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received by the observer can be shown to be obs = s w w ± s, (17.20) where s is the frequency of the source, s is the speed of the source along a line joining the source and observer, and w is the speed of sound. The minus sign is used for motion toward the observer and the plus sign for motion away from the observer, producing the appropriate shifts up and down in frequency. Note that the greater the speed of the source, the greater the effect. Similarly, for a stationary source and moving observer, the frequency received by the observer obs is given by obs = s w ± obs w, (17.21) where obs is the speed of the observer along a line joining the source and observer. Here the plus sign is for motion toward the source, and the minus is for motion away from the source. Example 17.4 Calculate Doppler Shift: A Train Horn Suppose a train that has a 150-Hz horn is moving at 35.0 m/s in still air on a day when the speed of sound is 340 m/s. (a) What frequencies are observed by a stationary person at the side of the tracks as the train approaches and after it passes? (b) What frequency is observed by the train’s engineer traveling on the train? Strategy To find the observed frequency in (a), obs = s w w ± s, must be used because the source is moving. The minus sign is used for the approaching train, and the plus sign for the receding train. In (b), there are two Doppler shifts—one for a moving source and the other for a moving observer. Solution for (a) 738 Chapter 17 \| Physics of Hearing (1) Enter known values into obs = s w w – s. obs = s w w − s = (150 Hz) 340 m/s 340 m/s – 35.0 m/s (2) Calculate the frequency observed by a stationary person as the train approaches. obs = (150 Hz)(1.11) = 167 Hz (3) Use the same equation with the plus sign to find the frequency heard by a stationary person as the train recedes. obs = s w w + s = (150 Hz) 340 m/s 340 m/s + 35.0 m/s (4) Calculate the second frequency. Discussion on (a) obs = (150 Hz)(0.907) = 136 Hz (17.22)
(17.23) (17.24) (17.25) The numbers calculated are valid when the train is far enough away that the motion is nearly along the line joining train and observer. In both cases, the shift is significant and easily noticed. Note that the shift is 17.0 Hz for motion toward and 14.0 Hz for motion away. The shifts are not symmetric. Solution for (b) (1) Identify knowns: • It seems reasonable that the engineer would receive the same frequency as emitted by the horn, because the relative velocity between them is zero. • Relative to the medium (air), the speeds are s = obs = 35.0 m/s. • The first Doppler shift is for the moving observer; the second is for the moving source. (2) Use the following equation: obs = s w ± obs w w w ± s. (17.26) The quantity in the square brackets is the Doppler-shifted frequency due to a moving observer. The factor on the right is the effect of the moving source. (3) Because the train engineer is moving in the direction toward the horn, we must use the plus sign for obs; however, because the horn is also moving in the direction away from the engineer, we also use the plus sign for s. But the train is carrying both the engineer and the horn at the same velocity, so s = obs. As a result, everything but s cancels, yielding obs = s. (17.27) Discussion for (b) We may expect that there is no change in frequency when source and observer move together because it fits your experience. For example, there is no Doppler shift in the frequency of conversations between driver and passenger on a motorcycle. People talking when a wind moves the air between them also observe no Doppler shift in their conversation. The crucial point is that source and observer are not moving relative to each other. Sonic Booms to Bow Wakes What happens to the sound produced by a moving source, such as a jet airplane, that approaches or even exceeds the speed of sound? The answer to this question applies not only to sound but to all other waves as well. Suppose a jet airplane is coming nearly straight at you, emitting a sound of frequency s. The greater the plane’s speed s, the greater the Doppler shift and the greater the value observed for approaches infinity, because the denominator in obs = s obs. Now,
as s approaches the speed of sound, w w ± s approaches zero. At the speed of sound, this result obs means that in front of the source, each successive wave is superimposed on the previous one because the source moves forward at the speed of sound. The observer gets them all at the same instant, and so the frequency is infinite. (Before airplanes exceeded the speed of sound, some people argued it would be impossible because such constructive superposition would produce pressures great enough to destroy the airplane.) If the source exceeds the speed of sound, no sound is received by the observer until the source has passed, so that the sounds from the approaching source are mixed with those from it when receding. This mixing appears messy, but something interesting happens—a sonic boom is created. (See Figure 17.18.) This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 17 \| Physics of Hearing 739 Figure 17.18 Sound waves from a source that moves faster than the speed of sound spread spherically from the point where they are emitted, but the source moves ahead of each. Constructive interference along the lines shown (actually a cone in three dimensions) creates a shock wave called a sonic boom. The faster the speed of the source, the smaller the angle. There is constructive interference along the lines shown (a cone in three dimensions) from similar sound waves arriving there simultaneously. This superposition forms a disturbance called a sonic boom, a constructive interference of sound created by an object moving faster than sound. Inside the cone, the interference is mostly destructive, and so the sound intensity there is much less than on the shock wave. An aircraft creates two sonic booms, one from its nose and one from its tail. (See Figure 17.19.) During television coverage of space shuttle landings, two distinct booms could often be heard. These were separated by exactly the time it would take the shuttle to pass by a point. Observers on the ground often do not see the aircraft creating the sonic boom, because it has passed by before the shock wave reaches them, as seen in Figure 17.19. If the aircraft flies close by at low altitude, pressures in the sonic boom can be destructive and break windows as well as rattle nerves. Because of how destructive sonic booms can be, supersonic flights are banned over populated areas of the United States. Figure 17.19 Two sonic booms, created by the nose and tail of an aircraft,
are observed on the ground after the plane has passed by. Sonic booms are one example of a broader phenomenon called bow wakes. A bow wake, such as the one in Figure 17.20, is created when the wave source moves faster than the wave propagation speed. Water waves spread out in circles from the point where created, and the bow wake is the familiar V-shaped wake trailing the source. A more exotic bow wake is created when a subatomic particle travels through a medium faster than the speed of light travels in that medium. (In a vacuum, the maximum speed of light will be = 3.00×108 m/s ; in the medium of water, the speed of light is closer to 0.75. If the particle creates light in its passage, that light spreads on a cone with an angle indicative of the speed of the particle, as illustrated in Figure 17.21. Such a bow wake is called Cerenkov radiation and is commonly observed in particle physics. Figure 17.20 Bow wake created by a duck. Constructive interference produces the rather structured wake, while there is relatively little wave action inside the wake, where interference is mostly destructive. (credit: Horia Varlan, Flickr) 740 Chapter 17 \| Physics of Hearing Figure 17.21 The blue glow in this research reactor pool is Cerenkov radiation caused by subatomic particles traveling faster than the speed of light in water. (credit: U.S. Nuclear Regulatory Commission) Doppler shifts and sonic booms are interesting sound phenomena that occur in all types of waves. They can be of considerable use. For example, the Doppler shift in ultrasound can be used to measure blood velocity, while police use the Doppler shift in radar (a microwave) to measure car velocities. In meteorology, the Doppler shift is used to track the motion of storm clouds; such “Doppler Radar” can give velocity and direction and rain or snow potential of imposing weather fronts. In astronomy, we can examine the light emitted from distant galaxies and determine their speed relative to ours. As galaxies move away from us, their light is shifted to a lower frequency, and so to a longer wavelength—the so-called red shift. Such information from galaxies far, far away has allowed us to estimate the age of the universe (from the Big Bang) as about 14 billion years. Check Your Understanding Why did scientist Christian Doppler observe musicians both on a moving train and also from a stationary point not
on the train? Solution Doppler needed to compare the perception of sound when the observer is stationary and the sound source moves, as well as when the sound source and the observer are both in motion. Check Your Understanding Describe a situation in your life when you might rely on the Doppler shift to help you either while driving a car or walking near traffic. Solution If I am driving and I hear Doppler shift in an ambulance siren, I would be able to tell when it was getting closer and also if it has passed by. This would help me to know whether I needed to pull over and let the ambulance through. 17.5 Sound Interference and Resonance: Standing Waves in Air Columns By the end of this section, you will be able to: Learning Objectives • Define antinode, node, fundamental, overtones, and harmonics. • Identify instances of sound interference in everyday situations. • Describe how sound interference occurring inside open and closed tubes changes the characteristics of the sound, and how this applies to sounds produced by musical instruments. • Calculate the length of a tube using sound wave measurements. The information presented in this section supports the following AP® learning objectives and science practices: • 6.D.1.1 The student is able to use representations of individual pulses and construct representations to model the interaction of two wave pulses to analyze the superposition of two pulses. (S.P. 1.1, 1.4) • 6.D.1.2 The student is able to design a suitable experiment and analyze data illustrating the superposition of mechanical waves (only for wave pulses or standing waves). (S.P. 4.2, 5.1) • 6.D.1.3 The student is able to design a plan for collecting data to quantify the amplitude variations when two or more traveling waves or wave pulses interact in a given medium. (S.P. 4.2) This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 17 \| Physics of Hearing 741 • 6.D.3.1 The student is able to refine a scientific question related to standing waves and design a detailed plan for the experiment that can be conducted to examine the phenomenon qualitatively or quantitatively. (S.P. 2.1, 2.2, 4.2) • 6.D.3.2 The student is able to predict properties of standing waves that result from the
addition of incident and reflected waves that are confined to a region and have nodes and antinodes. (S.P. 6.4) • 6.D.3.3 The student is able to plan data collection strategies, predict the outcome based on the relationship under test, perform data analysis, evaluate evidence compared to the prediction, explain any discrepancy and, if necessary, revise the relationship among variables responsible for establishing standing waves on a string or in a column of air. (S.P. 3.2, 4.1, 5.1, 5.2, 5.3) • 6.D.3.4 The student is able to describe representations and models of situations in which standing waves result from the addition of incident and reflected waves confined to a region. (S.P. 1.2) • 6.D.4.2 The student is able to calculate wavelengths and frequencies (if given wave speed) of standing waves based on boundary conditions and length of region within which the wave is confined, and calculate numerical values of wavelengths and frequencies. Examples should include musical instruments. (S.P. 2.2) Figure 17.22 Some types of headphones use the phenomena of constructive and destructive interference to cancel out outside noises. (credit: JVC America, Flickr) Interference is the hallmark of waves, all of which exhibit constructive and destructive interference exactly analogous to that seen for water waves. In fact, one way to prove something “is a wave” is to observe interference effects. So, sound being a wave, we expect it to exhibit interference; we have already mentioned a few such effects, such as the beats from two similar notes played simultaneously. Figure 17.23 shows a clever use of sound interference to cancel noise. Larger-scale applications of active noise reduction by destructive interference are contemplated for entire passenger compartments in commercial aircraft. To obtain destructive interference, a fast electronic analysis is performed, and a second sound is introduced with its maxima and minima exactly reversed from the incoming noise. Sound waves in fluids are pressure waves and consistent with Pascal’s principle; pressures from two different sources add and subtract like simple numbers; that is, positive and negative gauge pressures add to a much smaller pressure, producing a lower-intensity sound. Although completely destructive interference is possible only under the simplest conditions, it is possible to reduce noise levels by 30 dB or more using this technique. Figure 17.23 Headphones designed to cancel noise with destructive interference create a sound wave exactly opposite to
the incoming sound. These headphones can be more effective than the simple passive attenuation used in most ear protection. Such headphones were used on the recordsetting, around the world nonstop flight of the Voyager aircraft to protect the pilots’ hearing from engine noise. Where else can we observe sound interference? All sound resonances, such as in musical instruments, are due to constructive and destructive interference. Only the resonant frequencies interfere constructively to form standing waves, while others interfere destructively and are absent. From the toot made by blowing over a bottle, to the characteristic flavor of a violin’s sounding box, to the recognizability of a great singer’s voice, resonance and standing waves play a vital role. 742 Interference Chapter 17 \| Physics of Hearing Interference is such a fundamental aspect of waves that observing interference is proof that something is a wave. The wave nature of light was established by experiments showing interference. Similarly, when electrons scattered from crystals exhibited interference, their wave nature was confirmed to be exactly as predicted by symmetry with certain wave characteristics of light. Applying the Science Practices: Standing Wave Figure 17.24 The standing wave pattern of a rubber tube attached to a doorknob. Tie one end of a strip of long rubber tubing to a stable object (doorknob, fence post, etc.) and shake the other end up and down until a standing wave pattern is achieved. Devise a method to determine the frequency and wavelength generated by your arm shaking. Do your results align with the equation? Do you find that the velocity of the wave generated is consistent for each trial? If not, explain why this is the case. Answer This task will likely require two people. The frequency of the wave pattern can be found by timing how long it takes the student shaking the rubber tubing to move his or her hand up and down one full time. (It may be beneficial to time how long it takes the student to do this ten times, and then divide by ten to reduce error.) The wavelength of the standing wave can be measured with a meter stick by measuring the distance between two nodes and multiplying by two. This information should be gathered for standing wave patterns of multiple different wavelengths. As students collect their data, they can use the equation to determine if the wave velocity is consistent. There will likely be some error in the experiment yielding velocities of slightly different value. This error is probably due to an inaccuracy in the wavelength and/or frequency measurements. Suppose we hold a tuning fork near the
end of a tube that is closed at the other end, as shown in Figure 17.25, Figure 17.26, Figure 17.27, and Figure 17.28. If the tuning fork has just the right frequency, the air column in the tube resonates loudly, but at most frequencies it vibrates very little. This observation just means that the air column has only certain natural frequencies. The figures show how a resonance at the lowest of these natural frequencies is formed. A disturbance travels down the tube at the speed of sound and bounces off the closed end. If the tube is just the right length, the reflected sound arrives back at the tuning fork exactly half a cycle later, and it interferes constructively with the continuing sound produced by the tuning fork. The incoming and reflected sounds form a standing wave in the tube as shown. Figure 17.25 Resonance of air in a tube closed at one end, caused by a tuning fork. A disturbance moves down the tube. This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 17 \| Physics of Hearing 743 Figure 17.26 Resonance of air in a tube closed at one end, caused by a tuning fork. The disturbance reflects from the closed end of the tube. Figure 17.27 Resonance of air in a tube closed at one end, caused by a tuning fork. If the length of the tube is just right, the disturbance gets back to the tuning fork half a cycle later and interferes constructively with the continuing sound from the tuning fork. This interference forms a standing wave, and the air column resonates. Figure 17.28 Resonance of air in a tube closed at one end, caused by a tuning fork. A graph of air displacement along the length of the tube shows none at the closed end, where the motion is constrained, and a maximum at the open end. This standing wave has one-fourth of its wavelength in the tube, so that = 4. The standing wave formed in the tube has its maximum air displacement (an antinode) at the open end, where motion is unconstrained, and no displacement (a node) at the closed end, where air movement is halted. The distance from a node to an antinode is one-fourth of a wavelength, and this equals the length of the tube; thus, = 4. This same resonance can be produced by a vibration introduced at or near the closed end of the tube, as shown
in Figure 17.29. It is best to consider this a natural vibration of the air column independently of how it is induced. 744 Chapter 17 \| Physics of Hearing Figure 17.29 The same standing wave is created in the tube by a vibration introduced near its closed end. Given that maximum air displacements are possible at the open end and none at the closed end, there are other, shorter wavelengths that can resonate in the tube, such as the one shown in Figure 17.30. Here the standing wave has three-fourths of its wavelength in the tube, or = (3 / 4)′, so that ′ = 4 / 3. Continuing this process reveals a whole series of shorterwavelength and higher-frequency sounds that resonate in the tube. We use specific terms for the resonances in any system. The lowest resonant frequency is called the fundamental, while all higher resonant frequencies are called overtones. All resonant frequencies are integral multiples of the fundamental, and they are collectively called harmonics. The fundamental is the first harmonic, the first overtone is the second harmonic, and so on. Figure 17.31 shows the fundamental and the first three overtones (the first four harmonics) in a tube closed at one end. Figure 17.30 Another resonance for a tube closed at one end. This has maximum air displacements at the open end, and none at the closed end. The wavelength is shorter, with three-fourths ′ equaling the length of the tube, so that ′ = 4 / 3. This higher-frequency vibration is the first overtone. Figure 17.31 The fundamental and three lowest overtones for a tube closed at one end. All have maximum air displacements at the open end and none at the closed end. The fundamental and overtones can be present simultaneously in a variety of combinations. For example, middle C on a trumpet has a sound distinctively different from middle C on a clarinet, both instruments being modified versions of a tube closed at one end. The fundamental frequency is the same (and usually the most intense), but the overtones and their mix of intensities are different and subject to shading by the musician. This mix is what gives various musical instruments (and human voices) their distinctive characteristics, whether they have air columns, strings, sounding boxes, or drumheads. In fact, much of our speech is determined by shaping the cavity formed by the throat and mouth and positioning the tongue to adjust the fundamental and combination of overtones
. Simple resonant cavities can be made to resonate with the sound of the vowels, for example. (See Figure 17.32.) In boys, at puberty, the larynx grows and the shape of the resonant cavity changes giving rise to the difference in predominant frequencies in speech between men and women. This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 17 \| Physics of Hearing 745 Figure 17.32 The throat and mouth form an air column closed at one end that resonates in response to vibrations in the voice box. The spectrum of overtones and their intensities vary with mouth shaping and tongue position to form different sounds. The voice box can be replaced with a mechanical vibrator, and understandable speech is still possible. Variations in basic shapes make different voices recognizable. Now let us look for a pattern in the resonant frequencies for a simple tube that is closed at one end. The fundamental has = 4, and frequency is related to wavelength and the speed of sound as given by: Solving for in this equation gives w = = w = w 4, where w is the speed of sound in air. Similarly, the first overtone has ′ = 4 / 3 (see Figure 17.31), so that ′ = 3w 4 = 3. Because ′ = 3, we call the first overtone the third harmonic. Continuing this process, we see a pattern that can be generalized in a single expression. The resonant frequencies of a tube closed at one end are = w 4, = 1,3,5, (17.28) (17.29) (17.30) (17.31) where 1 is the fundamental, speed of sound and, hence, on temperature. This dependence poses a noticeable problem for organs in old unheated cathedrals, and it is also the reason why musicians commonly bring their wind instruments to room temperature before playing them. 3 is the first overtone, and so on. It is interesting that the resonant frequencies depend on the Example 17.5 Find the Length of a Tube with a 128 Hz Fundamental (a) What length should a tube closed at one end have on a day when the air temperature, is 22.0ºC, if its fundamental frequency is to be 128 Hz (C below middle C)? (b) What is the frequency of its fourth overtone? Strategy The length can be found from the relationship in = w 4 Solution for (
