a)). At microwave frequencies parillel-plate transmission hncs can be fabricated inexpensively on a dielectric substrate uslng printed-circuit tech- nology. They are often called striplines. 1 ' Principles of antennas and radlatmg systems will bd d~scussed In Chapter 11. 9-1 1 INTRODUCTION 371 t Fig. 9-1 Common types of ~ranslnission iines. We should note that other wave modes more complicated than the TEhl mode can propagate on 4 1 1 three ofthase types ofti:lnsmission lines when the sep;ir;ltiun hetivren the conductors is grci1tr.r ili;in ~crtiiili fri~cliolls of the oper:lLillg IV:IYCICII~III. TI~esc other transmiss.ion modes will be considered in the next chapter. We will shbw that the TEM wave solut~on ofMaxwell's equations for the parallel- plate guiding ptructure in Fig. 9-!(a) lends directly to a pair of rransmission-line equations. The'generd tra~~rmission-line eq~~;itions C;III also he deriicd ijoill ;I circ~~it model in terms of tile resistance. induclance. conductance, and cap;icit;ince per unit Ierlgth of a line. ~ h d transi~ion from the circuit model to the electromi~gneti~ model is elfected from a network with li~mped-p;iramerer elements (discrete resistors. in- (jllctors. :ind c;~p:~cito~~s) 10 onc \bi111 dis~rii~~~lecl p:ir:i1l~e~crs ( ~ ~ I ~ L ~ I I L I ~ L I ~ cIihtri[?~~~iot~~ of R. L, G, a@ k along ih; line). From the transmission-line equations all the chiir- acteristics of &ve propsg;ttion :,long ; , giYen line c;ln be derived ;ind siildied. The stody 1sJiinlc-li;11-1l?o1iic s~e;itiy-s~;i~c properties of Ira11snlissio11 lillcs is gr~~itly fiicilitaled by thi i~se of grilp1iic:il charls wbich avert the necessity of repe~ited cal- cuhtions with coinpler IIL/~:I~C~S. Thc hest know11 and most ividcly used graphical chart is the Smith churt. The use of Smith cfiart for determining wave characteristics on a transmission line and for impedance-.matching will be discussed. 372 THEORY AND APPLICATIONS OF TRANSMISSION LINES / 9 9-2 TRANSVERSE ELECTROMAGNETIC . WAVE ALONG A PARALLEL-PLATE T RANSMlSSlON LINE Let us consider a y-polarized TEM wave propagating in the +r direction along a uniform parallel-plate transmission line. Figure 9-2 shows the cross-sectional dimen- sions of such a line and the chosen coordinate system. For time-harmonic fields the wave equation to be satisfied in the sourceless dielectric region becomes the horno- geneous Helmhol:z's equation, Eq. (8-38). In the present case, the appropriate phasor solution is E = a,E, = a , . E , e ~ ~ ' . (9-la) The associated H field is, from Eq. (8-26) where y and 11 are, respectively, the propagation constant and the intrinsic impedance of the dielectric medium. Fringe fields at the edges of the-plaes are neglected. As- suming perfectly conducting plates and a lossless dielectric, we have, from Chapter 8, and . The boundary conditions to be satisfied at the interfaces of the dielectric and the perfectly conducting planes are, from Eqs. (7-52a, b, c. and d), as follows: At both y = 0 and y = d: H,, = 0, (9-5) which are obviously satisfied because Ex = E, = 0 and H, = 0. Fig. 9-2 Parallel-plate transmission line. )n along a nal dimen- : tidds the the homo- ,tw phasor n line, . . , ,, . . 4 . ! -!
$ . . \<!.;;.;-''y',': :; .' ,, ' , : I ' . . . '. '. ,> . , .: > _ . , . ?
. , < . . ' i'. . . . ; . . , , . . .. .
, .
, :,,. :! \ 9-2 1 TEM WAVE ALONG PARALLEL-PLATE LINE 373 At y = 0 (lower plate), a,, = a,: a, D = psc or psd = EE, = E E ~ ~ -~ ~ ~ (9-63) Eo aj, x H = J,, or J,, - -a,H, = a, - e-J". (9 -7a) '7 At y = d (uppet plate), a, = -a,: E - h y x H = J,, or J , , , = a z H x = - a z o r - j p i . (9-7b) r? Equations (9-6) add (9 -7) indicate thar surface charges and surface currents on rhc conducting planes var) sinusoidally with 2, as do E, and H,. This is illustrated schematically in Fib. 9--3. Field phasors E and H in Eqs. (9---12) and (9-1 b) satisfy the two Maxwell's curl equations: Since E = a,E, and H - a,EI,, Eqs. (9-8) and (9-9) become and Ordinary derivatives app~ilr above because phasors E, and H , are functions of .- only. f Fig. 9-3 TEM-mode fields, surface charges, and surface currents in parallel-plate transmission line. ' 374 THEORY AND APPLICATIONS OF TRANSMISSION LINES 1 9 Integrating Eq. (9-10) over y from 0 to d, we have '- = ~ L l ( : ) , (9 - 12) where v(r) = - J : E, dy = - E,fz)d is the potential difference or voltage between the upper and lower plates; ' r is the total current flowing in the +: direction in the upper plate: and I is the inductance per unit length of the parallel-plate transmission line. The depen- dence of phasors V(z) and I(z) on z is noted explicitly in Eq. (9-12) for emphasis. Siniilarly, wc intcgwtc Eq. (9-1 I ) nvcr .s from 0 to I- to obtain = joCV(z), where . is the capacitance per unit length of the parallel-plate transmission line. Equations (9-12) and (9-14) constitute a pair of time-harmonic trai~smissiorz-line ec1ucrtiot1.s for phnsors V ( z ) and I(:). Thcy may bc coinhincd to yield second-ordcr dilli.rcntial ccluutions ror V(z) and for I(=): The solutions of Eqb. (9-16a) and (9-16b) ate, for waves propagating in the +: direction, V(Z) = Voe - jsz - (9-17a) and I(z) = ~ , e - ~ ~ ' , (9-17b) where the phase conlltant is the same as that given in Eq. (9-2). The relation between V, and I , can be found by using either Eq. (9 1.1) or Eq. (9- 14) : which becomes, in view of the results of Eqs. (9-13) and (9-15). I I 2 , which, again, is the same as that of a TEM plahe wave in the dielectric medium 9-2.1 Lossy Parallel-Plate Transmission Lines ----. Wc l w c so far :i~sulhcd llic parallel-pl~~lc tramnission linc to be lossless. In actual situations loss may arise from two causes. First, the dielectric medium may have a nonvanishing loss tahger.1; and, second, the plates may not be perfectly conducting. To characterize these two effects we define two new parameters: G, the conductance ' This statement will be proved in Section 9-3 (see Eq. 9-87). 376 Ti4EORY AND APPLICATIONS OF TR-ANSMlSSlON LINES 1 9 , . per unit length across the two plates; and R, the resistance per unit length of the two plate conductors. The conductance between two conductors septlratcd by a dielectric medium having a permittivity E and a conductivity 0 can be determined readily by using Eq. (5-67) when the capacitance between the two conductors is known. We have Use of Eq. (9-15) directly yields If the parallel-plate conductors have a very large but finite conductivity a, (which must not be confused with the conductivity a of the dielectric medium), ohmic power will be dissipated in the plates. This necessitates the presence of a nonvanishing axial clcctric field a,E, at the plate sarbces. such that the avera$hynting vector = aspn =::.Xa(a,E, x a,Hz) (9 -24) has a y component and equals the average power per unit area dissipated in each of the conducting plates. (Obviously the cross product of ayE, and a,Hx does not result in a !
. component.) Consider the upper plate where the surface current density is JSu = H,. It is convenient to define a sur/ace impedance of an imperfect conductor. Z,, as the ratio of the tangential component of the electr'c field to the surface current density at the conductor surface. For the upper plate, we hive E E, =f = A = ~ C 9 (9 -26a) " S U Hx where qc is the intrinsic impedance of the plate conductor. Here we assume that both the conductivity 0, of the plate conductor and the operating frequency are sufficiently high that the current flows in a very thin surface layer and can be reprc- xnted by the surlacc current J,,,,, Thc intrinsic impcdvncc of ;l good conductor has been given in Eq. (8-46). We have - where the subscript c is used to indicate the properties of the conductor. length oi'the two . electric~medium lily by using Eq. . We have : ' (9 -23) ~chvity o, (which m), ohmic powcr ):.ivanishing axial ng vector (9 -23) r -' ipat - each of ' , doe3 not result s J,, = H,. It is '. Z,, as the ratio nt density at the \ve assume that 1 g frequency are nd can be repre- d c o w t o r has clor. I 9-2 1 TEM ~ A V E ALONG PARALLEL-PLATE LINE 377 I Substihi$ of;Eq. (9-26a) in Eq. (6-24) gives I (9-27) , The ohmic poker dissipated in a unit length of :he plate havlng a wldth vj IS spc, which can be expressed in terms of the total surface current. I = W J , .
, as -. Equation (b-28) is the power dissipated when a sinusoidal current ofdmplitude I flows through a resistance RJw. Thus, the effective serles resistance per unit length for both plates of a parallel-plate transmission line of wldth iv is . Table 9-1 lists the expressions for the four distributed purut~~eiers (R, L, G. 2nd C per unit length) of s parallel-plate transmission line of width L I and separation (1. We note from Eq. (9-26b) !hat surface impedance Z, has a positive renctanc- term X, that is nun3erically equal to R,. If the total complex power (insrend of its real Part, the ohmic Power P .
. only) associated with a unit length of the plate is con- sidered. X , will lend to an ititert?aiiniies inductance pcr wit lengtll L, = x,,~,, = A,, ,.,. At hi& frquedcics; Li is negligible in comparison with the exterllal inductlncu ,r. Table 9-1 Distributed Parameters of Parallel- Plate Transmission Line (Width = W. Stparation = d ) We note in the calculation of the power loss in the plate conductors of a finite conductivity o , that a nonvanishing electric field a,E, must exist. The very existence of this axial electric field makes the wave along a lossy transmission line strictly not TEM. However, this axial component is ordinarily very small compared to the transverse component E,. An estimate of their relative magnitudes can be made as follows: I (9-30) For copper plates [oc = 5.80 x lo7 (S/m)] in air [s = r0 = 10-'/36n (F/m)] at a frequency of 3 (GHz), IEJ ~ ~ 5 . 3 x lo-s/Eyl < < IE,~. Hence we retain the designation TEM as well as all its consequences. The introduction of a small E: in the calculation of p. and R is considered :I slight pcrturb;ltian. . -.
.
. --- Example 9-1 Striplines consisting of a thin metal strip separated from a conducting ground plane by a dielectric substrate are used extensively in microwave circuitry. Neglecting losses and assuming the substrate to have a thickness 0.4 (mm) and a dielectric constant 2.25, (a) determine the required width w of the metal strip in order for the stripline to have a characteristic resistance of 50 (R), (b) determine L and C of thc line. and (c) detcrminc 11,) along llic linc. (d) Kcpeat parts (;I). (b) ;~nd (c) lor a characteristic resistance of 75 (R). Solution a) We use Eq. (9-20) directly to find w. .
- - 0.4 x 377 50 = 2 x (m), or 2 (mm). a finite xistence ctly not 1 to the made as . (9 - 30) i l l ) ] at a , ~ ~ ~ U ~ t ~ O l l ti(?n. n ,r,db circuitry. in) and a ; , In order : I , and C d (c) for a d /? 1). 9-3 1 GENERAL TRANSMISSION-LINE EQUATIONS 379 1 . I d) Since w is inveksely proportional to,&, I we I have, for Zb = 75 (Q), $ 9-3 GENERAL TRANSMISSION-LINE EQUATIONS We will now derive the equations that govern general two-conductor uniform trans- mission lincs. Trdt~smission lincs clilrcr from ordinary clcctric nctworks in one essential feature. Whercas the physicai dimensions of electric networks are very much smaller than the operating wavelength, transmission lines are usually a considerable fraction ofa wavelength and may even be many wavelengths long. The circuit elements in an ordinary electric network can be considered discrete and as such may be de- scribed by lumped parameters. Currents flowing in lumped-circuit elements do not vary spatially over the elements, and no standing Waves exist. A transmission line, on the other hand, is a distributed-parameter network and must be described by . circuit parameters that are distributed throdghout its length. Except under matched conditions, standing waves exist in a transmission line. Consider a differential length Az of a transmission line which is described by the following four parameters : R, resistance per unit length (both conductors), in Qlm. L, inductance per unit length (both conductors), in Him. G, conductance per unit length, in S/m. C, capacitance per unit length, in F/m. Note that R and , !
, are series elements, and G and C are shunt elements. Figure 9-3 shows the equivalent clcctric circuit olsuch a line segment. The quantit~es o ( : , t ) .md v(z + Az, t) denote the instantkneous voltages at : and z + Az respectively. Similarly, i(z, t) and i(z + Az, t) denote the instantaneous currents at z and 2 + Az. Applying ~irchhas-voltage law, we obtain which leads to , - &(z, t) v(z + Ai, t) - 45 t) = Ri(z, t) + (9 -30a) A : at N i ( z + A r , r ) . . v(z + Az, 0 Fig. 9-4 Equivalent circuit of a I-----A~ -----I differential length Az o f a two-conductor transmission line. On the limit as Az -, 0, Eq. (9-30a) becomes Similarly, applying KirchhofYs current lilw to the node N in Fig. 9-4, we h:lvc 2o(z + A:, t) i(z, t) - G AZV(Z + A:, t) - C A: ---. - i ( : + A : , tj = 0. (9-32) (71 On dividing by Az and letting Az approach ;era, Eq. (9-32) becomes I I Equations (9-31) and (9-33) are a pair of first-order partial differential equations in v(z, t) and i(z, t). They are the ge~zeral transmission-line equations.' For harmonic time dependence, the use of phasors simplifies the transmission- line equations to ordinary differential equations. For a cosine reference we write where V(z) and I(z) arefinctions of the space coordinate z only, and both may be complex. Substitution of Eqs. (9-34a) and (9-34b) in Eqs. (9-31) and (9-33) yields the following ordinary differential equations for phason V(z) and I(z): ' Sometimes referred to as the telegraphist's equations. 9-3.1 Transr 1 nductof (9-31) wc l~ave (9 -33) n (9 -33) yuations mission- e write (9-34a) (9-34b) i may be 33) yields (9-35a) m (9 4 9-3 1 GENERAL TAANSMISSION-LINE EQUATIONS 381 . : Equations (9-3ja) dnd (9-35b) are timq-barionic transmission-line equarions, which - '.reduce to Eqs. (9-19) and (9-14) under lo&led conditions (R = 0, G = 0). c I ' 9-3.1 ' wave ~haractekistks on an Infinite Transmission Line I i The coupled time-harmonic transmission-line equations, Eqs. (9-35a) and (9-35b), can be combined rd solve for V(z) and ~ ( z ) . We obtain I and where is the propagation constccnt whose real and imaginary parts, cc and / , ' , are the artenrlurm constant (Np/m) and phase constant (radlm) of the line respectively. The nomenclature here is similar to that for plane-wave propagation in conducting media as defined -, in Section 8-3. THese quantities are not really constants because, in general, they depend on o in a complicated way. The solutions df E ~ L . (9-36a) and (9-36b).are where the plus and minus superscripts dendte waves traveling in the + z and - z directions respectiltely. Wave amplitudes, V;, V;, I:, and I ; are related by Eqs. ,(9-35:l) :~nd (9-35b), mi it is wsy to vcrify (Problcm P.9-5) thot V,+ V R+joL -- - ---- . lo' I , ' y ' I 1 8 . 382 THEORY AND APPLICATIONS OF TRANSMISSION LJNES 1 9 , . . , ~ b r an infinite line (actually a semi-infinitdline with the source at the leA end), the terms containing the eYz factor must vanish. There are no reflected waves; only the waves traveling in the + z direction exist. We have The ratio of the voltage and the current at any z for an infinitely long line is inde- pendent of z and is called the characteristic impedance of the line. Note that y and Z, are characteristic properties of a transmission line whether or not thc linc is infinitely long. They dcpcnd on R. L, G. C. and to-not on thc Icngth of the line. An inlinitc line simply implics that thcrc arc no rellcq~d waves. The general expressions for the characteristic impedance in ~<(9-41) and the propagation constant in Eq. (9-37) are relatively complicated. The following three limiting a w s hnvc special significance. I . Lassless Lirte (R = 0, G = 0). 3) Propagation constant: c i = O (9-42a) p = w a (a linear function of o). (9 -42b) b) Phase velocity: 0 1 u =- p . p=F (constant). c) Characteristic impedance: l - 7 ' - 2. ~ d w - L O S S Line (R < < oL, G cc wC). The 1.0~-loss condition is more easily sat- isfied at very high frequencies. I , IFt end), ; 1 :s; only I I is inde- asily sat- 4 . , ;t, " , ,, . , , & ;:;g , .., ,:- , . , , . ,, . . . ,. I .
. _ , z , - . . .. ! . i . , , < .!;' ' , 87 " , v : : , , , , > ; : F + : . .". ' , '\ 3 ,, ;. 9-3 / GENERAL TRANSM~SSION-LINE EQUATIONS 383 a) Propagation constant: $ +?$+ 2 G.;&)] P 2 ~ T c (approxinlately a linear function of o) b) Phase velocit): I 1 u = - - - % - - - I " /I JLc (approxi~natcly conbtanl). C) Characteristic impedance: 3. Distortionless Lhe (KjL = GIC). If the condition L . L is satisfied, the expressions for both y and 2, simplify. a) Propagation constant: P = o m (a linear function of w). b) Phase velocity: w 1 u p = - = - \ B rn (constant). (9 -50) c) Characteristic impedance: Ro = $ (constant) (9-51a) Xo= 0. (9-51b) Thus, except for a nonvanishing attenuation constant, the characteristics of a dis- tortionless line are the same as th6se of a lossless line; na'mely, a constant phase velocity (up = l/m) and a constant real characteristic impedance (2, = R, =j'LF). A constant phase velocity is a direct conscqtiencc of thc Iinc:~~. tlcpcntlcncc of IIw p l i w co~isl:~nl /I on (I),. Siwc :I si!:11;1I IISU:IIIJ' co~~sisb 01' ;I I X I I I ~ ol' I ' I ~ ~ ~ I c I I c ~ c ~ , i t is essential Illat thc dill'crcnt l'rcqucncy compoacnts travcl along 'ii-transmission line at the same velocity in order to avoid distortion. This condition is satisfied by a lossless line and is approximated by a line wirh very low losses. For a lossy line, wave amplitudes will be attentuated, and distortion will result when different frequency components attenuate differently, even when they travel with the same velocity. The condition specified in Eq. (9-48) leads to both a constant cr and a constant up- thus the name distortionless line. The phase constant of a lossy transmission line is determined by expanding the expression for y in Eq. (9-37). In general, the phase constant is not a linear function of o; thus, it will lead to a up, which depends on frequency. As the different fre- quency components of a signal propagate along the line with different velocities, the signal suffers dispersion. A general, lossy, transmission line is therefore dispersive, as is a lossy dielectric (see Subsection 8-3.1). Example 9-2 It is found that the attenuation on a 50-(Q) distortionless transmission line is 0.01 (dB/m). The line has ti capacitance of 0.1 (pF/m). a) Find the resistance, inductance, and conductance per meter of the line. b) Find the velocity of wave propagation. c) Determine the percentage to which the amplitude of a voltage traveling wave decreases in 1 (km) and in 5 (km). Solution a) For a distortionless line, (9-50) ' (9-51) 9-51a) 9-51bJ f ,i dis- I ;:!use --- ., I-, C). 2 of the i. I t IS on l i , % :d b) e, \4;lx'? 4'JL cloc1ty. onstant i ~ n g LIIC unction ent fre- locities, ,I)ersiue, mission 1g wave n \ 1 . The three relat?ons above are sufficient to solve for the three unknowns R, L, and G in terms bf the given C = 10-lo (F/m): b) The velocity of wave propagation on a distortionless line is the phase velocity given by Eq. (9-50). 1 1 = 2 x 10' (m/s). , L J(0.25 x c) Tho ratio of two voltagcs a distancc z apart along the linc is After 1 (km), (G/vl) = e-'Oooa = ,-1,15 = 0.317, or 31.7%. After 5 (km), (V'/V,) = e-5000a = e-5.75 = 0.0032, or 0.32%. 9-3.2 Transmission-Line Parameters The electrical properties of a transmission line at a given frequency are completely a characterized by its four distributed parameters R, L, G, and C. These parameters for a parallel-plate transmission line are listed in Table 9-1. We will now obtain them for ty;w!re and coaxial transmission lines. Our baslc premile is that the conductivity of the conductors in a transmission linc is asu;illy so !ligli Illill tho dTcct el. tbc wries rcsisli111cc oto lllc ~ ~ l l l p ~ l l i l t i ~ n 01 t 1 1 c propilgiltiot~ ct)llsti~ill is ~sgligihlc. Ihc is~plici~lion lsinp 1I1:11 ~ l l c wi~vcs on the lilx am approximate'g %M. We may write, ir) dropping R liom Eq. (9-37), 386 THEORY AND APPLICATIONS OF TRANSMISSION LINES / 9 t i i ' ' , [ - , . , , From Eq. (8-37) we know that the propagation constant for a TEM wave in a medium with constitutive parameters (p, E, Q) is I , But 1 G a , -- - - (9-54) i f C E i b in accordance with Eq. (5-67); hence comparison of Eqs. (9-52) and (9-53) yields L c (9-55) Equation (9-55) is a very useful relation, because if L is known 'or a,!ine with a given medium, C can be determined, and vice versa. Knowing C, we can find G I from Eq. (9-54). Series resistance R is determined by introducing a small axial El as a slight perturbation of the TEM wave and by finding theahmic power dissipated --. -7 in a unit length of the line, as was done in Subsection 9-2.1. Eq~iation (9-55), of course, also holds for a losslcss line. The velocity of. wave propugation on u 1o.ssle.s.s trun.srni.ssion line, u, = ]/,./LC, therefore, is ecluul to the velocity of propugation, I/&, o f unguided plane wove in the dielectric 06 the line. This fact has been pointed out in connection with Eq. (9-21) for parallel-pi: dte lines. I I 1. Two-wire tr.ansmission line. The capacitance per unit length of a two-wire trans- mission line, whose wires have a radius a and are separated by a distance D, has been found in Eq. (4-47). We have I (9-56)+ 1 cosh- ' ( D l 2 4 D I I From Eqs. (9-55) and (9-54), we obtain L = c o s h ' ( 5 ) (Him) 1 I and L . i '-7 (9-58)t i i To determine R, we go back to Eq. (9-27) and express the ohmic power I , dissipated per unit length of both wires in terms of p,. Assuming the current i + cosh - ( D l 2 4 z In (Dlu) if ( ~ 1 2 ~ ) ' > > 1. 1 medium. (9 -53) (9-54) I ) yields (9-55) line wth , i n Jimi G 11 m a 1 E, t ~ssipated / - 1 oj \VLlLV lrol he llrle. This lines. &Ire trans- nce D, has (9 -56)' (9-57)+ n ( " 8 ) ' mic power .he current I- . . , , . .'Z .
; ':F "'"" , , , . p .- . - r 1 . i : I . ' , 3 8 i ; , \ 4 " f ,! ' 9-3 1 GENERAL TRANSMISSION-LINE EQUATIONS 381 .
.
. . '. i :' I I 6 L 0 1 ) J , (Aim) to flow. in a very thin surface layer, the current in each wire is I = 2nal,, and . . (9-59) . I Hence the seriea resistance per unit length for both wires is In deriving ~ ~ l ( 9 - 5 9 ) and (9-60), we have assumed the surface current .
I , to be uniform over the circumference of both wires. This is an approximation. inasmach . as the proximity of the two wires tends to make the surface current nonun~form. 2. Couxiul rransntlssion fine. The external inductance per unit length of a coaxial transmission line with a center conductor of radius a and an outer conductor of inner radius b has b:xn found in Eq. (6-124): From Eq. (9-55), we obtain To determine K, we again return to Eq. (9-27), where JSi on the surface of the center conductor is different from J,. on the inner surface of the outer con- dudor. We mudt haye 388 THEORY AND' APPLICATIONS OF TRANSMISSION- LINES 1 9 Table 9-2 Distributed Parameters of TWO-Wire and . Coaxial Transmission Lines parameter Two-Wire Line Coaxial Line Unit I I.1 6 ~c0sh-l ( k ) -1n- t HIm L 2n a n Internal Inductance IS not included - 1 . From Eqs. (9-65a) and (9-65b), we obtain the resistance per unit length: , , I (9 -66) I i The R, L, G, C parameters for two-wire and coaxial transmission lines are listed 1 in Table 9-2. 9-3.3 Attenuation Constant from j Power Relations i The attenuation cqnstant dl a traveling wave on a transmission line is the red part I 1 of I ~ C propitgtion constant; it can be determined from the basic definition in Eq. ; , (9-37): a = &',(I) = : 1 ( [J(R + jmL)(G + jwC)]. . (9-67) ' I The attenuation constant can also be found from a power relationship. The / ? phasor voltage and phasor current distributions on an infinitely long transrnlssion line (no reflections) may be written as (Eqs. (9-4Un) and (9-40b) with thc plu\ snpcr- i script dropped for simplicity): i I v ( ~ ) = v o e - ( n + j P ) z . (9-68a) j I n llgth, (9 -66) ~cs are listed rhe real part ~ i t ~ o n in Eq. (9-67) ons ship. The ~r:~nsmission e ph . W e r - P - , , -68a) (9 -68 b) , : 9-3 / GENERAL TRA~SMISSION-LINE EQUATIONS 389 I . t r \ The time-average b o a r prbpagated along the bne at any z is ( 6 7 P(z) = +%[V(Z)~(Z)] The law of conservation of energy requires that the rate of decrease of P(I) with distance along the lide equals the time-avsrage.power loss PL per unit length. Thus, = 2ctP(2), from which we obtain the following formula: Example 9-3 a) Use Eq. (9-701 to lind thc attenuation constant of a lossy transmission line with distributed paradeters R, L, G and C. b) Specialize the result in part (a) to obtain the attenuation constants of a low-loss line and of a distortionless line. Solution a) For a lossy trahshission line the time-average power loss per unit length 1s PL(~) = j[lI(zj12R + p(z)12q - v 2 (R + GIZ~~')~-'". (9-71) -- 2lz0l2 ~ubstitutibn of Eqs. (0-69) and (9-71) in Eq. (9-70) gives (R + GIZOlZ) (NpIm). b) For a i$;toss line. Z , z R,, = Jq?, Eq. (9-72) becomes 390 THEORY AND APPLlcATloNs OF TRANSMISS~O'N I , which checks with Eq. (9-45). For a distortionless line, Zo = Ro = m. Eq. (9-72a) applies, and c x = h & + ; k ) , I 2 which, in view of the condition in Eq. (9-48), reduces to I I . = R & ,(9-72b) Equation (9-72b) is the same as Eq. (9-494. ! 9-4 WAVE CHARACTERISTICS ON' FINITE TRANSMISSION LINES In Subsection %3.l we indk~ted that the generd solutions for the time-harmonlc one-dimensional Helmholtz equations, Eqs (9-36a) andL9-36b), for transmission -. lines are (9-73a) where v : V L & 1 0 ' 1, For infinitely long lines there can be only forward waves traveling in the + .- direction, and the second terms on the right side of Eqs. (9-73a) and (9-73b). representing reflected waves, vanish. This is also true for finite lines terminated in a characteristic impedance; that is, when the lines are matched. From circuit theory we know that a maximum transjer of power fqom a given voltage source to a load occurs under "inuiciied co~~ditio~d' when the load impedarlce is the complex co~ljugate of the source i~npedn~lce (Problem P.9-1 I). In transmission line terminology, a line is inatci~ed d ~ e n il~r l o d ~m,wda~~re is rrlltcrl to thc c~hirrcrc/c~ris/ic irnpedencr (not the compicx corqilqalr [hi. charucteristic impedance) o f the line. I I il i i---------~. . z = o z = P z'= f z'= 0 I I Fig. 9-5 Finite transmission line terminated with load impedance ZL. i .