a) (1) Identify knowns: • • the fundamental frequency is 128 Hz the air temperature is 22.0ºC, but we will first need to find the speed of sound w. (2) Use = w 4 to find the fundamental frequency ( = 1 ). (3) Solve this equation for length17.32) (17.33) 746 Chapter 17 \| Physics of Hearing (4) Find the speed of sound using w = (331 m/s) 273 K. w = (331 m/s) 295 K 273 K = 344 m/s (5) Enter the values of the speed of sound and frequency into the expression for. = w 4 1 = 344 m/s 4(128 Hz) = 0.672 m Discussion on (a) (17.34) (17.35) Many wind instruments are modified tubes that have finger holes, valves, and other devices for changing the length of the resonating air column and hence, the frequency of the note played. Horns producing very low frequencies, such as tubas, require tubes so long that they are coiled into loops. Solution for (b) (1) Identify knowns: • • • • the first overtone has = 3 the second overtone has = 5 the third overtone has = 7 the fourth overtone has = 9 (2) Enter the value for the fourth overtone into = w 4. 9 = 9w 4 = 9 1 = 1.15 kHz (17.36) Discussion on (b) Whether this overtone occurs in a simple tube or a musical instrument depends on how it is stimulated to vibrate and the details of its shape. The trombone, for example, does not produce its fundamental frequency and only makes overtones. Another type of tube is one that is open at both ends. Examples are some organ pipes, flutes, and oboes. The resonances of tubes open at both ends can be analyzed in a very similar fashion to those for tubes closed at one end. The air columns in tubes open at both ends have maximum air displacements at both ends, as illustrated in Figure 17.33. Standing waves form as shown. Figure 17.33 The resonant frequencies of a tube open at both ends are shown, including the fundamental and the first three overtones. In all cases the maximum air displacements occur at both ends of the tube, giving it different natural frequencies than a tube closed at one end. Based on the
fact that a tube open at both ends has maximum air displacements at both ends, and using Figure 17.33 as a guide, we can see that the resonant frequencies of a tube open at both ends are: = w 2, = 1, 2, 3..., (17.37) 2 is the first overtone, where 1 is the fundamental, 3 is the second overtone, and so on. Note that a tube open at both ends has a fundamental frequency twice what it would have if closed at one end. It also has a different spectrum of overtones than a tube closed at one end. So if you had two tubes with the same fundamental frequency but one was open at both ends and the other was closed at one end, they would sound different when played because they have different overtones. Middle C, for example, would sound richer played on an open tube, because it has even multiples of the fundamental as well as odd. A closed tube has only odd multiples. This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 17 \| Physics of Hearing 747 Applying the Science Practices: Closed- and Open-Ended Tubes Strike an open-ended length of plastic pipe while holding it in the air. Now place one end of the pipe on a hard surface, sealing one opening, and strike it again. How does the sound change? Further investigate the sound created by the pipe by striking pipes of different lengths and composition. Answer When the pipe is placed on the ground, the standing wave within the pipe changes from being open on both ends to being closed on one end. As a result, the fundamental frequency will change from = 2 to = 4. This decrease in frequency results in a decrease in observed pitch. Real-World Applications: Resonance in Everyday Systems Resonance occurs in many different systems, including strings, air columns, and atoms. Resonance is the driven or forced oscillation of a system at its natural frequency. At resonance, energy is transferred rapidly to the oscillating system, and the amplitude of its oscillations grows until the system can no longer be described by Hooke’s law. An example of this is the distorted sound intentionally produced in certain types of rock music. Wind instruments use resonance in air columns to amplify tones made by lips or vibrating reeds. Other instruments also use air resonance in clever ways to amplify sound. Figure 17.34 shows a violin and a guitar, both of which have
sounding boxes but with different shapes, resulting in different overtone structures. The vibrating string creates a sound that resonates in the sounding box, greatly amplifying the sound and creating overtones that give the instrument its characteristic flavor. The more complex the shape of the sounding box, the greater its ability to resonate over a wide range of frequencies. The marimba, like the one shown in Figure 17.35 uses pots or gourds below the wooden slats to amplify their tones. The resonance of the pot can be adjusted by adding water. Figure 17.34 String instruments such as violins and guitars use resonance in their sounding boxes to amplify and enrich the sound created by their vibrating strings. The bridge and supports couple the string vibrations to the sounding boxes and air within. (credits: guitar, Feliciano Guimares, Fotopedia; violin, Steve Snodgrass, Flickr) 748 Chapter 17 \| Physics of Hearing Figure 17.35 Resonance has been used in musical instruments since prehistoric times. This marimba uses gourds as resonance chambers to amplify its sound. (credit: APC Events, Flickr) We have emphasized sound applications in our discussions of resonance and standing waves, but these ideas apply to any system that has wave characteristics. Vibrating strings, for example, are actually resonating and have fundamentals and overtones similar to those for air columns. More subtle are the resonances in atoms due to the wave character of their electrons. Their orbitals can be viewed as standing waves, which have a fundamental (ground state) and overtones (excited states). It is fascinating that wave characteristics apply to such a wide range of physical systems. Check Your Understanding Describe how noise-canceling headphones differ from standard headphones used to block outside sounds. Solution Regular headphones only block sound waves with a physical barrier. Noise-canceling headphones use destructive interference to reduce the loudness of outside sounds. Check Your Understanding How is it possible to use a standing wave's node and antinode to determine the length of a closed-end tube? Solution When the tube resonates at its natural frequency, the wave's node is located at the closed end of the tube, and the antinode is located at the open end. The length of the tube is equal to one-fourth of the wavelength of this wave. Thus, if we know the wavelength of the wave, we can determine the length of the tube. PhET Explorations: Sound This simulation lets you
see sound waves. Adjust the frequency or volume and you can see and hear how the wave changes. Move the listener around and hear what she hears. Figure 17.36 Sound (http://cnx.org/content/m55293/1.2/sound_en.jar) Applying the Science Practices: Variables Affecting Superposition In the PhET Interactive Simulation above, select the tab titled ‘Two Source Interference.’ Within this tab, manipulate the variables present (frequency, amplitude, and speaker separation) to investigate the relationship the variables have with the superposition pattern constructed on the screen. Record all observations. This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 17 \| Physics of Hearing 761 (3) Calculate to find the frequency returning to the source: 2,500,649 Hz. Solution for (c) (1) Identify knowns: • The beat frequency is simply the absolute value of the difference between s and obs, as stated in: (2) Substitute known values: B = ∣ obs − s ∣. ∣ 2,500649 Hz − 2,500000 Hz ∣ (3) Calculate to find the beat frequency: 649 Hz. Discussion (17.47) (17.48) The Doppler shifts are quite small compared with the original frequency of 2.50 MHz. It is far easier to measure the beat frequency than it is to measure the echo frequency with an accuracy great enough to see shifts of a few hundred hertz out of a couple of megahertz. Furthermore, variations in the source frequency do not greatly affect the beat frequency, because both s and obs would increase or decrease. Those changes subtract out in B = ∣ obs − s ∣. Industrial and Other Applications of Ultrasound Industrial, retail, and research applications of ultrasound are common. A few are discussed here. Ultrasonic cleaners have many uses. Jewelry, machined parts, and other objects that have odd shapes and crevices are immersed in a cleaning fluid that is agitated with ultrasound typically about 40 kHz in frequency. The intensity is great enough to cause cavitation, which is responsible for most of the cleansing action. Because cavitation-produced shock pressures are large and well transmitted in a fluid, they reach into small crevices where even a low-surface-tension cleaning fluid might not penetrate. Sonar is a familiar application of ultrasound. Son
ar typically employs ultrasonic frequencies in the range from 30.0 to 100 kHz. Bats, dolphins, submarines, and even some birds use ultrasonic sonar. Echoes are analyzed to give distance and size information both for guidance and finding prey. In most sonar applications, the sound reflects quite well because the objects of interest have significantly different density than the medium in which they travel. When the Doppler shift is observed, velocity information can also be obtained. Submarine sonar can be used to obtain such information, and there is evidence that some bats also sense velocity from their echoes. Similarly, there are a range of relatively inexpensive devices that measure distance by timing ultrasonic echoes. Many cameras, for example, use such information to focus automatically. Some doors open when their ultrasonic ranging devices detect a nearby object, and certain home security lights turn on when their ultrasonic rangers observe motion. Ultrasonic “measuring tapes” also exist to measure such things as room dimensions. Sinks in public restrooms are sometimes automated with ultrasound devices to turn faucets on and off when people wash their hands. These devices reduce the spread of germs and can conserve water. Ultrasound is used for nondestructive testing in industry and by the military. Because ultrasound reflects well from any large change in density, it can reveal cracks and voids in solids, such as aircraft wings, that are too small to be seen with x-rays. For similar reasons, ultrasound is also good for measuring the thickness of coatings, particularly where there are several layers involved. Basic research in solid state physics employs ultrasound. Its attenuation is related to a number of physical characteristics, making it a useful probe. Among these characteristics are structural changes such as those found in liquid crystals, the transition of a material to a superconducting phase, as well as density and other properties. These examples of the uses of ultrasound are meant to whet the appetites of the curious, as well as to illustrate the underlying physics of ultrasound. There are many more applications, as you can easily discover for yourself. Check Your Understanding Why is it possible to use ultrasound both to observe a fetus in the womb and also to destroy cancerous tumors in the body? Solution Ultrasound can be used medically at different intensities. Lower intensities do not cause damage and are used for medical imaging. Higher intensities can pulverize and destroy targeted substances in the body, such as tumors. Glossary acoustic impedance: property of medium that makes
the propagation of sound waves more difficult antinode: point of maximum displacement bow wake: V-shaped disturbance created when the wave source moves faster than the wave propagation speed 762 Chapter 17 \| Physics of Hearing Doppler effect: an alteration in the observed frequency of a sound due to motion of either the source or the observer Doppler shift: the actual change in frequency due to relative motion of source and observer Doppler-shifted ultrasound: a medical technique to detect motion and determine velocity through the Doppler shift of an echo fundamental: the lowest-frequency resonance harmonics: the term used to refer collectively to the fundamental and its overtones hearing: the perception of sound infrasound: sounds below 20 Hz intensity: the power per unit area carried by a wave intensity reflection coefficient: a measure of the ratio of the intensity of the wave reflected off a boundary between two media relative to the intensity of the incident wave loudness: the perception of sound intensity node: point of zero displacement note: basic unit of music with specific names, combined to generate tunes overtones: all resonant frequencies higher than the fundamental phon: the numerical unit of loudness pitch: the perception of the frequency of a sound sonic boom: a constructive interference of sound created by an object moving faster than sound sound: a disturbance of matter that is transmitted from its source outward sound intensity level: a unitless quantity telling you the level of the sound relative to a fixed standard sound pressure level: the ratio of the pressure amplitude to a reference pressure timbre: number and relative intensity of multiple sound frequencies tone: number and relative intensity of multiple sound frequencies ultrasound: sounds above 20,000 Hz Section Summary 17.1 Sound • Sound is a disturbance of matter that is transmitted from its source outward. • Sound is one type of wave. • Hearing is the perception of sound. 17.2 Speed of Sound, Frequency, and Wavelength The relationship of the speed of sound w, its frequency, and its wavelength is given by which is the same relationship given for all waves. In air, the speed of sound is related to air temperature by w =, w = (331 m/s) 273 K. w is the same for all frequencies and wavelengths. 17.3 Sound Intensity and Sound Level • Intensity is the same for a sound wave as was defined for all waves; it is This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 17 \| Physics of Hearing 763 where
is the power crossing area. The SI unit for is watts per meter squared. The intensity of a sound wave is also related to the pressure amplitude Δ =, = 2 Δ 2w, where is the density of the medium in which the sound wave travels and w is the speed of sound in the medium. • Sound intensity level in units of decibels (dB) is where 0 = 10–12 W/m2 is the threshold intensity of hearing. (dB) = 10 log10 0, 17.4 Doppler Effect and Sonic Booms • The Doppler effect is an alteration in the observed frequency of a sound due to motion of either the source or the observer. • The actual change in frequency is called the Doppler shift. • A sonic boom is constructive interference of sound created by an object moving faster than sound. • A sonic boom is a type of bow wake created when any wave source moves faster than the wave propagation speed. • For a stationary observer and a moving source, the observed frequency obs is: obs = s w w ± s, where s is the frequency of the source, s is the speed of the source, and w is the speed of sound. The minus sign is used for motion toward the observer and the plus sign for motion away. • For a stationary source and moving observer, the observed frequency is: w ± obs w obs = s, where obs is the speed of the observer. 17.5 Sound Interference and Resonance: Standing Waves in Air Columns • Sound interference and resonance have the same properties as defined for all waves. • In air columns, the lowest-frequency resonance is called the fundamental, whereas all higher resonant frequencies are called overtones. Collectively, they are called harmonics. • The resonant frequencies of a tube closed at one end are: w 4, = 1, 3, 5..., = 1 is the fundamental and is the length of the tube. • The resonant frequencies of a tube open at both ends are: w 2 =, = 1, 2, 3... 17.6 Hearing • The range of audible frequencies is 20 to 20,000 Hz. • Those sounds above 20,000 Hz are ultrasound, whereas those below 20 Hz are infrasound. • The perception of frequency is pitch. • The perception of intensity is loudness. • Loudness has units of phons. 17.7 Ultrasound • The acoustic impedance is defined as: =, is the density of a medium through which
the sound travels and is the speed of sound through that medium. • The intensity reflection coefficient, a measure of the ratio of the intensity of the wave reflected off a boundary between two media relative to the intensity of the incident wave, is given by 764 Chapter 17 \| Physics of Hearing • The intensity reflection coefficient is a unitless quantity. Conceptual Questions = 2 − 1 1 + 2 2 2. 17.2 Speed of Sound, Frequency, and Wavelength 1. How do sound vibrations of atoms differ from thermal motion? 2. When sound passes from one medium to another where its propagation speed is different, does its frequency or wavelength change? Explain your answer briefly. 17.3 Sound Intensity and Sound Level 3. Six members of a synchronized swim team wear earplugs to protect themselves against water pressure at depths, but they can still hear the music and perform the combinations in the water perfectly. One day, they were asked to leave the pool so the dive team could practice a few dives, and they tried to practice on a mat, but seemed to have a lot more difficulty. Why might this be? 4. A community is concerned about a plan to bring train service to their downtown from the town’s outskirts. The current sound intensity level, even though the rail yard is blocks away, is 70 dB downtown. The mayor assures the public that there will be a difference of only 30 dB in sound in the downtown area. Should the townspeople be concerned? Why? 17.4 Doppler Effect and Sonic Booms 5. Is the Doppler shift real or just a sensory illusion? 6. Due to efficiency considerations related to its bow wake, the supersonic transport aircraft must maintain a cruising speed that is a constant ratio to the speed of sound (a constant Mach number). If the aircraft flies from warm air into colder air, should it increase or decrease its speed? Explain your answer. 7. When you hear a sonic boom, you often cannot see the plane that made it. Why is that? 17.5 Sound Interference and Resonance: Standing Waves in Air Columns 8. How does an unamplified guitar produce sounds so much more intense than those of a plucked string held taut by a simple stick? 9. You are given two wind instruments of identical length. One is open at both ends, whereas the other is closed at one end. Which is able to produce the lowest frequency? 10. What is the difference between an overtone and a harmonic? Are
all harmonics overtones? Are all overtones harmonics? 17.6 Hearing 11. Why can a hearing test show that your threshold of hearing is 0 dB at 250 Hz, when Figure 17.39 implies that no one can hear such a frequency at less than 20 dB? 17.7 Ultrasound 12. If audible sound follows a rule of thumb similar to that for ultrasound, in terms of its absorption, would you expect the high or low frequencies from your neighbor’s stereo to penetrate into your house? How does this expectation compare with your experience? 13. Elephants and whales are known to use infrasound to communicate over very large distances. What are the advantages of infrasound for long distance communication? 14. It is more difficult to obtain a high-resolution ultrasound image in the abdominal region of someone who is overweight than for someone who has a slight build. Explain why this statement is accurate. 15. Suppose you read that 210-dB ultrasound is being used to pulverize cancerous tumors. You calculate the intensity in watts per centimeter squared and find it is unreasonably high ( 105 W/cm2 ). What is a possible explanation? This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 17 \| Physics of Hearing 765 Problems & Exercises 17.2 Speed of Sound, Frequency, and Wavelength 15. What intensity level does the sound in the preceding problem correspond to? 16. What sound intensity level in dB is produced by earphones that create an intensity of 4.00×10−2 W/m2? 1. When poked by a spear, an operatic soprano lets out a 1200-Hz shriek. What is its wavelength if the speed of sound is 345 m/s? 17. Show that an intensity of 10–12 W/m2 is the same as 10–16 W/cm2. 2. What frequency sound has a 0.10-m wavelength when the speed of sound is 340 m/s? 3. Calculate the speed of sound on a day when a 1500 Hz frequency has a wavelength of 0.221 m. 4. (a) What is the speed of sound in a medium where a 100-kHz frequency produces a 5.96-cm wavelength? (b) Which substance in Table 17.4 is this likely to be? 5. Show that the speed of sound in 20.0ºC air is 343 m/
s, as claimed in the text. 6. Air temperature in the Sahara Desert can reach 56.0ºC (about 134ºF ). What is the speed of sound in air at that temperature? 7. Dolphins make sounds in air and water. What is the ratio of the wavelength of a sound in air to its wavelength in seawater? Assume air temperature is 20.0ºC. 8. A sonar echo returns to a submarine 1.20 s after being emitted. What is the distance to the object creating the echo? (Assume that the submarine is in the ocean, not in fresh water.) 9. (a) If a submarine’s sonar can measure echo times with a precision of 0.0100 s, what is the smallest difference in distances it can detect? (Assume that the submarine is in the ocean, not in fresh water.) (b) Discuss the limits this time resolution imposes on the ability of the sonar system to detect the size and shape of the object creating the echo. 10. A physicist at a fireworks display times the lag between seeing an explosion and hearing its sound, and finds it to be 0.400 s. (a) How far away is the explosion if air temperature is 24.0ºC and if you neglect the time taken for light to reach the physicist? (b) Calculate the distance to the explosion taking the speed of light into account. Note that this distance is negligibly greater. 11. Suppose a bat uses sound echoes to locate its insect prey, 3.00 m away. (See Figure 17.11.) (a) Calculate the echo times for temperatures of 5.00ºC and 35.0ºC. (b) What percent uncertainty does this cause for the bat in locating the insect? (c) Discuss the significance of this uncertainty and whether it could cause difficulties for the bat. (In practice, the bat continues to use sound as it closes in, eliminating most of any difficulties imposed by this and other effects, such as motion of the prey.) 17.3 Sound Intensity and Sound Level 12. What is the intensity in watts per meter squared of 85.0-dB sound? 13. The warning tag on a lawn mower states that it produces noise at a level of 91.0 dB. What is this in watts per meter squared? 14. A sound wave traveling in 20ºC air has a pressure amplitude of 0.5 Pa. What is the intensity of the wave? 18.