I . Lct us now consider ihc gcneral c a u of it f h l c transmission line having a charac- teristic imp&ince-zil terminated in an arbitraty load impedance Z,, as depicted in Fig. 9-5. The lhgth Of the line is C. A sinusoidal voltage source V , E with an internal impedance 2, is conriected to the line at z = 0. In such a case, which obviously cadhot bc satisfied without the second terms on the right side of Eqs. (9-73a) and (9-33b) unless Z, = Z,. ~ h u s , reflected waves exist on unmatched lines. i . Given the characteristic y and Zo of the jine and its length &, there are four unknowns V : , V,', f,C, and I, in Eqs. (9-73aj and (9-73b). These four unknowns are not all independent because they are constrained by the relations at : = 0 m c l , at z = L. Both V(z) and Iiz) can be expressed either in terms of I/; and I, at the input end (Problem P.9-12), or in terms of the conditions at the load end. Conslder the latter case. Let z = C in Eqs, (9-Y3a) and (9-7313). We have VL = V:e-yt + V ; e Y C (9 -76a) v,+ J ' ; 1, = - e-yc - - Y/ e . z 0 (9-7Gb) 2 0 Solving Eqs. (9-76aj and (9-76b) for V, and V;, we Have V: = $(VL + I,z,)~Y' - (9-77a) V f = $(VL Ti ILZo)Ztt, . (9-77b) Substituting Eq. (9-75) in Eqs. (9-774 and (9-77b), and using the results in Eqs. (9-73a) and (9-73b), we 3btain Since L and z appear together in the combination (L - z), it is expedient to introduce a new variable z' = G - 2, which is the distance measured backward from the load. Equations (9-78a) aiid (9-78b) then become We note here that although the same symbols V and I are used in Eqs. (9-79a) and (9-79b) as in Eqs. (9-78a) and (9-78b);the dependence of V(zl) and I(zl) on z' is different from the dependence of V(z) and I(z) on z. 392 THEORY AND APPLICATIONS OF TRANSMISSION LINES 1 9 - : . ,: .
, , . ' . .. ,<. & > ' . I . . , : , ' ' ,,'. : . : '' 1 l' I 1 L. ransmission d end of the sees an input I finite trans- ~ltage I/, and t id Fig. 9-6. Of course, the volta& and current at any other location on line cannot be determined by using the equivalent circuit in Fig. 9-6. The average power delivered by the generator to the input terminals of the line is I i (PaV)~ = @~[JV?],=O, = I = / . - (9-85) The average power delivered to the load is Fur a lossless line, cons~xvation of power requires that (Pa,), = (Pa,),. A particularly impomnt special case is when a line is terminated wlth its charac- teristic impedance; that is, when Z L = Z,. The input impedance, Z , in Eq. (9-83), is seen to be equal to Z,. As a matter of fact, the impedance of the line loolung toward the load at any distance z' from the load is, from Eq. (9-82), Z(zf) = Z , (for Z , = Z,). (9-57) The - oltage and cdirent equations in Eqs. (9-7821) and (9-786) reduce to V ( z ) = (ILZ,eyL)e-Y" = Xe-?' (9-8Sa) I(z) = ( ~ ~ e ~ ' ) e - ~ ' = I,e-yz. (9-S8b) Equations (9-88a) and (3-88b) correspond to the pair of voltage and current equa- tions-Eqs. (9-40a) and (9-40b)-representing waves traveling in +I direction. and them are 110 rcllcctrd WL vcs. Hence, wiletl a jirtite trnnrplnsiurr litle is termitlured wtii its own characteristic itnpedunce (when a jinire transmission line is matched), rhe uoltuge und current dist~ibutions on the line are exactly the same as though the line had been extended to injinity. Example 9-4-A signal generator having an internal resistance 1 (Q) and an open- circuit voltage v&t) = 0.: cos 2n108t (V) is connected to a 50 (Q) lossless transmission line. The line is 4 (nl) locg and the velocity of wave propagation on the line is 2.5 x 10' (mh). For a matched load, find (a) the instantaneous expressions for the voltage and current at an arbitrary location on .the line, (b) the instantaneous expressions . . t , . 394 THEORY AND APPLICATIONS OF TRANSMISSION LINES / 9 for the voltage and current at the load, and (c) the average power transmitted to the load. Solution a) In order to find the voltage and current at an arbitrary location on the line, it is first necessary to obtain those at thc input end (z = 0, z' = I). The given quan- tities are as follows. 5 = 0.3/0" (V), a phasor with a cosine reference Z, = R, = 1 (n) Z0 = Ro = 50 (0) o = 2 7 7 x- lo8 (rad/s) up = 2.5 4 lo8 (m/s) f = 4 (m). Since the line is terminated with a matched load. Zi= Z , = 50 (R). The voltage I . and current at the input terminals can be evaluated from the equivalent circuit in Fig. 9-6. From Eqs. (9-84a) and (9-84b) we have # 50 v=- ' 1 + 5 0 x 0.3/0" = 0.294p (V) I.=-- 0'3/0" - O.0059p (A). ' 1 + 5 0 As only forward-traveling waves exist on a matched line, we use Eqs. (9-68a) and (9-68b) for, respectively, the voltage and current at an arbitrary location. For the given line, a = 0 and Thus, V(z) = 0.294e-j0.8nz (V) I(z) = 0.0059e-j0.8"z (A). These are phason. The corresponding instantaneous expressions are, from Eqs. (9 -34a) and (9 -34b), .
I ! tted to ihe . ' , , , I . , e line, it is ved quan- he voltage ent circuit / -' 1s. (9-68a) location. from Eqs. f7 9 Y i b) At the load, z = = 4 (m), I \ ut4, t) = 0.294 cos (2n!b8t - 3.2~~) (V) : ' i(4, t) = 0.0059 cos (2rr108t - 3 . 2 ~ ) (A). c) The average power transmitted to the load on a lossless line is equal to that at the input t e r h d l s , (PJJL = (P.Ji = :!%'L[v(z)1(2)] = j(0.294 x 0.0059) = 8.7 x (W) = 0.87 (mW). 9-4.1 Transmission Lines as Circuit Elements ' I , Not only can transdission lines be used zs wave-guiding structures for transkrring power and informatibn fium onc point to another, but at ultrahigh frequencies- UHF: from 300 ( M ~ z ) t.) 3 (GHz); wavelength, from I (m) to 0.1 (m)-they may scrvc 21s circuit clcmanls. A t thcsc rrcqucncics, ordinary lumped-circuit clcmcnts arc dilficult to make, and str.ly fields become important. Sections of transn~ission lines can be designed to gibe an ,inductive or capacitive impedance and are used :o match ' an arbitrary load td the internal impedance of a generator for maximum power transfer. The required length of such lines as circuit elements becomes practical in the UHF range. In most cases transmission-line segmenti can be considered lossless. y = j j l . Z , = R, and tanh 7L' = tanh (j/V) = j tan / I f . he formula in Eq. (9-83) for the input impedance Zi of a lossles line of length G terminated in 2, becomes ZL + jRo tad /?/ Zl = R, ------------ Ro + jZL tan /)/ (Lossless line) We now consider several important special cases. 1. Open-circuit rerntinatm (2, -t a). We have, from Eq. (9-89), 396 THEORY AND APPLICATIONS OF TRANSMISSLON LINES I 9 / Fig. 9-7 Input reactance of open-circuited transmission line. P Table 9-3 Input Reactance of Open-Circuited Line, X,, 1 1 K ( 2 n - 1 ) - < L < n - 4 2 (2n - 1) - < Pd < nn 2 Inductive I. IE (2n - 1) - 4 (2n - 1) - 2 0 When the length of an open-circuited line is very short in comparison with a wavelength, p .
4 < < 1, we can obtain a very simple formula for its capactive reac- tance by noting that tan P.4 2 pf. From Eq. (9-90) we have which is the impedance of a capacitance of C.4 farads. 2. Short-circuit termination (2, = 0). In this case, Eq. (9-89) reduces to Zi = jX,, = jRo tan b.4. (9-92) Since tan p .
4 can range from - co to + co, the input impedance of a short-circuited lossless line can also be either purely inductive or purely capacitive, depending on the value of p .
4 , Figure 9-8 is a graph of Xis versus 4 , and some important properties of Xis are listed in Table 9-4. , parison with pactive reac- ort-L i t e d : , depending le important Fig. 9-8 Idput reactance of short-circuited transmission line. A / . ( 2 n - 1 ) - < r ; < n - 4 2 Capacitive It is instructive to note that in the range where X,. is capacitive X,, is inductive, and vice versa. The input reactances of open-circuited and short-circuited lossless transmission lines are the same if their lengths differ by an odd multiple of 214. When the length of a short-circuited line is very short in comparison with a wavelength, D L < < 1 Eq. (9-92) becomes approximately which is the irnpcd~ncc of +n inductilncc of LL henries. 3. Quarter-wave secti~ns (f = 114, D L = ~12). When the length of a line is an odd mult$l+pf A/4, l = (2, - 1)1/4, ( n = 1,2,3, . . .), Hence, a quarter-wave lossless line transforms the load impedance to the input terminals as its inverse multiplied by the square of the characteristic resistance; it is often referred to as a quarter-wave transformer. An open-circuited, quarter-wave line appears as a short circuit at the input terminals, and a short-circuited quarter- wave line appears as an open circuit. Actually, if the series resistance of the line itself is not neglected, the input impedance of a short-circuited, quarter-wave line is an impedance of a very high value similar to that of a parallel resonant circuit. 4. Half-wuve sections (C = 142, P t = n). When the length of a line is an integral multiple of i.12, d = n1/2 (11 = 1, 2, 3, . . .), Equation (9-95) states that a half-wave lossless line transfers the load impedance 1 to the input terminals without change. I i tanpt =O, 1 and Eq. (9-89) reduces to t 1 Open- and short-circuit terminations are easily provided on a transmission line. j By measuring the input impedance of a line section under open- and short-circuit I conditions, we can determine the characteristic impedance and the propagation , constant of the line. The following expressions follow directly from Eq. (9-83). I Zi = ZL (Half-wave line). Open-circuitrd line, 2, -, co : (9-95) : Zi,, = Zo coth yd. Short-circuited line, ZL = 0: Z, = Zo tanh ye.. From Eqs. (9-96) and (9-97) we have (9 -94) J , r / ~ e input sistunce; it drter-wave 2d quarter- ol ihe line r-~ave line aut circuit. dn integral P .
' (9-95) 1 irnpcdarm nlsslon line. ,hort-circuit xopagation (9-83). (9 -96) h 9 7 ) (9 -98) and I , (9 -99) Equations (9-98) ahd (9.199) apply whether Or not the line is lossy. Example 9-5 The open-circuit and short-circuit impedances measured at the input terminals of a very low-loss transmission line of length 1.5 (m), which is less than a quarter wave~en~td, itre respectively -fi4.6 $2) andj103 (R). (a) Find Z, and y of the line. (b) Without ckanging the operating l'requency, lind ihe input impedance of a short-circuited lind that is twice the given length. (c) How long should the shon- circuited line be id ordeifor it to appear as an open circuit at the input terminals? Solution: The givbn quantities are Zit, = -j54.6, % , = jlO3, P = 1.5 , a) Using Eqs. (9-98) and (9-99), we find Z , = , / -j54.6(]103) = 75 (Q) 1 7 = - tanh-l /= = - 1,s j tan- ' 1.373 = jO.628 (rad/m). -j54.6 1.5 b) For a short-circuited line twice as long, I = 3.0 (m), ye = j0.628 x 3.0 = j1.884 (rad). The input impedance is, from Eq. (9-97), Z,, = 75 tanh (j1.884) = j75 tan 108" = j75( - 3.08) = -j231 (R). Note that Zis for the 3 (m) line is now A capactive reactance, whereas that for the 1.5 (m) line in p x t (a) is an inductive reactance. We may conclude from Table 9-4 that 1.5 (m) < 4 4 < 3.0 (rn). c) In order for a short-circuited line to appear as an open circuit at the input ter- minals, it should be an odd multiple of a quarter-wavelength long. . 2z 2n A = - - = - l o (m). p 0.628- Hence the required h e h g t h is 400, THEORY AND APPLICATIONS OF TRANS'MISSION ~ ~ N E S 1 9 , . ' , , , . , i : I 9-4.2 Lines with Resistive Termination a ! : , When a transmission line is terminated in a load impedance ZL different from the characteristic impedance Z , , both an incident wave (from the generator) and a re- :. !
flected wave (from the load) exist. Equation (9-79a) gives the phasor expression for the voltage at any distance z' = I - z from the load end. Note that, in Eq. (9-79a). the term with eYz' represents the jncident voltage wave and the term with e-"' re- presents the reflected voltage wave. We may write , where is the ratio of the complex amplitudes o f the reflected and incident voltage waves at I 1 i the load (z' = 0) and is called the voltage rejection coeficient of the load impedance Z , . It is of the same form as the delinition of the reflection coeficient in Eq. (8-93) 1 for a plane wave incident normally on a plane interface between two dielectric media. It is, in general, a complex quantity with a magnitude I r l 5 1. The current equation ! corresponding to V(zl) in Eq. (9-100a) is, from Eq. (9-79b). i The current reflection coefficient defined as the ratio of the complex amplitudes of the reflected and incident current waves, I ; / I , + , is different from the voltage re- flection coefficient. As a niatter of fact, the former is the negative of the latter, inasmuch as l;/l: = - V,/V:, as is evident from Eq. (9-74). In what follows, we shall refer only to the voltage reflection coelkicnt For a lossless transmission line, y = jfl, Eqs. (9-100a) and (9-100b) become I V(zl) = 4 2 ( Z , , + R,,)ejp"[l + re-jZ6"] rent from the tor) anii a re- :xpression for , 1 Eq. (9-79a), with e-Yz' re- (9-100b) : s amplitudes le voltage re- ter, inasmuch we shall refer I become b The voltage a n current phasors on a lossless line are more easily visualized : from Eqs. (9-80a) ahd (9-80b) by setting y =ID and V L = I,ZL. Noting that cosh j0 = cos 8, and sinh j8 31 j sin 8, we obtain -. . ,V(zl) = V L cos pz' 4- jILR, sin pz' (9-1033) I(,-') = IL cos p i ' + j - sin /3z1. (9-103b) L (Lossless line) If the terminating impedance is purely resistive, 2, = RL, V L = ILRL, the voltage and current magdituderi are given by , 1 V(z')J = vL Jcos2 O:' + (R,/R,)' s i n 2 F (9 - 1 04a) (I(:')l = I ~ J C O S ~ / I : ' + (RL/R0)2 s i n 2 p , - -. (9-104b) where R, = ~ L / C . Plots of IV(zt)l and )l(zr)I as functions of ; ' are standing waves with their maxima Bnd minima occurring at fixed locations along rhc ilnc. Analogously to t h ~ pl;il~c-wi~ve caw in Eq. (8-100). we dcfinc the r:ltio of thc rn:lrlmum to the minin~u~n voltages along a linite, ternlinated lille iis the rialding-mce ratio, S: (D~mcnsronlcss). d 1 - 11-1 (9- 105) The inverse relatiod of eq. (9-105) is (9-106) I It is clear from Eqs. (9-105) and (9-106) that on a lossless tr;lnsmission line r = 0, S = 1 when ZL = Z , (Matched load); I-= -1, S-60 when ZL = 0 (Short circuit); r = + 1, S -+ when ZL - m (Open circuit). Because of the wide range of S, it is customary to express it on a logarithmic scale: 20 log,, h!!dB). Sfbnding-wave ratio 5 defined in terms of lIma,l/~~,,,lnl results in the same expression as that defined in terms o f 1 V,,,//(V,~~,( in Eq. (9-105). A high standing- save ratio on iI line is i ~ ~ i d c s i ~ ~ h l c lxait~sc it results in a 1;lrgc power loss. Examination of Eqs. (9-lO2a) and (9-lO2b) reveals that /VmaX/ and II..,,/ occur ðer when (1 kz'( = 1, independent O~Z.): 402 THEORY AND APPLICATIONS OF T RANSMlSSlON LINES 1 9 On the other hand, Iv;,,~ and (I,,,I occur together when' 8,-2pzk= -(2n+ 1)n, ( n = O , 1,2,.:.). (9-108) ; .-. For resistive terminations on a lossless line, 2, = RL, 2, = R,, and Eq. (9-101) sim~lifies to RL - Ro (Resistive load). r = RL + Ro The voltage reflection coefficient is therefore purely real. Two cases are possible. 1. RL > R,. In this case, r is positive rcal and 0 , = 0. At the termination. 2' = 0, and condition (9-107) is satisfied (for II = 0). This mc:lns thi11 ti vo1t;igo ~n;lxi~num (current minimum) will occur at the terminating resistance. Other rnaxil~xi of tllc voltagc stunding wavc (minitnu of the currcnt st;~nding w;ivc) will hc locatcd iil 2/3z1 = 2rm, or z' = ni42 (11 = 1,2, . . .) from the load. 2. RL < R,. Equation (9-109) shows that r will be negative red and 0,. = -n. At the termination, : ' = 0, und co~~ditio~l (0 10s) is sntislicd (for 11 = 0 ) . A voltogc minimum (current maximum) will occur at the terminakng resistance. Other minima of the voltage standing wave (maxima of the current'standing wave) will be located at z' = rd/2 (11 = 1,2, . . .) from the load. The roles of the voltage and current standing waves are interchanged from those for the case of RL > R,. Figure 9-9 illustrates some typical standing waves for a lossless line with resistive terminaticn. . The standing waves on an open-circuited line are similar to those on a resistance- terminated line with RL > R,, except that the IV(zl)/ and Il(zl)l curves are now mag- nitudes of sinusoidal functions of the distance zr from the load. This is seen from Eqs. (9-104a) and (9-104b), by letting RL + co. Of course, I, = 0, but VL is finite. Wc have I v(zt)( = VL lcos Vz'( (9 - 1 lua) All the minima go to zero. For an open-circuited line, r = 1 and S -+ co. Fig. 9-9 Voltage and current standing waves on resistance-terminated lossless lines. ossible. ion, z' = 0, : maximum. xima of the : located at r = -rr.At i. X voltage ,lace. Other wave) will i o l t a q n d i I t , . d rcsistance- re now mag- is seen from t V, is finite. I V(z')I for open-c~rcu~ted line. - IMz1)l for short-circuited h e , L---- ll(z')l for open-c~rcuitcd line. ' 1 V(-zl)/ for short-circuiml h e . Fig. 9-10 Voltage h d current standing waver on open- and shortsircu~ted lossless lines. ,i . On the other and, the standing waves on a short-circuited line are simdar to i; those on a resistance-terminated line with RL < R,. Here RL = 0. VL = 0, but 1 , is finite. Equations (9-104a) and (9-104b) reddce to I V(zf)l = ILRo /sin Pz'l (9-1 1 la) Il(z')/ = I , /cm bz'l. (9-lllb) Typical standing Wzves for open- and short-circuited lossless lines are shown in Fig. 9- 10. Example 9-6 The staxling-wave ratio S on a transmiss~on line is an easily measur- able quantity. (a) Show how the value of a terminating resistance on a lossless line of known charactetistic impedance Ro can b determined by measuring S. (b) What is the impedance of the line looking toward the load at a distance equal to one quarter of the operating wavelength? Solutio)~ : a) Since the terminating impedance is purely resistive, ZL = R,, we can determine whether RL is greater than R, (if there are voltage maxima at z' = 0, i/2, i , etc.) or whether RL is less than R, (if there are voltage minima at ; ' = 0, i.12, 1, etc.). This can be easily ascertained by measurements. First, if RL > R,, 0, = 0. Both /VmaXI and (1.~~1 occur at pz' = 0; and 1~ ,,,,,I and ( 1 , , .
1 occuk at /Irt = n/2. We have, from Eqs. (9-102a) and (9-1OZb). 404 THEORY AND APPLICATIONS OF TRANSMISSION Second, if RL < R , , 8 , = -A. Both (Vmi,I and / 1 , , ; I occur at pz' = 0; and /vmaxl and IImi,I occur at pz' = n/2. We have b) The operating wavelength, A, can be determined from twice the distance between two neighboring voltage (or current) maxima or minima. At 2' = 214, 1 1 : ' = nI2. cos /kt = 0, and sin / I : ' = 1. Equations (9-103a) and (9-1UbJ become V(jL/4) = jILRo (Question: What is the significance of thc j in thcsc cquations?) The ratio of V(1.14) to 1(1/4) is the input impedance of a quarter-wavelength, resistively ter- minated, lossless line. This result is anticipated because of the impedance-transformation property of a quarter-wave line given in Eq. (9-94). 9-4.3 Lines with Arbitrary Termination . In the preceding subsection we not:d4hat the standing wave on a resistively terminated lossless transmission line is such that a voltage maximum (a current minimum) occurs at the terminatiorwhere z' = 0 if RL > R,, and a voltage minimum (a current maxi- mum) occurs there if RL < R,. What will happen if the terminating impedance is not a pure resistance? It is intuitively correct to expect that a voltage maximum or minimum will not occur at the termination, and that both will be shifted away from the termination. In this subsection we will.show that information on the direction and amount of this shift can be used to determine the terminating impedance. le ratio of brivel~f ler- operty of a rzmtirrared ti[,) O F 'rent 1- .iIlCC is 1 m t x i or away from : direction mce. Let the termindting (or load) impedand be 2, = RL + jXL, and assume the voltage standing wdve on the line to look like that depicted in Fig. 9-11. We note that neither a voltage maximum noca voltage minimum appears at the load at z' = 0. If we let the standing wave continue, say, by an extra distance f,,,, it will reach a minimum. Thc volthge minimum is where it should he if the original termmating impedance ZL is replaced by a line section of lehgth l,,, terminated by a pure resistance . R , < R,, as shown in the figure. The voltage distribution on the line to the left of the actual termination (where z' > 0) is not changed by this replacement. The fact that any complex impedance can be obtained as the input impedance of a section of loul&s line terminated in a resfstive load can be seen from Eq. (9-89). Using R, for 2 , add dm for d, we have ' I The rcll u d im:&ar) pails of Eq (9-114) form two equations, Sroin w h ~ h t l x two unknowns, R,, hnd I,,, can bc solved (sce Problem P.9-34). Thc load impcd:ln~c % , can bc determined expcrimcntally by musunng the stand~ng-wave ratio S and the distance 2: in Fig. 9-11. (Remember that 1 : . + i, = ; -, Q .
) The procedure is as follows: S - l 1. Find l r l from S. Usu I l -1 = -- from Eq. (9-106). S + l 2. Find 0, from 6 Use 0, = 2P::, - n forjl = 0 from Eq. (9-10s). t I ' Fig. 9-11 Voltage standing wave on line ------I t;n z'Z o I a terminated by arbitrary im~edance and -. 1.- equivalent line section with pure resistive -7 load. A Find ZL, which is the ratio of Eqs. (9-102a) and (9 - 102b) at I ' = 0: 1 + ITlejer ZL = RL + FL = Ro I - Irl ,or - (9-115) The value ;f Rm that, if terminated on a line of length em, will yield an input impedance ZL can be found easily from Eq. (9-1 14). Since R, < Ro. R,,! = R&. The procedure leading to Eq. (9-1 15) is used to determine Z, from a measurement of S and of 2 ; . the distance from the termination to the first voltage minimum. Of course, the distance from the termination to a voltage maximum could be used instead of 2 : . However, the voltage minima of a standing wave are sharper than the voltage maxima. The former, therefore, can be located more accurately than the latter, and it is preferable to find unknown quantities in terms of S and &. Example 9-7 The standing-wale ratio on a lossless 50-(Q) trvnsmission line ter- minated in an unknown load impedance is found to be 3.0. Thc dintan& betwccn successive voltage minima is 20 (cm), and the first minimum is loc;~tcd i l l 5 ( ~ 1 1 1 ) fro111 the load. D ~ ~ ~ I I I I I I ~ (:\I ~ I I C ~cllcctio~\ cocIlicic111 l', : I I I ~ (b) [IIC l ~ d UII~CLIJIICC Z , . In addition, find (c) the equivalent.length and termina&@-resistance of a line such that the input impedance is equal to 2,. Solution a) The distance between successive voltage minima is half a wavelength. 2x 2x - A = 2 x 0.2 = 0.4 (m), /? = - = - = 5 x (rad/m) . i, 0.4 Step 1 : We find the magnitude of the reflection coefficient, lrl, from the standing- wave ratio S = 3. Step 2: Find the angle of the reflection coefficient, B,, from 0, = 2& - n = 2 x 5n x 0.05 - n = - 0 . 5 ~ (rad) r = (rider = 0.5e-j0.5n = -j0.5 b) The load impedance ZL is determined from Eq. (9-115): / c) Now we find R , and C , , , in Fig. 9-1 1. We may use Eq. (9-1 14) 30 - j4O = 50 Rm + j50 tan p t , 50 + j R , tan p t , (9 IS] 5 ) an input RoIS. lsurement irnum. o f 1 be used r than the the lattei-, n line ter- c bctwccn < i t 5 (crnr mpedancc : of 4 4 e : standing- /7 1 9-4 1 WAVE CHARACTERISIICS Ohl FINITE TRANSMISSION LINES 40i I and solve the airndtaneous equations obtained from the real and imaginary pans for R; and Pt,,,, Actually, we know i, + t , , , = 112 and Rm = R,/S. Hence,' 1 em=-- 2 zk = 0.2 - 0.05 = 0.15 (m) and 50 Rm=- = 16.7 (R). 3 9-4.4 Transmission-Line Circuits Our discussions on the properties of trmsrnission lines so far have been restricted primarily to the effects of the load on the input impedance and on the characteristics of voltage and current waves. No attention has been paid to the generator at the "other cnd," which is thc sourcc of Lhc wavcs. Just as the constraint (the boundary condition), V, = I&,, which the voltage V , and the current I , must satisfy at the load end (z = l, z' i 0), a constraint exists at the generator end where 2 = 0 and 2' = i . Let a voltage generator V , with an internal impedance 2, represent the source connected to a finite transmission line of length L that is terminated in a load im- pcdance Z,, as shown ill Fig. 9-5. Thc additional constraint at : = 0 will cnable the voltage and current anywhere on the line to be expressed in terms of the source characteristics (V,, Z,), the line characteristics (1, Z,, 4, and the load impedance iZ,). The constraint Bt z = 0 is = V, -IiZ,. But, from Eqs. (9-i00aj and (9-IOOb), IL = y (ZL + Z,)eiLII + re-'?' 1 (9-117a) and Substitution of Eqg. (9- 117a) and (9-1 17b) in Eq. (9-116) enables us to find ' Another set of solutiofls to pirt (c) is fm = t',,, - 1 1 4 = 0 .
0 5 (m) and Rm = SR, = 150 (a). Do you see why? 408 THEORY AND APPLICATIONS OF TRANSMISSION +' is the voltage reflection coefficient of the generator end. Using Eq. (9-118) in Eqs. (9-100a) and (9-100b), we obtain Similarly, f Equations (9-120a) and (9-120b) are analytical phasor expressions for the volt- age and current at any point on a finite line fed by a sinusoidal voltage s'ource V , . These are rather complicated expressions, but their significance can be interpreted in the following way. Let us concentrate our attention on the voltage equation (9-120a); obviously the interpretation of the current equa&&-(9-120b) is quite similar. We expand Eq. (9-120a) as follows: where V; = r(VMe-yd)e-yz' (9-121b) V z = rg(TVMe-2Yd)e-Yz. (9-121c) - The quantity v - v , z o -z0 + Zg , (9-122) is the complex amplitude of the voltage wave initially sent down the transmission line from the generator. It is obtained directly from the simple circuit shown in Fig. 9-12(a). The phasor V : in Eq. (9-121a) represents the initial wave traveling in the Lrr thc volt- 2 source Vg. interpreted &e equation 3b) is quite - sansr~~~ssion iown in Fig. Fig. 9-12 A transffiission4ne circuit and travelitlg waves. + z direction. Before this wave reaches the load impedance Z,. it sees Z, of the line as if the line were hnitely long. When the first wave VT = Vb,e-'' reaches 2, at 2 = d', it is reflected because of mismatch, resulting in a wave V ; with a complex amplitude T(V,,,e-:IC) travelifig in the - 2 direction. As the wave V ; returns to the generator at z = 0, it is again re- flected for 2, # Z , , giving rise to a second wave V: with a complex amplitude T,,ITV,,:,,C-~"') traVclin~ i n 4: direction. This proccss continues indefinitely with refections at both ends, and the resulting standing wave I / ( : ' ) is the sum of all the waves traveling in both dircctions. This is illustrated schematically in Fig. 9-lZ(b). In practice, 7 = ct + jp has a real part, and the attenuation effect of r - " l diminishes the amplitude of rl reflected wave each time the wave transverses the length af the line. When the lirle is terminated with a nlatched load, 2 , = Z,, r = 0, only V ; exists, and it stops at the matched load with no reflections. If 2, # 2, but Z, = 2, (if the internal idpedance of the generator is matched to the line), then r f 0 and r, = 0. As a consequence, both V : and V ; exist, and V ; , V ; and all higher-order reflections vanish, Esnmple 9-8 A 100-(Mllr,) gcnerutor with V, = 1 0 0 - (V) and ~nternal resistance 50 (R) is connected to J lassless 50 (R) air line that is 3.6 ( r n ) long and term~narsd in a 25 + j25 (R) load. Find (a) V(z) at a locatiod z Irom the generator, (b) I.; at the input terminals and V , idt tht load, tc) the vo!tage standing-wave ratio on the line, and (d) -. ' the average power delivered to the load. -1 - Ro = 50 (Q), ZL = 25 + j25 = 35.36/450 (R). t' = 3.6 (m). 410 THEORY AND APPLICATIONS OF TRANSMISSION LINES 1 9 Thus, w 2n1O8 2n p=-=-- - - (rad/m), p/ = 2.4n (rad) c 3 x lo8 3 - - r = ZL - Zo - - (25 + j25) - 50 - -25 + j25 - 35.36/135" - Z, + Z, (25 +'j25) + 50 - 100 + j25 103.1/14" = 0.343/121" = 0.34310.672~ 1 -, = 0. . a) From Eq. (9-120a), we have V(z) = - z, + z , ; - -- e- j2n:/3[l + 0.343~j(0.672-&8)n j4nzI3 100 e I We see that, because r, = 0, V(z) is the superposition of only two traveling waves, V: and V;, as defined in Eq. (9-121).: b) At the input terminals, Vi = V(0) = 5(1 + 0.343e-J0.'28n ) = 5(1.316 - jO.134) = 6.611- 5.82" (V) . At the load, V L = V(3.6) = 5[e-j0.4n + 0.343ej0.272] = 5(0.534 - j0.692) = 3.461- 52.3" (V). c) The voltage standing-wave ratio is d) The average power delivered to the load is It is interesting to compare this result with the case of a matchcd load when ZL = ZO = 50 + j O (a). In that case, l- = 0, V IV,l = Iq ='--I = 5 (V), 2 9-5 / THE SMITH CHART 41 1 / and a maximum avirage:bower is delivered td the load: I v : 5 t Mlximum P,, = - = -- = 0.25 (W), 'I 2R, 2 x 50 which is consideradly larger than the Pa. calculated for the unmatched load in part (d). ThE SMITH CHART , b Transmission-line ~alculations-such as the determination of input impedance by Eq. (9-89), reflection coefficient by Eq. (9-101), and load impedance by Eq. (9-1 15)-often invdhe tedious miinipulations of complex numbcrr This lcdl~rm c.in be alleviated by using a grilphicd mec!i~d of solut~on. The beit known m d most widely used graphicel chart is the Siriirh cbnrt devlsed by P. H. Smlrh.' Stated suc- cinctly, a Sn~itll chilrt I: a graphical plot of norn~alized resistance and reactance functions in the reflection-coefficient plane. In order to understlnd how the Smith chart for a l o ~ ~ l c s ~ transmlision h e 15 constructed, let us examine the voltage reflection coefficient of the load impedance defined in Eq. (9-101): Let the load impedance ZL be normalized with respect to the characteristic imped- ance R, = of the line. where r and x are the normalized resistance and normalized reactance respectively. Equation (9-101) cdn be rewritten as where r, and r, are the real and imaginary parts of the voltage reflection coefficient r respectively. The inverse relation of Eq. (9-124) is - P. H. Smith. "Transmission-linc calculator," Electronlo. vol. 12, p 29. January 1939: m d "An improved transmission-line calculator," Electronics, vol. 17, p. 130, January 1944. 412 THEORY .AND APPLICATIONS OF TRANSMISSION LINES I 9 Multiplying both the numerator and the denominator of Eq. (9-126) by the complex conjugate of the denominator, and separating the real and imaginary parts, we obtain If Eq. (9-127a) is plotted in the?, - Ti plane for a given value of r, the resulting graph is the locus for this r. The locus can be recognized when the equation is re- arranged as . Fig. 9-13 Smith chart with rectangular coordinates. , . 8 .
: . , i ; ; , , .., ! . : complex . . ' .
/ . , , .
; I : weobtain . 1 ; , : ! : / . (Y-I&) .
, , , , . . : I (9-l27b) . ' 5 : redulting , ;, 1 .
, . tion is re- , . : 1 . . I .. , fi 9-5 1 THE SMITH CHART 413 1 Several sa~iknt properties of the r-dircleslare noted as follows: 1. The centers oh11 rhrcles lie on the'ir-$xis. 2. Thc r = ~,.circ~c, hwiog ;I unity radi'us and ccntcrcd at thc origin, is thc largcst. - 4 ' I 3. The r-cmles becoAe progressively 'smiller as r increases from 0 toward m, ending at the (r, = 1, Ti = 0) point. 4. All r-circles pdss through the (I-, = 1, Ti = 0) point. Similarly, Eq. (9- l2lb) may be rearranged as This is the equation for a circle having radius l/[x( and centered at T, = 1 and T, = l/x. Different values ofx yield circles of different radii with centers at different posi- tions on thc r, = 1 line. A family of the portions of x-circles lying inside thc /I-/ = 1 boundary are shown in dashcd lines in Fig. 9-13. The following is a list of several salient properties of the u-circles. 1. The centers of ali s-circles lie on the Tr = 1 line; those for x > 0 (inductive reactance) lie above the r,-axis, and those for x < 0 (capacitive reactance) lie below the r,-hxis. 2. The x = 0 circle bzcomes the rr-axis. 3. The x-circles become progressively smaller as 1 x 1 increases from 0 toward a, ending at the [r, = 1, Ti = 0) point. 4. All x-circles pass through the (T, = 1, rl = 0) point. A Smith chart is a chart of r- and x-circles in the T, - I-i plane for I r l 5 1. It can be proved thi; the r- and s-circles are everywhere orthogonal to one another. The intersection of an r-circle and an x-circle defines a point that represents a normalized load impedance z, = 1. + ,js. Thc actual load impedance is Z , = R,(r + js). Since a Smith chart plots the normalized impedance, it can be used for cn1cul;~tions concern- ing : I Inssicss i~x~lklllissitl~l linc wit11 ; I I ~ ;~~.l>iilx~'y ~ I ; I I . : I C ~ C S ~ X ~ ~ L ' ~ ! ~ ~ ~ C L I : I I ~ C C . As an illustration, point P in Fig. 9 -13 is the intersection of the r = 1.7 circle and the x = 0.6 circle. Herice it represents 2,. = 1.7 + j0.6. The point P,, at (T, = - 1, Ti = 0) corresponds toi r = 0 and x = 0 nhd, therefore, represents a short-circuit. The point Po, at (T, = 1, Ti A 0) corresponds to an infinite impedance and repre- sents an open-circuit. The'Smith chart ic Fig. 9-13 is marked with T, and Ti rectangular coordinates. The same chart can be marked withpolar coordinates, such that every point in the r-plane is specified by a magnitude If 1 and a phase angle 8,. This is illustrated in Fig. 9-14, where several ]TI-circles are shown in dotted lines and some Or-angles are marked aroufid the I r l = 1 circler The Irl-circles are normally not shown on commercially available Smith charts; but once the point representing a certain . ,. ' I , I : I: ' -41 4 THEORY AND APPLICATIONS OF TRANSMISSION 9n0 270' Fig. 9-14 Smith chart with polar coordinates. z , = r + jx is located, it is a simple matter to draw a circle centered at the origin , through the point. The fractional distance from the center to the point (compared i I I with the unity radius to the edge of the chart) is equal to the magnitude I r l of the load reflection coefficient; and the angle that the line to the point makes with the real axis is 0,. This graphical determination circumvents the need for computing by Eq. (9-124). Each Irl-circle intersects thc real axis at two points. In Fig. 9--14 we designate the point on the positive-rcal axis (OfJ,,,) as I', and thc point on thc: ncgativc-rcal axis (OP,,) as P,. Since x = 0 along the real axis, l', and P, both represent situations with a purely resistive load, Z , = R,.. Obviously RL > R, at P,,, where r > 1 : and I RL < I<, at l',,, where r < 1. ln Eq. (9-1 19) we found that S = K,/R, = r Tor KL > R,. This relation enables us to say immediately, without using Eq. (9-105), that the value of the r-circle passing through the point P, is numerically eqlial to the stand- ing-wave ratio. Similarly, we conclude from Eq. (9-113) that the value of the r-circle I i A passing through the point P , on the negative-real axisis numerically equal to 11s. For thc : , = 1.7 + jO.6 point, markcd P in Fig. 9 14, wc find I r l = atid 0,. = 28 . At I , P,, r = S = 2.0. These results can be verified analytically. ' I In summary, we note the following: 1. All Il-1-circles are centered at the origin, and their radii vary uniformly from 0 to 1. I I .i i.,. . . 9-5 1 THE SMITH CHART 1415 /- he origin :.~mpdred ,I'l of the , ~11th the iputing r designate a u k e-real i~tuatlOnS > 1 : and R , > R,. . that the he uartd- 'le r-circle I 1,s. For = 1 8 ' 7 3mOto 1. , 2. The angle, meisured from the positive r h l axis, of the line drawn from the origin through the pdint representing z, equah 0,. 3. The value of the r-circle passing thrdugh the intersection of the irl-circle and tHe positive-real akis eqhals the standing-whve ratio S. I So far we have based the construction of the Smith chart on the definition of the voltage reflection c o e a e n t of the load irnpdtiance, as given in Eq. (9-101). The input impedance looking toward the load at a distance z' from the load is the ratio of V(t') and I(z'9. From Eqs. (9-100a) and (9-100b) we have, by writing jp for y for a lossless - ! , I I line. , The normalized inbut impedance is We note that Eq. (9-130) relating z, and = /rleJ9 is of exactly the same form as Eq. (9-125) rcliitinf ; , and = IT[eJnr. In fact, the laltcr 1s a s p c c ~ l case ol. the former for z' = 0(4 = Or). The magnitude, ~rl, of the reflection coefficient and, there- fore, the standing-have ratio S, are not changed by the additional line length ?. Thus, just as we can use the Smith chart to Rnd I r I and Or for a given ;,at the load, we can keep I r l constant and subtract (rotatz in the clockwise direction) from Or an angle equal to 2pz' = 4Rz1/ik This will locate the point for lTIeJ\ which determines z,, the normalized input iinpedance looking into a lossless line of characteristic impedance R,, length z', and a nofmalized load impedance z,. Two additional scales in Az'li. are usually provided along the perimeterof the I r l = 1 circle for easy reading of the phase change 2p(Ar1) due to a change in line length A:': the outer scale 1 s marked "wavelengths tow&-d generator" in the clockwise direction (increasing 2'); and the inner scale is marked "wavelengths toward l o a d in the counterclockwlse directlon (decreasing z'). Eigure 9-15 is a typical Smith chart, which is commercially available.' It has a complicatid appearance, but actually it consists merely of constant-r and constant-x circles. We note that a change ofhalf-a-wavelength In line length (A:' = i/2) corresponds?~ a ifl(Azl) = 2n change in 4. A complete revolution around a /TI-circle rettlrns to thesame point and results in no change in impedance, as was asserted in Eq. (9-95). ' All of the Srni!h charts used in th~s book are ~eprinted wlth perrnlsslon of Ernelold Industries, Inc., New Jersey. 416 THEORY AND APPLICATIONS OF TRANSMISSION LINES 1 9 ! . I Fig. 9-15 The Smith chart. i i I In the following we shall illustrate the use of the Smith chart for solving some j typical transmission-line problems by several examples. 1 i r d + $ Use the Smith chart to find the input impedance of a section of a I 50-(Q) lossless transmission line which is 0.1. wavelength long and is terminated in F b a short-circuit. i 1. Enter the Smith chart at the intersection of r = 0 and x = 0 (Point P, on the extreme left of thart. See Fig. 9-16.) 2. Move along the pehmeter of the chart ( 1 ' 1 = 1) by 0.1 "wavelengths toward generator" in a clockwise direction to P,, 3. At P, read r = 0 and x 2 0.725, or z, = jO.725. Thus, Zi = Rozi = 50(~0.725) = j36.3 (0). (The input impedance is purely inductive.) This result can be checked readily by using Eq. (9-92). Z, = jR,, tan /I/ = j5O tan = j5O tan 36 = j36.4 (Q). A lossless transmlssion line of length 0.4341 and characteristic im- pedance 100 ( 1 1 ) is terminated in an impedance 260 + j180 (Q). Find (a) the vohape reflection coefficient, (b) the standing-wave ratio, (c) rhe input impedance, and (d) the location of a voltage maximum on the line. . .
I .
. Solution: Given . 'e find tlie voitage reflection coefficient in several steps: Enter the Smith chart at r , = Z L I R o = 2.6 + jl.8 (Point P , in Fig 9-16.) With the cedtw at the origin, draw a circle of radius W, = T J = 0.60. (The radids of the chart OFsc equals unity.) Draw ,~e.straight line OP, and extend it to P2 on the periphery. Read 0.220 on "wavelen&hs towardgenerator" scale. The phase angle 6, of the reflectmn coeffi~ient is (0.250 - 0.220) x 4n = 0 . 1 2 ~ (rad) or 21'. (We multlply the change in wavelengths by 4n because angles on the Smlth chart are measured in 2p?c: in.-'/I,. A hiill-w:lveleogth change in line lengtll corresponds :o a complck ~aro~uiion on the Suit11 chart.) The answer to part (a) is then r = jrleJ8>= 0.60/21". h) The (rl = 0.60 circle intersects wit6 the positive-real axis OP.. at r = S = 4. Thus the voltage stwndi~~~-wavc ratio is 4. , ,418 THEORY AND APPLICATIONS OF TRANSMISSION LINES / 9 . Fig. 9-16 Smith-chart calculations for Examples 9-9 and 9-10. c) To find the input impedance, we proceed as follows: 1. Move P; at 0.220 by a total of 0.434 "wavelengths toward generator," first to 0.500 (same as 0.000) and then further to 0.1 %[(OSOO - 0.220) + 0.1 54 = 0.4341 to P;. I .- 2. Join 0 and P; by a straight line which intcrsects the I r l = 0.60 circle at P,. rcle at P,. ., L , ; r 1 '1 i 9-5 / THE SMITH CHART 419 f 1 , 3. Read r = 0.64 and x = 1.2 at P,. Hencd, . .. ; z ; = ROz, = lOO(q69 + j1.2) = 69 + j120 (R). d) In going from P i to P,, the lrl = 0.60 circle intersects the positive-real axis OPoc at P, where thd voltage is a maximum. Thus, a voltage maximum appears at (0.250 - 0.220)A or'0.030A from the load. Example 9-11 ~ o h e E~ample 9-7 by using the Smith chart. Given 4 R o .? 50 (Q) w S=3.0 1 , . , , 2 = 2 x 0.2 = 0.4 (m) I First voltage minimum at rk = 0.05 (m), find (a) a), (b) Z,, (c)'&, 2nd R,,, (Fig. 9-1 1). Solution a) On the positive-rcill dris OP,. locale ~ h c po~nt I>,,, .lr wlilcb = s = 3.0 (see F , ~ . 9-17). Then DM = /r/ = 0.5 (mot = 1.0). We cannot find 0, until we have located the point that represents the normalized load imped;ince. b) We use the following procedure to find the load ihpedance on the Smith &art: 1. Draw a circle centered at the origin with radius m,\,, which intersects with the negative-r a1 anis OP,. at P,,, where there will be a voltage minimum. 2 Since &/i = J05/0.4 = 0.125, move from P,. 0.125 '6wav&ngths toward load7' - in the counterclosewise direction to P;. 3. Join 0 and P ; by a straight line, intersecting the (TI = O.j circle at P,. This 1s the point rebrescnling the normalized load impedance. 4. Read the angle i POCOP; = 90' = n/2 (rad). There is no need to use a pro- tractor, because L POCOP; = 4n(0.250 - 0.125) = n/2. Hence 6, = - @ (nd), or r = 0.51-40" = -jO.j. 5- Read at PL, 2, = 0.60 - j0.80, which gives . All the above results are the same as those obtained in Example 9-7, but no calculations with Complex numben are needed in using the Smith chart. 420 THEORY AND APPLICATIONS OF TRANSMISSION LINES 1 9 Fig. 9-17 Smith-chart calculations for Example 9-11. 9-5.1 Smith-Chart Calculations for Lossy Lines In discussing the use of the Smith chart for transmission-line calculations, we hav.e assumed the line to be lossless. This is normally a satisfactory approximation for we generally deal with relatively short sectioxis of low-loss lines. The lossless assumption enablcs us to say, following Eq. (9-130), that the magnitude of the Te-12"' term S' ? 1 I does not change with line length z' and that we can find z, from z,, and vice versa, 1 1 I 8 by moving along the lrikircle by an angle eqhal to 2p.2'. I For a lossy line of a sufficient length 8, sdch that 2 d is not negligible compared i to unity, Eq. (9-136) must be amended to read . r 1 + re - 2 a ~ ' ~ - j2pz' 2, = 1 - re- 2az' -12pr' e ' l + ( r ( e - 2 a - ' e ~ 9 - - 1 - 6 = or - 2 w . (9-132) ,r Hence, to find I, @om z , , we cannot aimply move along the (TI-circle; auxiliary calciilations are necessary in order to account for the e-'"' factor. The following example illustrates What has ta be done. . we nave on for we sumption 12i7:' 4 term Example 9-12 Tlle input impedance of n short-circuited loshy transmission line of length 2 (m) and characteristic impedance 75 $2) (approximately real) is 45 + j225 (0). (a) Find a and / 3 of the line. (b) Determine the input impedance if the short-circuit is replaced by a lo:ld impc~lilncc Z , = 67.5 - j45 (Q). Solution a) The short-circuit load is represented by the point P, on the extreme left of the Smith impedance chart. 1. Enter zil = (45 + j225)/75 = 0.60 + j3.O in the'chart as P, (Fig. 9-18), 2. Draw a straight line from the origin OthrouSh P, to P;. -- 3. Measure OP,/OP', = 0.89 = c-". It follows that 4. Record that the arc P,P; is 0.20 "wavelengths toward generator." We have //A = 0.20 and 286 = 4n//A = 0.8~. Thus, b) To find the input impedance for Z , = 67.5 - j45 (R): - I. hnkr ZL = %,/2, = (67.5 - /45)/75 = 0.9 - jO.6 on thc Smith chart as P,. 2. Draw a straight line from 0 through P, to P2 where the "wavelengths toward generator" reading is 0.364. - 3. Drawa/r(-circle centered at 0 with radius m2. 4 Move Pi alohg the perimeter by 0.20 "wavelengths toward generator.' to Pi at 0.364 + 0.20 = 0.564 or 0.064. , 5. Join P3 and d by a straight line, intersecting the /r/-circle at P,. 6. Mark on line OP, a point Pi such that 0P1/CF3 = e-2a' = 0.89. 7. At Pi, read zi 0.64 + j0.27. Hence, 2 , = 75(0.64 + j0.271 = 48.0 + j20.3 (R). \ 422 THEORY AND APPLlcATloNs OF TRANsMlssloN LINES 1 9 e Fig. 9-18 Smith-chart calculations for lossy transmission line (Example 9-12). 9-6 TRANSMISSION-LINE IMPEDANCE MATCHING Transmission lines are used for the transmission of power and information. For radio-frequency power transmission it is highly desirable that as much power as possible is transmitted from the generator to the load and as little power as possible is lost on the line itself. This will require that the loztd be matched to the characteristic i I 9-6.1 Quarter /. > I: 9-6.1 Impedance ~ a k h l h ~ by Quarter-Wave Transformer ' Kb = ,/=. (9-133) Since the length of the quarter-wave line depends on wavelengrh, this matching is frequency-~eniitive. as are a]! the other methods to be discussed. Example 9-13 A &pal generator is to feid equal power through a lossless air transmission line with a characteristic impedance 50 (Q) to two separate resistive loads, 64 and 23 (n). Quarter-wave transformers are used to match the ]oa& to the 50 c n ) line, as shouo in Fig. 9 -1% (a) Determine the required charactenstic impedances of the quarler-wave lines. (b) Find the standing-wave ratios on the matching line scctiohs. , . fig. 9-19 impedance matching by quarter-wave lines (Ex:irnplr 9- 13). i 424 THEORY AND APPLICATIONS OF TRANSMISSION LINES 1 9 Solution a) To feed equal power to the two loads, the input rehstance at the junction with the main line' looking toward each load must be equal to 2Ro. Ril = Ri2 = 2Ro = 100 (Q). ~b~ = J ~ = J i W Z i = 8 o ( n ) G2 = Jm = 4TGE-Z = 50 (Q). b) Under matched conditions, there are no standing waves on the main transmission line (S = 1). The standing-wave ratios on the two matching line sections are Matching section No. 1 . Matching section No. 2 Ordinarily the main transmission line and the matching line sections are essen- tially lossless. In that case both Ro and Ro are purely real and Eq. (9-133) will have no solution if RL is replaced by a complex ZL. Hence quarter-wave transformers are not useful for matching a complex load impedance to a low-loss line. In the following subsection we will discuss a method for matching an arbitrary load impedance to a line by using a single open- or short-circuited line section (a single stub) in parallel with the main line and at an appropriate distance from the load. Since it is more convenient to use admittances instead of impedances for parallel connections, we first examine how the Smith chart can be used to make admittance . calculations. Let Y L = l/ZL denote the load admittance. The normalized load impedance is where YL = YLP0 = YLIGo . = ROYL = g + jb (Dimensionless), (9-135) ion with = Rtz = smission .S are re essen- will have mers are , ubitrary sction (a from the r'parallel mittance is the normalized load admittance having normalized conductance g and normalized susceptance h as its teal and imagin;~ry pdrts respectively Equ;ltion (9-134) suggcsts 111:11 ; I qtmrlcr-w;tvt li:w will, ;I tt~l~ly ilornlilli~d char:~~tert~tic impedance will lj'mh-rn 2, lo y,, iltiU vice versa. On the Smith chart we need only to move the . point representing T L along the IT(-circle by a quarter-wavelength in order to locate thc point rcpres~ntih~ , . Since a 214-change in line length (AzlN = i) corresponds to a change of n radians (2PAzf = x) on the Smith chart. the poir~ts rrpreset~titlg Z, and y, are then diametrically opposite to eachuther on the Jr/-circle. This observation enables US to find yi. from r,, and z, from y,, on the Smith chart in a very simple manner. Solution: This problem has nothing to do with any transmission line. In order to use the Smith chart, we can choose an arbitrary normalizing constant; for instance, RO = 50 (R). Thus, Enter z, as point P, on the smith chart in Fig. 9-20. The point P, on the other side of the line joining PI and 0 represents y,: 07, = m1. -1 - Example 9-15 ~ i n d the input admittance of an open-circuited line of characteristic impedance 300 (a) and leapth 0.042. 426 THEORY AND APPLICATIONS OF TRANSMISSION LINES 1 9 a - Fig. 9-21 Finding input admittance of open-circuited line (Example 9-15). Solution 1. For an open-circuited line, we start from the point Po, 6nYthcextreme right of the impedance Smith chart, at 0.25 in Fig. 9-21. 2. Move along the perimeter of the chart by 0.04 "wavelengths toward generator" to P3 (at 0.29). 3. Draw a straight line from P3 through 0, intersecting at P; on the opposite side. 4. Read at P; y, = 0 + jO.26. Thus, 1 x=- 300 (0 + j0.26) = j0.87 (mS). In the preceding two examples we have made admittance calculations by using the Smith chart as an impedance chart. The Smith chart can also be used as an admittance chart, in which case the r and x circles would be g and h circles. The points representing an open- rind short-circuit termination would be the points on the extreme left and t%e extreme right, respectively, on an admittance chart. For Example 9-15, we could then start from extreme left point on the chart, at 0.00 in Fig. 9-21, and move 0.04 "wavelengths toward generator" to P; directly. 9-6.2 Single-Stub Matching We now tackle the problem of matching a load impedance 2, to a lossless line that has a characteristic impedance R, by placing a single short-circuited stub in parallel with the line, as shown in Fig. 9-22. This is the single-stub method for impedance matching. We need to determine the length of the stub, L', and the distance from thc load, z', such that the impedance of the pariillel combination to the right of points B-B' equals R,. Short-circuited stubs are usually used in preference to open-circuited ttance 9-15). I rght of r ' eratc tte stae. y using i as an es. The 1111h fill l l 1'111 0.W in n ne th, 3ara" ' ledan, om the points rcuited $ 2 I stubs ~CC;IUSC i l n inlinitc tormin:~ling imadi~nu: IS more difficult to re;~lim (hiin il zero t o d i n a t h i irdpcdancc for reasons of radiation from an open end and coupling effects with neighbbring,objects. Moreover, a short-circuited stub of an adjustable length and a constant qharacteristic resistdhce is much easier to construct than an open-circuited one. Of course, the difference in the required length for an open- circuited stub and,that for a short-circuited stub is an odd multiple of a quarter- wavclcncth. - The parallel cdhbination of a line terminated in Z, and a stub at points B-B in Fig. 9-22 sugsddts that it is advantageous to analyze the matching requirements in terms of admitt~hces. The basic requirement is > - In terms of normalized Lidmittances. Eq. (9- 136) becomes 1 = + J,, (9-137) where y , = ROY, is for tne load sectloll and y, = ROY, is for the short-circurted stub. However, since thdi input admtttance of a short-circuited stub is purely susceptive. y, is purely imaginary. As a consequencg.Eq. (9-137) can be satisfied only if and YII = 1 + jb, . (9-138a) Y B = - J ~ B , (9-138b) where b, can be either positive or negative. Our objectives, then, are to find the length d such that the admittance, y,, of the load section looking to the rlght of ter- minals B-E' has a hnity r e d purt and to find the length fB of the stub required to cancel the imuginury purr. Fig: 9-22 Impedance matching Y by ringicrtvb method. 428 THEORY AND APPLICATIONS OF'TRANSMISSION~LINES / 9 Using the Smith chart as an admittance chart, we proceed as follows for single- , t stub matching: 1. Enter the point representing the normalized load admittance y,. 2. Draw the Irl-circle for y,, which will intersect the g = 1 circle at two points. At these points y,, = 1 + jbBl and y,, = 1 + jb,,. Both are possible solutions. 3. Determine load-section lengths dl and d2 from the angles between the point representing y, and the points representing y,, andly,,. 4. Determine stub lengths t,, and tB2 from the angles between the short-circuit point on the extreme right of the chart to the points representing -jb,, and - jh,], respcctively. The following example will illustrate the necessary steps. Example 9-16 A 50-(R) transmission line is connected to a load impedaice Z, = 35 - j47.5 (Q). Find the position and length of a short-circuitcd stub rcquircd to march rhc linc. -- Solution: Given R o = 50(Q) .