(a) What is the decibel level of a sound that is twice as intense as a 90.0-dB sound? (b) What is the decibel level of a sound that is one-fifth as intense as a 90.0-dB sound? 19. (a) What is the intensity of a sound that has a level 7.00 dB lower than a 4.00×10–9 W/m2 sound? (b) What is the intensity of a sound that is 3.00 dB higher than a 4.00×10–9 W/m2 sound? 20. (a) How much more intense is a sound that has a level 17.0 dB higher than another? (b) If one sound has a level 23.0 dB less than another, what is the ratio of their intensities? 21. People with good hearing can perceive sounds as low in level as –8.00 dB at a frequency of 3000 Hz. What is the intensity of this sound in watts per meter squared? 22. If a large housefly 3.0 m away from you makes a noise of 40.0 dB, what is the noise level of 1000 flies at that distance, assuming interference has a negligible effect? 23. Ten cars in a circle at a boom box competition produce a 120-dB sound intensity level at the center of the circle. What is the average sound intensity level produced there by each stereo, assuming interference effects can be neglected? 24. The amplitude of a sound wave is measured in terms of its maximum gauge pressure. By what factor does the amplitude of a sound wave increase if the sound intensity level goes up by 40.0 dB? 25. If a sound intensity level of 0 dB at 1000 Hz corresponds to a maximum gauge pressure (sound amplitude) of 10–9 atm, what is the maximum gauge pressure in a 60-dB sound? What is the maximum gauge pressure in a 120-dB sound? 26. An 8-hour exposure to a sound intensity level of 90.0 dB may cause hearing damage. What energy in joules falls on a 0.800-cm-diameter eardrum so exposed? 27. (a) Ear trumpets were never very common, but they did aid people with hearing losses by gathering sound over a large area and concentrating it on the smaller area of the eardrum. What decibel increase does an ear trumpet produce if its sound gathering area is 900 cm2 and the area of the eardrum is
0.500 cm2, but the trumpet only has an efficiency of 5.00% in transmitting the sound to the eardrum? (b) Comment on the usefulness of the decibel increase found in part (a). 28. Sound is more effectively transmitted into a stethoscope by direct contact than through the air, and it is further intensified by being concentrated on the smaller area of the eardrum. It is reasonable to assume that sound is transmitted into a stethoscope 100 times as effectively compared with transmission though the air. What, then, is the gain in decibels produced by a stethoscope that has a sound gathering area of 15.0 cm2, and concentrates the sound 766 Chapter 17 \| Physics of Hearing onto two eardrums with a total area of 0.900 cm2 with an efficiency of 40.0%? 29. Loudspeakers can produce intense sounds with surprisingly small energy input in spite of their low efficiencies. Calculate the power input needed to produce a 90.0-dB sound intensity level for a 12.0-cm-diameter speaker that has an efficiency of 1.00%. (This value is the sound intensity level right at the speaker.) 17.4 Doppler Effect and Sonic Booms 30. (a) What frequency is received by a person watching an oncoming ambulance moving at 110 km/h and emitting a steady 800-Hz sound from its siren? The speed of sound on this day is 345 m/s. (b) What frequency does she receive after the ambulance has passed? 31. (a) At an air show a jet flies directly toward the stands at a speed of 1200 km/h, emitting a frequency of 3500 Hz, on a day when the speed of sound is 342 m/s. What frequency is received by the observers? (b) What frequency do they receive as the plane flies directly away from them? 32. What frequency is received by a mouse just before being dispatched by a hawk flying at it at 25.0 m/s and emitting a screech of frequency 3500 Hz? Take the speed of sound to be 331 m/s. 33. A spectator at a parade receives an 888-Hz tone from an oncoming trumpeter who is playing an 880-Hz note. At what speed is the musician approaching if the speed of sound is 338 m/s? 34. A commuter train blows its 200-Hz horn as it approaches a crossing. The speed of
sound is 335 m/s. (a) An observer waiting at the crossing receives a frequency of 208 Hz. What is the speed of the train? (b) What frequency does the observer receive as the train moves away? 35. Can you perceive the shift in frequency produced when you pull a tuning fork toward you at 10.0 m/s on a day when the speed of sound is 344 m/s? To answer this question, calculate the factor by which the frequency shifts and see if it is greater than 0.300%. 36. Two eagles fly directly toward one another, the first at 15.0 m/s and the second at 20.0 m/s. Both screech, the first one emitting a frequency of 3200 Hz and the second one emitting a frequency of 3800 Hz. What frequencies do they receive if the speed of sound is 330 m/s? 37. What is the minimum speed at which a source must travel toward you for you to be able to hear that its frequency is Doppler shifted? That is, what speed produces a shift of 0.300% on a day when the speed of sound is 331 m/s? 17.5 Sound Interference and Resonance: Standing Waves in Air Columns 38. A “showy” custom-built car has two brass horns that are supposed to produce the same frequency but actually emit 263.8 and 264.5 Hz. What beat frequency is produced? 39. What beat frequencies will be present: (a) If the musical notes A and C are played together (frequencies of 220 and 264 Hz)? (b) If D and F are played together (frequencies of 297 and 352 Hz)? (c) If all four are played together? 40. What beat frequencies result if a piano hammer hits three strings that emit frequencies of 127.8, 128.1, and 128.3 Hz? 41. A piano tuner hears a beat every 2.00 s when listening to a 264.0-Hz tuning fork and a single piano string. What are the two possible frequencies of the string? This content is available for free at http://cnx.org/content/col11844/1.13 42. (a) What is the fundamental frequency of a 0.672-m-long tube, open at both ends, on a day when the speed of sound is 344 m/s? (b) What is the frequency of its second harmonic? 43. If
a wind instrument, such as a tuba, has a fundamental frequency of 32.0 Hz, what are its first three overtones? It is closed at one end. (The overtones of a real tuba are more complex than this example, because it is a tapered tube.) 44. What are the first three overtones of a bassoon that has a fundamental frequency of 90.0 Hz? It is open at both ends. (The overtones of a real bassoon are more complex than this example, because its double reed makes it act more like a tube closed at one end.) 45. How long must a flute be in order to have a fundamental frequency of 262 Hz (this frequency corresponds to middle C on the evenly tempered chromatic scale) on a day when air temperature is 20.0ºC? It is open at both ends. 46. What length should an oboe have to produce a fundamental frequency of 110 Hz on a day when the speed of sound is 343 m/s? It is open at both ends. 47. What is the length of a tube that has a fundamental frequency of 176 Hz and a first overtone of 352 Hz if the speed of sound is 343 m/s? 48. (a) Find the length of an organ pipe closed at one end that produces a fundamental frequency of 256 Hz when air temperature is 18.0ºC. (b) What is its fundamental frequency at 25.0ºC? 49. By what fraction will the frequencies produced by a wind instrument change when air temperature goes from 10.0ºC to 30.0ºC? That is, find the ratio of the frequencies at those temperatures. 50. The ear canal resonates like a tube closed at one end. (See Figure 17.41.) If ear canals range in length from 1.80 to 2.60 cm in an average population, what is the range of fundamental resonant frequencies? Take air temperature to be 37.0ºC, which is the same as body temperature. How does this result correlate with the intensity versus frequency graph (Figure 17.39 of the human ear? 51. Calculate the first overtone in an ear canal, which resonates like a 2.40-cm-long tube closed at one end, by taking air temperature to be 37.0ºC. Is the ear particularly sensitive to such a frequency? (The resonances of the ear canal are complicated by its nonuniform shape, which we shall
ignore.) 52. A crude approximation of voice production is to consider the breathing passages and mouth to be a resonating tube closed at one end. (See Figure 17.32.) (a) What is the fundamental frequency if the tube is 0.240-m long, by taking air temperature to be 37.0ºC? (b) What would this frequency become if the person replaced the air with helium? Assume the same temperature dependence for helium as for air. 53. (a) Students in a physics lab are asked to find the length of an air column in a tube closed at one end that has a fundamental frequency of 256 Hz. They hold the tube vertically and fill it with water to the top, then lower the water while a 256-Hz tuning fork is rung and listen for the first resonance. What is the air temperature if the resonance occurs for a length of 0.336 m? (b) At what length will they observe the second resonance (first overtone)? Chapter 17 \| Physics of Hearing 767 54. What frequencies will a 1.80-m-long tube produce in the audible range at 20.0ºC if: (a) The tube is closed at one end? (b) It is open at both ends? 17.6 Hearing 55. The factor of 10−12 in the range of intensities to which the ear can respond, from threshold to that causing damage after brief exposure, is truly remarkable. If you could measure distances over the same range with a single instrument and the smallest distance you could measure was 1 mm, what would the largest be? 56. The frequencies to which the ear responds vary by a factor of 103. Suppose the speedometer on your car measured speeds differing by the same factor of 103, and the greatest speed it reads is 90.0 mi/h. What would be the slowest nonzero speed it could read? 57. What are the closest frequencies to 500 Hz that an average person can clearly distinguish as being different in frequency from 500 Hz? The sounds are not present simultaneously. 58. Can the average person tell that a 2002-Hz sound has a different frequency than a 1999-Hz sound without playing them simultaneously? 59. If your radio is producing an average sound intensity level of 85 dB, what is the next lowest sound intensity level that is clearly less intense? 60. Can you tell that your roommate turned up the sound on the TV if its average sound intensity level goes from 70 to 73 dB? 61. Based on the
graph in Figure 17.38, what is the threshold of hearing in decibels for frequencies of 60, 400, 1000, 4000, and 15,000 Hz? Note that many AC electrical appliances produce 60 Hz, music is commonly 400 Hz, a reference frequency is 1000 Hz, your maximum sensitivity is near 4000 Hz, and many older TVs produce a 15,750 Hz whine. 62. What sound intensity levels must sounds of frequencies 60, 3000, and 8000 Hz have in order to have the same loudness as a 40-dB sound of frequency 1000 Hz (that is, to have a loudness of 40 phons)? 63. What is the approximate sound intensity level in decibels of a 600-Hz tone if it has a loudness of 20 phons? If it has a loudness of 70 phons? 64. (a) What are the loudnesses in phons of sounds having frequencies of 200, 1000, 5000, and 10,000 Hz, if they are all at the same 60.0-dB sound intensity level? (b) If they are all at 110 dB? (c) If they are all at 20.0 dB? 65. Suppose a person has a 50-dB hearing loss at all frequencies. By how many factors of 10 will low-intensity sounds need to be amplified to seem normal to this person? Note that smaller amplification is appropriate for more intense sounds to avoid further hearing damage. 66. If a woman needs an amplification of 5.0×1012 times the threshold intensity to enable her to hear at all frequencies, what is her overall hearing loss in dB? Note that smaller amplification is appropriate for more intense sounds to avoid further damage to her hearing from levels above 90 dB. 67. (a) What is the intensity in watts per meter squared of a just barely audible 200-Hz sound? (b) What is the intensity in watts per meter squared of a barely audible 4000-Hz sound? 68. (a) Find the intensity in watts per meter squared of a 60.0-Hz sound having a loudness of 60 phons. (b) Find the intensity in watts per meter squared of a 10,000-Hz sound having a loudness of 60 phons. 69. A person has a hearing threshold 10 dB above normal at 100 Hz and 50 dB above normal at 4000 Hz. How much more intense must a 100-Hz tone be than a 4000-Hz tone if they are both barely audible to this person? 70. A child has
a hearing loss of 60 dB near 5000 Hz, due to noise exposure, and normal hearing elsewhere. How much more intense is a 5000-Hz tone than a 400-Hz tone if they are both barely audible to the child? 71. What is the ratio of intensities of two sounds of identical frequency if the first is just barely discernible as louder to a person than the second? 17.7 Ultrasound Unless otherwise indicated, for problems in this section, assume that the speed of sound through human tissues is 1540 m/s. 72. What is the sound intensity level in decibels of ultrasound of intensity 105 W/m2, used to pulverize tissue during surgery? 73. Is 155-dB ultrasound in the range of intensities used for deep heating? Calculate the intensity of this ultrasound and compare this intensity with values quoted in the text. 74. Find the sound intensity level in decibels of 2.00×10–2 W/m2 ultrasound used in medical diagnostics. 75. The time delay between transmission and the arrival of the reflected wave of a signal using ultrasound traveling through a piece of fat tissue was 0.13 ms. At what depth did this reflection occur? 76. In the clinical use of ultrasound, transducers are always coupled to the skin by a thin layer of gel or oil, replacing the air that would otherwise exist between the transducer and the skin. (a) Using the values of acoustic impedance given in Table 17.8 calculate the intensity reflection coefficient between transducer material and air. (b) Calculate the intensity reflection coefficient between transducer material and gel (assuming for this problem that its acoustic impedance is identical to that of water). (c) Based on the results of your calculations, explain why the gel is used. 77. (a) Calculate the minimum frequency of ultrasound that will allow you to see details as small as 0.250 mm in human tissue. (b) What is the effective depth to which this sound is effective as a diagnostic probe? 78. (a) Find the size of the smallest detail observable in human tissue with 20.0-MHz ultrasound. (b) Is its effective penetration depth great enough to examine the entire eye (about 3.00 cm is needed)? (c) What is the wavelength of such ultrasound in 0ºC air? 79. (a) Echo times are measured by diagnostic ultrasound scanners to determine distances to reflecting surfaces in a patient. What is the difference in echo times for
tissues that are 3.50 and 3.60 cm beneath the surface? (This difference is the minimum resolving time for the scanner to see details as small as 0.100 cm, or 1.00 mm. Discrimination of smaller time differences is needed to see smaller details.) (b) Discuss whether the period of this ultrasound must be smaller than the minimum time resolution. If so, what is the minimum 768 Chapter 17 \| Physics of Hearing frequency of the ultrasound and is that out of the normal range for diagnostic ultrasound? 80. (a) How far apart are two layers of tissue that produce echoes having round-trip times (used to measure distances) that differ by 0.750 μs? (b) What minimum frequency must the ultrasound have to see detail this small? 81. (a) A bat uses ultrasound to find its way among trees. If this bat can detect echoes 1.00 ms apart, what minimum distance between objects can it detect? (b) Could this distance explain the difficulty that bats have finding an open door when they accidentally get into a house? 82. A dolphin is able to tell in the dark that the ultrasound echoes received from two sharks come from two different objects only if the sharks are separated by 3.50 m, one being that much farther away than the other. (a) If the ultrasound has a frequency of 100 kHz, show this ability is not limited by its wavelength. (b) If this ability is due to the dolphin’s ability to detect the arrival times of echoes, what is the minimum time difference the dolphin can perceive? 83. A diagnostic ultrasound echo is reflected from moving blood and returns with a frequency 500 Hz higher than its original 2.00 MHz. What is the velocity of the blood? (Assume that the frequency of 2.00 MHz is accurate to seven significant figures and 500 Hz is accurate to three significant figures.) 84. Ultrasound reflected from an oncoming bloodstream that is moving at 30.0 cm/s is mixed with the original frequency of 2.50 MHz to produce beats. What is the beat frequency? (Assume that the frequency of 2.50 MHz is accurate to seven significant figures.) Test Prep for AP® Courses 17.2 Speed of Sound, Frequency, and Wavelength 1. A teacher wants to demonstrate that the speed of sound is not a constant value. Considering her regular classroom voice as the control, which of the following will increase the speed of sound leaving her mouth? I. Submerge her mouth underwater and
speak at the same volume. Increase the temperature of the room and speak at the same volume. Increase the pitch of her voice and speak at the same volume. I only I and II only I, II and III II and III III only II. III. a. b. c. d. e. 2. All members of an orchestra begin tuning their instruments at the same time. While some woodwind instruments play high frequency notes, other stringed instruments play notes of lower frequency. Yet an audience member will hear all notes simultaneously, in apparent contrast to the equation. Explain how a student could demonstrate the flaw in the above logic, using a slinky, stopwatch, and meter stick. Make sure to explain what relationship is truly demonstrated in the above equation, in addition to what would be necessary to get the speed of the slinky to actually change. You may include diagrams and equations as part of your explanation. 17.3 Sound Intensity and Sound Level This content is available for free at http://cnx.org/content/col11844/1.13 3. In order to waken a sleeping child, the volume on an alarm clock is tripled. Under this new scenario, how much more energy will be striking the child’s ear drums each second? twice as much three times as much a. b. c. approximately 4.8 times as much d. six times as much e. nine times as much 4. A musician strikes the strings of a guitar such that they vibrate with twice the amplitude. a. Explain why this requires an energy input greater than twice the original value. b. Explain why the sound leaving the string will not result in a decibel level that is twice as great. 17.4 Doppler Effect and Sonic Booms 5. A baggage handler stands on the edge of a runway as a landing plane approaches. Compared to the pitch of the plane as heard by the plane’s pilot, which of the following correctly describes the sensation experienced by the handler? a. The frequency of the plane will be lower pitched according to the baggage handler and will become even lower pitched as the plane slows to a stop. b. The frequency of the plane will be lower pitched according to the baggage handler but will increase in pitch as the plane slows to a stop. c. The frequency of the plane will be higher pitched according to the baggage handler but will decrease in pitch as the plane slows to a stop. d. The frequency of the plane will be higher pitched according
to the baggage handler and will further increase in pitch as the plane slows to a stop. 6. The following graph represents the perceived frequency of a car as it passes a student. Chapter 17 \| Physics of Hearing 769 c. Figure 17.55 d. Figure 17.56 9. A student sends a transverse wave pulse of amplitude A along a rope attached at one end. As the pulse returns to the student, a second pulse of amplitude 3A is sent along the opposite side of the rope. What is the resulting amplitude when the two pulses interact? a. 4A b. A c. 2A, on the side of the original wave pulse d. 2A, on the side of the second wave pulse 10. A student would like to demonstrate destructive interference using two sound sources. Explain how the student could set up this demonstration and what restrictions they would need to place upon their sources. Be sure to consider both the layout of space and the sounds created in your explanation. 11. A student is shaking a flexible string attached to a wooden board in a rhythmic manner. Which of the following choices will decrease the wavelength within the rope? I. The student could shake her hand back and forth with greater frequency. II. The student could shake her hand back in forth with a greater amplitude. III. The student could increase the tension within the rope by stepping backwards from the board. I only I and II I and III II and III I, II, and III a. b. c. d. e. 12. A ripple tank has two locations (L1 and L2) that vibrate in tandem as shown below. Both L1 and L2 vibrate in a plane perpendicular to the page, creating a two-dimensional interference pattern. Figure 17.51 Plot of time versus perceived frequency to illustrate the Doppler effect. a. If the true frequency of the car’s horn is 200 Hz, how fast was the car traveling? b. On the graph above, draw a line demonstrating the perceived frequency for a car traveling twice as fast. Label all intercepts, maximums, and minimums on the graph. 17.5 Sound Interference and Resonance: Standing Waves in Air Columns 7. A common misconception is that two wave pulses traveling in opposite directions will reflect off each other. Outline a procedure that you would use to convince someone that the two wave pulses do not reflect off each other, but instead travel through each other. You may use sketches to represent your understanding. Be
sure to provide evidence to not only refute the original claim, but to support yours as well. 8. Two wave pulses are traveling toward each other on a string, as shown below. Which of the following representations correctly shows the string as the two pulses overlap? Figure 17.52 a. b. Figure 17.53 Figure 17.54 770 Chapter 17 \| Physics of Hearing c. Using information from the graph, determine the speed of sound within the student’s classroom, and explain what characteristic of the graph provides this evidence. d. Determine the temperature of the classroom. 15. A tube is open at one end. If the fundamental frequency f is created by a wavelength λ, then which of the following describes the frequency and wavelength associated with the tube’s fourth overtone? f λ (a) 4f λ/4 (b) 4f λ (c) 9f λ/9 (d) 9f λ 16. A group of students were tasked with collecting information about standing waves. Table 17.10 a series of their data, showing the length of an air column and a resonant frequency present when the column is struck. Table 17.10 Length (m) Resonant Frequency (Hz) 1 2 3 4 85.75 43 29 21.5 a. From their data, determine whether the air column was open or closed on each end. b. Predict the resonant frequency of the column at a length of 2.5 meters. 17. When a student blows across a glass half-full of water, a resonant frequency is created within the air column remaining in the glass. Which of the following can the student do to increase this resonant frequency? I. Add more water to the glass. II. Replace the water with a more dense fluid. Increase the temperature of the room. III. I only a. I and III b. c. II and III d. all of the above 18. A wooden ruler rests on a desk with half of its length protruding off the desk edge. A student holds one end in place and strikes the protruding end with his other hand, creating a musical sound. a. Explain, without using a sound meter, how the student could experimentally determine the speed of sound that travels within the ruler. b. A sound meter is then used to measure the true frequency of the ruler. It is found that the experimental result is lower than the true value. Explain a factor that may have caused this difference. Also
explain what affect this result has on the calculated speed of sound. 19. A musician stands outside in a field and plucks a string on an acoustic guitar. Standing waves will most likely occur in which of the following media? Select two answers. a. The guitar string b. The air inside the guitar c. The air surrounding the guitar Figure 17.57 Describe an experimental procedure to determine the speed of the waves created within the water, including all additional equipment that you would need. You may use the diagram below to help your description, or you may create one of your own. Include enough detail so that another student could carry out your experiment. 13. A string is vibrating between two posts as shown above. Students are to determine the speed of the wave within this string. They have already measured the amount of time necessary for the wave to oscillate up and down. The students must also take what other measurements to determine the speed of the wave? a. The distance between the two posts. b. The amplitude of the wave c. The tension in the string d. The amplitude of the wave and the tension in the string e. The distance between the two posts, the amplitude of the wave, and the tension in the string 14. The accepted speed of sound in room temperature air is 346 m/s. Knowing that their school is colder than usual, a group of students is asked to determine the speed of sound in their room. They are permitted to use any materials necessary; however, their lab procedure must utilize standing wave patterns. The students collect the information Table 17.9. Table 17.9 Trial Number Wavelength (m) Frequency (Hz) 1 2 3 4 5 3.45 2.32 1.70 1.45 1.08 95 135 190 240 305 a. Describe an experimental procedure the group of students could have used to obtain this data. Include diagrams of the experimental setup and any equipment used in the process. b. Select a set of data points from the table and plot those points on a graph to determine the speed of sound within the classroom. Fill in the blank column in the table for any quantities you graph other than the given data. Label the axes and indicate the scale for each. Draw a best-fit line or curve through your data points. This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 17 \| Physics of Hearing 771 d. The ground beneath the musician a. Based on the information above,
what is the speed of 20. the wave within the string? b. The guitarist then slides her finger along the neck of the guitar, changing the string length as a result. Calculate the fundamental frequency of the string and wave speed present if the string length is reduced to 2/3 L. Figure 17.58 This figure shows two tubes that are identical except for their slightly different lengths. Both tubes have one open end and one closed end. A speaker connected to a variable frequency generator is placed in front of the tubes, as shown. The speaker is set to produce a note of very low frequency when turned on. The frequency is then slowly increased to produce resonances in the tubes. Students observe that at first only one of the tubes resonates at a time. Later, as the frequency gets very high, there are times when both tubes resonate. In a clear, coherent, paragraph-length answer, explain why there are some high frequencies, but no low frequencies, at which both tubes resonate. You may include diagrams and/or equations as part of your explanation. Figure 17.59 21. A student connects one end of a string with negligible mass to an oscillator. The other end of the string is passed over a pulley and attached to a suspended weight, as shown above. The student finds that a standing wave with one antinode is formed on the string when the frequency of the oscillator is f0. The student then moves the oscillator to shorten the horizontal segment of string to half its original length. At what frequency will a standing wave with one antinode now be formed on the string? f0/2 f0 a. b. c. 2f0 d. There is no frequency at which a standing wave will be formed. 22. A guitar string of length L is bound at both ends. Table 17.11 shows the string’s harmonic frequencies when struck. Table 17.11 Harmonic Number Frequency 1 2 3 4 225/L 450/L 675/L 900/L Chapter 18 \| Electric Charge and Electric Field 773 18 ELECTRIC CHARGE AND ELECTRIC FIELD Figure 18.1 Static electricity from this plastic slide causes the child's hair to stand on end. The sliding motion stripped electrons away from the child's body, leaving an excess of positive charges, which repel each other along each strand of hair. (credit: Ken Bosma/Wikimedia Commons) Chapter Outline 18.1. Static Electricity and Charge: Conservation of Charge 18.