, 2, = 35 - j47.5 (R) z, = Z,/R, = 0.70 - j0.95. 1. Enter z, on the Smith chart as P , (Fig. 9-23). 2.. Draw a Irl-circle centered at 0 with radius m,. 3. Draw a straight line from P, through 0 to point P i on the perimeter, intersecting the Il-1-circle at P,, which represents y,. Note 0.109 at P; on the "wavelengths toward generator" scale. 4. Note the two points of intersection of the Irl-circle with the g = 1 circle. At P,: y,, = 1 + j1.2 = 1 + jb,,; AtP,: y,,= 1 -j1.2= 1 +jbBz. 5. Solutions for the position 'of the stub: For P, (from P; to P;): dl = (0.168 - 0.109)A = 0.0591.; For P4 (from P; to fi): d, = (0.332 - 0.109)A = 0.2231. , 6. Solutions for the length of short-circuited stub to provide y, = -jb,: For P , (from P,, on the extreme right of chart to P ; ' , which represents -jh,, = -j1.2): For P4 (from P , , to P i , which represents -jb,, = j1.2): r single- t . .
. < t i points. : - Jutions. e point sccting lengths Fig. 9-23 Construction for single-stub matching. In general,. the solution with the shorter lengths is preferred unless there are other practical constraints. The exact length, t'B, of the short-circuited stub mey require fine adjustments in the actual matching procedure; hence the shorted match- ing sections are sometimes called stub tuners. The use of Smith chart in solving impedance-matching problems avoids the manipulation of complex numbers and the computation of tangent and arc-tangent .?. _. I 430 THEORY AND APPLICATIONS OF TRANSMISSIQN LINES / 9 1 functions; but graphical c~nstructions are needed, and graphical methods have limited accuracy. Actually the analytical solutions of jmpedance-matching problems are relatively simple, and ebsy access to a computer'may diminish the reliance on the Smith chart and, at the same time, yield more accurate results. For the single-stub matching problem illustrated in Fig. 9-22, we have, from Eq. (9-89), where t = tanpd. (9 - 140) The normalized input admittance to the right of points B-B' is where and A perfect match requires the simultaneous satisfarion of Eqs. (9-138a) and (9-l3Sb). Equating g, in Eq. (9-142a) to unity, we lii~vc . (r: - I)t2 - 2sLt + (I., - 1 .
; - .yl) = 0. (9-133) Solving Eq. (9-143), we obtqin The required length d can be found from Eqs. (?-140), (9-144a), and (9-1 44b): - tan-'t, t 2 0 , -(n + tan-It), t < 0. Similarly, from Eqs. (9-138b) and (9-142b), we obtain ds have roblems ance on re, from (9-139) (9-140) (9-141) 1 - l32a) /-- 14%) I %b), 9-143) - 1 Ma) -144b) I 4b): Y - 145a) - 145b) / I - 1464,- -146b) 1 , ~, - For a given load [mpdi\nw, both , 1 1 2 itn?;f/~ can bc determined easily on a scientific calculato~. It$ also a simple matter ta write a general computer program for the single-stub mhtchidg problem. Mbre &curate answers to the problem in Example 9-16 (r, = 0170 and s , = -0.95) are Of course, such accuracies are seldom needkd in dn actual problem; but these answers have been obtained easily without a Smithkhart, rl '. Double-Stub ~atching The method of impedance matching by mean8 of a single stub described in the preceding subsection can be used to match any atbitrary, nonzero. finite load imped- ance to the characteristic resistance of a line However, the single-stub method requires that the stub be nrtnchcd to the main lidc at a specific point which varies as thc loild impedance is changed. This requirement &en presents practical difficulties because the specified junction point may occur at an undesirable location from a mechanical viewpoint, Furthcrmorc. it is very dificult to build : 1 variable-lengh C L X I X ~ ~ I ~ linc will1 ;I conslu~~t:ch;m~clcristic impcdilncc. In such c;~scs, ttn altcmatire mclhod for irnpcd;~nc~-m;ltcbing is to us, two short-ciccuitcd stubs i~ttilchcd to the main line a1 lined posilion:.., i s shown in Yig.'9:24. Here, the distance do is fined and arbitrarily chosen (such as L/16. JL/8, 3/:/16,3i/8, etc.), and the lengths of the two stub tuners are adjusted to match a given load impedahce Z , to the main line. This scheme is the double-.stub melhid for impedanm matchink. In the arrangement in Fig. 9-24, a stub of length e , is connected directly in parallel with the load itnped&nce Z , at terminals A-A', and a second stub of length I Fig. 9-24 Impedance matching by double-stub method. I 432 THEORY~AND APPLICATIONS OF TRANSMISSION LINES 1 9 lB is attached at terminals B-B' at a fixed distance do away. For impedance matching I w i t h a main line that has a characteristic resistance R,, we demand the total input admittance at terminals B-B', looking toward the load, to equal the characteristic conductance of the line; that is, In terms of normalized admittances, Eq. (9-147) becomes Now, since the input admittance y,, of a short-circuited stub is purely imaginary, Eq. (9-148) can be satisfied only'if .v,, = I + jh, (9 - 140:1) and Note that these requirements are exactly the same as those for single-stub matching. On the Smith admittance chart, the point representing y, must lie on the g = 1 circle. This requirement must be translated by a distance d,/L "wavelengths toward load"; that is, y, at terminals A-A' must lie on the g = 1 circle rotated by an angle 4xd,/L in the counterclockwise direction. Again, since the input admittance y,, of the short-circuited stub is purely imaginary, the real part of y, must be solely con- tributed by the real part of the normalized load admittance, g,. The solution (or solutions) of the double-stub matching problem is then determined by the inter- section (or intersections) of the g,-circle with the rotated g = 1 circle. The procedure for solving a dou'ble-stub matching problem on the Smith admittance chart is as follows. 1. Draw the g = 1 circle. This is where the point representing y, should be located. 2. Draw this circle rotated in the counterclockwise direction by d,JA "wavelengths toward load." This is where the point representing y, should be located. 3. Enter the y, = g, + jb, point. 4. Draw the g = g, circle, intersecting the rotated g = 1 circle at one or two points where y, = g~ + jbA. . .
.
.
" 5. Mark the corresponding y,-points on the g = 1 circle: y, = I f jb,. 6. Determine stub length t ' , from the angle between the point representing y, and the point representing y,. 7. Determine stub length t ' , from the angle between the point representing -jbB and P,, on the extreme right. e matching , 1 l i total input f . - aracteri~tic , , 1; I ' J 1 ' i imaginary, I , l~l,ltcllill& , I ? ' , = I t'lncc t,, of wlely con- iolution (or y the mter- IZ procedure ! chart is as 1 be located. [:lo points /Itinr and ' enting -jb, .' t line is connected to a load impedance Z , = i 60 + j80 (a). A an e&hth of a wavelength apart is used to 9-24. Find the required lengths of the L 1 , < ' ? I Solution: Given R, = 50 (R) and Z, = a + 180 (R), it is easy to calculate t' (We could find y~ on the Smith chart by locating the point diametrically opposite to z, = (60 + j80)/50 = 1.40 + j1.60, but this wouldclutter up the chart too much.) We follow the procedure outlined above. ' 1. Draw the g = 1 dircle (Fig. 9-25). circle by $ "wavelengths toward load" in the counterclockwise of rotation is 4x18 (rad) or 90'. 3. Enter y, = 0.30 - jO 40 as P,. 4. Mark the two pgints of intersection, PA, and PA,, of the g, = 0.30 circle w ~ t h Ihc rol;ltctl (1 -- I c1rc.1~. At PA,, read y,, = 0.30 i jQ.29 ; At PA,, ready,, = 0.30 + jl.75. 5. Use a com$stss ccntcrbd at the origin 0 to mark the points P,, and P,, on the g = 1 circle correspoqding, respectively, to the points PA, and PA,. At P,,, read y,, = 1 + jl.38; , At P,,, read y,, = i - j3.4. 5 ) 6. Determine the reguiri;d stub lengths t,, axid tA2 from \ (.Vsn)l =1',,, -j1,=j0.69, =(0.097+0.250)%=0.3471.(PointA,), (y;,), yd2 L YL = j2.11, tA2 = (0.17'9 + 0.250)/1 = 0.4291, (Point A,). 2 .' ' 7. Determini-the required stub lengths / , , and fB2 from: (&)I = -jl 38, Ll = (0.350 - 0.250)iL = O.lOOit (Point B,), a - -(jj;;J2 =~,2.4, f,{, = (0.205 t . 0.3O)i. = 0.4551. (Point B?). ~xamidiibtl 0; {he ~cnstrwtion in Fig. 9-25 reveals that if the point P,, repre- senting the n&~r@liied lead ahittance yt = g, + jb, lies within the g = 2 circle (if gL > 2), then t& 3 = gi circle does notjntersect with the rotated g = 1 circle and no solution exists for double-stub matching with do = R/8. This region for no solution varies with the chosen distanced, between the stubs (Problem P.9-38). In such cases Fig. 9-25 Construction for double-s~ub matching. impedance matching by the double-stub method can be achieved by adding an appropriate line section between Z, and terminals A-A', as illustrated in Fig. 9-26 (Problem P.9-37). An analytical solution of the double-stub impedance matching problem is, of course, also possible, albeit more involved than that of the single-stub problem P diiing an Fig. . '6 . - lem is, of problem I Fig. 9-26 ~ouble-sthb irlpedhnce matching with added load-line section. developed in the preceding subsection. The more ambitious reader may wish to obtain such an analytical solution and write a computer program for determining dJA //A, and lB/A in terms of z, and dJi. REVIEW QUESTIONS R.9-1 Discuss the similarities and dissimilarities of uniform plane waves in an unbounded media and TEM waves along transmission lines. H.9-2 What are the Ihree diost common typesof guiding structures that support TEM wava? R.9-3 Compare the advantages and disadvantages of coaxial cables and two-wire transmission lines. R-9-4 Write the trad~mk~on-line equations for a lossless parallel-plate line aupport~ng TEM waves. R.9-5 What are str~phnes ? R.9-6 Describe how the character~stm Impedance of a parallel-plate transmiss~on l~nc depcnda on plate w~dth and dielectric thickni~s. H.9-7 C w ~ p ; ~ r c lhv ~ e l u r ~ l y 01' TEM-wave propagation along a parallel-plate transmasion line with that i32n unbounded medium. . R.9-8 Dcfin~ ,q,hce imp&nce. How is surface itnpedance related to the power dissipated i , n a plate condpcror? R.9-13 State the differtnoe between the surfac6 resistance and the resistance per u n ~ t length of a parallel-plate transmislon line. . > , . 436 . THEORY AND APPLICATIONS OF TRANSMISSION LINES 1 9 R.9-11 What is the essential dikerence between a transmission line and an ordinary electric network? R.9-12 Explain why waves along a lossy transmission line cannot be purely TEM. R.9-13 Write the general transmission-line equations for arbitrary time dependence and for time-harmonic time dependence. R.9-14 Define propagation constant and characteristic impedance of a transmission line. Write their general expressions in terms of R, L, G, and C for sinusoidal excitation. R.9-15 What is the phase relationship between the voltage and current waves on an infinitely long transmission line? R.9-16 What is meant b ; a "distortionless line"? What relation must the distributed parameters of a line satisfy in order for the line to be distortionlcss? R.9-17 Outline the procedure for'determining the distributed parameters of a transmission line. R.9-20 On wh;~t factors does the input i~npcdance fa transmission line depend'! ,A. R.9-21 What is the input impedance of an open-circuited lossless transmission line if the length of the line is (a) %/4, (b) 1/2, and (c) 3i./4? R.9-22 What is the input impedance of a short-circuited lossless transmission line if the length of the line is (a) A/4, (b) 142, and (c) 31./4? R.9-23. Is the input reactance of a transmission line 1/8 long inductive or capacitive if it is (a) open-circuited, and (b) short-circuited? R.9-24 On a line of length C, what is the relation between the line's characteristic impedance and propagation constant and its open- and short-circuit input impedances? R.9-25 What is a "quarter-wave transformer"? Why is it not useful for matching a cornplcx load impedance to a low-loss line? R.9-26 What is the input impedance b f a lossless transmission line of length C that is terminated in a load impedance Z, if (aJ C = 142, and (b) f = R? R.9-27 Define voltage reflection coeficient. Is it the same as "current reflection coetEcien!"'? Explain. R.9-28 Define standing-wave ratio. How is it related to voltage and current reflection coefficients? R.9-29 What are r and S for a line with an open-circuit tcrmination? A short-circuit tcrmination? R.9-30 Where do the minima of the voltage standing wave on a lossless line with a resistive termination occur (a) if RL > R,, and (b) if R, < R,? R.9-31 Explain how the value of a terminating resis!ancc can be determined by measuring the standing-wave ratio on a lossless transmission line. . Write finitely meters ' sdance :ient'"! :]ems? a t i o n 7 p Gstive - ng the H.9-32 Explain value of an arbitrary temidting impedance on a lossless transmission standirig-wave measbIemedh on the line. R.9-33 A voltage genekitor having an internal impe&me 2 . is connected at I = 0 to the input terminals of a lossless t dsmigsion line of length t. h e line has a characteristic impedance Z, and is terminated with i~pedance Z,. At what t h e will a steady state on the line be reached if (a) Z,, = 2, and Z, = e,, (b) Z, = Z, but Z, # Z,, (c) Z, = Z, but 2, # Z,, and (d) Z, # 2, and Z, # Z,? I R.9-34 What is a smith chad>and why is it u6eful id making transmission-line calculations? ! R.9-35 Wlfere is the pb nt rep'resking a matchcd lodd on a Smith chart? , R.9-36 For a given loaa impdance Z , on a lossless l k e of characteristic impedance Z,, how do we use a Smith chart to, hetermine (a) the reflect& coefficient anti (b) the standing-wave ratlo? R.9-37 Why does a chdnge of half-a-wavelength in line length correspond to a complete rcvo- lution on o Smith chart I R.9-38 Given an impedance Z = k + jX, what procedure do we follow to find the admittance Y = 1/Z on a Smith chart? K.9-39 Given an admittance Y = G + JB, how do we use a Smith chart to find the impedance z = IIY? R.9-40 Where is the poifit representing a short;circuit on a Smith admittance chart? R.9-41 Is the standing-have xtio constant on a trdd9mission.line even when the line is lossy? Explain. R.9-42 Can a Smith c h t b: used for impedance ccilculations on a lossy transmlsslon line'! Explain. R.9-43 Why is it more';convenient to use a Smith chart as an admittance chart for solving impedance-matching problems than to use it as an impedance chart? R.9-44 Explain the single-stu!, method for impedance matching on a tranarn~ssion line. 1t.O-45 Bxpliiin lhc doublc-sit~h rncthod for impcdancc matching on a transmission Line. R.9-46 Compare the relative ddvantages and disadvantages of the single-stub and the double- stub methods of impedance matching. PROBLEMS P.9-1 Neglecting fringe fields, prove analytically that B y-polarized TEM wave that propagates along a parall&piate tratlsmission line in + z direction has the following properties: dE,/dx = 0 and dH,/dy = 0. P.9-2 The electric and hagnetic fields of a gcneral TEM wave traveling in the + z direction alohg a transmission line may have both x and y components, and both components may be functions of the transverse dimensions. .! a) Find the relatioh amang EJx. y), E,(s, y), H,,(.t, y), and H,(s, y). 438 THEORY AND APPLICATIONS OF TRANSMISSION LINES / 9 b) Verify that all the four field components in part (a) satisfy the two-dimensional Laplace's equation for static fields. P.9-3 Consider lossless stripline designs for a given characteristic impedance. a) How shodld the dielectric thickness, d, be changed for a given plate width, w, if the di- electric constant, E,, is doubled? b) How should w be changed for a given d if c, is doubled? c) How should w be changed for a given E, if d is doubled? d) Will the velocity of propagation remain the same as that for the original line after the changes specified in parts (a), (b), and (c)? Explain. P.9-4 Consider a transmission line made of two parallel brass strips:-a, = 1.6 x 10' (S/m)-- of width 20 (mm) and separated by a lossy dielectric slab-p = p,, r, = 3, a = (S/m)-of thickness 2.5 (mm). The operating frequency is 500 MHz. a) Calculate the R, L, G, and C p& unit length. 1)) Compnrc the magnitudes of thc nxinl ant1 Ir;~nsvcrsc components of the electric field. c) Find 7 and Z,,. P.9-5 Verify Eq. (9 -39). -1 - P.9-6 Show that the attenuation and phase constants for a transmission line with perfect' conductors separated by a lossy dielectric tlii~t.,llas u complex prnmittivity E = c' - jc" arc, resncctivclv. P.9-7 In the derivation of the approximate formulas of y and 2, for low-loss lines in Subsection 9-3.1, all terms containing the second and higher powers of (RIoL) and (G/wC) were neglected in comparison with unity. At lower frequencies better approximations than those given in Eqs. (9-45) and (9-47) may be required. Find new formulas for y and Z , for low-loss lines that retain terms containing (RIwL)' and (GIwC)'. Obtain the corresponding expression for phase velocity. P.9-8 Obtain approximate expressions for y and Z, for a lossy transmission line at very low frequencies such that wL < < R and o C < < G. P.9-9 The following characteristics have been measured on a lossy transmission line at 100 MHz: Determine R, L, G, and C for the line. \ P.9-10 It is desired to construct uniform transmission lines using polyethylene (E, = 2.25) as the dielectric medium. Assuming negligible losses, (a) find the distance of separatiqn for a 300-(R), two-wire line, where the radius of'the conducting wires is 0.6 (mm); and (b) findthe inner radius ofthe outer conductor for a 7542) coaxial line, where thc radius ofthc ccnter conductor is 0 .
G (mm). ne after the ctric ficld. .,vitb perfect m : e , ,n Subsection ere neglected glven in Eqs. es that retain hase velocity. e at very low eat 100 MHz: - 2' s the for a >JO-(R), ie inner radius tor is 0.6 (mm). 8 : 8 PROBLEMS 439 I . .
' ! f P.9-11 Prove that a maximum power is ,transferred from a voltage source with an internal impedance Z, to aload impedance 2 , over a~!ossle$s transmission line when 2, = 2:. What is the , . , . maximum power-transfer efficiency? I r -. 1 P.9-12 Express V ( Z ) and I(z) in terms of the voitage and current I, at the input end and y 'and Z , of a transmisdion line (a) in exponential fohn and (b) in hyperbolic form. P.9-13 A DC generhtor oi voltage C , and internal resistance R, n connected to a lossy trans- mission line characterized by a resistance per unit length R and a conductance per unit length G. a) Write the governing voltage and current ttansmission-line equations. b) Find the gedkral solutions for V(z) and I@). c) Specialize tH'e solutions in part (b) to those for an infinite line. d) Spccialim tht solutions in part (b) to thosc for a finite line of length L that 1 s termmated in a load redstance RL. P .
9 -1 4 A gcncrator with n.n opcn-circuit volwge v,(t) = 10 sin X O O O n l (V) and intcrnal impedance %,, .- 40 +-,j301!1) is conilclftcd 1 0 :I 50-(<1) ~ I i ~ ~ o i ~ ~ i c m l s ? ; ~ linc. 'l.hl: lir~c I W S :I rcslslancc of03 (f>/rn). and i t s lossy dielscLtk rncdi;tm has a loss tangent of 0.18%. The line is 50 (km) long and is tsrmi- nated in a matched load. Find (a) the instantaneous expressions for the voltage and current at an arbitrary location on the line, (b) the instantaneous expressions for the voltage and current at the load, and (c) the av&e power transmitted to the load. P.9-15 The Input impedance of an open- or short-circuited l o ~ y transmmion h e ha5 both d resistwe and, a reactlve component. Prove that the input mpedance of a very short section C of a shghtly lossy line (a4 c 1 and P/ << I ) 1 s approximately ' a) Z,, = i (R + fd)/ w~th a short-c~rcu~t teimlnat~on. b) Z,, = (G -I~c)/[G + (UC)~]~' with an open-circuit tennation P.9-16 A 2-(m) lossfess transmission line having a characteristic impedance 50 (R) is termmated with an irnwdance 40 + j301R) at an operaticg frequency of 200 (MHz). Find the input impedance. P.9-17 The open-circuit and short-circuit impedhnces measured at the input terminals of a transmissio$he 4 (m) long are, respectively, 250- ((R and 360/20" (R). a) Det~finine Z,, a, and p of the line. b) Detebnine R, L, G: and C. P.9-18 A lossless quarter-wave h e section of characteristic impedance R, is terminated with an indwtive load impedance ZL 5 RL + jXL. a) ~ r o v g that.the input impedance is effectively a resistance Ri in parallel with a capacitive reactance X,. Determine R? and Xi in terms of R,, R,, and X,. b) F i ~ d tcs -?.ti0 of the magnitude of the voltage at the input to that at the load (volrage tr(i;i:>?:rt~ti~ti/~t~ r(~tio! l~,,l/l~,l) in tcms of 2, and ZL, P.9-19 A 7 5 -( R ) lossless lj& is terminated in a load impedance Z, = RL + jXL. a) What must be the relation between RL and X, in order that the standing-wave ratio on !$e line be 31 b) Find X,, if dt = 150 (Q). c) Where does the voltage minimum &arest to the load occur on the line for part (b)? 440 THEORY AND APPLICATIONS OF TRANSMISSION LINES I 9 P.9-20 Consider a lossless transmission line. s . . . a) Determine the line's characteristic resistance so that it will have a minimum possible standing-wave ratio for a load impedance 40 + j30 (Q). b) Find this minimum standing-wave ratio and the corresponding voltage reflection coefficient. c) Find the location of the voltage minimum nearest to the load. P9-21 A lossy transmission line with characteristic impedance Zo is terminated in an arbitrary load impedance ZL. a) Express the standing-wave ratio S on the line in terms of Z0 and ZL. b) Find in terms of S and Zo the impedance looking toward the load at the location of a voltage maximum. C) Find the impedance looking toward the load at a location of a voltage minimum. P.9-23 The standing-wave ratio on a lossless 300-(R) transmission linet'bnhated in an unknown load impedance is 2 .
0 , and the nearest voltage minimum is at a distance 0 .
3 1 . from the load. Determine (a) the reflection coefficient l - of the load, (b) the unknown load impedance Z L , and (c) the equivalent length and terminating resistance of a line, such that the input impedance is equal to 2 , . P.9-24 Obtain from Eq. (9-114) the formulas for finding the length t,,, and the terminating resistance R, of a lossless line having a characteristic impedance Ro such that the input impedance equals Z i = R, + jXi. P.9-25 Obtain an analytical expression for the load impedance ZL connected to a line of char- acteristic impedance Zo in terms of standing-wave ratio $ and the distance, z u i .
, of the voltage minimum closest to the load. P.9-26 A sinusoidal voltage generator with V, = O . l p (V) and internal impedance Z , = Ro is connected to a lossless transmission line having a characteristic impedance Ro = 50 (R). The line is f meters long and is terminated in a load resistance R , = 25 (R). Find (a) v, Ii, VL, and 1,; (b) the standing-wave ratio on the line: and (c) the average power delivered to the load. Compare the result in part (c) with the case where R , = 5 0 (R). P .
9 -2 7 A sinusoidal voltage generator u, =, ll0sinwt (V) and internal impedance 2, = 50 (R) . is connected to a quarter-wave lossless line having a characteristic impedance Ro = 50 (R) that is terminated in a purely reactive load ZL = j50 (R). a) Obtain voltage and current phasor expressions V(zl) and I(zl). b) Write the instantaneous voltage and current expressions u(zi, t) and i(z1, t). c) Obtain the instantaneous power and the average power delivered to the load. P.9-28 The characteristic impedance of a given lossless transmission line is 75 (R). Use a Smith chart to find the input impedance a n 0 0 (MHz) of such a line that is (a) 1 (m) long and open- circuited, and (b) 0.8 (m) long and short-circuited. Then (c) determine the corresponding input admittances for the lines in parts (a) and (b). ssible :don jitrary n o f a n. a load crlstic mown . I s a L. 4 in indtlng edance )f char- 4 oltage = Ro is The line (b) the )are the = so (n) (R) that P S m m d open- ~g lnput ? P.9-29 A load impedadce 30 + 110 (Q) is connected d ( b a l~ssless transmission line of length 0.1011. and characteristic im&danke 50 (a). Use a Smith. chart to find (a) the standing-wave ratio, (b) the voltage re~ectlbn coefficient, (c) the jnput impedance, (d) the input admittance, and (e) the location of the $b~tagdm~nimum on thiline.$ ~ 9 - 3 0 Repeat probl+ P.9%29 for a load i m p d a n k 30 - jlO (a). P.9-31 In a laborator;, exp&ment conducted d ; l a SO-($2) lossless transmlss~on hne ternmated in an unknown load Ypedance, it is found that the standing-wave ratio is 2.0. The successive voltage minima are 2$(cm) apatt and. the first minknum occurs at 5 (cm) from the load. Find (a) the load impdand, and (b) the reflection meffiEient of the load. (c) Where would the fint voltage minimum be Idtated'if the load were replaced by a short-circuit? P.9-32 Thc input imdduncc of a short-circuited lose transmission line of length 1.5 (m) (ci/2) and characteristic impedance.100 (R) (approximately real) is 40 - j280 (R). a) Find a: and / I of the line. b) Determine the input impedance if the short-circuit is replaced by a load impedance ZL = 50 + j50 (Q). c) Find the input impedance of the short-circuifed line for a line leneth 0.15i.. - P.9-33 A dipole antenna having an i n p ~ ~ t iili~diince of 73 (0) is fed by i 200-(MHz) source through a 3W-(n) t w o 4 r e trsnsmission line. Desidn a quarter-wave two-wire air line with a 2-(cm) spacing to matcg the antenna to the 300-(Q) line. P.9-34 The single-stub melhod is used to match a load impedance 25 + j25 (Q) to a 50-(R) transmission line. I' a) Find the required length and position of a ihort-circuited stub made of a section of the same 50-(R) linc. ' b) Repeat part (a) asstming the short-circuited stub is made of a section of a line that has a characteristic! impedance of 75 (Q). ' P.9-35 A load impedance can be hatched to a transmission line also by using a single stub placed in series with the load I t an appropriate location, as shown in Fig. 9-27. Assuming ZL = 25 + j25 (Q), Ro = 50 (R), and KO = 35 (0). find d and I required for matching. Fig. 9-27 Impedancg hatching by a series stub. P.9-36 The double-stub method is used to match a load impedance 100 + j103 (Q) to a losrless transmission line of characteristic impedance 300 ($2). The spacing between the stubs is 3218, with I ', . " , ,,,one stub connected directly in parallel with the load. ~etennind the lengths of ihe siub tuners if , (a) they are both short-circuited, and (b) if they are both opensircuited. " .. 6 ,)l - - P9-37 If the load impedance in Problem P.9-36 is changed to 100 + j50 (Q), one discovers that a perfect match using the double-stub method with do = 3118 and one stub connected directly across the load is not possible. However, the modified arrangement shown in Fig. 9-26 can be used to match this load with the line. a) Find the minimum required additional line length dL. b) Find the required lengths of the short-circuited stub tuners, using the minimum dL found L, . in part (a). P9-38 The double-stub method shown in Fig. 9-24 cannot be used to match certain loads to a line with a given characteristic impedance. Determine the regions of load admittances on a Smith admittance chart for which the double-stub arrangement in Fig. 9-24 cannot lead to a match for do = A/16,1/4, 3118, and 71/16. d, found ' I lbads to , , :ces on a ! 1, .ead to a '.\ I 10-1 - INTRODUCTION In the preceding cha&er we studied the characteristic properties of transverse slec- tromagnetic ( T E ~ ) waves guided by transmission lines. The TEM mode of guided waves is one in which the tlectric and magnetic fields are perpendicular to each other and both are transvkrfe tc. the direction of propagation along the guiding line. One of the salient properties i ~ f TEM waves guided by conducting lines of negligible resistance is that the velocity of propagation of a wave of any frequency is the same . as that in an unbounded dielectric medium. This was pointed out in connection with Eq. (9d1) and was-reinforced by Eq. (9-55). TEM waves, however, are not the only mode of guided waves that can propagate on transmission lines;. nor are the three types .of transmission lines (parallel-platz, two-wire, and coaxial) mentioned in Section 9-1 the only possible wave-guiding structures. As a mattcr of hct, wc scc from Eqs. (9-45a) and (9-49a) that thc Litteri- ualion constant resulting iiom thc linite conductivity of the lines increases with R. the resistance per unit line length, that, in turn, is proportional to fi in accordance with Tables 9-1 and 9-2. Hence the attenuation of TEM waves tends to increase monotonically with frequzncy and would be prohibitively high in the microwave range. In this chapter we first present a general analysis of the characteristics of the waves propagating along uniform guiding structures. Waveguiding structures are ' called waveguides, of which ,the three types of ttansmission lines are special cases. The basic governini equations will be examined. We will see that, in addition to transverse electromagrfetic (TEM) waves, which have no field components in rhe direction of propagation, both tru'nscerse r~tnqrzitic (TM) waves with a longitudinal electric-field cornponefit and tmr~sverse electric (TE) \caws with a longitndir.~! magnetic-tield.eomponent crin also exist. Both TM and TE modes have characterisric cutofl fi.cyuerf~?c.s. Waves of frcquencics below thc cutolT frequency of a particillar lnode cannot propagate, atid power and signal transmission at that mode is possible only for frequencies highe;. than the cutoff frequency. Thus, waveguides operating in TM and TE modes are like high-pass filters. Also in this chapter we will reexamine the. field and wave characteristics of parallel-plate waveguides with emphasis on TM and TE modes and show that all , .,. ti transvers~~field components can be expressed in teims of~E,'(z being thk diredkn ' ., &.> , . , a , , , , 1 $ .
< . of propagation) for TM waves, and in ternis 6 f Hz fbr TE wave's. The attenuation f . . . . . . I .
, .
. . - 1 . . . . . . b- t. .., - .,.#-> . , ., constants resulting from imperfectly conddctihg prates will be detkmiined for TM , . and TE waves, and we will find that the attenuation cbnstant depends, in a com- I ' I I' plicated way, on the mode of the propagating wave, as well as on frequency. For some modes the attenuation may decrease as the frequency increases; for other modes, the attenuation may reach a minimum as the frequency exceeds the cutoff frequency by a certain amount. I I Electromagnetic waves can propagate through hollow'metal pipes of an arbitrary cross section. Without electromagnetic theory it would not be possible to explain '?. the properties of hollow waveguides. We will see that single-conductor waveguides cannot support TEM waves. We will examine in detail the fields. the current and charge distributions, and the propagation characteristics of rectangular waveguides. Both TM and TE modes will be discussed. An analysis of the propertie$ of circu- lar waveguides requires a familiarity with Bessel functions as a consequence of manipulating Maxwell's equations III cylindrical coordinates. Circular w:~voguidcs will not be studied in this book. In millly applic:ltions wave psopagation in a rcctan- ,oular waveguide in the dominant (TE,,) mode is desirable because the electric field in the guide is polarized in a fixed direction. Electromagnetic waves can also be guided by an open dielectric-slab waveguide. The fields are essentially confined within the dielectric region and decay rapidly away from the slab surface in the transverse plane. For this reason, the waves sup- ported by a dielectric-slab waveguide are called sudtrce waces. Both TM and TE modes are possible. We will examine the field characteristics and cutoff frequencies of those surface waves. At microwave frequencies, ordinary lumped-parameter elements (such as in- ductances and capacitances) connected by wires are no longer practical as resonant circuits because thc dimcnsions of thc clcrncnts would hove to bc cxtrcmcly small, because the resistance of the wire circuits becomes very high as a result of the skin effect, and because of radiation. All of these difficulties are alleviated if a hollow conducting box is used as a resonant device. Because the box is enclosed by conducting walls, electromagnetic fields are confined inside the box and no radiation can occur. Moreover, since the box walls provide large areas for current flow, losses are extremely small. Consequently, an enclosed conducting box can be a resonator of a very high Q. Such a box, which is essentially a segment of a waveguide with closed end faces, is . called a cavity resonator. We will discuss the different mode patterns of the fields inside rectangular cavity resonators. 10-2 GENERAL WAVE BEHAVIORS ALONG. UNIFORM GUIDING STRUCTURES In this section we examine some general chatactcristics for waves propagating along straight guiding structures with a uniform cross section. We will assume that the waves propagate in the +r direction with a propagation constant -J = ci + jB that firection , 8 . ' is yet to be aetermined. For harmonic time dependence with an angular frequency o, muation the depende&e on r ahd t for all field components can be described by the exponential for TM . , L factor :acorn- \, . n i : I i- = e ( j ~ t - Y Z ) = -ee j(ut - BZ) (10-la) x y . For 1 . I AS an exa.mpl; for a. bosine reference we may write the instantaneous expression for or other the E field as I ne cutoff i a E(x, y, z; t) = %[EO(x, y)e(jU'-Y')], (10-lb) ) explain veguides rent and - of circu- veguides a rcctan- x i : field n :ve: -e. rapidly ives sup- and TE ~quencies :h as in- resonant -:y small, :he skin .I ho!low nduc~ing In occur. : s tremely -y high Q. ; faces, is :he fields Ing along rhat the - j/3 that where EO(x, y) is il two-dimensional vector phasor that depends only on the cross- sectional coordinated. The instantaneous expression for the H field can be written in a similar way. Hence. in using a phasor representation in cquations relating field quantities, we may replacc partial derivatives with respect to i and z simply by products witki ( j o ) and ( - 7 ) respectively; the common factor r'jw'-"' can be dropped: We consider a 9tra.ght waveguide in the form of a dielectric-filled metal tube having an arbitrary nots section and lying along the .- axis, as shown in Fig 10-1. According to Eqs. (7-8.6) and (7-87), the electric and magnetic field intensities in the charge-free diclcktr:~ region inside satisiy the following homogeneous vector l-lclmholtz's equations: and V'E + k'E = 0 V2H + k " = 0, (10-2b) where E and H are three-dimensional vector phasors; and k is the wavenumber r -- k = Wd,LLE. (10-3) The three-dimensiona! Laplncian operator V2 may be broken illto two parts: 2 FuIu2 for the cross-sectional coordinates and Vf for the longitudinal coordinate. For waveguides with a rectangular cross section, we use Cartesian coordinates: = V.<,E + ;q2E. Combination of Eqs. (lb-2a) and (10-4) gives Fig. 10-1 A uniform wave- guide with an arbitrary cross section. ....... - .tl , . . I . . . . 1 -. . , 446 WAVEGUIDES AND CAVITY RESONATORS / 10. : - .
.
.
. Similarly, from Eq. (10-2b) we have v$H + (yf + k 2 ) ~ = 0 . (10-6) We note that each of Eqs. (10-5) and (10-6) is really three second-order partial differential equations, one for each component of E and H. The exact solutibn of these component equations depends on the cross-sectional geometry and the bound- ary conditions that a particular field component must satisfy at conductor-dielectric interfaces. We note further that by writing V : + for the transversal operator V:,, Eqs. (10-5) and (10-6) become the governing equations for waveguides with a circular cross section. t Of course, the various components of E and H are not all independent, and it is not necessary to solve all six second-order partial differential equations for the six components of E and H. Let us edamine the interrelationships among the six com- poncnts in Ca rtcsian coordinates hy cspnnding thc two sourcc-free curl ctptions. Ecp. (7-851) atid (7 -8.517): From V x E = -jo,uH: From V x-W-,jocE: Note that partial derivatives with respect to z have been replaced by multiplications by (-y). All the component field quantities in the equations above are phasors that depend only on x and y, the common c-'' factor for z-dependence having been omitted. By manipulating these equations, we can express the transverse field com- ponents H;. H .
: , E .
!
, and E : in terms of the two longitudinal components Ep and H : . For instance. Eqs. (10-la) and (10-8b) can be combined to eliminate E; and obtain H : in terms of E: and HP. We have wund- ~ C M ~ C 2 Vxp il,ith a :d it is ihe six . \ com- J , ltions. stions .s tha: : been cotn- and : : and where The wave behavior in a waveguide can be analyzed by solving Eqs. (10-5) and (10-6) respectively for the longitudinal components, and HZ subject to the required boundary conditions, and then by using Eqs. {lo-9) through (10-12) to determine the other components. It is convenient to classify the propagating waves in a uniform waveguide into three types according to whether E, or H , exists. 1. ~rnnsvers~'~lect~oma~netic ( T E M ) Waves. These are waves that contain neither Ez nor H;: We eehoontered TEM waves in Chapter 8 when we discussed plane waves and in Chlpter 9 on waves along tr&cimission lines. 2. Transverse Maglati: (TIM) Wows. These are waves that contain a nonzero E;.. ' but H, = 0. 3. Tm~nuerse Electric (TEE) W a m . These are waves that contain a nonzero ?I=. hut IS, = 0. The propagation charac:eristics of the vorious types of waves are different; they will be discussed in,subsequc;nt subsections. 10-2.1 Transverse ~lectrotna~netic Waves Since E, = 0 and Hz = 0 for TEM waves within a guidc, we see that Eqs. (10-9) through (10-12) constitute a set of trivial sol~tions (all field components van~sh) unless the denominator h ' also equals zero. In other words, TEM waves exist oniy when + k ' = 0 (10-14) or ;lTEM = jk =ju JF, (10-15) which is exactly the same expression for the propagation constant of a uniform pime wave in an unbounded inedium characterized by constitutive parameters s and 11. We recall that Eq. (10-i5) also holds for a TEM wave on a lossless :ransmission line. It follows that the .elocity of propagation (phase velocity) for TEM waves is We can obtain the ratio between E:! and H; from Eqs. (10-7b) and (10-8a) by setting E, and H, to zero. This ratio is called the wcive impedance. We have r ! .. , . . 448 . WAVEGUIDES AND CAVITY RESONATORS 1 10 which becomes, in view of Eq. (10-15), . , . > (10-18) We note that Z , , is the same as the intrinsic impedance of the'dielectric medium, as given in Eq. (8-25). Equations (10-16) and (10-18) assert that the phase uelocity and the wave impedance for TEA4 waves ore iirdependent of rl~e frequency of the ivnves. Letting EP = 0 in Eq. (10-7a) and Hp = 0 in Eq. (10-8b), we obtain Equations (10-17) and (10-19) can be combined to obtain the following formula for a TEM wave propagating in the + 2 direction: which, again, reminds us of a similar relation for a uniform plane wave in an un- bounded mcdium -see Eq. (8 -24). Single-conductor waveguides cannot support TEM waves. In Section 6-2 we pointed out that magnetic flux lines always close upon themselves. Hence. if a TEM wave were to exist in a waveguide, the field lines of B and H would form closed loops in a transverse plane. However, the generalized Ampere's circuital law, Eq. (7-38b), requires that the line integral of the magnetic'field (the magnetomotive force) around any closed loop in a transverse plane must equal the sum of the longitudinal conduc- tion and displacement currents through the loop. Without an inner conductor, there is no longitudinal conduction current inside the waveguide. By definition, a TEM wave does not have an E, component; consequently, there is no longitudinal dis- placement current. The total absence of a longitudinal current inside a waveguide leads to the conclusion that there can be no closed loops of magnetic field lines in any transverse plane. Therefore,. we conclude that TEM waves cannot exist in a single-condttcror hollow (or die1ectric;filled) waceguide of any shape. On the other hand, assuining pei$ect conductors, a coaxial transmission line having an inner con- ductor can support TEM waves: so can a two-conductor stripline and a two-wire ,transmission linc. Wllcn (17c conductors havc losscs, waves along transmission iirics arc strictly n o lo~igcr 'I'LM. 11o1cd i l l Scctio~~ 9 2. 10-2.2 Transverse Magnetic Waves Transverse magnetic (TM) waves do not have a component of the magnetic field in the direction of propagation. H, = 0. The behavior of TM waves can be analyzed by solving Eq. (I0 .5) l i ) ~ E, subject to tlic hountl:lry conditions or tlic guiclc :lnd using : d i u n i , , i-locity waves. I . . ula for -2 we . T E M ! loops -38b), :round mduc- : . there . TEM .a1 dis- eguide lnes in ,r pl a 2 other : r con- :o-wire In lines P field in :zcd by i i using Eqs. (10-9) ihrough (10-12) to determine the other components. Writing Eq. (10-5) for E;, we have L ViyE: + (y2 + k?)~: = 0 , .
; . (10-21) - or Equation (10-22),is a second-order partial diffdrential equation, which can be solved for E:. In thitsection we wish only to discuss,the general properties of the various wave types. he actual solution of Eq. (10-22) will wait until subsequent sections when we examine patticu!ar wavcguidcs. For TM waves, we set Hz = 0 m Eqs. (10-9) through (10-12) to obtain It is convenient to combi1.e Eqs. (10-23c) and (10-23d) and write L where denotes the grkdient of 1:: in the tmnsvcrse plane. Equation (10-24) is a concise formula for finding E: an.i E , O from Ep. -. The transverse components of magnetic field intensity, H, and H,, can be deter- mined simply from Ex ard E, on the introduction of the wave impedance for the TM mode. We have, from Eqs. (10-23), --.- It is important to note thai Z.,. is not cquil? to jw,u/y, becausc y for TM waves, unlike YTEM, is not equal to j o ,/a. The following relation between the electric and magnetic field intensities holds for TM waves: 1 L J Equation (10-27) is seen to be of the same form as Eq. (10-20) for TEM waves. When we undertake to solve the two-dimensional homogeneous Helmholtz equation, Eq. (10-22), subject to the boundary conditions of a given waveguide, we will discover that solutions are possible only for discrete ralues o f h. There may be an infinity of these discretc valucs. hut solutions ilre not possible for all values of 11. The values of11 for which a solution of Eq. (10-22) exists are called the chifrirct~ri~fi~ uolllrr or eioe~~crilues of the boundary-value problem. Each ol the rigenvalues deter- mines the characteristic properties of a particular TM mode of the glven waveguide. In thc following sections we will also discover that the ei~envalues of the various wilveguidc problcms are real numhcrs. From Eq. (10-1:) we have 7 = ,/I,: - 1;: --- r? 7 --- = ,jh- - (?)-/A€. (10-3) Two distinct ranges of the values for the propagation constant are noted, the dividing point being y = 0, where w:,u E = 112 (10-29) The frequency,.j,, at which 7 = O is called a cutolj'jrcq~~~~ncy. Ti" uuhe ($,I; for a partieuior mode in a wuueyuidr dcpeads o r 1 lie eige~lcalur uj this mode. Using Eq. (10-30), we can write Eq. (10-28) as (10-31) The two distinct ranges of p can be defined in terms of the ratio (f/jd2 as compared to unity. ) ( > 1 , > . in this rrngr, wzyr ; /I' and 7 is imaginary. w e have. from Eq. (10-2% . - - - -- . - = jp = j k dl - (!) I< =,k Jm. (10-32) It is a propagating mode with a phase consrant 0: I : s . llholtz :de. we nay be 2; oi 11. terisfic , detcr- 't cguide. -- various (10-28) divieg (16 , ) ( I 0 - 30) J; /or u sing Eq. (10-31) -0111 p;11.ctl :ax, from , r 3 2 ) (ld-33) - \ b . % .
s r r .
: r-?.;r....a,s~.'.L... %$ .+ I +\I, L + , , 1- .sf P a h .P. j l ~ l l ? l u ~ ~ ~ 4 W ? - ~ .llTr - j ' bWFC ' . m'''qL'wwpr 1::. +, 10-2 GENERAL WAVE BEHAVIORS ALONG UNIFORM QU~D~NG STRUCTURES 451 4 , : : 'i'. , ; The corresponding wavelength in the guide ifi where 2 n 1 ' u i a m c m p l - (10-35) "+....k,. , / z a f 2 is the wavelength of a plane wave with a frequency f in an unbounded dielectric medium characterized by p and s, and LL = I/& is the velocity of light in the mcdiurn. The phase velocity of the propagating wave in the guide is We sce from Eq. (1036) thilt thc phase vclocity within a wavcguidc is always higher thzn that in a.1 unbounded mcdiurn and is frequency-dependent. Hence single-conductor wuucguides ure dispersive rrut~smissiot~ systems. The group velocity for a propagating waie in a waveguide can be determined by using Eq. (8-j2): Substitution of Eq. (10-32) in Eq. (10-26) yields L I The wuae irqwduizce 1 f'p'apuyotirig TAM tnoddrs i~ u wur.eg.de is picrely resistive am! i s uiwuys less tho.i the ir?rrinsi ii~lpiutlee of the dieiecri-ic rnedietn. The vnri- ation of Zm versa j:/; for f > /, is sketched in Fig. 10-2. Fig. 19-2 Normalized wave irnped- ances for propagating TM and TE waves. b) ($1' < 1, or f < /,. When the operating frequency is lower than the cutoff frequency, y is real and Eq. (10-31) can be written as which is, in fact, an attenuation constant. Since all field components contain the propagation factor e-?' = e-", the wave diminishes rapidly with r and is said to be evanescent. Therefore a waveguide exhibits the property o f a high-pass jlter. For a given mode, only waves with a frequency higher than the cutoff \, frequency of the mode can propagate in the guide. Substitution of Eq. (10-39) in Eq. (10-26) gives the wave impedance of TM modes for f < f ; .
: -1 Thus, the wave impedance of evanescent TM modes st freq>encies below cutoK is purely reactive. indicating that tlicrc is no powcr llow associ:~tcd wit11 cv;inusccnt wa\'cs. 10-2.3 Transverse Electric Waves Transverse electric (TE) waves do not have a component of the electric field in the direction of propagation, Ez = 0. The behavior of TE waves can be analyzed by first solving Eq. (10-6) for H,: Proper boundary conditions at the guide walls must be satisfied. The transverse field components can then be found by substituting H z into the reduced Eqs. (10-9) through (10-12) with 4 , set to zero. We have SHP tr; = -L- h2 C ) , . . . , ; .
. . ,.. !i 1!$i' 7e' cutoff ' : .