2. Conductors and Insulators 18.3. Conductors and Electric Fields in Static Equilibrium 18.4. Coulomb’s Law 18.5. Electric Field: Concept of a Field Revisited 18.6. Electric Field Lines: Multiple Charges 18.7. Electric Forces in Biology 18.8. Applications of Electrostatics Connection for AP® Courses The image of American politician and scientist Benjamin Franklin (1706–1790) flying a kite in a thunderstorm (shown in Figure 18.2) is familiar to every schoolchild. In this experiment, Franklin demonstrated a connection between lightning and static electricity. Sparks were drawn from a key hung on a kite string during an electrical storm. These sparks were like those produced by static electricity, such as the spark that jumps from your finger to a metal doorknob after you walk across a wool carpet. Much has been written about Franklin. His experiments were only part of the life of a man who was a scientist, inventor, revolutionary, statesman, and writer. Franklin's experiments were not performed in isolation, nor were they the only ones to reveal connections. 774 Chapter 18 \| Electric Charge and Electric Field Figure 18.2 Benjamin Franklin, his kite, and electricity. When Benjamin Franklin demonstrated that lightning was related to static electricity, he made a connection that is now part of the evidence that all directly experienced forces (except gravitational force) are manifestations of the electromagnetic force. For example, the Italian scientist Luigi Galvani (1737-1798) performed a series of experiments in which static electricity was used to stimulate contractions of leg muscles of dead frogs, an effect already known in humans subjected to static discharges. But Galvani also found that if he joined one end of two metal wires (say copper and zinc) and touched the other ends of the wires to muscles; he produced the same effect in frogs as static discharge. Alessandro Volta (1745-1827), partly inspired by Galvani's work, experimented with various combinations of metals and developed the battery. During the same era, other scientists made progress in discovering fundamental connections. The periodic table was developed as systematic properties of the elements were discovered. This influenced the development and refinement of the concept of atoms as the basis of matter. Such submicroscopic descriptions of matter also help explain a great deal more. Atomic and molecular interactions, such as the forces of friction, cohesion, and adhesion, are now known to be manifestations of the electromagnetic force.
Static electricity is just one aspect of the electromagnetic force, which also includes moving electricity and magnetism. All the macroscopic forces that we experience directly, such as the sensations of touch and the tension in a rope, are due to the electromagnetic force, one of the four fundamental forces in nature. The gravitational force, another fundamental force, is actually sensed through the electromagnetic interaction of molecules, such as between those in our feet and those on the top of a bathroom scale. (The other two fundamental forces, the strong nuclear force and the weak nuclear force, cannot be sensed on the human scale.) This chapter begins the study of electromagnetic phenomena at a fundamental level. The next several chapters will cover static electricity, moving electricity, and magnetism – collectively known as electromagnetism. In this chapter, we begin with the study of electric phenomena due to charges that are at least temporarily stationary, called electrostatics, or static electricity. The chapter introduces several very important concepts of charge, electric force, and electric field, as well as defining the relationships between these concepts. The charge is defined as a property of a system (Big Idea 1) that can affect its interaction with other charged systems (Enduring Understanding 1.B). The law of conservation of electric charge is also discussed (Essential Knowledge 1.B.1). The two kinds of electric charge are defined as positive and negative, providing an explanation for having positively charged, negatively charged, or neutral objects (containing equal quantities of positive and negative charges) (Essential Knowledge 1.B.2). The discrete nature of the electric charge is introduced in this chapter by defining the elementary charge as the smallest observed unit of charge that can be isolated, which is the electron charge (Essential Knowledge 1.B.3). The concepts of a system (having internal structure) and of an object (having no internal structure) are implicitly introduced to explain charges carried by the electron and proton (Enduring Understanding 1.A, Essential Knowledge 1.A.1). An electric field is caused by the presence of charged objects (Enduring Understanding 2.C) and can be used to explain interactions between electrically charged objects (Big Idea 2). The electric force represents the effect of an electric field on a charge placed in the field. The magnitude and direction of the electric force are defined by the magnitude and direction of the electric field and magnitude and sign of the charge (Essential Knowledge 2.C.1). The magnitude of the electric field is proportional to the net charge of the
objects that created that field (Essential Knowledge 2.C.2). For the special case of a spherically symmetric charged object, the electric field outside the object is radial, and its magnitude varies as the inverse square of the radial distance from the center of that object (Essential Knowledge 2.C.3). The chapter provides examples of vector field maps for various charged systems, including point charges, spherically symmetric charge distributions, and uniformly charged parallel plates (Essential Knowledge 2.C.1, Essential Knowledge 2.C.2). For multiple point charges, the chapter explains how to This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 18 \| Electric Charge and Electric Field 775 find the vector field map by adding the electric field vectors of each individual object, including the special case of two equal charges having opposite signs, known as an electric dipole (Essential Knowledge 2.C.4). The special case of two oppositely charged parallel plates with uniformly distributed electric charge when the electric field is perpendicular to the plates and is constant in both magnitude and direction is described in detail, providing many opportunities for problem solving and applications (Essential Knowledge 2.C.5). The idea that interactions can be described by forces is also reinforced in this chapter (Big Idea 3). Like all other forces that you have learned about so far, electric force is a vector that affects the motion according to Newton's laws (Enduring Understanding 3.A). It is clearly stated in the chapter that electric force appears as a result of interactions between two charged objects (Essential Knowledge 3.A.3, Essential Knowledge 3.C.2). At the macroscopic level the electric force is a long-range force (Enduring Understanding 3.C); however, at the microscopic level many contact forces, such as friction, can be explained by interatomic electric forces (Essential Knowledge 3.C.4). This understanding of friction is helpful when considering properties of conductors and insulators and the transfer of charge by conduction. Interactions between systems can result in changes in those systems (Big Idea 4). In the case of charged systems, such interactions can lead to changes of electric properties (Enduring Understanding 4.E), such as charge distribution (Essential Knowledge 4.E.3). Any changes are governed by conservation laws (Big Idea 5). Depending on whether the system is closed or open, certain quantities of the system remain the same
or changes in those quantities are equal to the amount of transfer of this quantity from or to the system (Enduring Understanding 5.A). The electric charge is one of these quantities (Essential Knowledge 5.A.2). Therefore, the electric charge of a system is conserved (Enduring Understanding 5.C) and the exchange of electric charge between objects in a system does not change the total electric charge of the system (Essential Knowledge 5.C.2). Big Idea 1 Objects and systems have properties such as mass and charge. Systems may have internal structure. Enduring Understanding 1.A The internal structure of a system determines many properties of the system. Essential Knowledge 1.A.1 A system is an object or a collection of objects. Objects are treated as having no internal structure. Enduring Understanding 1.B Electric charge is a property of an object or system that affects its interactions with other objects or systems containing charge. Essential Knowledge 1.B.1 Electric charge is conserved. The net charge of a system is equal to the sum of the charges of all the objects in the system. Essential Knowledge 1.B.2 There are only two kinds of electric charge. Neutral objects or systems contain equal quantities of positive and negative charge, with the exception of some fundamental particles that have no electric charge. Essential Knowledge 1.B.3 The smallest observed unit of charge that can be isolated is the electron charge, also known as the elementary charge. Big Idea 2 Fields existing in space can be used to explain interactions. Enduring Understanding 2.C An electric field is caused by an object with electric charge. Essential Knowledge 2.C.1 The magnitude of the electric force F exerted on an object with electric charge q by an electric field ( →. The direction of the force is determined by the direction of the field and the sign of the charge, with → = → is positively charged objects accelerating in the direction of the field and negatively charged objects accelerating in the direction opposite the field. This should include a vector field map for positive point charges, negative point charges, spherically symmetric charge distribution, and uniformly charged parallel plates. Essential Knowledge 2.C.2 The magnitude of the electric field vector is proportional to the net electric charge of the object(s) creating that field. This includes positive point charges, negative point charges, spherically symmetric charge distributions, and uniformly charged parallel plates. Essential Knowledge 2.C.3 The electric field outside a spherically symmetric charged object is radial
, and its magnitude varies as the inverse square of the radial distance from the center of that object. Electric field lines are not in the curriculum. Students will be expected to rely only on the rough intuitive sense underlying field lines, wherein the field is viewed as analogous to something emanating uniformly from a source. Essential Knowledge 2.C.4 The electric field around dipoles and other systems of electrically charged objects (that can be modeled as point objects) is found by vector addition of the field of each individual object. Electric dipoles are treated qualitatively in this course as a teaching analogy to facilitate student understanding of magnetic dipoles. Essential Knowledge 2.C.5 Between two oppositely charged parallel plates with uniformly distributed electric charge, at points far from the edges of the plates, the electric field is perpendicular to the plates and is constant in both magnitude and direction. Big Idea 3 The interactions of an object with other objects can be described by forces. Enduring Understanding 3.A All forces share certain common characteristics when considered by observers in inertial reference frames. Essential Knowledge 3.A.3 A force exerted on an object is always due to the interaction of that object with another object. Enduring Understanding 3.C At the macroscopic level, forces can be categorized as either long-range (action-at-a-distance) forces or contact forces. Essential Knowledge 3.C.2 Electric force results from the interaction of one object that has an electric charge with another object that has an electric charge. 776 Chapter 18 \| Electric Charge and Electric Field Essential Knowledge 3.C.4 Contact forces result from the interaction of one object touching another object, and they arise from interatomic electric forces. These forces include tension, friction, normal, spring (Physics 1), and buoyant (Physics 2). Big Idea 4 Interactions between systems can result in changes in those systems. Enduring Understanding 4.E The electric and magnetic properties of a system can change in response to the presence of, or changes in, other objects or systems. Essential Knowledge 4.E.3 The charge distribution in a system can be altered by the effects of electric forces produced by a charged object. Big Idea 5 Changes that occur as a result of interactions are constrained by conservation laws. Enduring Understanding 5.A Certain quantities are conserved, in the sense that the changes of those quantities in a given system are always equal to the transfer of that quantity to or from the system by all possible interactions with other systems. Essential Knowledge 5.A.
2 For all systems under all circumstances, energy, charge, linear momentum, and angular momentum are conserved. Enduring Understanding 5.C The electric charge of a system is conserved. Essential Knowledge 5.C.2 The exchange of electric charges among a set of objects in a system conserves electric charge. 18.1 Static Electricity and Charge: Conservation of Charge Learning Objectives By the end of this section, you will be able to: • Define electric charge, and describe how the two types of charge interact. • Describe three common situations that generate static electricity. • State the law of conservation of charge. The information presented in this section supports the following AP® learning objectives and science practices: • 1.B.1.1 The student is able to make claims about natural phenomena based on conservation of electric charge. (S.P. 6.4) • 1.B.1.2 The student is able to make predictions, using the conservation of electric charge, about the sign and relative quantity of net charge of objects or systems after various charging processes, including conservation of charge in simple circuits. (S.P. 6.4, 7.2) • 1.B.2.1 The student is able to construct an explanation of the two-charge model of electric charge based on evidence produced through scientific practices. (S.P. 6.4) • 1.B.3.1 The student is able to challenge the claim that an electric charge smaller than the elementary charge has been isolated. (S.P. 1.5, 6.1, 7.2) • 5.A.2.1 The student is able to define open and closed systems for everyday situations and apply conservation concepts for energy, charge, and linear momentum to those situations. (S.P. 6.4, 7.2) • 5.C.2.1 The student is able to predict electric charges on objects within a system by application of the principle of charge conservation within a system. (S.P. 6.4) • 5.C.2.2 The student is able to design a plan to collect data on the electrical charging of objects and electric charge induction on neutral objects and qualitatively analyze that data. (S.P. 4.2, 5.1) • 5.C.2.3 The student is able to justify the selection of data relevant to an investigation of the electrical charging of objects and electric charge induction on neutral objects. (S.