, ,, , (10-39) ntuin the id is said high-pass :he cutoff ' >. :dance of (10-40) 3 W c p f f ;meL ~ l t -Id in tlic Jyzed by (10-41) xansverse . p . (10-9) ( 10-42a) ( l 0 P b ) ( 10 -k) ( 10 -43d) + : -. . 10-2 1 GENERAC W A J ~ BEHAVIORS ALONG UW~FORM GUIDING STRUCTURES 453 .i' . c . _ ).- '. . , I Combining Eqs. (10-42a) and (10-42b). we obtain We note that Eq. (10-43) is entirely similar to Eq. (10-24) for TM modes. The transverse components of eiectric field intensity, E O and E;, are related to those of magnetic field intensity through the wave impedance. We have, from Eqs. (10-42% b, c, and d), Note that Z,, in Eq. (10-44) is quite different from Z,, I n Eq. (10-36) bemuse ; ) for TE waves. unlike ynln i s ,lor cql~al to ,or fi. Equations ( 1 0 4 2 ~ ) . (10 -42di. ad ( I 0 44) c;in t w w bc c o ~ ~ l l ) i ~ l ~ d IO give LIIC SOIIOWIIIC VCCLOI lil.~:iuIa: Inasmuch as we hahc not changed the relation between 7 and 11. Eqs. (10-28) through (10-31) pertaining to TM waves also apply to TE waves. There are also two distinct ranges of 7, depending on whethdr the operating frequency is higher or lower than thc cutort'frcqu~nc~, J C , givcn in Eq. (10-30). a) ( > 1, or j' . In this range 7 is imaginary. and we have a propagaring mode. The expression for 7 is the same as that given in Eq. (10-32): (1 0-46) Consequently, the formulas for 1, )L,, up, and u, in Eqs. (10-33). (10-34). (10-36). and (10-37), respectively. also hold for TE waves. Using Eq. (10-46) in Eq. (10-44), we obtain I . I b) ($)2 < 1, or / <I:. In this case, 7 is real and we have an evanescent or non- propaghting mode . , Substitution of Eq. (10-48) in Eq. (10-47) gives the wave impedance of TE modes for f <f,. which is purely reactive, indicating again that there is no power flow for evanescent waves at f < h. Example 10-1 (a) Determine the wave impedance and guide wavelength at a fre- quency equal to twice the cutoff frequency in a waveguide for TEM. TAM, and TE modes. (b) Repeat part (a) for a frequency equal to one-linlf-olrhe cutoff frequency. a) At f = 2A. which is above the cutoff frequency, we have propagating modes. The appropriate formuias are listed in Table 10-1. Table 10-1 Wave Impedances and Guide Wavelengths for j' > J ' : 1 TEM j/ n = ~ ~ E fre- i TE v 0 " Fig. 10-3 An w-b diagram for waveguide. b) At f = J,/2 < f,, the waveguide modes are evanescent and guide wavelength - has no significance. We now have We note that Z,,, does not change with frequency'bewuse TEM waves do not exhibit a cutoff propeity. Both Z,, and Z& become imagnary for evanescent modes at f c /;; their values depend on the eigenvaluc 11, whlch is a characteristic of the particular TM or TE mode. For propagating modes, 7 = j/i and the variation of /j' versus frcquency deter- mines the characteristics of a wave along a guide. It is therefore useful to plot 2nd examine an W-fl diagram.' Figure 10-3 is such a diagram in which the dashed line through the origin reprcsds the w-8 relationship for TEM mods. The constant slbpe of this stmight iine is iu/b = u = I,'&, which is the same as the velonty of light in an unbounded dielectric medium with cbnstitutive parameters p and 6. The solid curve abovr the dashed line depicts a typical w-P relation for either :I TM o r ;I Ti7 ~ ~ l ' O ~ ~ : l g i l ~ i l l r ~ I I I O ( ~ C , givc11 l)y J{CJ, (10 33). We GIII writc NII The a-,/3 curve intersects ihe W-axls ([i = 0) at o = m , . The slope of the line jo~nlng the origin and any point. L U C ~ as P. on the curve is equal to the phase velocity, up, for a particular mode having 3 cutoff frequency /; and operating at a particular ' Also referred to as a Biillouin drayram. Fig. 10-4 Relation between attenuation constant and operating frequency for evanescent modes (Example 10-2). frequency. The local slope of the w-/3 curve at P is the group velocity, u,. We note that, for propagating TM and TPwaves in a waveguide. up z u and ug < u. In fact, Eqs. (10-36) and (10-37) show that As the operating frequency increases much above the cutoff frequency, both 11, and ug approach u asymptotically. The exact value of o . depends on the eigenvalue h in Eq. (10-30)- that is. on the particular TM or TE mode in a waveguide of a given cross section. Methods for determining h will be discussed when we examine different types of waveguides. Example 10-2 Obtain a graph sliowin$ thc relation between the attenuation con- stant n and the operating frequency f for evanescent modes. Solution: For evanescent T M or TE modes, f < and Eq. (10-39) or (10-48) applies. We have Hcncc thc graph of(j.r/h) plotted versus/ is a circle ccntcicd at the origin and having a radiusl,. This is shown in Fig. 10-4. The value of n for any/' < j; can be found from this quarter of a circle. 10-3 PARALLEL-PLATE WAVEGUIDE In Section 9-2 we discussed the characteristics of TEM waves propagating along a parallel-plate transmission line. It was then pointed out. and again emphasized in subsection 10-2.1, that the field behavior for TEM modes bears a very close re- semblance to that for uniform plane waves in an unbounded dielectric medium. However, TEM modes are not the only type of waves that can propagate along Je note ?
. In fact, I , and u, L i e /r In con- having : d from r' I a , I Z U , . ~n IOSC re- irdium. : along perfectly conduc& p~mllel-plates separated by a dielectric. A parallel-plate wave- '" ' 'guide can also support' TM and TE waves. ?he characteristics of these waves are examined separately in following subsections. + 10-3.1 TM Waves between parallel Plates I ' Consider the paallel-plate waveguide of two perfectly conducting plates separated by a dielectric medium with constitutive parameters 6 and p, as shown in Fig. 10-5. The plates arc assbmed to bc infinitc in extent in thc x-direction. This is tantamount ' to assuming that the Aelds do not vary in the x-direction and that edge effects are negligible. Let us suppose that TM waves (Hz = 0) propagate in the + z direction. For harmonic time dependence, it is expedient to work with equations relating field quantities with the common factor ej(''"-yz) omitted. We write the phasor Ez(y, 2) as . ~:(y)e-~'. Equation (10-22) then becomes d 2 ~ ; ( y ) + h2E;(y) = 0. cly2 (10-53) The solution of Eq. (1b-5:3) must satisfy the boundary conditions E:'(y)=O a t y = O and y = h . From Section 4-5 we cor-clude that E:(y) musr be of the Sollowing form (h = n q b ) : where the a m p h d e A,, depends on the strength of excitation of the particular TM wave. The only other nonzero field components are obtained from Eqs. (10-23a) and (10-23d). Keeping in mind that ZE,/?x = 0 and omitting the e - p factor, we have jar (n:) H ~ ( Y ) =- A, cos - h 'Fig. 10-5 . waveguide. Ad infinite parallel-plate , , .;. . ----- . , #,:" . a , .
. . . - . . - . . . . . . . . . . , - . . . \ . . . . < . -. . . - : , . . , .
; , : : : . :. . , . , ' " ' .. . . . . "., ... -,..+. . ...:. ::'.. , ..'.. ' -'-: . . . _ . .. - . . . . . . , . . . - , - - .I&~.l?.;< .
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. -.. , V:,J; - : ,, .. , : ,;-. . , .,... . , :, .: , ,,.,,&$ , .
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, +%>! .,,.... A,"",.! :i: .,.. , , . - .- : , ' . .\ . . 6 8 . : WAVEGUIDES AND C A V l n R E S ~ ~ ~ ~ ~ ~ & 1 10 , . 2'. , The y in Eq. (10-54c) is the propagation constant that can be determined from Eq. (10-28): Cutoff frequency is the frequency that makes y = 0. We have which, of course, checks with Eq. (10-30). Waves with f > propagate with : I phase constant A given in Eq. (10-33): and waves with / 5 J; are evanescent. Depending on the value of n . here are different possible propa,. o~ting TM modes (e~genmodes) corresponding to ll~c iiilhrent cigcll~;~lucs /I. Tllar. ~llcrc ;lrc.lllc Tkl, mode (11 = I) with cutolYfrrquency (j;), = l/ll~,/~a. the TMl.mode (11 = 2) with = iC I/hdli+ and so on. Each mode has its own characteristic phase constant, guide wavelength, phase velocity. group velocity, and wave impedance; they can be deter- mined from. respectively, Eqs. (10-33), (10-34). (10-36), (10-3i), and (10-38). When n = 0, E, = 0, and only the transverse components H, and E,, exist. Hence TIM, mode is the TEM mode, for which j, = 0. The mode having the lowest cutoff frequency is called the dornindnt mode of the waveguide. For parallel-phte waveguides, the domi- nanr rnorie is rlw TEM inode. Example 10-3 (a) Write the instantaneous field expressions for TM, mode in a parallel-plate waveguide. (b) Sketch the electric and magnetic field lines in the yz-plane. Solution a) The instantaneous field expressions for the TM, mode are obtained by multi- plying the phasor expressions in Eqs. (10-54a), (10-54b), and (10-54c) with eJ("'-a' and taking the real part of the product. We have, for 11 = 1, where ncd Ii-onl i ; . (10-55) --(10-56) h a phasw- .
M modes the TM, 1th (I;), = :nt, gutdc be @r- cn M, jde qu$ncy is the domi- node in a I pz-plane. 5y multl- -54~) with (10-57a) (10-57b) r I ! 7 4 (10-58) . " , . . b ) In the y-z E has both a y and a z component, the equation of the electric , : i -l .,$eId lines w a given t can be found from the relation: . ,- %. I. , - , , . .: dy d d z - = -. (10-59) E, E L For example, at t = 0, Eq. (10-59) can be Written as d ~ ' E,(Y, I; 0) - - = - -@ cot ( y ) tan fiz, dz E,(y,z;O) le which gives the dope of the electric field lincs. Equation (10-60) can be integrated to give which is the equhtion of the electric field line for a particular yo at ; = 0. DiLrent values of y,, givc dilfcrcnt loci. Severill ilch electric licld lincs ;ire d r ~ w n in Fig. 10--6. The ficld lines rcpwt tl~cmaclvcs iur every cliangc of4r by ?n rad. Since H ha3 only an s component, tNc magnetic field lincs are everywhere perpendicular to the y-; pime. For thc TM, mode at t = 0, Eq. (10-57c) becomes The density of h, lines varier as cos ini/b) in the y direction and as sin 0 : in the z direction. This is also skctched in Fig. 10-6. At the conducting plates (y = 0 and y = h). there arr. surface currcnu because of a discontinuity in the tangential magnetic field and : urface charges because of the presence of a normal e1ec:ric field. (Problem 10-1). - Electric t':c!d lines. a 8 Mngnctic field lincs (.--asis inld thc paper). Fig. 10-6 Field llnes IJr TM, mode in parallel-plate waveguide. . . . / . .
/ - 460 . WAVEGUIDES AND CAVITY RESONATORS / 10 . . , . .
& .- r . . ,. ! 9 Y Example 10-4 Show that the field solution of 9 propagating TM, wave in a parallel- .!, , , , ,piate waveguide can be interpreted as the superposition of two plane waves bouncing back and forth obliquely between the two conducting plates. Solution: This can be seen readily by writing the phasor expression of EP(y) from Eq. (10-54a) for n = 1 and with the factor e-jB' restored. We have A1 = - [ e - j ( B : - n ~ l b ) - - j ( P : + n ~ l h t I. (10-G3) 2j 4 ' . From Chapter 8 we recognize that the first term on the right side of Eq. (10-63) represents a plane wave propagating obliquely in the + z and -y directions with phase constants , !
3 and n/b respectively. Similarly, the second term represents, a plane wave propapting obliquely in the +: and +y directions with the same phase con- stants and d h as those of the first plane wave. Thus, a propagating TM, wave in a parallel-plate waveguide can be regarded as the superposition of two planc waves. as dcpictcd in Fig. 10-7. In Subsection 8-6.2 on reflection of a ~arallelly polarized plane wave incident obliquely at a conducting boundary plane, we obtained an expression for the lon- gitudinal component of the total El field that is the sum of the longitudinal com- ponents of the incidcnt Ei and the retlectcd E,. To adapt the coordinate designations of Fig. 8-10 to those of Fig. 10-5, .
.
c and z must be changed to z and - y respectively. Wc rcwritc E, of Eq. (8--86a) as Comparing the exponents of the tcrms in this equation with those in Eq. (10-63), we obtain two equations: /3, sin Bi = p (10-64a) X . p, cos Oi = -- b Fig. 10-7 Propagating wave in parallel-plate waveguide as superposition of two plane waves. t.10-63) i 10-63) ns with "- a plane se con- aave in M'3 V C S , n i C l r .L he .- :I corn- xations ;~ively. p= which is the same as Eq. (10-58), and n /Z COS 0, = - - P,b - 26' (10-65) where I = 2n//3, is the wavelength in the unbounded dielectric medium. We observe thLt B solution ol Eq. (10-65) for Oi exists only whcn L/Zb i I. At /./Zb = I. or /' = I I / ~ = l/?b,/~. whicll is the cutoff frcqucncy in Eq. (10-56) for 1l = 1, cos Oi = I. and Oi = 0. This corresponds to the case when the wves bounce . back and forth in the j9 direction, normal to thc prailei plates. ilnd there is no prop- agation in the : direction (b = 11, sin Oi = 0). Propagation of TM, mode is possibie only when / . c i , = 2b or f > ji.. Both cos Oi and sin Oi can be expressed in terms of cutoff frequency /:.. FPorn Eqs. 11 0 - 65) and (1 0-64a) wc htvc and sin Qi =5=2=/%. / .
g I$ (10-66b) Equation (LO-66bj is in agreement with Eqs. (10-34) and (10-36). 10-3.2 TE Waves between Parallel Plates I For transverse eiectnc waves. Ez = 0, we solve the following equation for HP()), which is a simplified versim of Eq. (10-41) with no -dependence. We note that H,(R : ) = H : ( y)e-". The boundary conditions to be satisfied by H:[)I) are obtained from Eq. (10-42c). Since Ex must vanish at the surfaces of the con- ducting plates, we require H : ( g) = B,, fos (F) , a . i <. c .
, . .. . / .. I . , '. . L ' . 462 WAVEGUIDES AND CAVITY RESONATORS I 10 ' 5 . . , where the amplitude B,, depends,on the strength of excitation of the particular TE . . % I _ wave. We ~ b t a i n the only other nonzero fidd components from Eqs. (10-42b) and (10-42c), keeping in mind that dHz/2x = 0: The propagation constant y in.Eq. (10-68b) is the same a3 that for TM waves given 3 ' . in Eq. (10-55). Inasmuch as cutoff frequency is the frequency that makes y = 0, tlre cuto]'jicqua~j~ for rlic TE, rlrotlc iir ( 1 patrllcl-pltrrc wreguidc is estrcrl!i the strrnr as tllutjbr the TM, mode gicen irt Ey. (10-56). For n = 0, both H , and E, vanish; hence the TE, mode does not exist'ln a parallel-plate waveguide. Example 10-5 (a1 Write the instantaneous field expressions for the TE, mbde in a parallel-plate waveguide. (b) Skctch thc elcctric m d ma~~ictic field lincs in the J-: plane. .-. Solution a) The instantaneous field expressions for ;he TE, mode are obtained by taking the real part of the products of the phasor expressions in Eqs. (10-68a), (10-68b), and (10-6Sc) with ej(""-4''. We have. for n = 1, H,(y, z : t ) = B1 cos - cos (at - Pz) (lid) (10-69a) where the phase constant P is given by Eq. (10-58), same as that for the TM, mode. b) In the y-z plane E has only an x component. At t = 0, Eq. (10-69c) becomes ' Thus the density of Ex lines varies as sin (xylh) in the y direction and as sin / ?
i in the i direction; Ex lines are sketchcd as dots and crosses in Fig. 10-8. The magnetic field has both a y and a z component. The equation of the magnetic field lines at t = 0 can be round from the following relation: d y I! ( v 2 : 0) /lh - = - tan - tan llz. dz H z ( y , z ; 0 ) n ( 7 ) . 0-~SC) :s given = 0 , rhr 1 e same vanish; , , .
, ~ d e in a the y-z P %In, e 0-boo), I 0-69a) 10-69 b) 10-69~) 2e TM, .ornes i 10 -70) 1 >In / k f- I L ' : : \ 10-71) ---- Magnetic flcld lines. O 8 ~lcctric fieid lines (x-axis into the paper). Fig. 10-8 Field line; for TE, mode in parallel-plate waveguide. Upon integration, Eq. (10-71) gives sin (nb 11)) = , -2 % SIII (ny/h) ' which is the equation of thc magnetic field line for a particular yo at z = 0. Several such lifles are drawn in Fig. 10-8 for different values of yo. The field lines repeat themselves for every change of flz by 2x.rad. 10-3.3 Attenuation in Parallel-Plate Waveguides Attenuation in any wa~c.guide.(not just the parallel-plate waveguide) arises i'rorn two sources: lossy dielectric and imperfectly conducting walls. Losses modify the electric and magnetic fields within the guide, making exact solutions difficult to obtain. However, in practical wa1:eguides the losses are usually very small, and we will assume that the transverse field patterns of the propagating modes are not affected by them. A real of thc propagation constant now appears as the attenuation constant, which accounts for power losses. The attenuation constant consists of two parts : .,u = ud + z , , (10-73) TEM Modes The attenuat.on constant for-TEM modes on a parallel-plate trans- mission line has been discwed in Subsection 9-33. From Eq. (9-72) and Table 9-1 we have approximately , < . . , , I where E, , u , and.a are, respectively, the permittivity, permeability, and conductivity of the dielectric medium. In Eq. (10-73a) if = is the intrinsic impedance of the dielectric if the dielectric is lossless. Also from Eq. (9-72) and Table 9-1 we have where a, is the conductivity of the metal plates. We note that. for TEM modes, r, is independent of frequency. and cc, is proportional io J?. We note further that a,, -. 0 as a + 0 and that r, -+ 0 as a , 4 cc. as expected. TM Modes The attenuation constant due to losses in the dielectric at frequencies above ji can be found from Eq. (10-55) by substituting E,, =-e.+(aijw) for 6. We have Only the first two terns in the binomial expansion for the second line in Eq. (10-74) are retained in the third line under the assumption that From Eq. (10-56) wc scc that With this relation, Eq. (10-74) becomes from which we obtain -- -- .
.
.
. -ES- . . . - ? I ) - - - ..... : . ..:-:,:<.._ . ( < L a ".',: 6 . - .. 8 -.: .., .&fmJq' . . . . ' . 1, .
. .....,... :r ... ~rr'r:y~uthi--u\~- Lr'su'ii.' .A'‘..'.. 10-3 I%IRALLEL-PLATE .wAvEciwx 465 . . . ' 3 . >-.+;:.+ ': : - I.. s = w & J - iradlm). (10-76) -. I a Thus ad for TM mod& decreases when keqienky increases. .
. t r To find the atterluation constant due ta losses in the imperfectly conducting plates, we use Eq. (9-70), which was derived from the law of conservation of energy. Thus, . (10-73a) ; 1 , , - :<'I: ,ndudtivity - - . ' !
: " " ; ' ' mce of the we have .- (10-73b) modes, a, xther th8i , 'requencies e. Wc have fi (10-74) - kq. (10-74) J1 (10-75) P k ) S(, = - 2 ~ ( $ (10-77) where P(r) is the time-average power flowing through a cross section (say, of wldth w) of the waveguide, Bnd P,(3 is the time-average power lost in the two plates per unit length. For TM modes we use Eqs. (10-54b) and (10-54c): . I'hc surfxc curicnt dwsil:cs on illv aplicl- ;md lower plates hilve thc silnie n,agniiudc. On the lower plate where y = 0, wi: havc The total power loss per unit length in two plates of width bv is Substitution of Eqs. (10-Xa) and (10-78b) in Eq. (10-77) yields where, from Eq. (9-26b), The use of Eq. (10-801 in Eq. (10-79) gives the explicit dependence of r, on j for TM 11locit.s7- . - A sketch of the no&alize(lz, is shown in Eig. 10-9, which reveals the existence of a minimum. i attcn- finite :es in .
.
. I 10-4 RECTANGULAR ~ A ~ E G U I D E S - The analysis of pdnhel-plate waveguides in Section 10-3 assumed the plates to be of an infinite extent it the transverse x direction; that is, the fields do not vary with x. In practice, these plates are always finite in width, with fringing fields at the edges. Electromagnetic endgy will leak through the sides of the guide and create undesirable stray couplings to other circuits and systems. Thus, practical waveguides are usually uniform structures of a cross section of the enclosed variety. The simplest of such cross sections, in ternis of ease both in analysis and in manufacture, are rectanguhr and circular. In this section we will analyze the wave behavior in hollow rectangular waveguides. Circular waveguides will not be treated in this book, because to do so requires a knowledge of the properties of Bessel functions. Renders possessing such - knowledge. however; would have little difficulty following an .~n;iiysis of circular . sevquidr.~ in more advanced books. bcc;luse the procedure is the same as dem~bed here. Rectangular waveguides are much more commonly used in practlce than circular waveguides. In the following diw.r:i~iotl. wc .lr:iw 011 ~iie n ~ i i ~ c r ~ i ~ l 111 S C C I I , ~ ~ 10 1 con'crnlng ~ C I ~ C I X ~ wavc bcliaviorr along unrllml guiding struclurcs. l'rupapation of nrnc- harmonlc waves in the + z direction wirh a propagation constant 7 is considered. -.
. I M and TE modcs will be discussed scpamtcly. As wc h;~vc notcd pmv~ously, TE\I wave, cannot exist B a single-conductor hollow or dielectric-tilled wnvegulde. 10-4.1 TM Waves in Rectangular Waveguides Consider the waveguide ketched in Fig. 10-10, with its rectangular cross sec:ion of sides u and b. The enclosid dielectric medium is assumed to have constitutive param- eters r and p. For TM waves, H, = 0 and Ez is to be solved from Eq. (10-22). Writing E&, Y 9 4 as . . _ . I , , . I .
. 7. , " , :. ' .
: . . -. . . .. .. , ,'. ?. .y . . .
. . ' .. , .. I ,. . . . . . . . . , . I . . . a , . . . . i . > : , ' . . : - . ' ., .-.;,. ,a ?iT .-... 'l.,'.. I;: . . ; .. , . .
; - , , . . . 468 " WAVEGUIDES AND CAVITY RESONATORS 1 10 . . , : - c . - . . - . ," . . ,. - . . we solve the following second-order partial differential equation: % C , J a2 aZ ( + + h Z (10-85) it f 4 Here we use the method of separation of variables discussed in Section 4-5 by I ?C letting s' Substituting Eq. (10-86) in Eq. (10-85) and dividing the resulting equation by X(x) Y ( y), we have I t Now we argue that, since the left side of Eq. (10-87) is a function of x only and the 1 I right side is a function of y only. b6th sides must equal a constant in order for the equation to hold for all values of .K and y. Calling this constant k : , we obtain two 4 separate ordinary diffcrcntinl equations: 2 Y ' ) - 7 - ddF +~;YO.)=O. (10-89) - where k2 = p - ,p 'c' (10-90) The possible solutions of Eqs. (10-88) and (10-89) are listed in Table 4-1. Section 4-5. The appropriate forms to be chosen must satisfy the following boundary F , conditions. 1. In the .
K direction: EP(0, y) = 0 . EP(a, y) = 0. 2. In the y direction: E~(.Y. 0) = 0 Ey(s, 6) = 0. Obviously, then, we must choose: X ( x ) 111 the form of sin li,.~, Y ( y ) in the form of sin k,y, 7 . ,. - d . 466' WAVEGUIDES AND CAVITY RESONATORS / 10 f I I TE modes Fig. 10-9 NormaIized atten- uation constant due to finite conductiJity of the plates in parallel-plate waveguide. TE Modes In Subsection 10-3.2 we noted that the expression for the propa_eation constant for TE waves between parallel plates is the same as that for TM waves. It follows that the formula for a, in Eq. (10-75) holds for TE mode~iiwell. In order to determine the attenuation constant a, due to losses in the imperfectly conducting plates, we again apply Eq. (10-77). Of course, the field expressions in Eqs. (10-68a). (10-68bh and (10-68c) for TE modcs must now bc used. We have A normalized a, curve based on Eq. (10-53) is also sketched x, for TM modes, a, for TE modes does not have a minimum tonically as f increases. in Fig. 10-9. Unlike but decreases mono- 10-4.1 Rectan and the proper so~utibn for E;(X, y) is E:(x, y ) ' = E, sin (10-92) \ and the . : for the :sin two I . , From Eq. (10-go), we have - The other field components arc obtained from Eqs. (10-23a) through (10-23d): where , y mn E , o ( ~ , yi = -2 (7) E, cos x ) sin (y )) . r y nx ~ e ( x . 1.: = -p ( , ) E, sin x) cos ( 7 y ) . , j o c nn H ~ L Y , y = ( T ) ~ , , sin (yx) cos ( 7 y ) Every combination of the integers n i and 11 defines a possibic mode that may be designated as the TM,,, nlode; thus there are a double infinite number of TM n o d s . The first subscript denotes the number of half-cycle variations of the fields in the x-direction, and the second subscript denotes the number of half-cyc!e variations of the fields in they direction. The cutoff of a particular mode is the condition that makes y vanish. For the TM,,,,, mode, the cutoR frequency is L which checks with Eq. (10-30). Alternatively, we may writo ---- where i , is the curoj"~uve1enyth. , For TM modes, neither m nor n can be zero. (Do you know why?) Hence, TM,, , I , . , . . mode has the lowest cutoff frequency of all TM modes in a rectangular waveguide. , . ' The expressions for the phase constant j 3 and the wave impedance Z , for propagating modes in Eqs. (10-33) and (10-38), respectively, apply here directly. Example 10-6 (a) Write the instantaneous field expressions for the TM,, mode in a rectangular waveguide of sides a and 6. (b) Sketch the electric and magnetic field lines in a typical x-y plane and in a typical y-z plane. a) The instantaneous field expressions for the TM,, mode are obtained by multi- plying the phasor expressions in.Eqs. (10-92) and (10-94a) through (10-94d) with e ~ ( u t - P : ) and then taking the real part ofthe product. We have, for in = n = 1 . ----- E,(x, y. 3; t) = $-(;)TO sin ( : x).-cos (% y ) sin (wt - pi) (10-97b) . . E-(x, y, z; t) = E, sin - .
u sin - y cos (ot - pz) (r ) ( ; (lo-97c) HJx, y, z; t) = -- ma (?) E,, sin ( : X) cos (F g) sin (or - Bz) (10-97d) h ' b H,(.x, y, z ; t) = T E, cos - x sin - y sin (ot - jz) (10-97e) 1 1 - ' ( ( ) ( ; ) where b) In a typica I x-y plane, the slopes of the electric field and magnetic field lines are @IE = 9 tan ( : x) cot ( ; .
y ) h ( ; I H = -; cot (z x) tan (i y) - These equations are quite similar to Eq. (10-60) and can be used to sketch the E and H lines shown in Fig. 10-ll(a). Note that from Eqs. (10-99a) and (10-99b) :c field '\ multi- -- ) 94cl) 17 = 1, .ines are I \ ' ; .
: : . ? -. 3 .. -- , : . i r' ' ; J o ~ /. RECTANGULAR WAVEGUIDES 471 ... 1 - Electric field li,m -- - - - Magnetic field lines Fig. 10-11 Field lines for TM,, mode in rectangular waveguide. . i ~ n d u t i n g that E and H lines are everywhere perpendicular to one mother. Note also that E liiles arc normal and that H lines are pir:illel to conducting guldc walls. Similarly, in a ijpiad y r p h c , say. Br .Y = u/2 or sin (n.i/(r) = 1 sod cos (7rs/(1) = 0. we h;.vc (?)[: = $ (i) Cot (i y ) tan cwt - pzi, and ti has only an s-component. Some typical'^ and H lines nrs drawn in Fig. 10-1 l(b) for r = 0. 10-4.2 TE Waves in Rectangular Waveguides For transverse electric waves, .!iz = 0, we solve Eq. (10-11) for I f z . We write H,(x, y, z) = H;(X, ~ l ) e - ~ " , (10-100) where H:(x, 2) satisfies the following second-order partial differential equation: Equation (10-101) is seen to be of exactly the same form as Eq. (10-85). The solutron for R;(I. fl must s m f y :he fcllcwing !xmdari cuiidit~onb. .. - -. 1. I11 the s-dircction: 2. In the y direction: ,.-. -= aH;o O & O ~ i t y = o ZY (10-102c) aHp -= JY 0 (Ex = 0) at y = h. (10-102d) It is readily verified that the appropriate soiution for H-?(~, )-) is (10-103) 4 . The relation between the eigenvalue hand (mn/a) and (nnib) is the same as that given in Eq. (10-93) for TM modes. The other field components are obti~ined from Eqs. (10-42a) tlirough (10-42d): j o p nn E'(-y. Y ) = - ( .
) Ho COS (~p ) sin (y ,)-\ (10- 1 h ' . , E,:(-Y,J) = -- "o" ( I : ) - HO sili r.r - X ) cos (y y ) 1 10 - 1 04t1) hZ y i m H.x~, 1)) = 2 (4 H , sin ( y .y) cos (s y ) (10-IOJC) . , 7 nn " " 9 Y ) = ( ; ; : I H~ cos (F .
) sin (y .), where y has the same expression as that given in Eq. (10-95) for TM modes. Equation (10-96a) for cutoff frequency also applies here. For TE modes, either rrz or r1 (but not both) can be zero. If a > h, the cutoff frequency is the l o w c ~ whcn nt = 1 and 11 = 0: The corresponding cutoff wavelength is Hence the TE,, mode is the dorninunt mode oj u rectangular wuueguide with a > b. Because the TE,, mode has the lowest attenuation of all modes in a rectangular . waveguide and its electric field is definitely polarized in one direction everywhere, it is of particular practical importance. 10-4 RECTANGULAR WAVEGUIDES 473 ' I ' . , Example 10-7 (h) write the instantaneous field expressions for the TE,, mode in a rectangular waveguide having sides u and b. (b) Sketch the electric and magnetic field lines i n typical;-y. y-r, and x-r planeii(c1 Sketch the surface currents on the - guide walls. . f Solution l a) The instantaneous field expressions for the dominant TE,, mode are obta~ned by multiplying the haso or expressions in Eqs. (10-103) and (10-104a) through (10-104d) with e"".'"' and then taking the real pan of the product. We have, for m = 1 and 1 1 = 0, EJx, .
, : , z ; t) = 0 (10-107a) h) We scc from k ; r . 1 l;) IO72 j iliro~~gl~ ( 10 - 1 O?f) i11;it [hi. TE, ,, mu uni? three nonzero fie!d -omponcnts-namely. E,, H,, and Hz. In a typicnl I-) plane, say, when sin (or - p:) = 1, both E, and H, vary as sin j q i ; i ) and arc independent of y, as jhown in Fig. 10-l?(a). . . ln a typical y-z plane, for example, at r = u/Z or sin (n.x/a) = 1 and cos (nx/a) = 0, we o~ily have E, and H,, both of which vary sinusoidally with QI. A sketch of E, an3 i l , ;I[ i = 0 is givcn in Fig. 10-1?(b). The sketch in an x-; plane will show all three nonzero field components - Ey. FI,. and 1-1;. Thc i l ~ p c of h e I 1 lines : ~ t i = O is govcrncd by t l ~ a kiiIo\iinp equation: $ ("1 tan (" s) tan p : , which can be ~iscd to dciw thc H lincs in Fig. 10-i7(c). Thcc lina ;lit ini1i.pr.n- dcllL 0I,jn. e) The surface curreni ~ h s i t y an guide ivalls, J,, is related to the magnetic field intensity by Eq. (7-5Ob): ------ Magnetic field lines Fig. 10-12 Field lines for TE,, mode in rectangular waveguide. 10-4. Wave. - Fig. 10-13 Surface currents on guide walls for TE:, mode in rectangular waveguide. . . .:.P Q .I \: . 1 0 4 RECTANGULAR WAVEGUIDES 475 -. . - 1 , . 4 - .
. I .
, . . . . .- $ , , ' ? " . . : b ' . I . . where a, is the outward normal to the will surface and H is the magnetic field . . intensity at the &all. We have, at t = O7 ,,'" . ' .
. . ,' I JS(x 0 ) = -4N,(O, y, z; 0) = -a;&@ cos pr (10-Ilia) J~(x = a) = a,,h,(u, y, z; 0) = J,(x = 0) (10-lllbj J,(y -0) = a.H,(x,O,r;O) - n,H,(x, 0,z; 0) (10-111c) J,(Y = b) = -J,(y = 0). (10-1 1 Id) The surface currents on the inside walls at x = 0 and at y = h are sketched in Fig. 10-13. 10-4.3 Attenuation in ~ e c t a n ~ u l a r Waveguides Allcnuntion for propagating modts rodits whm there are losbes in the dielec~ric and in the imperfectly conducting guide walls. I3ecause these losses are uscilly very snull, we will assume, a? In the casc of parallel-plate waveguides, that thc trnnsversc field patterns are not appreciably affected by the losses. The attenuation constant ,Y - due to losses in the dielxtric s:io be obtained by sobstitutin, - E + ( ~ , ' J ( : J J for c in Eq. (10-95). The result is exactly the same as that given in Eq. (10-751, which is rcpeated below: , where o and ii are the conductivity and intrinsic impedance of the dielectric m L' d' lum respectively, and J ' , is given by Eq. (!0-96a1. - i o determine the attenuation constant due to wall losses, we make use of Eq. (10-77). The derivation:; of r, for the general TM.,,. and TE,.,, modes tend to be tedious. Below we obtain the formula for the dominant TE,, mode, which is the most important of all propagaiing modes in a rectangular waveguide. For the TE,, mode the only nonzero field components are E,, H,, and Hz. Letting m = 1, n = 0 . and h = (lr/uj in Eqs. (10-1046) and (10-104c), we calculate the time-average power i-lowing through a cross section of the waveguide: In order to calculate the time-average power lost in the conducting walls per , .
c ' . L k , I . . unit length. we must consider all four' waIls. From E q s . (10-110), (10-1031, and . .- b (10-104c) we see that q I J:(.u = 0) = J:(x = a) = -a,HP(s = 0) = -a,Ho (10-114a) and D pa = , , 0 s ( 2 .) - a. - H . sin ( 2 .). 7T The total power loss is then double the sum of the losses in the walls at s = 0 and b at y = 0. We have and Substitution of Eqs. (10-116a) and (10-116b) in Eq. (10-115) yields The last expression is the result df recognizing that Inserting Eqs. (10-1 13) and (10-1 17) in Eq. (10-77), we obtain i14a) 114b) j and -115) !16a) r 1 l6b) 1-1 17) i-118) .. -, . >~-.kp... &if A..'~+.,.'. ' 3 .
. .
. . . - . -a- 1 '.--t>- . . , 4 . R~CTANGUUR WAVEGUIDES 477 r. .
. . '< . . "' P'" "-'C ' : - - \ . I - I .. : Equation (10-118) reveals a rather complicated dependence of (o,h,, on the + -, ratio (M). It tends to infinity when f is ddse to the cutoff frequency, decrezses toward a m i n i m 6 as f increases, and increapes again steadily for further increases in f. v.: 7' For a g i a n guide width u, the attenvatian decreases as b increases. However, increasing b also decreases the cutoff frequency of the next higher-order mode TE, (or TM,,), with the consequence that thc available bandwidth for the dominant TE,, mode (the range of frequencies over which TE,, is the only possible propagating mode) is reduced. The uiual compromise is td choose the ratio b/a in the neighbor- hood of +. .
I Example 10-8 A TE,, wave at 10 (GHz) propagates in a brass - o , = 1.57 x 10' (S/m)-rectangular waveguide with inner dimensions a = 1.5 (cm) and b = . 0.6 (cm), which is filled with polyethylene-c, = 2.25, p, = I, loss tangent = 4 x lo-'. Determine (a) the phase constant, (b) the guide wavelength, (c) the phase velocity, (d) the wave impedance, (e) the attenuation constant due to loss in the dielectric, and (f) the attenuation constant due to loss in the guide walls. Solution: At f' = 101° (Liz), the wavelength in unbounded polyethylene is The cutoff frequency for the TE1, mode is, from Eq. (10-105), U j;=- - - 2 x loH - = 0.667 x 10" ( H z ) 2~ 2 x (1.5 x lo-') a) The phase constant is, from Eq. (10-1 IS), = 7 4 . 5 ~ = 234 (rad/m) b) TIle guide wavelength is, from Eq. (10-34), c) The phase velocity is, irom Eq. (10-36), . , -. t , .
' .
.
' , , . 6 8 4 . . r t "" -478 WAVEGUIDES AND CAVITY RESONATORS / 10 > < , . 8 , .
, . a d) The wave impedance is, from Eq. (10-47), .. . , . , , 2 - _ ' I . t L -, 8 (~n),, = " JEi; 3 7 7 7 ~ . J = 0.745 = 337.4 (R). e) The attenuation constant due to loss in dielectric is obtained from Eq. (10-112). The effective conductivity for polyethylene at 10 (GHz) can be determined from the given loss tangent by using Eq. (7-93): = 5 x (Slm). Thus. d C( --ZTE= 5 1 0 - ~ d - 2 2 x 337.4 = 0.084 (Nplm) f ) Theattenuation constant due to loss in Lhc guidc wdls is Ibittid Srvm Eq. ((0-1 I()). . - We have, from Eq. (9-26b). . --. 10-5 DIELECTRIC WAVEGUlDES In previous sections we discussed the behavior of electromagnetic waves propagating , - along waveguides with conducting wails. We now show that dielectric slabs and rods without conducting walls can also support guided-wave modes that are confined essentially within the dielectric medium. Figure 10-14 shows a lon_gitudinai cross section of a dielectric-slab waveguide of thickness d. For simplicity we consider this a problem with no dependence on €0, PO dielectric-slab wavepuide. 0-112). L'LI from ' . i 0-1 19). ,-- ';p In) -B!m). Fagating and rods confined avcguidc ,&nce on r- .n G ' 1 ! - i . . - % . ( 4 . . - - . 'L'.,,,. I ; the x coordinate. Let ed and9,ud be, respectively, the permittivity and permeability of the dielectric slab,.which is situated in free space (a,, p,). We assume that the dielectric is lossless and that waves propagate in the +z direction. The behavior of TM and , TE modes will now be halyzed separately. , , 10-5.1 TM Waves alohg a Dlelectric Slab For transvcrsc tnagnctic waves, M, = 0. Sincc thcrc is no x-dcpendcncc, Eq. (10 -53) applies. We have where 7 7 11- = y- + w2/J€. Solutions of Eq. (10-120) must be considered in both the slab and the free-space regions, and they must b? matched at the boundaries. In the slab region we Jssume that the wavcs propagate in thc +:direction without attenuation (lossless dielectric); that is, we assume The solution of Eq. (10-120) in the dielestri~ slab may connin both a sine term and a cosine term. which are respectively an odd and an even ful~ction of y: EP( y) = E, sin k,y + E, cos k,y, d lyj I - > 2. (10-123) where A : = &pdcd - lj2 = hJ. (10-124) In the free-space regions I y > d/2. and y < - d/2), the waves must decay exponentially so that they are guided along the slab and do not radiate away from it. We have 10.' - dl 2) d Y 2 - i (10-12ja) Lq( y) = c , ~ z ( P + ~ / ~ ) re- ' ~ 1 5 - - ri 110-12jb) -- . 2' where Equations (10-124) and (10-126) are called dispersion relations because they show the nonlinear dependence of the phase constant /3 on w. At this stage we have not yet determined the values of k, and a; nor have we found the relationships among the amplitudes E,, E,, C , . and C,. In the following, we will consider the odd and even TM modes separately. a) Odd TM Modes. For odd TM modes, E : ( !
) is described by a sine function that is antisymmetric with respect to the J = 0 plane. The only other field components, E;(y) and I I : ( ~ ) , are obtained from Eqs. (10-23d) and (10-23a) respectively. i) In the dielectric region, 1 j .
I I dl?: 41. EP(y) = E, sin kyy jB E:(y) = -- li, E, cos k,y (10-127b) jed H;(y) = - E, cos k,y. (10-137~) - k?. - _ -. --. ii) In the upper free-space region, y >_ d/2: where C , in Eq. (10-125a) has been set to cqual E, sin (k,d/2), which is the i d u e of Ez(y) in Eq. (10-127a) at the upper interface. y = d/2. iii) In the lower free-space region. y I -d/2: ( k ; d ) e z O + d i I i E P ( ~ ) = - E, sin - (10-129a) where C1 in Eq. (10-125b) has becn set to equal - E,, sin (kyd/2), which is the value of E:(y) in Eq. (10-127a) at the lower interface y = -42. Now we must determine k , and TX for a given angular frequency of excitation w. The continuity of H , at thi dielectric surface requires that Hl(d/2) computed . . I I . , . , ' . . .
: . ' . ..q ?,. ., "." p... ' 1 2 . avu we , . I . - , , ' . '''i.~ , A - (Odd TM modes). (10-130) I .
. / -. , I By adding dispersion relations Eqs. (10-124) and (10- 126), we find \ Equations (10-130) m d (10-131b) can be combined to glve an expression in ' which k, is thc only cnhno\vn: Unfortunately 1 1 1 .
: tsanscendentri! quation. Eq. (10-132). caniiot be sol'-ed analytically. But for ;L given o and given values of E,, p,, and ti of the dielectric slab, both the left and the right sidcs of Eq. (10-132) can be plotted versus k,.. The intersections of ..he two curves give the values of k , for odd Th4 modes. of which there are o h y 2 finite number, indicating that there are only a finite number of possible modes. This is in contrast with the infinite number of modes possible ir. waveguides with exlosed conducting walls. We note from Eq. (1 0- 1 X'a) that EZ = 0 for !
. = 0. Hence, a perfecdy son- (111cli1ip ~ I ; I I ~ c m y ,>c ~IIII.O(III~:(YI to coi~i~,i[lt: w i ~ l ~ ~ l i c 1, . ii pl.111~: ~ ~ ~ l / i t ) ( ~ t I I : I I I I I I l o v I I I i i ; i 01' odii 1',U w a v o propagating along a dielectric-slab waveguide of thickness d are the same 2s those of the corre~pocdin~ Th4 modes supported by a dielectric slab of a thickness dl2 that is backed by a perfectly conducting plane. The surjuce impedance looking down from above on the surface or ciclectric slab is El ; : , , = - -- - Y - j - (TM modes), N,O we, (10-133) 482 '."WAVEGUIDES AND CAVITY RESONATORS / 10 ,. , , - The other nonzero field components, E; and H : , both inside and outside the i . , dielectric slab can be obtained in exactiy the same manner as in the case of odd I TM modes (see Problem P.lO-25). Instead of Eq. (10-130). the characteristic relation between and a now becomes a -= -- k,d E0 cot - 7 (Even TM modes), (10-135) 1 ; ,. Ed - I I which can be used in conjunction with Eq. (10-131b) to determine the transverse wavenumber k, and the transverse attenuation constant a. The several solutions a \ correspond to the several even Thf modes that can exist in the dielectric slab waveguide of thickness d. Of course. in this case a conducting plane cunnot be placed at !
: = 0 without disturbiw the whole field structure. From Eqs. (10-124) and (10-126), it is easy to see that the phase constant, P, of propagating TM waves lies between the intrinsic phase constant of the free space, - -- k, = o J,LL,,E~. and that of the dielectric, k, = Q Jpdcd; that is. As approaches the value of w&, Eq. (10-126) indicates that a approaches zero. An absence of attenuation means that tlic waves are no longer bound to the siab. The limiting frequencies under this condition are called the cutof frequencies of the dielectric waveguide. From Eq. (10-124) we have k,. = o,,/L~,c, -.poco at cutoff. Substitution into Eqs. (10-132) and (10-135) with cc set to zero yields the following relations for TM modes. At cut-off: Odd TM Modes tan r+ = 0 I I Even TM Modes I I o d I c o t ( + , / 3 ) = 0 I It is seen that j,, = 0 for n = 1. This means that the lowest-order odd TM mode can propagate along a die!ectric-slab waveguide regardless of the thickness of the siab. As the frequency of a given T X I wave increases beyond the corresponding cutoff frequency, a increases and the wave clings more tightly to the slab. . . , " . . , , . .-.. ,,.,.., " . .
) . . , . . . : : . , , . ;,.: i .
!
; , : , ; .:h:; " . . : . : . - . ,. ,. ie the ' : : a . , i ,f odd , . . . .: / :. svcrse utions c slab lorbe .
I : . p, of space, /? s zero. 2 s. of the cutoff. iowing A d 7 - ness of .on2 . . 30-5.2 TE Waves along a Dielectric Slab For transverse electric waves, Ez = 0 , and Eq. (10-67) applies wherc k, has been defined in Eq. (10-121). The solution for H:(y) may also contain both a sine term and a cosine term: (1 H,;(Y) == H . Sill /i;J'-t- He COS k,$, b.1 7. (10-138) & In the free-space regions (,v > dl2 and : r < -d/2), the waves must decay exponentially We write where u is defined in Eq. (10-126). Following the same procedure dr used for TLI waves, we consider the odd and even TE modes separately. Besides H;(J). the only other field components are H:iy) and E:(),). which can be obtained from Eqi. I 10- 42b) and f 10-42c). ii) In the upper free-<pace region, J- 2 (112: .
.
& .
.
' " . . . .
A '.. . I . , "484 WAVEGUIDES AND CAVITY RESONATORS 1 10 iii) In the lower free-space region, y I - 4 2 : - , ~ : ( y ) = - (H. sin y) 8 ( Y t d 1 2 ) ' (10-141a) A relation between k, and a can be obtained by equating EP(y). given in ' .
, Eqs. (10-139d and (10-140c), at J . = d/2. Thus, ' l o tan - (Odd TE modes), which is seen to be closely analogous to the characteristic equation, Eq. (10- 1301, for odd TM modes. Equations (10-131b) and (10-147) can- bcqmbined in the manner of Eq. (10-132) to find k,. graphically. Froin li,.. x can bc found from Eq. (10-131b). From a position of looking down from above. the surface impedance of the dielectric slab is EO - -IT- - -.
( J ' P o H : IT (TE modes). (10-143) which is a c:~p;~citivc rc:lct;~ncc. Tlcnct:. ir TE .srir./irc.r. w t r r v cetrrl lw . s ~ r p p ~ r . / , ~ r l /)I rr c . u p ~ c . i / I ' r . ~ , S I I I ~ \ I I ~ ~ P . b) Even TE Modes. For even TE modes, is described by a cosine function that is symmetric with respect to the y = 0 plane. H;(.V) = H,, cos k , .
!
.
, 1j.l 5 dl?. (10-144) The other nonzero field components, H: and E,O, both inside and outside the dielectric slab can be obtained in the same manner as for odd TE modes (see Problem P.lO-27). The characteristic relation between k,. and a is ciosely anal- ogous to that for even TM modes as given in Eq. (10-135): 1t is easy to see that the expressions for the cutolf frequencies given in Eqs. (10-136a, b) apply also to TE modes. The characteristic relations for all the propagating modes along a dielectric-slab waveguide of a thickness d are listed in Table 10 -2. 141 b) -141~) ven in ' . 1-142) -130). n the lrom /- of the 3- 143) ~d by u .nctlon (J- 144) ide the x s (see , tln~11- 0 145) f -- .n Efl.; , 1 1 1 :e listed Example 10-9 A dielectric-slab waveguide with constitutive parameters p, = p,, and ed = 3 . 5 0 ~ ~ is situated in free space. Determ~ne the minimum thickness of the slab so that a TIM br TE wave of the even lype at a frequency 20 GHz may propagate along the guide. Solution: The lowest T !
/ ! and TE waws of the even type have the same cutoff frequency along a dielectric-slab waveguide: Therefore, C d",," = -- 7- Example 10-10 (a) Obtain an :~pproximatc espression Sor the decaying r;ltc: of the ~Io~ni~i:~nl TM SLII~:ICC w.ivc oi~isi~lc 0 U il very ~hin ~liclcctric-s1:1> \v:~vquidc. ( b j Find the time-a;.ciage pow: per unit width transmitted in the transverse direction along the guide. Solution a) Tile dominant TM wave is the odd mode having a zero cutoff frequency- f , , = 0 for n = 1, independent of the slab thickness (see Table 10-2). With a slab . . < , . .-. , 'dk".486 ' I 3) . . ' . , WAVEGUIDES AND CAVITY RESONATORS 1 10 I , .... - . . i - that is very thin compared to the operating wavelength; k,,d/2 < < 1, tan (kyd/2) z k#2, and Eq. (10-130) bepmes . Z a s - k , d . €0 2 (10-146) 2% Using Eq. (10-131a), Eq. (10-146) can be written approximately as In Eq. (10-147), it his been assumed that rd/2 < < ed/eO ?. b) The time-average Poynting vector in the +: direction in the dielectric slab is Pa, = $h(- aYEy x a,H,). Using Eqs. (10-127b) and (10-117~). we have P,,,. = n,P,,,:and (1; 2 Pa. = 2 J' iP,, d!. = , 0 cos' ( k ~ ) dy k, where and In this section we have studied the characteristics of TM and TE waves guided by dielectric slabs. The same principles govern the transmission of light waves along round quartz fibers that form optical waveguides. Optical waveguides are of great importance as transmission media for communication systems because of their low- loss and large-bandwidth properties. Their analysis requires the knowledge of Bessel functions which we do not assume in this book. 10-6 CAVITY RESONATORS We have previously pointed out that at UHF (300 MHz to 3 GHz) and higher fre- quencies, ordinary lumped-circuit elements such as R. L, and C are difficult to make. and stray fields become important. Circuits with dimensions comparable to the opcr:lt ing w:lvclcnplh hccomc cllicicnr r:~tli:l tors and will interfere with othcr circ~~ifs and systems. Furthermore, corivcritiolial wire circuits tend to have a high elYcctive resistance both because of energy loss through radiation and as a result of skin effect. To provide a resonant circuit at UHF and higher frequencies, we look to an enclosure (a ca\,ity) completely surrounded by conducting walls. Such a shielded enclosure confines electromagnetic lields inside and furnishes large areas for current flow. thus eliminating radiation and high-resistance effects. These enclosures have natural Long great :r low- Basel .: : r f : e - nuhe. io the . . ":cJlts kc:{- . ciiect. 2:os - i0:;UrZ : How, xtural (a) Probe c.sci:at~on. (b) Loop excitation. Fig. 10-13 Excitation of cavity modcs by a coaxial line. rcsonant frequencies anc: it vcry high Q tquitlity factor), i~nd itre callcd ~ . i l c i r j ~ r C . w nutors. In this section WL will.study the properties of rectangular cavity resonators. Consider a rectanghr waveguide with both ends c!osed by a conductin. wiill. Thc interior dimensions 01' tllc ci~vity ;iru n, b, ;~nd (1. its shown i n frig I0 - 15. Sinm both TM and TE modei can exist in a rectangular guide, we expect Tkl and TE modes in a rectdn:u~ar resonator too. However, the designation of TM and TE modes in a resonator is .iot m i p r because we are free to choose i or y or r as the "direction of propap~ti~m": that is, tllere is no unique "longitudinal direction." For example, a TE mode with respect to the : axis could be a TM mode with rzspect to the y axis. For our purposes. we cbnorr the : uxis us rbr rejetwice "direction of propaq3tion." In actuality, the existent: of conuucting end walls at .- = 0 and r = d gives rise to multiple reflections and bets up standing waves; no wave propagates in an enclosed cavity. A three-symbol (~nnp) subscript is needed to designate a T I M or TE stand- ing-wave pattern in a cavity resonator. 10-6.1 T M , , , , Modes The expressions for the rransverse variations of the field components for TM,,, modes in ii iv:ivspide ls~vc hccn givcn in Eqs ( 10--92) ilntl ( 1 ii %I:I, b. c. ~ 1 ) . h ~ t c that the longiti1dik11 v:inilios lor 2 waic traveling in the +: direction is dcscribeil 1.. ",, -1 facLOr e-:'= O r r-;,:': . as indichted in Eq. (10-54). This wave will be reflected by the end wall at .- = d ; and ihe reflected wave, going in the -: direction. is described by a factor rjD'. he si~perposition of a term with r-I#z and another of the same amplitudet with r " ' resu!is in a standin: wave of the sin pi or cos P; type. Which should it be'? The answer i o this question depends on the particular field component. ' The reflection coefficient at a lrrfcct cocductor is - I. 3 .'.-,. . I ., , ' . . 488 WAVEGUIDES AND C A V I N RESONATORS 1 10 Consider the transverse component E,(x, y, z). Boundary conditions at the . conducting surfaces require that it be zero at z = 0 and z = d. This means that (1) its ,--dependence be of the sin Br type and that (2) P = pnld. The same argument applies to the other transverse electric field component E,(x, y; z). Recalling that the appearance of the factor (- y) in Eqs. (10-94a) and (10-94b) is the result of a differentiation with respect to z, we conclude that the other com- ponents E,(x, y, z), H,(x, y, z), and H,(x, y, z), which do not contain the factor (-;)), dust vary according to cos /3z. We have then. from Eqs. (10-92) and (10-94a, b. c, d). the following p11asor.s of thc ficld components for TM,ll,lp modcs ill a rcct;~npl.~r cavity resonator. ~ ( x . y, 2) = E~ sin r : x) sin (y y) cos (7 z) H,(x, y, 2) = -- "'(7) 12 - Eo cos > z ) ) - x s i n - v cos ("$;), (10-149e) where From Eq. (10-95), we obtain the following expression for the resonant frequency for TM,,, modes: f m n p 2 t I 10-5.2 T E , , , Modes For TE,,, modes (E, = 0). the phasor expressions for the standing-wave field com- .