P. 4.1) Figure 18.3 Borneo amber was mined in Sabah, Malaysia, from shale-sandstone-mudstone veins. When a piece of amber is rubbed with a piece of silk, the amber gains more electrons, giving it a net negative charge. At the same time, the silk, having lost electrons, becomes positively charged. (credit: Sebakoamber, Wikimedia Commons) This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 18 \| Electric Charge and Electric Field 777 What makes plastic wrap cling? Static electricity. Not only are applications of static electricity common these days, its existence has been known since ancient times. The first record of its effects dates to ancient Greeks who noted more than 500 years B.C. that polishing amber temporarily enabled it to attract bits of straw (see Figure 18.3). The very word electric derives from the Greek word for amber (electron). Many of the characteristics of static electricity can be explored by rubbing things together. Rubbing creates the spark you get from walking across a wool carpet, for example. Static cling generated in a clothes dryer and the attraction of straw to recently polished amber also result from rubbing. Similarly, lightning results from air movements under certain weather conditions. You can also rub a balloon on your hair, and the static electricity created can then make the balloon cling to a wall. We also have to be cautious of static electricity, especially in dry climates. When we pump gasoline, we are warned to discharge ourselves (after sliding across the seat) on a metal surface before grabbing the gas nozzle. Attendants in hospital operating rooms must wear booties with aluminum foil on the bottoms to avoid creating sparks which may ignite the oxygen being used. Some of the most basic characteristics of static electricity include: • The effects of static electricity are explained by a physical quantity not previously introduced, called electric charge. • There are only two types of charge, one called positive and the other called negative. • Like charges repel, whereas unlike charges attract. • The force between charges decreases with distance. How do we know there are two types of electric charge? When various materials are rubbed together in controlled ways, certain combinations of materials always produce one type of charge on one material and the opposite type on the other. By convention, we call one type of charge “positive”, and the other type “negative.” For example, when glass is rubbed with silk, the glass
becomes positively charged and the silk negatively charged. Since the glass and silk have opposite charges, they attract one another like clothes that have rubbed together in a dryer. Two glass rods rubbed with silk in this manner will repel one another, since each rod has positive charge on it. Similarly, two silk cloths so rubbed will repel, since both cloths have negative charge. Figure 18.4 shows how these simple materials can be used to explore the nature of the force between charges. Figure 18.4 A glass rod becomes positively charged when rubbed with silk, while the silk becomes negatively charged. (a) The glass rod is attracted to the silk because their charges are opposite. (b) Two similarly charged glass rods repel. (c) Two similarly charged silk cloths repel. More sophisticated questions arise. Where do these charges come from? Can you create or destroy charge? Is there a smallest unit of charge? Exactly how does the force depend on the amount of charge and the distance between charges? Such questions obviously occurred to Benjamin Franklin and other early researchers, and they interest us even today. Charge Carried by Electrons and Protons Franklin wrote in his letters and books that he could see the effects of electric charge but did not understand what caused the phenomenon. Today we have the advantage of knowing that normal matter is made of atoms, and that atoms contain positive and negative charges, usually in equal amounts. Figure 18.5 shows a simple model of an atom with negative electrons orbiting its positive nucleus. The nucleus is positive due to the presence of positively charged protons. Nearly all charge in nature is due to electrons and protons, which are two of the three building blocks of most matter. (The third is the neutron, which is neutral, carrying no charge.) Other charge-carrying particles are observed in cosmic rays and nuclear decay, and are created in particle accelerators. All but the electron and proton survive only a short time and are quite rare by comparison. 778 Chapter 18 \| Electric Charge and Electric Field Figure 18.5 This simplified (and not to scale) view of an atom is called the planetary model of the atom. Negative electrons orbit a much heavier positive nucleus, as the planets orbit the much heavier sun. There the similarity ends, because forces in the atom are electromagnetic, whereas those in the planetary system are gravitational. Normal macroscopic amounts of matter contain immense numbers of atoms and molecules and, hence, even greater numbers of individual negative and positive charges. The charges of electrons and prot
ons are identical in magnitude but opposite in sign. Furthermore, all charged objects in nature are integral multiples of this basic quantity of charge, meaning that all charges are made of combinations of a basic unit of charge. Usually, charges are formed by combinations of electrons and protons. The magnitude of this basic charge is The symbol is commonly used for charge and the subscript indicates the charge of a single electron (or proton). The SI unit of charge is the coulomb (C). The number of protons needed to make a charge of 1.00 C is ∣ ∣ = 1.60×10−19 C. 1.00 C× 1 proton 1.60×10−19 C = 6.25×1018 protons. (18.1) (18.2) Similarly, 6.25×1018 atom), there is a smallest bit of charge. There is no directly observed charge smaller than ∣ ∣ Small: The Submicroscopic Origin of Charge), and all observed charges are integral multiples of electrons have a combined charge of −1.00 coulomb. Just as there is a smallest bit of an element (an (see Things Great and ∣ ∣. Things Great and Small: The Submicroscopic Origin of Charge With the exception of exotic, short-lived particles, all charge in nature is carried by electrons and protons. Electrons carry the charge we have named negative. Protons carry an equal-magnitude charge that we call positive. (See Figure 18.6.) Electron and proton charges are considered fundamental building blocks, since all other charges are integral multiples of those carried by electrons and protons. Electrons and protons are also two of the three fundamental building blocks of ordinary matter. The neutron is the third and has zero total charge. Figure 18.6 shows a person touching a Van de Graaff generator and receiving excess positive charge. The expanded view of a hair shows the existence of both types of charges but an excess of positive. The repulsion of these positive like charges causes the strands of hair to repel other strands of hair and to stand up. The further blowup shows an artist's conception of an electron and a proton perhaps found in an atom in a strand of hair. This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 18 \| Electric Charge and Electric Field 779 Figure 18.6 When this person touches a Van
de Graaff generator, she receives an excess of positive charge, causing her hair to stand on end. The charges in one hair are shown. An artist's conception of an electron and a proton illustrate the particles carrying the negative and positive charges. We cannot really see these particles with visible light because they are so small (the electron seems to be an infinitesimal point), but we know a great deal about their measurable properties, such as the charges they carry. The electron seems to have no substructure; in contrast, when the substructure of protons is explored by scattering extremely energetic electrons from them, it appears that there are point-like particles inside the proton. These sub-particles, named quarks, have never been directly observed, but they are believed to carry fractional charges as seen in Figure 18.7. Charges on electrons and protons and all other directly observable particles are unitary, but these quark substructures carry charges of either − 1 3. There are continuing attempts to observe fractional charge directly and to learn of the properties of quarks, which or + 2 3 are perhaps the ultimate substructure of matter. Figure 18.7 Artist's conception of fractional quark charges inside a proton. A group of three quark charges add up to the single positive charge on the proton1. Separation of Charge in Atoms Charges in atoms and molecules can be separated—for example, by rubbing materials together. Some atoms and molecules have a greater affinity for electrons than others and will become negatively charged by close contact in rubbing, leaving the other material positively charged. (See Figure 18.8.) Positive charge can similarly be induced by rubbing. Methods other than rubbing can also separate charges. Batteries, for example, use combinations of substances that interact in such a way as to separate charges. Chemical interactions may transfer negative charge from one substance to the other, making one battery terminal negative and leaving the first one positive. 780 Chapter 18 \| Electric Charge and Electric Field Figure 18.8 When materials are rubbed together, charges can be separated, particularly if one material has a greater affinity for electrons than another. (a) Both the amber and cloth are originally neutral, with equal positive and negative charges. Only a tiny fraction of the charges are involved, and only a few of them are shown here. (b) When rubbed together, some negative charge is transferred to the amber, leaving the cloth with a net positive charge. (c) When separated, the amber and cloth
now have net charges, but the absolute value of the net positive and negative charges will be equal. No charge is actually created or destroyed when charges are separated as we have been discussing. Rather, existing charges are moved about. In fact, in all situations the total amount of charge is always constant. This universally obeyed law of nature is called the law of conservation of charge. Law of Conservation of Charge Total charge is constant in any process. Making Connections: Net Charge Hence if a closed system is neutral, it will remain neutral. Similarly, if a closed system has a charge, say, −10e, it will always have that charge. The only way to change the charge of a system is to transfer charge outside, either by bringing in charge or removing charge. If it is possible to transfer charge outside, the system is no longer closed/isolated and is known as an open system. However, charge is always conserved, for both open and closed systems. Consequently, the charge transferred to/from an open system is equal to the change in the system's charge. For example, each of the two materials (amber and cloth) discussed in Figure 18.8 have no net charge initially. The only way to change their charge is to transfer charge from outside each object. When they are rubbed together, negative charge is transferred to the amber and the final charge of the amber is the sum of the initial charge and the charge transferred to it. On the other hand, the final charge on the cloth is equal to its initial charge minus the charge transferred out. Similarly when glass is rubbed with silk, the net charge on the silk is its initial charge plus the incoming charge and the charge on the glass is the initial charge minus the outgoing charge. Also the charge gained by the silk will be equal to the charge lost by the glass, which means that if the silk gains –5e charge, the glass would have lost −5e charge. In more exotic situations, such as in particle accelerators, mass, Δ, can be created from energy in the amount Δ = 2. Sometimes, the created mass is charged, such as when an electron is created. Whenever a charged particle is created, another having an opposite charge is always created along with it, so that the total charge created is zero. Usually, the two particles are “matter-antimatter” counterparts. For example, an antielectron would usually be created at the same time as an electron. The antielectron has a positive
charge (it is called a positron), and so the total charge created is zero. (See Figure 18.9.) All particles have antimatter counterparts with opposite signs. When matter and antimatter counterparts are brought together, they completely annihilate one another. By annihilate, we mean that the mass of the two particles is converted to energy E, again obeying the relationship Δ = 2 annihilation; thus, total charge is conserved.. Since the two particles have equal and opposite charge, the total charge is zero before and after the Making Connections: Conservation Laws Only a limited number of physical quantities are universally conserved. Charge is one—energy, momentum, and angular momentum are others. Because they are conserved, these physical quantities are used to explain more phenomena and form more connections than other, less basic quantities. We find that conserved quantities give us great insight into the rules followed by nature and hints to the organization of nature. Discoveries of conservation laws have led to further discoveries, such as the weak nuclear force and the quark substructure of protons and other particles. This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 18 \| Electric Charge and Electric Field 781 Figure 18.9 (a) When enough energy is present, it can be converted into matter. Here the matter created is an electron–antielectron pair. ( is the electron's mass.) The total charge before and after this event is zero. (b) When matter and antimatter collide, they annihilate each other; the total charge is conserved at zero before and after the annihilation. The law of conservation of charge is absolute—it has never been observed to be violated. Charge, then, is a special physical quantity, joining a very short list of other quantities in nature that are always conserved. Other conserved quantities include energy, momentum, and angular momentum. PhET Explorations: Balloons and Static Electricity Why does a balloon stick to your sweater? Rub a balloon on a sweater, then let go of the balloon and it flies over and sticks to the sweater. View the charges in the sweater, balloons, and the wall. Figure 18.10 Balloons and Static Electricity (http://cnx.org/content/m55300/1.2/balloons_en.jar) Applying the Science Practices: Electrical Charging Design an experiment to demonstrate the electrical charging of objects, by using a glass rod, a balloon
, small bits of paper, and different pieces of cloth (like silk, wool, or nylon). Also show that like charges repel each other whereas unlike charges attract each other. 18.2 Conductors and Insulators Learning Objectives By the end of this section, you will be able to: • Define conductor and insulator, explain the difference, and give examples of each. • Describe three methods for charging an object. • Explain what happens to an electric force as you move farther from the source. • Define polarization. The information presented in this section supports the following AP® learning objectives and science practices: 782 Chapter 18 \| Electric Charge and Electric Field • 1.B.2.2 The student is able to make a qualitative prediction about the distribution of positive and negative electric charges within neutral systems as they undergo various processes. (S.P. 6.4, 7.2) • 1.B.2.3 The student is able to challenge claims that polarization of electric charge or separation of charge must result in a net charge on the object. (S.P. 6.1) • 4.E.3.1 The student is able to make predictions about the redistribution of charge during charging by friction, conduction, and induction. (S.P. 6.4) • 4.E.3.2 The student is able to make predictions about the redistribution of charge caused by the electric field due to other systems, resulting in charged or polarized objects. (S.P. 6.4, 7.2) • 4.E.3.3 The student is able to construct a representation of the distribution of fixed and mobile charge in insulators and conductors. (S.P. 1.1, 1.4, 6.4) • 4.E.3.4 The student is able to construct a representation of the distribution of fixed and mobile charge in insulators and conductors that predicts charge distribution in processes involving induction or conduction. (S.P. 1.1, 1.4, 6.4) • 4.E.3.5 The student is able to plan and/or analyze the results of experiments in which electric charge rearrangement occurs by electrostatic induction, or is able to refine a scientific question relating to such an experiment by identifying anomalies in a data set or procedure. (S.P. 3.2, 4.1, 4.2, 5.1, 5.3) Figure 18.11
This power adapter uses metal wires and connectors to conduct electricity from the wall socket to a laptop computer. The conducting wires allow electrons to move freely through the cables, which are shielded by rubber and plastic. These materials act as insulators that don't allow electric charge to escape outward. (credit: Evan-Amos, Wikimedia Commons) Some substances, such as metals and salty water, allow charges to move through them with relative ease. Some of the electrons in metals and similar conductors are not bound to individual atoms or sites in the material. These free electrons can move through the material much as air moves through loose sand. Any substance that has free electrons and allows charge to move relatively freely through it is called a conductor. The moving electrons may collide with fixed atoms and molecules, losing some energy, but they can move in a conductor. Superconductors allow the movement of charge without any loss of energy. Salty water and other similar conducting materials contain free ions that can move through them. An ion is an atom or molecule having a positive or negative (nonzero) total charge. In other words, the total number of electrons is not equal to the total number of protons. Other substances, such as glass, do not allow charges to move through them. These are called insulators. Electrons and ions in insulators are bound in the structure and cannot move easily—as much as 1023 water and dry table salt are insulators, for example, whereas molten salt and salty water are conductors. times more slowly than in conductors. Pure Figure 18.12 An electroscope is a favorite instrument in physics demonstrations and student laboratories. It is typically made with gold foil leaves hung from a (conducting) metal stem and is insulated from the room air in a glass-walled container. (a) A positively charged glass rod is brought near the tip of the electroscope, attracting electrons to the top and leaving a net positive charge on the leaves. Like charges in the light flexible gold leaves repel, separating them. (b) When the rod is touched against the ball, electrons are attracted and transferred, reducing the net charge on the glass rod but leaving the electroscope positively charged. (c) The excess charges are evenly distributed in the stem and leaves of the electroscope once the glass rod is removed. Charging by Contact Figure 18.12 shows an electroscope being charged by touching it with a positively charged glass rod. Because the glass rod is an insulator, it must actually touch the electroscope to transfer
charge to or from it. (Note that the extra positive charges reside on the surface of the glass rod as a result of rubbing it with silk before starting the experiment.) Since only electrons move in This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 18 \| Electric Charge and Electric Field 783 metals, we see that they are attracted to the top of the electroscope. There, some are transferred to the positive rod by touch, leaving the electroscope with a net positive charge. Electrostatic repulsion in the leaves of the charged electroscope separates them. The electrostatic force has a horizontal component that results in the leaves moving apart as well as a vertical component that is balanced by the gravitational force. Similarly, the electroscope can be negatively charged by contact with a negatively charged object. Charging by Induction It is not necessary to transfer excess charge directly to an object in order to charge it. Figure 18.13 shows a method of induction wherein a charge is created in a nearby object, without direct contact. Here we see two neutral metal spheres in contact with one another but insulated from the rest of the world. A positively charged rod is brought near one of them, attracting negative charge to that side, leaving the other sphere positively charged. This is an example of induced polarization of neutral objects. Polarization is the separation of charges in an object that remains neutral. If the spheres are now separated (before the rod is pulled away), each sphere will have a net charge. Note that the object closest to the charged rod receives an opposite charge when charged by induction. Note also that no charge is removed from the charged rod, so that this process can be repeated without depleting the supply of excess charge. Another method of charging by induction is shown in Figure 18.14. The neutral metal sphere is polarized when a charged rod is brought near it. The sphere is then grounded, meaning that a conducting wire is run from the sphere to the ground. Since the earth is large and most ground is a good conductor, it can supply or accept excess charge easily. In this case, electrons are attracted to the sphere through a wire called the ground wire, because it supplies a conducting path to the ground. The ground connection is broken before the charged rod is removed, leaving the sphere with an excess charge opposite to that of the rod. Again, an opposite charge is achieved when charging by induction and the charged rod loses none of its excess charge. Figure 18.13 Charging by induction
. (a) Two uncharged or neutral metal spheres are in contact with each other but insulated from the rest of the world. (b) A positively charged glass rod is brought near the sphere on the left, attracting negative charge and leaving the other sphere positively charged. (c) The spheres are separated before the rod is removed, thus separating negative and positive charge. (d) The spheres retain net charges after the inducing rod is removed—without ever having been touched by a charged object. 784 Chapter 18 \| Electric Charge and Electric Field Figure 18.14 Charging by induction, using a ground connection. (a) A positively charged rod is brought near a neutral metal sphere, polarizing it. (b) The sphere is grounded, allowing electrons to be attracted from the earth's ample supply. (c) The ground connection is broken. (d) The positive rod is removed, leaving the sphere with an induced negative charge. Figure 18.15 Both positive and negative objects attract a neutral object by polarizing its molecules. (a) A positive object brought near a neutral insulator polarizes its molecules. There is a slight shift in the distribution of the electrons orbiting the molecule, with unlike charges being brought nearer and like charges moved away. Since the electrostatic force decreases with distance, there is a net attraction. (b) A negative object produces the opposite polarization, but again attracts the neutral object. (c) The same effect occurs for a conductor; since the unlike charges are closer, there is a net attraction. Neutral objects can be attracted to any charged object. The pieces of straw attracted to polished amber are neutral, for example. If you run a plastic comb through your hair, the charged comb can pick up neutral pieces of paper. Figure 18.15 shows how the polarization of atoms and molecules in neutral objects results in their attraction to a charged object. When a charged rod is brought near a neutral substance, an insulator in this case, the distribution of charge in atoms and molecules is shifted slightly. Opposite charge is attracted nearer the external charged rod, while like charge is repelled. Since the electrostatic force decreases with distance, the repulsion of like charges is weaker than the attraction of unlike charges, and so there is a net attraction. Thus a positively charged glass rod attracts neutral pieces of paper, as will a negatively charged rubber rod. Some molecules, like water, are polar molecules. Polar molecules have a natural or inherent separation of charge, although they are neutral overall. Polar molecules are particularly
affected by other charged objects and show greater polarization effects than molecules with naturally uniform charge distributions. Check Your Understanding Can you explain the attraction of water to the charged rod in the figure below? This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 18 \| Electric Charge and Electric Field 785 Figure 18.16 Solution Water molecules are polarized, giving them slightly positive and slightly negative sides. This makes water even more susceptible to a charged rod's attraction. As the water flows downward, due to the force of gravity, the charged conductor exerts a net attraction to the opposite charges in the stream of water, pulling it closer. Applying the Science Practices: Electrostatic Induction Plan an experiment to demonstrate electrostatic induction using household items, like balloons, woolen cloth, aluminum drink cans, or foam cups. Explain the process of induction in your experiment by discussing details of (and making diagrams relating to) the movement and alignment of charges. PhET Explorations: John Travoltage Make sparks fly with John Travoltage. Wiggle Johnnie's foot and he picks up charges from the carpet. Bring his hand close to the door knob and get rid of the excess charge. Figure 18.17 John Travoltage (http://cnx.org/content/m55301/1.2/travoltage_en.jar) 18.3 Conductors and Electric Fields in Static Equilibrium Learning Objectives By the end of this section, you will be able to: • List the three properties of a conductor in electrostatic equilibrium. • Explain the effect of an electric field on free charges in a conductor. • Explain why no electric field may exist inside a conductor. • Describe the electric field surrounding Earth. • Explain what happens to an electric field applied to an irregular conductor. • Describe how a lightning rod works. • Explain how a metal car may protect passengers inside from the dangerous electric fields caused by a downed line touching the car. The information presented in this section supports the following AP learning objectives: • 2.C.3.1 The student is able to explain the inverse square dependence of the electric field surrounding a spherically symmetric electrically charged object. 786 Chapter 18 \| Electric Charge and Electric Field • 2.C.5.1 The student is able to create representations of the magnitude and direction of the electric field at various distances (small compared to plate size) from two electrically charged plates of equal
magnitude and opposite signs and is able to recognize that the assumption of uniform field is not appropriate near edges of plates. Conductors contain free charges that move easily. When excess charge is placed on a conductor or the conductor is put into a static electric field, charges in the conductor quickly respond to reach a steady state called electrostatic equilibrium. Figure 18.18 shows the effect of an electric field on free charges in a conductor. The free charges move until the field is perpendicular to the conductor's surface. There can be no component of the field parallel to the surface in electrostatic equilibrium, since, if there were, it would produce further movement of charge. A positive free charge is shown, but free charges can be either positive or negative and are, in fact, negative in metals. The motion of a positive charge is equivalent to the motion of a negative charge in the opposite direction. Figure 18.18 When an electric field E is applied to a conductor, free charges inside the conductor move until the field is perpendicular to the surface. (a) The electric field is a vector quantity, with both parallel and perpendicular components. The parallel component ( E∥ ) exerts a force ( F∥ ) on the free charge, which moves the charge until F∥ = 0. (b) The resulting field is perpendicular to the surface. The free charge has been brought to the conductor's surface, leaving electrostatic forces in equilibrium. A conductor placed in an electric field will be polarized. Figure 18.19 shows the result of placing a neutral conductor in an originally uniform electric field. The field becomes stronger near the conductor but entirely disappears inside it. Figure 18.19 This illustration shows a spherical conductor in static equilibrium with an originally uniform electric field. Free charges move within the conductor, polarizing it, until the electric field lines are perpendicular to the surface. The field lines end on excess negative charge on one section of the surface and begin again on excess positive charge on the opposite side. No electric field exists inside the conductor, since free charges in the conductor would continue moving in response to any field until it was neutralized. Misconception Alert: Electric Field inside a Conductor Excess charges placed on a spherical conductor repel and move until they are evenly distributed, as shown in Figure 18.20. Excess charge is forced to the surface until the field inside the conductor is zero. Outside the conductor, the field is exactly the same as if the conductor were replaced by a point charge at its center equal to the excess charge.