. . ponents can be written from Eqs. (10-103) and (10-104a, b, c, d). We follow the same rules as those we used for TM,,, modes; namely, (1) the transverse (tangential) electric field components must vanish at z = 0 and z = d, and (2) the factory indicates m : st the :r com- ~r ('v), b. c, d), m&u 1-149a) i ; . J-149b) I-149~) n J-1-, -) O-l4e) 0-149f) quency ,lo-i50) /--. .dJ iI- :nc same qentlal) indicates .. : . . . : .' . ,. . .. , ,!., ,. , ,... :. ,!-, . .
: ; !
. . (.' . . . ' ,( ": ' '.,'," ' -7 . . !
; ? f . ... . . .. : . . ' , .. . . . . . , , , , ; , .. '. ' ...; I ! . . . . . . ' . ,: r. .. . , \ L. -. . f + 1 . . ' ' 10-6 [ CAVITY RESONATORS 489 .# ,: % . < i I a ncgative partid .difkrcntialion with rcspcct to z. Thc first rulc rcquircs ;l sin (p.rr:/tl) factor i n E,(x; y; z).knd Ey(x, y, z), as w e l l i n H,(x, y,z); and the second rule . ' indicates a cos (&Id) factor in H,(x, y,:,'z) and Hy(x, y, z), and the replacement of y a by -(psl/d). ~ h u s , jup (y) Ho cos ( y x) sin (y y) sin (7 : ) EJx, Y , 4 = - h2 Ey(x, y, Z ) = ' ) H sin :) 0 s ( ) sin ) (1 0- 1 5 i c) h2 \ a where I?has been given in Eq. (10-143f). The expression for resonant frequency. f , , , , remains the sahe ;IS that obtained for TM,,, modes in Eq. (10-150). Different modes having the same resonant frequency are called dcgenrrate rnorles. The mode with the lowest resonant frequency for a given cavity size is referred to as the riorni~lanr mode. A particular mode n a cavity resonaror.(or a waveguide) may be txcited from a coaxial line by means of a small probz or loop antenna. In Fig. 10-13a) a probe is shown that is the tip o f the inner conductor of a coaxial cable and protrudes into a cavity at a location whcrc thc clectric ficld is a maximum for thc dcsircd modc. Tiic probe is, in fact, an anttnna that coupies electromagnetic energy into the resou~tar. Alternatively, a cavity r:sonator may be excited through thz introduction of a small loop at a piace where the magnetic flux of the desired mode linking the lcop is a maximum. Figure 10-15(b) illustrates such an arrangement. Of course, the source frequency from the coa ikll line must be the same as the rcsonant frequency of the desired mode in the cavlty. AS an example, for the TE,,, mode in an a x b x d rectangular cavity, there are only three nonzero held componznts : . j ( ! ) l l ~ E,. = -4- I !
, , sin 11-'1 .
. --. 7i - Fix = -- H,, sin l r a d H= = H, cos !
: s) sin /: : ) , This mode may be exclted by a probe inserted in the center region of the top or bottom face where E,, is maximum, as shown in Fig. 10-15a, or by a loop to couple _ . , , a maximum H, placed inside the front or back face, as shown in Fig. 10-15b. The 4 , . , best location of a probe or a loop is affected by the impedance-matching requirements of the microwave circuit of which the resonator is a part. 'J ci A commonly used method for coupling energy from a waveguide to a cavity resonator is the introduction of a hole or iris at an appropriate location in the cavity wall. The field in the waveguide at the hole must have a component that is favorable in exciting the desired mode in the resonator. . . Example 10-11 Determine the dominant modes and their frequencies in an air- filled rectangular cavity resonator for (a) a > b > d. (b) n > d > b, and (c) a = b = d. a : . where a. b, and d are the dimensions in the .
u , y, and z directions respectively. Solutiou: With the z axis chosen as .the reference "direction of propagation": First, for TM,,,,, modes. Eqs. (10-14% b, c, d, c) show that ncirhcr 111 nor rr can bl; zero, but that p can be zero; second, for TE,,,,,, modes, Eqs. (10-151~1, b, c, d. e) show that either n r or 11 (1x11 not both rn xnd 11) can hc zcro. hut that p cinnot be zcro. Thus. the modes of the lowest orders arc I--.. TM,,,, TEoll. and TElo,. The resonant frequency for both TM and?^ modes is given by Eq. (10-150). a) For a > h > d: Thc lowest resonant Srcqucncy is where c is the velocity of light in free space. Therefore TM, ,, is the dominant modc. 1 b) For u > ti > b: The iowcst rcsonant frcqucncy is i and TE,,, is the dominant mode. i c) For a = b = d, all three of the lowest-order modes (namely, TM,,,, TE,,,, and i TE, ,,,) have the same ficld patterns. The resonant frequency of these degenerate i 1 modcs is ! I 10-6.3 Quality Factor of Cavity Resonator A cavity resonator stores energy ir? the electric and magnetic fields for any particular mode pattern. In any practical cavity the walls have a finite conductivity; that is, a nonzero surface resistance. and the resulting powcr loss causes a decay of the stored i . The Gents A ; < ; . . : : a+-ytt,- -< -.. avity i-able -I- air- ' = d ; '.\ . . First, zzro, that Y, the , - 50). inant . and ::rate /-- :cular i I S , .tored i energy. The quality fdcror, or Q, of a resonator, like that of any resonant circuit. is a measure of the bandwidth of the resonator and is defined as 1 lme-average energy stored at a resonant frequency Q = 2n . (10-153) Energy dissipated in one period of this freauencv (Dimensionless) . + Let W be the total time-average energy in a cavity resonator. We write where We and Wn, dencte the energes stored in the electric and magnetic field, respertively. If P, is the .ime-average power dissipated in the cav~ty, then the energy dissipated in one period li P, diridsd by frequency, and Eq. (10-1 53) can be wr~rten i s In determining the Q of .
I cavity at a resonant frequency, ir is cus:omary to nssume that the loss is smail cnoqh to allow the use of the field patterns without loss. We will now find thi Q of an n x b x d cavity for the TE,,, mode that has three nonzero field compoaenrs $ivm in Eqs. (10-1524 b. and c). The time-average stored electric energy is where we have used /I-' := ja,n)' from Eq. (10-1491). The total time-a~ernge stored magnetic energy is - - - Po ff; ' 7 ' ) - b I:) - + ( !
: 7 , h ( ?
$ I - =- b + I . (10-l56b) 4 d - ( 2 , From Eq. (10-150), the resonant frequency for the TE,,, mode is I 5 ?
. f l o l = - (10-157) \ ; & 4 . Substitution of fl0, from Eq. (10-157) in Eq. (10-156a) proves that at the resonant frequency We = W , . Thus, w=2w,=2wm=- 8 (10-158) To find P , , we note that the power loss unit area is 'i .you = f (J,/~R, = )IH,~~R,, (10-159) where ~H,I denotes the magnitude of the tangential component of the magnetic field at the cavity w;~lls. The pnwcr l o t ill the : -- (1 (h:~chl a ; ~ l l is tlic .;:line ;IS t11:il .ill 1I1c 1 = 0 (front) wall. Similarly, the power loss ill ille r = (Icfl) wall is tlm s;1111c ;IS t h i in ihc Y = 0 (riphll ~v:~ll: i ~ n d ~lic lxn\cl- Inst ill tllc 1. - 1) ( t ~ ~ ~ , ~ c r ) ill is I I I C >:\tl~c as that in the y = 0 (lowcr) wall. LVc ha\e 1 . 1 . PL = $ ~ a . ds = R , { J : J ; IH.(Z = o)12 b d l + J'Sb JH(X = o)l2 dy dz 0 0 -+ Sod S ; iH,12 dx d~ i kd J ; I H = J ~ d~ d~ - -- Using Eqs. (10-158) and (10- 160) In Eq. (10-155). we obtain T -. ,- ' . .' ~ f l o l ~ o a b d ( a ~ + d2) r- Q l o l = (For TE,,, mode), (10-161) A I . -a I - - - ~ , [ 2 b ( a ~ + d3) + nd(a2 + d2)] 1,- -. . - i e where Jlo, 1x1s been given in Eq. (10-157). Example 10-12 (a) What should be the size of a hollow cubic cavity made of copper s ' I in order for it to have a dominant resonant frequency of 10 (GHz)? (b) Find the Q at that frcqucncy. 'Solution , a) For a cubic cavity, a = b = d: From Example 10-11, we know that TM,,,, 9 TEol 1, and TElol are degenerate dominant modes having the same field pat- I terns. and that 3 x lo8 f i o l = - - - 101° (HZ). 4 2 ' 2 1:rt in the s;me as :he same of copper -1nd the Q b) T h e expression of Q in Eq. (10-161) for a cubic cavity reduces to xi1 0 I P d ~ = f J - 0 1 0 1 = 3R, ' i i i a l ~ o o . (10-162) For copper, o = 5.80 x lo7 (S/m), we have The Q ola cavity resonator is. thus. cstremcly high comparcd with that obt;linable liom lumped L-C resonant circuits. In practice, the preceding value is sornen hat lower due to losse: through feed connections a n d surface irregularities. REVIEW QUESTIONS R.10-I Why ar? the conmon typcs of ir.~nsmisiion lines not useful ior the long-distance sional trrm.mission of n~icrowiiv,: frequencies in the TEbI mode'! . - R.10-2 What is memt by a a ~ r o f l ' n / l . q u n i c ~ of awaveguide? RJO-3 Why 2re lumped-panmeter elements connected by wires not useiul as resonant c~rcuits at microwave frequencies? R.10-4 What is the govrrning equation for slectric m d magnetic field intensity phasors in the dielectric region of a sttaight waveguide with a uniform cross section'? R-10-5 What -re the rhrze basic types of propagating waves in a uniform waveguide? R.10-6 Define wuce inlpt dance. R.10-7 Explain why single-conductor hollow or dielectric-filled waveguides cannot support TEM waves. R.10-8 Discuss the analytical procedure for studying the chamcteristics of Tb1 waves in a waveguide. R.10-9 Discuss the rna1ytic:ll procedure ior studying the chamiteniiiis "f TE u;liei in i w;lvcgllltlc. .. R.10-I0 What are eiqoic~l!~.~ of a boundair-raise problem? 8.10-11 Can a waveguide have more than one cutoff frequency? On what factors does the cut- off frequency of a waveguiae depend. ' R.10-I3 Is the guide wavelength of a propagating wave in a waveguide longer or shorter than the wavelength in the corresponding unbounded dielectric medium? R.lO-14 In what way does the wave impedance in a waveguide depend on frequency: a) For a propagating TEM wave? b) For a propagating TM wave? c) For a propagating TE wave'? R.lO-15 What is the significance of a purely reactive wave impedance? R.10-16 Can one tell from an w-p diagram whether a certain propagating mode in a waveguide is dispersive? Explain. R.10-17 Explain how one determines the phase velocity and the group velocity of a propagating mode from ~ t s w-/j' diagram. R.10-18 What is meant by an eigenmode? R.10-19 On what factors does the cutoff frequency of a parallel-plate waveguide depend? R.lO-20 What is meant by the dominant mode of a waveguide? What is the dominant mode of : I parallci-plate wnvcguide? -I -. R.10-21 Can a TM or TE wave with a wavelength 3 (cm) propagate in a parallel-plate waveguide whose plate separation is 1 (cm)? 2 (cm)? Explain. R.10-22 Compare the cutol-l' frequencies of TM,, TM,, TM,, (m > n), and TE, modes in a parallel-plate waveguide. R.lO-23 Does the attenuation constant due to dielectric losses increase or decrease with fre- quency.for TM and TE modes in a parallel-plate waveguide'? R.10-24 Discuss the essential differecces in the frequency behavior of the attenuation caused by finite plate conductivity in a parallel-plate waveguide for TEM. TM, and TE modes. R.10-25 State thc boundary conditions to bc satislicd by E, for TM wavcs in n rectangular waveguide. R.lO-26 Which TM mode has the lowest cutoff frequency of all the TM modes in a rectangular waveguide? K.10-27 State the boundafi conditions to bc satislicd by H, for TE waves in a rectangular waveguide. R.lO-23 Which mode is the dominant mcde in a rectangular waveguide if (a) o > h, (b) n < h. and (c) a = h? R.10-29 What is thc cutor wavelength of the TE,, mode in a rectangular wavcguiuc? R.10-30 Which are the nonzero ficld componcnts for the TE,, mode in n rectangular waveguide? - R.10-31 Discuss the general attenuation bchavior caused by wall losses as a function of fre- quency for the TE,, mode in a rectangular waveguide. mode r- I"UIC R.lO-32 Discuss the factors that affect the choice of the linear dimensions n and b for the cross section of a rectangular paveguide. . . . . R.10-33 W h y is it necessary that the permittivity d t h e dielectric slab in a dielectric waveguide . . be larger than that O f the surrounding medium? R.10-34 What are dispersion relations? R.10-35 Can a dielectric-slab waveguide support an infinite number of discrete TM and TE modes? Explain. R.lO-36 What kind of shrfxe can support a TM surfnce wave? A TE surhce wave? R.10-37 What is the dominant, mode in a dielectric-slab waveguide? What is its cutodfrequency '? R.10-38 Does the attenuation of the waves outside a dielectric slab waveguide increase or . decrease with slab thlckhess? R.10-39 What are cavity resonators'? What are their most desirable properties'! R.10-40 Are the field patterns in a cavity resonator traveling waves or standing waves'? How do they differ from those in .I waveguide'! < R.10- 11 In terms of field pmerns what does the TM, , , mode signify? The TE, 2, mode? R.10-42 What is the e%preision f a the resonmt frequency of T M , , , , , , modes in n rectanjulx cavity resonator of dimensicns a x h x d? Of T E , " , , , modes'! . R.10-43 Whar is meant by te~~clrel.c~te t~lodes? - k.10-4 What are the mod:s of the lowest orders in a rectangular cavity resonator? R.10-4 Define the quality factor, Q, of a resonator. R.10-46 Explain why the measured Q of a cavity resonator is lower ~ h a n rhe calculated vnius PROBLEMS P.10-1 Starting from the iwo time-harmonic Maxwell's curl equations in cylindrical zoordi- nates, Eqs. (7-8ja) and ( 7 4 5b), express the transverse field components E,, E,, H,, and H, in terms of the longitudinal corlponents E, and H,. What equations must E: and H, satlsfy'? P.10-2 In studying the wale behavior in a straight waveguide having a uniform bur arbitrary cross section, it is txpedient :o find general formulas expressing the transverse field components in terr~is oftheir longitudinal components. We write 1 . " , . \ i .
. . , where the subscript T denotes "transverse." Prove the followi& relatipns for time-harmonic excitation: - 1 a) Er = -p (y VTE, - a , j q x VTHz) (10-163a) where it2 is that given in Eq. (10-13). P .
l O -3 For rectangular waveguides. ' . a) plot the universal circle diagrams relating tr,/~r and P / k versus],jj, b) plot the universal graphs of uh,. B/k, and ;.,,lib versus fjf,, C) find u,j11, u,/u, P/k, and i.,iiL ' ~ t 4 = 1.251,. P .
1 0 -J Skctch thc c t ~ - - P tliugr;~m of 3 p:~rallcl-pl:~tc waveguide separated by 3 dielect~ic slab of thickness 6 and constitutive pa~uncters ( E . p ) for TbI ,. Th4,. and Tb13 modcs. Discuss a) how b and the constitutive parameters affect the diagram. b) whether the same curves apply to TE modes. - ----.
. P.lO-5 Obtain the expressions for the surface charge density and the surface current density for TM, modes on tllc conducting piatcs o f ;I pa~~liei-phtc w;~vcg~~iclc. Do the currcnts on the two plates flow in the same direction or in opposite directions? P.lO-6 Obtain the expressions for the surface current density for TE, modes on the conducting plates of a parallel-plate waveguide. Do the currents on the two plates flow in the same direction or ii' opposite directions'? P .
l O -7 Sketch the electric and magnetjc field !ines for (a) the TM2 modc and (b) the TE, modc in a parallel-plate waveguide. P.10-8 A waveguide is formed by two parallel copper sheets-0, = 5.80 x 10' (S/m)-sepa- rated by a 54cm) thick lossy Jiclectric - E , = 7.25, ii, = 1. a = 10- l o (S/nl). For :in operllting frequency of 10 (GHz), find /I, z , , , qC, up, u,, and i .
, for (a) thc TEM modc. (b) the TM, modc. and (c) the TM, mode. P .
1 0 -9 Repeat problem P.10-8 for (a) the TE, mode and (b) the TE, mode. P .
l O -1 0 For a parallel-plate waveguide, a) find he frequency (in tel'iils of the cutoff frequency 1;) at which the attenuation constant due to conductor losses for the TM, mode is a minimum, b) obrzin thc ror~nulil for this mini~ri~~rn ~~t~:nu;~lion constant. c) calculate this minimum 2, h r the T M , nwdc it' ~llc p:~r;dlcl plalcs arc niadc ol'coppcr and spaced 5 (cm) apart in air. P .
l O -1 1 A parallel-plate waveguide made of two perfectly conduciing infinite planes spaced 3 (cm) apart in air operates at ; 1 frequency I0 (GHz). Find the maximum time-average power that can bc propagltcd pcr unit width of the pitic without o voltagc hrcnkdown for a) the TEM rnodc. !I) thc 'fM , nlotlc, c) tllc 'l.ll, I I I C ~ C . ipsct..' power PROBLEMS 497 , P .
l O -1 2 Prove that the foflowing wavclcngth rclation holds for n uniform wavcguide: . ; - '% & ? I . I where i., = guide wavelength, i = wavelength in unbounded dielectric medium, and i , = u/f, = cutoff wavelength. P.10-13 For an (1 x b re~tas~gular waveguide operating at the TM,, mode. a) derive the exprcssicns for the surfacc current dcnsitics on the conducting walls, 11) skctch illc surfacc c ~ ~ r r c n ~ s un Lhc walls at s ; = 13 and at ) = 6. P .
1 0 -1 .
1 Calcuiare and list i 1 ascsndiny order the cutoff ticqucncics (in icmms ol' the cutolf frc- qucncy of the dominant midz) of an u x b rcc~angular waveguidt: for tht Ijl!owlng modes: TE,,, TEl,7TEl,,TE,2. TE c: TM, TM12. :mi TM,? [a) i f a - 3h. anti ~ h ) i T t ~ = h 1'.10-15 An air-lillsd ' r x b I!, < .I < 36) rec~angular wavsguidz 1s to b\: consrrucred to opcmrc at 3 ICHz) irl the dornin;ill~ i:~oclc. :I/c clcsirc t l ~ opmrting Srcqtlcncy 10 hc ;it 1e:lat 20'1; h~ghcr Lhall ll~c culoif lrcqucncy ol't.ic clu~nina~i~ nlodc .1nd :dso at lens^ 20:'" below ~112 cutotr rrquencq of the next higher-ordcr mod-. a) Give a typicril dtsiy ; for the dimensions u and b. b) Calculate for your d-sign 1 , I(,,, i .
, , and .he ~c'.tve impedance at thi opcratlng frequency. P.lO-16 Calculate and cornrxs the values of /, u,, u,, i,, :lnd ZT,,,, for .
t 2.5 (cm) x 1.5 (cm) rcctangular wavcgu~tlc opcratng at 7.5 ICHz) P) ~f the Lvavegu~de 1s lioilow. h) if thc waveguide is t~lled with .I d~electnc mcdiurn chnractcrizcd by E, = 2. p, = 1 2nd a = 0. P.10-17 An air-filled rcctangular wavcguide made oi copper and having transverse dimensions u = 7.20 (cm) and b = 3.40 (cm) operates at a frequency 3 (GHz) in the dominant mode. Find (a) fc, (b) i.,, (c) a,, and (d) the distance over which the field intensities of the propagating wave will be attenuated by 50%. P.10-18 An avcrdgc ppowcr o, 1 (kW) at I0 (GHz) is to be delivered to an antenna a: the TE,, mode by an air-filled rectangular copper waveguide l (m) long and having sides o = 1.35 (cm) and h = I .OO (cm). Finti P.10-19 Find the maximum amount of 10-(GHz) hverage power that can be transmitted through an air-filled rectangular wavegl~de-u = 2.25 (cm), b = 1 .OO (cm)-at the TE,, node without .I breakdown. P.10-20 Determine the value,of (fl') at which the attenuation constant due to conductor losses in an a x b rectangular waveguide for the TE;, mbdeis a minimum ". P.lO-21 Find the formula for the attenuation constant due to conductor losses in an a x b rectangular waveguide for the TM , , mode. P.lO-22 Show that electromagnetic waves propagate along a dielectric waveguide with a velocity between that of plane-wave propagation in the dielectric medium and that in the medium outside. P.10-23 Find the solutions of Eq. (10-132) for k, by plottmg qd versus k,.tI for d = 1 (cm) and E, = 3.25 if (a) f = 200 (MHz) and (b) f = 500 (MHz). Determine / 3 and a for the lowest-order i . odd TM modes at the two frequencies. P.10-21 Repeat problem P.lO-23 using Eq. (10-135) for the lowest-order even TM modes. P.lO-25 For an intinite dielectric-slab waveguide of thickness d situated in air, obtain the in- stantaneous expressions of all the nonzero field components for even TM modes in the slab, as well as in the upper and lower free-space regions. P.lO-26 When the slab thickness of a dielectric-slab waveguide is very small in terms of the operating wavelength, the ficld intensities decay very slowly away from the-slab -- surface. and the propagation constant is nearly equal to that of the surrounding medium. a) Show that if k,d <c 1, the following relations hold approximately for the dominant TE rnodc: 1 1 2 k,, I ' where k, = w,/pded and k, = o , , , % , . h) For a slab of thickncss 5 (mm) and dielectric constant 3. estimatc the distance from the slab surface at which the ficld intensities have dccaycd to 30.8% of their valucs at ~ h c surface for an operating frequency of 300 (MHz). P.lO-27 For an infinite dielectric-slab waveguide of thickness d situated in free space, obtain the instantaneous expressions of all the nonzero field components for even TE modes in the slab, as well as In the upper and lower free-space regions. Derive Eq. (10-145). P.10-28 A waveguide consists of a ~ i inlin~tc diclcctr~c slab (ti. 11,) ol' tllickncss ti 111at is sitting on a perfect conductor. . 3) What are the propagating modes and what are their cutoff frequencies? b) Obtain the phasor expressions for the surface current and surface charge densities on the conducting base for the propagating modes. P.lO-29 Given an air-filled lossless rectangular cavity resonator with dimensions 8 Icm) x 6 (cm) x 5 (cm). find the first twelve lowest-order modes and their resonant frequencies. P.lO-30 An air-filled rectangular cavity with brass walls-E". po, a = 1.57 x lo7 (S/m)--has the following dimensions: a = 4 (cm), b = 3 (cm). and ti = 5 (cm). a) Determine the dominant mode and its resonant frequency for this cavity. b) Find the Q and the time-average stored electric and magnetic energies at the resonant frequency, assuming H , to be 0.1 (Aim). -- .... .
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" . , . PROBLEMS '499 . ' , ' ' . . . . . ,. . , :ctor losses 1 ' 1 I (an) and )west-order .. modes. .Ice from the .:;llucs at the pacc, obtain : s in he slab, the resonant P.lO-31 If the rectangular cavity in Problem P .
1 0 -3 0 is filled with a lossless dielectric matenal having a dielectric constant 2.5, find a) the resonant frequency of tht: dominant mode, b) the Q, I c) the time-aver:\ge stored clcctric and magnetic energies at the rcsonanr frequency, assuming II,, l o bc O.1 (A/m). -.. P.lO-32 For an air-filled rectangular cavity resonator, a) calculate its Q for the TE,,,, mode if its dimensions are u = d = I.Sh, I)) dcicrminc how much h \llould bc rncrcascd In ordcr to make Q 70Zhighcr. P.10-33 Derive an expression for the Q of an air-filled a x b x d rectangular resohator for the TM , , , mode. Fig. 10-16 A ring-shaped resonator with a narrow center part (Problem P.lO-34). P.10-34 In some microwave applicsrions ring-ihaped cavity resonators with II vzry narrow center part are used. A crxs section ofsuch a resonator is shown in Fig. 10-16. in which d is Lery small compared with the resonimt waveiengrh. Assuming that this resonator can be represented approximately by a purallsl cotnhination of the capacitance of the n:lrrow center parr and the inductance of the rest of :he structure, find a ) thc ;~pproxim;:t : rcsonant I'rcqucncy. b) the appro xi mat^: resonant wavelength. 11-1 INTRODUCTION In Cllaptcr 8 we studied the propag:ltion chnractcristics of plane electromitgnetic wavcs iii sourcc-(roc ~ilcrlia witliout co~~sidcri~i$ ho\v tlic W:IVCS \VCI.C ~ C I I C ~ : I ( C ~ . OI' course. the waves must origina~c I'rotii sourccs, which in clcctrolna~~~ctic terms ;~rc time-varying charges and currents. In order to radiate e l e ~ t r ~ m a ~ n e t i c energ efficiently in prescribed directions, the charges and currents must be distributed in specific ways. Airtennns are structures designed for radiating electromagnetic energy effectively in a prescribed manncr. Without an eficient antcnna, clectrom~~gnctic energy would be localized, and wireless transmission of information over long distances would be impossible. An antenna may be a single straight wire or a conducting loop excited by a voltase source, an aperture at the end of a waveguide, or a complex array of these properly arranged radiating elements. Reflectors and lenses may be used to accentuate certain radiation characteristics. Among radiation characteristics of importance are field pattern, directivity, impedance, and bandwidth. These parameters\,will be exam- ined when particular antenna types are studied in this chapter. To study electromagnetic radiation we must call upon our knowledge of Max- well's equations and relate electric' and magnetic fields to time-varying charge and current distributions. A primary difficulty of this task is that the charge and current distributions on antenna structures resulting from given excitations are generally unknown and very difficult to determine. In fact. the geometrically simple case of a straight conducting wire (linear antenna) excited by a voltage source in the middlet ij;ls hccn a sut?jcct o!'c.u!cnsivc ~.ci,::~t.cll li,r rll:llly y ~ 1 1 . 5 , ;lrlcl 1111: cx:ict c11;11.pc :~ntl currcnt distributions o n ;I wirc: of :I lini~c ~ ~ c l i u s ;lrc cxtrcnicly cotnplicatctl cvcn when the wire is assumed to be perfectly conducting. Fortunately. thc radiation licld of such an antenna is relatively insensitive to slight deviations in the current distri- bution, and a physically plausibie approximate current on the wire yields useful results for ncarly all practical purposes. Wc will exanline the radiation properties of linear antennas with assumed currents. |
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