This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 18 \| Electric Charge and Electric Field 787 Figure 18.20 The mutual repulsion of excess positive charges on a spherical conductor distributes them uniformly on its surface. The resulting electric field is perpendicular to the surface and zero inside. Outside the conductor, the field is identical to that of a point charge at the center equal to the excess charge. Properties of a Conductor in Electrostatic Equilibrium 1. The electric field is zero inside a conductor. 2. Just outside a conductor, the electric field lines are perpendicular to its surface, ending or beginning on charges on the surface. 3. Any excess charge resides entirely on the surface or surfaces of a conductor. The properties of a conductor are consistent with the situations already discussed and can be used to analyze any conductor in electrostatic equilibrium. This can lead to some interesting new insights, such as described below. How can a very uniform electric field be created? Consider a system of two metal plates with opposite charges on them, as shown in Figure 18.21. The properties of conductors in electrostatic equilibrium indicate that the electric field between the plates will be uniform in strength and direction. Except near the edges, the excess charges distribute themselves uniformly, producing field lines that are uniformly spaced (hence uniform in strength) and perpendicular to the surfaces (hence uniform in direction, since the plates are flat). The edge effects are less important when the plates are close together. Figure 18.21 Two metal plates with equal, but opposite, excess charges. The field between them is uniform in strength and direction except near the edges. One use of such a field is to produce uniform acceleration of charges between the plates, such as in the electron gun of a TV tube. Earth's Electric Field A near uniform electric field of approximately 150 N/C, directed downward, surrounds Earth, with the magnitude increasing slightly as we get closer to the surface. What causes the electric field? At around 100 km above the surface of Earth we have a layer of charged particles, called the ionosphere. The ionosphere is responsible for a range of phenomena including the electric field surrounding Earth. In fair weather the ionosphere is positive and the Earth largely negative, maintaining the electric field (Figure 18.22(a)). In storm conditions clouds form and localized electric fields can be larger and reversed in direction (Figure 18.22(b)). The exact charge distributions depend on the local conditions, and variations of Figure
18.22(b) are possible. If the electric field is sufficiently large, the insulating properties of the surrounding material break down and it becomes conducting. For air this occurs at around 3×106 form of lightning sparks and corona discharge. N/C. Air ionizes ions and electrons recombine, and we get discharge in the 788 Chapter 18 \| Electric Charge and Electric Field Figure 18.22 Earth's electric field. (a) Fair weather field. Earth and the ionosphere (a layer of charged particles) are both conductors. They produce a uniform electric field of about 150 N/C. (credit: D. H. Parks) (b) Storm fields. In the presence of storm clouds, the local electric fields can be larger. At very high fields, the insulating properties of the air break down and lightning can occur. (credit: Jan-Joost Verhoef) Electric Fields on Uneven Surfaces So far we have considered excess charges on a smooth, symmetrical conductor surface. What happens if a conductor has sharp corners or is pointed? Excess charges on a nonuniform conductor become concentrated at the sharpest points. Additionally, excess charge may move on or off the conductor at the sharpest points. To see how and why this happens, consider the charged conductor in Figure 18.23. The electrostatic repulsion of like charges is most effective in moving them apart on the flattest surface, and so they become least concentrated there. This is because the forces between identical pairs of charges at either end of the conductor are identical, but the components of the forces parallel to the surfaces are different. The component parallel to the surface is greatest on the flattest surface and, hence, more effective in moving the charge. The same effect is produced on a conductor by an externally applied electric field, as seen in Figure 18.23 (c). Since the field lines must be perpendicular to the surface, more of them are concentrated on the most curved parts. Figure 18.23 Excess charge on a nonuniform conductor becomes most concentrated at the location of greatest curvature. (a) The forces between identical pairs of charges at either end of the conductor are identical, but the components of the forces parallel to the surface are different. It is F∥ that moves the charges apart once they have reached the surface. (b) F∥ producing the electric field shown. (c) An uncharged conductor in an originally uniform electric field is polarized, with the most concentrated charge
at its most pointed end. is smallest at the more pointed end, the charges are left closer together, Applications of Conductors On a very sharply curved surface, such as shown in Figure 18.24, the charges are so concentrated at the point that the resulting electric field can be great enough to remove them from the surface. This can be useful. Lightning rods work best when they are most pointed. The large charges created in storm clouds induce an opposite charge on a building that can result in a lightning bolt hitting the building. The induced charge is bled away continually by a lightning rod, preventing the more dramatic lightning strike. Of course, we sometimes wish to prevent the transfer of charge rather than to facilitate it. In that case, the conductor should be very smooth and have as large a radius of curvature as possible. (See Figure 18.25.) Smooth surfaces are used on high-voltage transmission lines, for example, to avoid leakage of charge into the air. Another device that makes use of some of these principles is a Faraday cage. This is a metal shield that encloses a volume. All electrical charges will reside on the outside surface of this shield, and there will be no electrical field inside. A Faraday cage is used to prohibit stray electrical fields in the environment from interfering with sensitive measurements, such as the electrical signals inside a nerve cell. During electrical storms if you are driving a car, it is best to stay inside the car as its metal body acts as a Faraday cage with zero electrical field inside. If in the vicinity of a lightning strike, its effect is felt on the outside of the car and the inside is unaffected, provided you remain totally inside. This is also true if an active (“hot”) electrical wire was broken (in a storm or an accident) and fell on your car. This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 18 \| Electric Charge and Electric Field 789 Figure 18.24 A very pointed conductor has a large charge concentration at the point. The electric field is very strong at the point and can exert a force large enough to transfer charge on or off the conductor. Lightning rods are used to prevent the buildup of large excess charges on structures and, thus, are pointed. Figure 18.25 (a) A lightning rod is pointed to facilitate the transfer of charge. (credit: Romaine, Wikimedia Commons) (b) This Van de Graaff generator has a smooth surface with a
large radius of curvature to prevent the transfer of charge and allow a large voltage to be generated. The mutual repulsion of like charges is evident in the person's hair while touching the metal sphere. (credit: Jon ‘ShakataGaNai' Davis/Wikimedia Commons). 18.4 Coulomb’s Law By the end of this section, you will be able to: Learning Objectives • State Coulomb's law in terms of how the electrostatic force changes with the distance between two objects. • Calculate the electrostatic force between two point charges, such as electrons or protons. • Compare the electrostatic force to the gravitational attraction for a proton and an electron; for a human and the Earth. The information presented in this section supports the following AP® learning objectives and science practices: • 3.A.3.3 The student is able to describe a force as an interaction between two objects and identify both objects for any force. (S.P. 1.4) • 3.A.3.4 The student is able to make claims about the force on an object due to the presence of other objects with the same property: mass, electric charge. (S.P. 6.1, 6.4) • 3.C.2.1 The student is able to use Coulomb's law qualitatively and quantitatively to make predictions about the interaction between two electric point charges (interactions between collections of electric point charges are not covered in Physics 1 and instead are restricted to Physics 2). (S.P. 2.2, 6.4) • 3.C.2.2 The student is able to connect the concepts of gravitational force and electric force to compare similarities and differences between the forces. (S.P. 7.2) 790 Chapter 18 \| Electric Charge and Electric Field Figure 18.26 This NASA image of Arp 87 shows the result of a strong gravitational attraction between two galaxies. In contrast, at the subatomic level, the electrostatic attraction between two objects, such as an electron and a proton, is far greater than their mutual attraction due to gravity. (credit: NASA/HST) Through the work of scientists in the late 18th century, the main features of the electrostatic force—the existence of two types of charge, the observation that like charges repel, unlike charges attract, and the decrease of force with distance—were eventually refined, and expressed as a mathematical formula. The mathematical formula for the electro
static force is called Coulomb's law after the French physicist Charles Coulomb (1736–1806), who performed experiments and first proposed a formula to calculate it. Coulomb's Law = \|1 2\| 2 Coulomb's law calculates the magnitude of the force between two point charges, 1 and 2, separated by a distance. In SI units, the constant is equal to. (18.3) = 8.988×109N ⋅ m2 C2 ≈ 8.99×109N ⋅ m2 C2. (18.4) The electrostatic force is a vector quantity and is expressed in units of newtons. The force is understood to be along the line joining the two charges. (See Figure 18.27.) Although the formula for Coulomb's law is simple, it was no mean task to prove it. The experiments Coulomb did, with the primitive equipment then available, were difficult. Modern experiments have verified Coulomb's law to great precision. For ∝ 1 / 2 example, it has been shown that the force is inversely proportional to distance between two objects squared to an accuracy of 1 part in 1016. No exceptions have ever been found, even at the small distances within the atom. Figure 18.27 The magnitude of the electrostatic force between point charges 1 and 2 separated by a distance is given by Coulomb's law. Note that Newton's third law (every force exerted creates an equal and opposite force) applies as usual—the force on 1 is equal in magnitude and opposite in direction to the force it exerts on 2. (a) Like charges. (b) Unlike charges. Making Connections: Comparing Gravitational and Electrostatic Forces Recall that the gravitational force (Newton's law of gravitation) quantifies force as = 2. The comparison between the two forces—gravitational and electrostatic—shows some similarities and differences. Gravitational force is proportional to the masses of interacting objects, and the electrostatic force is proportional to the magnitudes of the charges of interacting objects. Hence both forces are proportional to a property that represents the strength of interaction for a given field. In addition, both forces are inversely proportional to the square of the distances between them. It may seem that the two forces are related but that is not the case. In fact, there are huge variations in the magnitudes of the two forces as they depend on different parameters and different mechanisms. For electrons (or protons), electrostatic force is
dominant and is much greater than the gravitational force. On the other hand, gravitational force is This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 18 \| Electric Charge and Electric Field 791 generally dominant for objects with large masses. Another major difference between the two forces is that gravitational force can only be attractive, whereas electrostatic could be attractive or repulsive (depending on the sign of charges; unlike charges attract and like charges repel). Example 18.1 How Strong is the Coulomb Force Relative to the Gravitational Force? Compare the electrostatic force between an electron and proton separated by 0.530×10−10 m with the gravitational force between them. This distance is their average separation in a hydrogen atom. Strategy To compare the two forces, we first compute the electrostatic force using Coulomb's law, = \|1 2\| 2. We then calculate the gravitational force using Newton's universal law of gravitation. Finally, we take a ratio to see how the forces compare in magnitude. Solution Entering the given and known information about the charges and separation of the electron and proton into the expression of Coulomb's law yields = \|1 2\| 2 = 8.99×109 N ⋅ m2 / C2 × (1.60×10–19 C)(1.60×10–19 C) (0.530×10–10 m)2 (18.5) (18.6) Thus the Coulomb force is = 8.19×10–8 N. The charges are opposite in sign, so this is an attractive force. This is a very large force for an electron—it would cause an acceleration of 8.99×1022 m / s2 (verification is left as an end-of-section problem).The gravitational force is given by Newton's law of gravitation as: (18.7) = 2 where = 6.67×10−11 N ⋅ m2 / kg2. Here and represent the electron and proton masses, which can be found in the appendices. Entering values for the knowns yields, (18.8) = (6.67×10 – 11 N ⋅ m2 / kg2)× (9.11×10–31 kg)(1.67×10–27 kg) (0.530×10–10 m)2 = 3.61×10–47 N (
18.9) This is also an attractive force, although it is traditionally shown as positive since gravitational force is always attractive. The ratio of the magnitude of the electrostatic force to gravitational force in this case is, thus, = 2.27×1039. (18.10) Discussion This is a remarkably large ratio! Note that this will be the ratio of electrostatic force to gravitational force for an electron and a proton at any distance (taking the ratio before entering numerical values shows that the distance cancels). This ratio gives some indication of just how much larger the Coulomb force is than the gravitational force between two of the most common particles in nature. As the example implies, gravitational force is completely negligible on a small scale, where the interactions of individual charged particles are important. On a large scale, such as between the Earth and a person, the reverse is true. Most objects are nearly electrically neutral, and so attractive and repulsive Coulomb forces nearly cancel. Gravitational force on a large scale dominates interactions between large objects because it is always attractive, while Coulomb forces tend to cancel. Chapter 18 \| Electric Charge and Electric Field 801 The Van de Graaff Generator Van de Graaff generators (or Van de Graaffs) are not only spectacular devices used to demonstrate high voltage due to static electricity—they are also used for serious research. The first was built by Robert Van de Graaff in 1931 (based on original suggestions by Lord Kelvin) for use in nuclear physics research. Figure 18.38 shows a schematic of a large research version. Van de Graaffs utilize both smooth and pointed surfaces, and conductors and insulators to generate large static charges and, hence, large voltages. A very large excess charge can be deposited on the sphere, because it moves quickly to the outer surface. Practical limits arise because the large electric fields polarize and eventually ionize surrounding materials, creating free charges that neutralize excess charge or allow it to escape. Nevertheless, voltages of 15 million volts are well within practical limits. Figure 18.38 Schematic of Van de Graaff generator. A battery (A) supplies excess positive charge to a pointed conductor, the points of which spray the charge onto a moving insulating belt near the bottom. The pointed conductor (B) on top in the large sphere picks up the charge. (The induced electric field at the points is so large that it removes the charge from the belt.) This can be done because the charge does not remain inside the conducting sphere but
moves to its outside surface. An ion source inside the sphere produces positive ions, which are accelerated away from the positive sphere to high velocities. Take-Home Experiment: Electrostatics and Humidity Rub a comb through your hair and use it to lift pieces of paper. It may help to tear the pieces of paper rather than cut them neatly. Repeat the exercise in your bathroom after you have had a long shower and the air in the bathroom is moist. Is it easier to get electrostatic effects in dry or moist air? Why would torn paper be more attractive to the comb than cut paper? Explain your observations. Xerography Most copy machines use an electrostatic process called xerography—a word coined from the Greek words xeros for dry and graphos for writing. The heart of the process is shown in simplified form in Figure 18.39. A selenium-coated aluminum drum is sprayed with positive charge from points on a device called a corotron. Selenium is a substance with an interesting property—it is a photoconductor. That is, selenium is an insulator when in the dark and a conductor when exposed to light. In the first stage of the xerography process, the conducting aluminum drum is grounded so that a negative charge is induced under the thin layer of uniformly positively charged selenium. In the second stage, the surface of the drum is exposed to the image of whatever is to be copied. Where the image is light, the selenium becomes conducting, and the positive charge is neutralized. In dark areas, the positive charge remains, and so the image has been transferred to the drum. The third stage takes a dry black powder, called toner, and sprays it with a negative charge so that it will be attracted to the positive regions of the drum. Next, a blank piece of paper is given a greater positive charge than on the drum so that it will pull Chapter 18 \| Electric Charge and Electric Field 805 electric field is given to be upward, the electric force is upward. We thus have a one-dimensional (vertical direction) problem, and we can state Newton's second law as where net = −. Entering this and the known values into the expression for Newton's second law yields = net. = − = 9.60×10−14 N − 3.92×10−14 N 4.00×10−15 kg = 14.2 m/s2. (18.24) (18.
25) Discussion for (c) This is an upward acceleration great enough to carry the drop to places where you might not wish to have gasoline. This worked example illustrates how to apply problem-solving strategies to situations that include topics in different chapters. The first step is to identify the physical principles involved in the problem. The second step is to solve for the unknown using familiar problem-solving strategies. These are found throughout the text, and many worked examples show how to use them for single topics. In this integrated concepts example, you can see how to apply them across several topics. You will find these techniques useful in applications of physics outside a physics course, such as in your profession, in other science disciplines, and in everyday life. The following problems will build your skills in the broad application of physical principles. Unreasonable Results The Unreasonable Results exercises for this module have results that are unreasonable because some premise is unreasonable or because certain of the premises are inconsistent with one another. Physical principles applied correctly then produce unreasonable results. The purpose of these problems is to give practice in assessing whether nature is being accurately described, and if it is not to trace the source of difficulty. Problem-Solving Strategy To determine if an answer is reasonable, and to determine the cause if it is not, do the following. 1. Solve the problem using strategies as outlined above. Use the format followed in the worked examples in the text to solve the problem as usual. 2. Check to see if the answer is reasonable. Is it too large or too small, or does it have the wrong sign, improper units, and so on? 3. If the answer is unreasonable, look for what specifically could cause the identified difficulty. Usually, the manner in which the answer is unreasonable is an indication of the difficulty. For example, an extremely large Coulomb force could be due to the assumption of an excessively large separated charge. Glossary conductor: a material that allows electrons to move separately from their atomic orbits conductor: an object with properties that allow charges to move about freely within it Coulomb force: another term for the electrostatic force Coulomb interaction: the interaction between two charged particles generated by the Coulomb forces they exert on one another Coulomb's law: the mathematical equation calculating the electrostatic force vector between two charged particles dipole: a molecule's lack of symmetrical charge distribution, causing one side to be more positive and another to be more negative electric charge: a physical property of an object that causes it to be attracted toward or repelled from
another charged object; each charged object generates and is influenced by a force called an electromagnetic force electric field: a three-dimensional map of the electric force extended out into space from a point charge electric field lines: a series of lines drawn from a point charge representing the magnitude and direction of force exerted by that charge 806 Chapter 18 \| Electric Charge and Electric Field electromagnetic force: one of the four fundamental forces of nature; the electromagnetic force consists of static electricity, moving electricity and magnetism electron: a particle orbiting the nucleus of an atom and carrying the smallest unit of negative charge electrostatic equilibrium: an electrostatically balanced state in which all free electrical charges have stopped moving about electrostatic force: the amount and direction of attraction or repulsion between two charged bodies electrostatic precipitators: filters that apply charges to particles in the air, then attract those charges to a filter, removing them from the airstream electrostatic repulsion: the phenomenon of two objects with like charges repelling each other electrostatics: the study of electric forces that are static or slow-moving Faraday cage: a metal shield which prevents electric charge from penetrating its surface field: a map of the amount and direction of a force acting on other objects, extending out into space free charge: an electrical charge (either positive or negative) which can move about separately from its base molecule free electron: an electron that is free to move away from its atomic orbit grounded: when a conductor is connected to the Earth, allowing charge to freely flow to and from Earth's unlimited reservoir grounded: connected to the ground with a conductor, so that charge flows freely to and from the Earth to the grounded object induction: the process by which an electrically charged object brought near a neutral object creates a charge in that object ink-jet printer: small ink droplets sprayed with an electric charge are controlled by electrostatic plates to create images on paper insulator: a material that holds electrons securely within their atomic orbits ionosphere: a layer of charged particles located around 100 km above the surface of Earth, which is responsible for a range of phenomena including the electric field surrounding Earth laser printer: uses a laser to create a photoconductive image on a drum, which attracts dry ink particles that are then rolled onto a sheet of paper to print a high-quality copy of the image law of conservation of charge: is created simultaneously states that whenever a charge is created, an equal amount of charge with the opposite sign photoconductor: a substance that is an insulator until it is exposed to light,
when it becomes a conductor point charge: A charged particle, designated, generating an electric field polar molecule: a molecule with an asymmetrical distribution of positive and negative charge polarization: slight shifting of positive and negative charges to opposite sides of an atom or molecule polarized: a state in which the positive and negative charges within an object have collected in separate locations proton: a particle in the nucleus of an atom and carrying a positive charge equal in magnitude and opposite in sign to the amount of negative charge carried by an electron screening: the dilution or blocking of an electrostatic force on a charged object by the presence of other charges nearby static electricity: a buildup of electric charge on the surface of an object test charge: A particle (designated ) with either a positive or negative charge set down within an electric field generated by a point charge Van de Graaff generator: a machine that produces a large amount of excess charge, used for experiments with high voltage vector: a quantity with both magnitude and direction vector addition: mathematical combination of two or more vectors, including their magnitudes, directions, and positions xerography: a dry copying process based on electrostatics This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 18 \| Electric Charge and Electric Field 807 Section Summary 18.1 Static Electricity and Charge: Conservation of Charge • There are only two types of charge, which we call positive and negative. • Like charges repel, unlike charges attract, and the force between charges decreases with the square of the distance. • The vast majority of positive charge in nature is carried by protons, while the vast majority of negative charge is carried by electrons. • The electric charge of one electron is equal in magnitude and opposite in sign to the charge of one proton. • An ion is an atom or molecule that has nonzero total charge due to having unequal numbers of electrons and protons. • The SI unit for charge is the coulomb (C), with protons and electrons having charges of opposite sign but equal magnitude; the magnitude of this basic charge ∣ ∣ is • Whenever charge is created or destroyed, equal amounts of positive and negative are involved. • Most often, existing charges are separated from neutral objects to obtain some net charge. • Both positive and negative charges exist in neutral objects and can be separated by rubbing one object with another. For macroscopic objects, negatively charged means an excess of electrons and positively charged means a depletion of electrons. ∣ ∣
= 1.60×10−19 C. • The law of conservation of charge ensures that whenever a charge is created, an equal charge of the opposite sign is created at the same time. 18.2 Conductors and Insulators • Polarization is the separation of positive and negative charges in a neutral object. • A conductor is a substance that allows charge to flow freely through its atomic structure. • An insulator holds charge within its atomic structure. • Objects with like charges repel each other, while those with unlike charges attract each other. • A conducting object is said to be grounded if it is connected to the Earth through a conductor. Grounding allows transfer of charge to and from the earth's large reservoir. • Objects can be charged by contact with another charged object and obtain the same sign charge. • • Polarized objects have their positive and negative charges concentrated in different areas, giving them a non-symmetrical If an object is temporarily grounded, it can be charged by induction, and obtains the opposite sign charge. charge. • Polar molecules have an inherent separation of charge. 18.3 Conductors and Electric Fields in Static Equilibrium • A conductor allows free charges to move about within it. • The electrical forces around a conductor will cause free charges to move around inside the conductor until static equilibrium is reached. • Any excess charge will collect along the surface of a conductor. • Conductors with sharp corners or points will collect more charge at those points. • A lightning rod is a conductor with sharply pointed ends that collect excess charge on the building caused by an electrical storm and allow it to dissipate back into the air. • Electrical storms result when the electrical field of Earth's surface in certain locations becomes more strongly charged, due to changes in the insulating effect of the air. • A Faraday cage acts like a shield around an object, preventing electric charge from penetrating inside. 18.4 Coulomb’s Law • Frenchman Charles Coulomb was the first to publish the mathematical equation that describes the electrostatic force between two objects. • Coulomb's law gives the magnitude of the force between point charges. It is = \|1 2\| 2, where 1 and 2 are two point charges separated by a distance, and ≈ 8.99×109 N · m2/ C2 • This Coulomb force is extremely basic, since most charges are due to point-like particles. It is responsible for all electrostatic effects and underlies most macroscopic forces. • The Coulomb force is extraordinarily
strong compared with the gravitational force, another basic force—but unlike gravitational force it can cancel, since it can be either attractive or repulsive. • The electrostatic force between two subatomic particles is far greater than the gravitational force between the same two particles. 18.5 Electric Field: Concept of a Field Revisited • The electrostatic force field surrounding a charged object extends out into space in all directions. 808 Chapter 18 \| Electric Charge and Electric Field • The electrostatic force exerted by a point charge on a test charge at a distance depends on the charge of both charges, as well as the distance between the two. • The electric field E is defined to be E = F where F is the Coulomb or electrostatic force exerted on a small positive test charge. E has units of N/C. • The magnitude of the electric field E created by a point charge is E = \|\| 2. where is the distance from. The electric field E is a vector and fields due to multiple charges add like vectors. 18.6 Electric Field Lines: Multiple Charges • Drawings of electric field lines are useful visual tools. The properties of electric field lines for any charge distribution are that: • Field lines must begin on positive charges and terminate on negative charges, or at infinity in the hypothetical case of isolated charges. • The number of field lines leaving a positive charge or entering a negative charge is proportional to the magnitude of the charge. • The strength of the field is proportional to the closeness of the field lines—more precisely, it is proportional to the number of lines per unit area perpendicular to the lines. • The direction of the electric field is tangent to the field line at any point in space. • Field lines can never cross. 18.7 Electric Forces in Biology • Many molecules in living organisms, such as DNA, carry a charge. • An uneven distribution of the positive and negative charges within a polar molecule produces a dipole. • The effect of a Coulomb field generated by a charged object may be reduced or blocked by other nearby charged objects. • Biological systems contain water, and because water molecules are polar, they have a strong effect on other molecules in living systems. 18.8 Applications of Electrostatics • Electrostatics is the study of electric fields in static equilibrium. • In addition to research using equipment such as a Van de Graaff generator, many practical applications of electrostatics exist, including photocopiers, laser printers, ink-jet printers and electrostatic air filters. Conceptual Questions 18.
1 Static Electricity and Charge: Conservation of Charge 1. There are very large numbers of charged particles in most objects. Why, then, don't most objects exhibit static electricity? 2. Why do most objects tend to contain nearly equal numbers of positive and negative charges? 18.2 Conductors and Insulators 3. An eccentric inventor attempts to levitate by first placing a large negative charge on himself and then putting a large positive charge on the ceiling of his workshop. Instead, while attempting to place a large negative charge on himself, his clothes fly off. Explain. 4. If you have charged an electroscope by contact with a positively charged object, describe how you could use it to determine the charge of other objects. Specifically, what would the leaves of the electroscope do if other charged objects were brought near its knob? 5. When a glass rod is rubbed with silk, it becomes positive and the silk becomes negative—yet both attract dust. Does the dust have a third type of charge that is attracted to both positive and negative? Explain. 6. Why does a car always attract dust right after it is polished? (Note that car wax and car tires are insulators.) 7. Describe how a positively charged object can be used to give another object a negative charge. What is the name of this process? 8. What is grounding? What effect does it have on a charged conductor? On a charged insulator? 18.3 Conductors and Electric Fields in Static Equilibrium 9. Is the object in a conductor or an insulator? Justify your answer. This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 18 \| Electric Charge and Electric Field 809 Figure 18.43 10. If the electric field lines in the figure above were perpendicular to the object, would it necessarily be a conductor? Explain. 11. The discussion of the electric field between two parallel conducting plates, in this module states that edge effects are less important if the plates are close together. What does close mean? That is, is the actual plate separation crucial, or is the ratio of plate separation to plate area crucial? 12. Would the self-created electric field at the end of a pointed conductor, such as a lightning rod, remove positive or negative charge from the conductor? Would the same sign charge be removed from a neutral pointed conductor by the application of a similar externally created electric field? (The answers to both questions have implications for charge transfer utilizing points.) 13. Why is
a golfer with a metal club over her shoulder vulnerable to lightning in an open fairway? Would she be any safer under a tree? 14. Can the belt of a Van de Graaff accelerator be a conductor? Explain. 15. Are you relatively safe from lightning inside an automobile? Give two reasons. 16. Discuss pros and cons of a lightning rod being grounded versus simply being attached to a building. 17. Using the symmetry of the arrangement, show that the net Coulomb force on the charge at the center of the square below (Figure 18.44) is zero if the charges on the four corners are exactly equal. Figure 18.44 Four point charges,,, and lie on the corners of a square and is located at its center. 18. (a) Using the symmetry of the arrangement, show that the electric field at the center of the square in Figure 18.44 is zero if the charges on the four corners are exactly equal. (b) Show that this is also true for any combination of charges in which = and = 19. (a) What is the direction of the total Coulomb force on in Figure 18.44 if is negative, = and both are negative, and = and both are positive? (b) What is the direction of the electric field at the center of the square in this situation? 20. Considering Figure 18.44, suppose that = and =. First show that is in static equilibrium. (You may neglect the gravitational force.) Then discuss whether the equilibrium is stable or unstable, noting that this may depend on the signs of the charges and the direction of displacement of from the center of the square. 21. If = 0 in Figure 18.44, under what conditions will there be no net Coulomb force on? 22. In regions of low humidity, one develops a special “grip” when opening car doors, or touching metal door knobs. This involves placing as much of the hand on the device as possible, not just the ends of one's fingers. Discuss the induced charge and explain why this is done. 23. Tollbooth stations on roadways and bridges usually have a piece of wire stuck in the pavement before them that will touch a car as it approaches. Why is this done? 810 Chapter 18 \| Electric Charge and Electric Field 24. Suppose a woman carries an excess charge. To maintain her charged status can she be standing on ground wearing just any pair of shoes? How would you discharge her? What are the consequences if she simply
walks away? 18.4 Coulomb’s Law 25. Figure 18.45 shows the charge distribution in a water molecule, which is called a polar molecule because it has an inherent separation of charge. Given water's polar character, explain what effect humidity has on removing excess charge from objects. Figure 18.45 Schematic representation of the outer electron cloud of a neutral water molecule. The electrons spend more time near the oxygen than the hydrogens, giving a permanent charge separation as shown. Water is thus a polar molecule. It is more easily affected by electrostatic forces than molecules with uniform charge distributions. 26. Using Figure 18.45, explain, in terms of Coulomb's law, why a polar molecule (such as in Figure 18.45) is attracted by both positive and negative charges. 27. Given the polar character of water molecules, explain how ions in the air form nucleation centers for rain droplets. 18.5 Electric Field: Concept of a Field Revisited 28. Why must the test charge in the definition of the electric field be vanishingly small? 29. Are the direction and magnitude of the Coulomb force unique at a given point in space? What about the electric field? 18.6 Electric Field Lines: Multiple Charges 30. Compare and contrast the Coulomb force field and the electric field. To do this, make a list of five properties for the Coulomb force field analogous to the five properties listed for electric field lines. Compare each item in your list of Coulomb force field properties with those of the electric field—are they the same or different? (For example, electric field lines cannot cross. Is the same true for Coulomb field lines?) 31. Figure 18.46 shows an electric field extending over three regions, labeled I, II, and III. Answer the following questions. (a) Are there any isolated charges? If so, in what region and what are their signs? (b) Where is the field strongest? (c) Where is it weakest? (d) Where is the field the most uniform? Figure 18.46 18.7 Electric Forces in Biology This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 18 \| Electric Charge and Electric Field 811 32. A cell membrane is a thin layer enveloping a cell. The thickness of the membrane is much less than the size of the cell. In a static situation the membrane has a charge distribution of −2.5×10−6 C
/m2 on its outer surface. Draw a diagram of the cell and the surrounding cell membrane. Include on this diagram the charge distribution and the corresponding electric field. Is there any electric field inside the cell? Is there any electric field outside the cell? C/m 2 on its inner surface and +2.5×10−6 812 Chapter 18 \| Electric Charge and Electric Field Problems & Exercises 18.1 Static Electricity and Charge: Conservation of Charge 1. Common static electricity involves charges ranging from nanocoulombs to microcoulombs. (a) How many electrons are needed to form a charge of –2.00 nC (b) How many electrons must be removed from a neutral object to leave a net charge of 0.500 C? electrons move through a pocket calculator 2. If 1.80×1020 during a full day's operation, how many coulombs of charge moved through it? 3. To start a car engine, the car battery moves 3.75×1021 electrons through the starter motor. How many coulombs of charge were moved? 4. A certain lightning bolt moves 40.0 C of charge. How many fundamental units of charge ∣ ∣ is this? 18.2 Conductors and Insulators 5. Suppose a speck of dust in an electrostatic precipitator has 1.0000×1012 protons in it and has a net charge of –5.00 nC (a very large charge for a small speck). How many electrons does it have? 6. An amoeba has 1.00×1016 0.300 pC. (a) How many fewer electrons are there than protons? (b) If you paired them up, what fraction of the protons would have no electrons? 7. A 50.0 g ball of copper has a net charge of 2.00 C. What fraction of the copper's electrons has been removed? (Each copper atom has 29 protons, and copper has an atomic mass of 63.5.) protons and a net charge of 8. What net charge would you place on a 100 g piece of sulfur if you put an extra electron on 1 in 1012 of its atoms? (Sulfur has an atomic mass of 32.1.) 9. How many coulombs of positive charge are there in 4.00 kg of plutonium, given its atomic mass is 244 and that each plutonium atom has 94 protons? 18.3 Conductors and Electric Fields
in Static Equilibrium 10. Sketch the electric field lines in the vicinity of the conductor in Figure 18.47 given the field was originally uniform and parallel to the object's long axis. Is the resulting field small near the long side of the object? Figure 18.48 12. Sketch the electric field between the two conducting plates shown in Figure 18.49, given the top plate is positive and an equal amount of negative charge is on the bottom plate. Be certain to indicate the distribution of charge on the plates. Figure 18.49 13. Sketch the electric field lines in the vicinity of the charged insulator in Figure 18.50 noting its nonuniform charge distribution. Figure 18.50 A charged insulating rod such as might be used in a classroom demonstration. 14. What is the force on the charge located at = 8.00 cm in Figure 18.51(a) given that = 1.00 μC? Figure 18.47 11. Sketch the electric field lines in the vicinity of the conductor in Figure 18.48 given the field was originally uniform and parallel to the object's long axis. Is the resulting field small near the long side of the object? Figure 18.51 (a) Point charges located at 3.00, 8.00, and 11.0 cm along the x-axis. (b) Point charges located at 1.00, 5.00, 8.00, and 14.0 cm along the x-axis. 15. (a) Find the total electric field at = 1.00 cm in Figure 18.51(b) given that = 5.00 nC. (b) Find the total electric field at = 11.00 cm in Figure 18.51(b). (c) If the charges are allowed to move and eventually be brought to rest by friction, what will the final charge configuration be? (That is, This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 18 \| Electric Charge and Electric Field 813 will there be a single charge, double charge, etc., and what will its value(s) be?) 16. (a) Find the electric field at = 5.00 cm in Figure 18.51(a), given that = 1.00 μC. (b) At what position between 3.00 and 8.00 cm is the total electric field the same as that for –2 alone? (c) Can the electric field be zero anywhere between 0
.00 and 8.00 cm? (d) At very large positive or negative values of x, the electric field approaches zero in both (a) and (b). In which does it most rapidly approach zero and why? (e) At what position to the right of 11.0 cm is the total electric field zero, other than at infinity? (Hint: A graphing calculator can yield considerable insight in this problem.) 17. (a) Find the total Coulomb force on a charge of 2.00 nC located at = 4.00 cm in Figure 18.51 (b), given that = 1.00 μC. (b) Find the x-position at which the electric field is zero in Figure 18.51 (b). 18. Using the symmetry of the arrangement, determine the direction of the force on in the figure below, given that = =+7.50 μC and = = −7.50 μC. (b) Calculate the magnitude of the force on the charge, given that the square is 10.0 cm on a side and = 2.00 μC. Figure 18.52 19. (a) Using the symmetry of the arrangement, determine the direction of the electric field at the center of the square in Figure 18.52, given that = = −1.00 μC and = =+1.00 μC. (b) Calculate the magnitude of the electric field at the location of, given that the square is 5.00 cm on a side. 20. Find the electric field at the location of in Figure 18.52 given that = = =+2.00 nC, = −1.00 nC, and the square is 20.0 cm on a side. 21. Find the total Coulomb force on the charge in Figure 18.52, given that = 1.00 μC, = 2.00 μC, = −3.00 μC, = −4.00 μC, and =+1.00 μC. The square is 50.0 cm on a side. 22. (a) Find the electric field at the location of in Figure 18.53, given that b = +10.00 C and c = –5.00 C. (b) What is the force on, given that a = +1.50 nC? Figure 18.53 Point charges located at the corners of an equilateral triangle 25.0 cm on a side.
23. (a) Find the electric field at the center of the triangular configuration of charges in Figure 18.53, given that =+2.50 nC, = −8.00 nC, and =+1.50 nC. (b) Is there any combination of charges, other than = =, that will produce a zero strength electric field at the center of the triangular configuration? 18.4 Coulomb’s Law 24. What is the repulsive force between two pith balls that are 8.00 cm apart and have equal charges of – 30.0 nC? 25. (a) How strong is the attractive force between a glass rod with a 0.700 C charge and a silk cloth with a –0.600 C charge, which are 12.0 cm apart, using the approximation that they act like point charges? (b) Discuss how the answer to this problem might be affected if the charges are distributed over some area and do not act like point charges. 26. Two point charges exert a 5.00 N force on each other. What will the force become if the distance between them is increased by a factor of three? 27. Two point charges are brought closer together, increasing the force between them by a factor of 25. By what factor was their separation decreased? 28. How far apart must two point charges of 75.0 nC (typical of static electricity) be to have a force of 1.00 N between them? 29. If two equal charges each of 1 C each are separated in air by a distance of 1 km, what is the magnitude of the force acting between them? You will see that even at a distance as large as 1 km, the repulsive force is substantial because 1 C is a very significant amount of charge. 30. A test charge of +2 C is placed halfway between a charge of +6 C and another of +4 C separated by 10 cm. (a) What is the magnitude of the force on the test charge? (b) What is the direction of this force (away from or toward the +6 C charge)? 31. Bare free charges do not remain stationary when close together. To illustrate this, calculate the acceleration of two isolated protons separated by 2.00 nm (a typical distance between gas atoms). Explicitly show how you follow the steps in the Problem-Solving Strategy for electrostatics. 32. (a) By what factor must you change the distance between two point charges to change
the force between them by a factor of 10? (b) Explain how the distance can either increase or decrease by this factor and still cause a factor of 10 change in the force. 814 Chapter 18 \| Electric Charge and Electric Field (b) What magnitude and direction force does this field exert on a proton? 18.6 Electric Field Lines: Multiple Charges 47. (a) Sketch the electric field lines near a point charge +. (b) Do the same for a point charge –3.00. 48. Sketch the electric field lines a long distance from the charge distributions shown in Figure 18.34 (a) and (b) 49. Figure 18.54 shows the electric field lines near two charges 1 and 2. What is the ratio of their magnitudes? (b) Sketch the electric field lines a long distance from the charges shown in the figure. Figure 18.54 The electric field near two charges. 50. Sketch the electric field lines in the vicinity of two opposite charges, where the negative charge is three times greater in magnitude than the positive. (See Figure 18.54 for a similar situation). 18.8 Applications of Electrostatics 51. (a) What is the electric field 5.00 m from the center of the terminal of a Van de Graaff with a 3.00 mC charge, noting that the field is equivalent to that of a point charge at the center of the terminal? (b) At this distance, what force does the field exert on a 2.00 C charge on the Van de Graaff's belt? 52. (a) What is the direction and magnitude of an electric field that supports the weight of a free electron near the surface of Earth? (b) Discuss what the small value for this field implies regarding the relative strength of the gravitational and electrostatic forces. 53. A simple and common technique for accelerating electrons is shown in Figure 18.55, where there is a uniform electric field between two plates. Electrons are released, usually from a hot filament, near the negative plate, and there is a small hole in the positive plate that allows the electrons to continue moving. (a) Calculate the acceleration of the electron if the field strength is 2.50×104 N/C. (b) Explain why the electron will not be pulled back to the positive plate once it moves through the hole. 33. Suppose you have a total charge tot that you can split in any manner. Once split, the separation distance is fixed. How do you
split the charge to achieve the greatest force? 34. (a) Common transparent tape becomes charged when pulled from a dispenser. If one piece is placed above another, the repulsive force can be great enough to support the top piece's weight. Assuming equal point charges (only an approximation), calculate the magnitude of the charge if electrostatic force is great enough to support the weight of a 10.0 mg piece of tape held 1.00 cm above another. (b) Discuss whether the magnitude of this charge is consistent with what is typical of static electricity. 35. (a) Find the ratio of the electrostatic to gravitational force between two electrons. (b) What is this ratio for two protons? (c) Why is the ratio different for electrons and protons? 36. At what distance is the electrostatic force between two protons equal to the weight of one proton? 37. A certain five cent coin contains 5.00 g of nickel. What fraction of the nickel atoms' electrons, removed and placed 1.00 m above it, would support the weight of this coin? The atomic mass of nickel is 58.7, and each nickel atom contains 28 electrons and 28 protons. 38. (a) Two point charges totaling 8.00 C exert a repulsive force of 0.150 N on one another when separated by 0.500 m. What is the charge on each? (b) What is the charge on each if the force is attractive? 39. Point charges of 5.00 C and –3.00 C are placed 0.250 m apart. (a) Where can a third charge be placed so that the net force on it is zero? (b) What if both charges are positive? 40. Two point charges 1 and 2 are 3.00 m apart, and their total charge is 20 C. (a) If the force of repulsion between them is 0.075N, what are magnitudes of the two charges? (b) If one charge attracts the other with a force of 0.525N, what are the magnitudes of the two charges? Note that you may need to solve a quadratic equation to reach your answer. 18.5 Electric Field: Concept of a Field Revisited 41. What is the magnitude and direction of an electric field that exerts a 2.00×10-5 N upward force on a –1.75 C charge? 42. What is the magnitude and direction of the force exerted on a 3.
50 C charge by a 250 N/C electric field that points due east? 43. Calculate the magnitude of the electric field 2.00 m from a point charge of 5.00 mC (such as found on the terminal of a Van de Graaff). 44. (a) What magnitude point charge creates a 10,000 N/C electric field at a distance of 0.250 m? (b) How large is the field at 10.0 m? 45. Calculate the initial (from rest) acceleration of a proton in a 5.00×106 N/C electric field (such as created by a research Van de Graaff). Explicitly show how you follow the steps in the Problem-Solving Strategy for electrostatics. 46. (a) Find the direction and magnitude of an electric field that exerts a 4.80×10−17 N westward force on an electron. This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 18 \| Electric Charge and Electric Field 815 because air begins to ionize and charges flow, reducing the field. (a) Calculate the distance a free proton must travel in this field to reach 3.00% of the speed of light, starting from rest. (b) Is this practical in air, or must it occur in a vacuum? 60. Integrated Concepts A 5.00 g charged insulating ball hangs on a 30.0 cm long string in a uniform horizontal electric field as shown in Figure 18.56. Given the charge on the ball is 1.00 C, find the strength of the field. Figure 18.55 Parallel conducting plates with opposite charges on them create a relatively uniform electric field used to accelerate electrons to the right. Those that go through the hole can be used to make a TV or computer screen glow or to produce X-rays. 54. Earth has a net charge that produces an electric field of approximately 150 N/C downward at its surface. (a) What is the magnitude and sign of the excess charge, noting the electric field of a conducting sphere is equivalent to a point charge at its center? (b) What acceleration will the field produce on a free electron near Earth's surface? (c) What mass object with a single extra electron will have its weight supported by this field? 55. Point charges of 25.0 C and 45.0 C are placed 0.500 m apart. (a) At what point along
the line between them is the electric field zero? (b) What is the electric field halfway between them? 56. What can you say about two charges 1 and 2, if the electric field one-fourth of the way from 1 to 2 is zero? 57. Integrated Concepts Calculate the angular velocity ω of an electron orbiting a proton in the hydrogen atom, given the radius of the orbit is 0.530×10–10 m. You may assume that the proton is stationary and the centripetal force is supplied by Coulomb attraction. 58. Integrated Concepts An electron has an initial velocity of 5.00×106 m/s in a uniform 2.00×105 N/C strength electric field. The field accelerates the electron in the direction opposite to its initial velocity. (a) What is the direction of the electric field? (b) How far does the electron travel before coming to rest? (c) How long does it take the electron to come to rest? (d) What is the electron's velocity when it returns to its starting point? 59. Integrated Concepts The practical limit to an electric field in air is about 3.00×106 N/C. Above this strength, sparking takes place Figure 18.56 A horizontal electric field causes the charged ball to hang at an angle of 8.00º. 61. Integrated Concepts Figure 18.57 shows an electron passing between two charged metal plates that create an 100 N/C vertical electric field perpendicular to the electron's original horizontal velocity. (These can be used to change the electron's direction, such as in an oscilloscope.) The initial speed of the electron is 3.00×106 m/s, and the horizontal distance it travels in the uniform field is 4.00 cm. (a) What is its vertical deflection? (b) What is the vertical component of its final velocity? (c) At what angle does it exit? Neglect any edge effects. Figure 18.57 62. Integrated Concepts The classic Millikan oil drop experiment was the first to obtain an accurate measurement of the charge on an electron. In it, oil drops were suspended against the gravitational force by a vertical electric field. (See Figure 18.58.) Given the oil drop to be 1.00 m in radius and have a density of 920 kg/m3 (a) Find the weight of the drop. (b) If the drop has a single excess electron, find the electric field strength needed to balance its weight.
: 816 Chapter 18 \| Electric Charge and Electric Field 67. Construct Your Own Problem Consider two insulating balls with evenly distributed equal and opposite charges on their surfaces, held with a certain distance between the centers of the balls. Construct a problem in which you calculate the electric field (magnitude and direction) due to the balls at various points along a line running through the centers of the balls and extending to infinity on either side. Choose interesting points and comment on the meaning of the field at those points. For example, at what points might the field be just that due to one ball and where does the field become negligibly small? Among the things to be considered are the magnitudes of the charges and the distance between the centers of the balls. Your instructor may wish for you to consider the electric field off axis or for a more complex array of charges, such as those in a water molecule. 68. Construct Your Own Problem Consider identical spherical conducting space ships in deep space where gravitational fields from other bodies are negligible compared to the gravitational attraction between the ships. Construct a problem in which you place identical excess charges on the space ships to exactly counter their gravitational attraction. Calculate the amount of excess charge needed. Examine whether that charge depends on the distance between the centers of the ships, the masses of the ships, or any other factors. Discuss whether this would be an easy, difficult, or even impossible thing to do in practice. Figure 18.58 In the Millikan oil drop experiment, small drops can be suspended in an electric field by the force exerted on a single excess electron. Classically, this experiment was used to determine the electron charge e by measuring the electric field and mass of the drop. 63. Integrated Concepts (a) In Figure 18.59, four equal charges lie on the corners of a square. A fifth charge is on a mass directly above the center of the square, at a height equal to the length of one side of the square. Determine the magnitude of in terms of,, and, if the Coulomb force is to equal the weight of. (b) Is this equilibrium stable or unstable? Discuss. Figure 18.59 Four equal charges on the corners of a horizontal square support the weight of a fifth charge located directly above the center of the square. 64. Unreasonable Results (a) Calculate the electric field strength near a 10.0 cm diameter conducting sphere that has 1.00 C of excess charge on it. (b) What is unreasonable about this result? (c)
Which assumptions are responsible? 65. Unreasonable Results (a) Two 0.500 g raindrops in a thunderhead are 1.00 cm apart when they each acquire 1.00 mC charges. Find their acceleration. (b) What is unreasonable about this result? (c) Which premise or assumption is responsible? 66. Unreasonable Results A wrecking yard inventor wants to pick up cars by charging a 0.400 m diameter ball and inducing an equal and opposite charge on the car. If a car has a 1000 kg mass and the ball is to be able to lift it from a distance of 1.00 m: (a) What minimum charge must be used? (b) What is the electric field near the surface of the ball? (c) Why are these results unreasonable? (d) Which premise or assumption is responsible? This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 18 \| Electric Charge and Electric Field 817 Test Prep for AP® Courses Z is attracted to balloon Y. Which of the following can be the charge on Z? Select two answers. 18.1 Static Electricity and Charge: Conservation of Charge 1. When a glass rod is rubbed against silk, which of the following statements is true? a. Electrons are removed from the silk. b. Electrons are removed from the rod. c. Protons are removed from the silk. d. Protons are removed from the rod. 2. In an experiment, three microscopic latex spheres are sprayed into a chamber and become charged with +3e, +5e, and −3e, respectively. Later, all three spheres collide simultaneously and then separate. Which of the following are possible values for the final charges on the spheres? Select two answers. X Y Z (a) +4e −4e +5e (b) −4e +4.5e +5.5e (c) +5e −8e (d) +6e +6e +7e −7e 3. If objects X and Y attract each other, which of the following will be false? a. X has positive charge and Y has negative charge. b. X has negative charge and Y has positive charge. c. X and Y both have positive charge. d. X is neutral and Y has a charge. 4. Suppose a positively charged object A is brought in contact with an uncharged object B in a closed system. What type
of charge will be left on object B? a. negative b. positive c. neutral d. cannot be determined 5. What will be the net charge on an object which attracts neutral pieces of paper but repels a negatively charged balloon? a. negative b. positive c. neutral d. cannot be determined 6. When two neutral objects are rubbed against each other, the first one gains a net charge of 3e. Which of the following statements is true? a. The second object gains 3e and is negatively charged. b. The second object loses 3e and is negatively charged. c. The second object gains 3e and is positively charged. d. The second object loses 3e and is positively charged. 7. In an experiment, a student runs a comb through his hair several times and brings it close to small pieces of paper. Which of the following will he observe? a. Pieces of paper repel the comb. b. Pieces of paper are attracted to the comb. c. Some pieces of paper are attracted and some repel the comb. d. There is no attraction or repulsion between the pieces of paper and the comb. 8. In an experiment a negatively charged balloon (balloon X) is repelled by another charged balloon Y. However, an object a. negative b. positive c. neutral d. cannot be determined 9. Suppose an object has a charge of 1 C and gains 6.88×1018 electrons. a. What will be the net charge of the object? b. If the object has gained electrons from a neutral object, what will be the charge on the neutral object? c. Find and explain the relationship between the total charges of the two objects before and after the transfer. d. When a third object is brought in contact with the first object (after it gains the electrons), the resulting charge on the third object is 0.4 C. What was its initial charge? 10. The charges on two identical metal spheres (placed in a closed system) are -2.4×10−17 C and -4.8×10−17 C. a. How many electrons will be equivalent to the charge on b. each sphere? If the two spheres are brought in contact and then separated, find the charge on each sphere. c. Calculate the number of electrons that would be equivalent to the resulting charge on each sphere. 11. In an experiment the following observations are made by a student for four charged objects W, X, Y, and Z: •
A glass rod rubbed with silk attracts W. • W attracts Z but repels X. • X attracts Z but repels Y. • Y attracts W and Z. Estimate whether the charges on each of the four objects are positive, negative, or neutral. 18.2 Conductors and Insulators 12. Some students experimenting with an uncharged metal sphere want to give the sphere a net charge using a charged aluminum pie plate. Which of the following steps would give the sphere a net charge of the same sign as the pie plate? a. bringing the pie plate close to, but not touching, the metal sphere, then moving the pie plate away. b. bringing the pie plate close to, but not touching, the metal sphere, then momentarily touching a grounding wire to the metal sphere. c. bringing the pie plate close to, but not touching, the metal sphere, then momentarily touching a grounding wire to the pie plate. touching the pie plate to the metal sphere. d. 13. Figure 18.60 Balloon and sphere. When the balloon is brought closer to the sphere, there will be a redistribution of charges. What is this phenomenon called? a. electrostatic repulsion 818 Chapter 18 \| Electric Charge and Electric Field b. conduction c. polarization d. none of the above 14. What will be the charge at Y (i.e., the part of the sphere furthest from the balloon)? a. positive b. negative c. zero d. It can be positive or negative depending on the material. 15. What will be the net charge on the sphere? a. positive b. negative c. zero d. It can be positive or negative depending on the material. second experiment the rod is only brought close to the electroscope but not in contact. However, while the rod is close, the electroscope is momentarily grounded and then the rod is removed. In both experiments the needles of the electroscopes deflect, which indicates the presence of charges. a. What is the charging method in each of the two experiments? b. What is the net charge on the electroscope in the first c. experiment? Explain how the electroscope obtains that charge. Is the net charge on the electroscope in the second experiment different from that of the first experiment? Explain why. 16. If Y is grounded while the balloon is still close to X, which of the following will be true? 18.3 Conductors and Electric Fields in Static Equilibrium a. Electrons will flow from the sphere to the
ground. b. Electrons will flow from the ground to the sphere. c. Protons will flow from the sphere to the ground. d. Protons will flow from the ground to the sphere. 21. 17. If the balloon is moved away after grounding, what will be the net charge on the sphere? a. positive b. negative c. zero d. It can be positive or negative depending on the material. 18. A positively charged rod is used to charge a sphere by induction. Which of the following is true? a. The sphere must be a conductor. b. The sphere must be an insulator. c. The sphere can be a conductor or insulator but must be connected to ground. d. The sphere can be a conductor or insulator but must be already charged. 19. Figure 18.62 A sphere conductor. An electric field due to a positively charged spherical conductor is shown above. Where will the electric field be weakest? a. Point A b. Point B c. Point C d. Same at all points 22. Figure 18.63 Electric field between two parallel metal plates. The electric field created by two parallel metal plates is shown above. Where will the electric field be strongest? a. Point A b. Point B c. Point C d. Same at all points 23. Suppose that the electric field experienced due to a positively charged small spherical conductor at a certain distance is E. What will be the percentage change in electric field experienced at thrice the distance if the charge on the conductor is doubled? 24. Figure 18.61 Rod and metal balls. As shown in the figure above, two metal balls are suspended and a negatively charged rod is brought close to them. a. If the two balls are in contact with each other what will be the charges on each ball? b. Explain how the balls get these charges. c. What will happen to the charge on the second ball (i.e., the ball further away from the rod) if it is momentarily grounded while the rod is still there? If (instead of grounding) the second ball is moved away and then the rod is removed from the first ball, will the two balls have induced charges? If yes, what will be the charges? If no, why not? d. 20. Two experiments are performed using positively charged glass rods and neutral electroscopes. In the first experiment the rod is brought in contact with the electroscope. In the This content is available for free at http://cnx.org/content
/col11844/1.13 Chapter 18 \| Electric Charge and Electric Field 819 b. Will this ratio change if the two electrons are replaced by protons? If yes, find the new ratio. 18.5 Electric Field: Concept of a Field Revisited 31. Two particles with charges +2q and +q are separated by a distance r. The +2q particle has an electric field E at distance r and exerts a force F on the +q particle. Use this information to answer questions 31–32. What is the electric field of the +q particle at the same distance and what force does it exert on the +2q particle? a. E/2, F/2 b. E, F/2 c. E/2, F d. E, F 32. When the +q particle is replaced by a +3q particle, what will be the electric field and force from the +2q particle experienced by the +3q particle? a. E/3, 3F b. E, 3F c. E/3, F d. E, F 33. The direction of the electric field of a negative charge is inward for both positive and negative charges. a. b. outward for both positive and negative charges. c. inward for other positive charges and outward for other negative charges. d. outward for other positive charges and inward for other negative charges. 34. The force responsible for holding an atom together is frictional a. b. electric c. gravitational d. magnetic 35. When a positively charged particle exerts an inward force on another particle P, what will be the charge of P? a. positive b. negative c. neutral d. cannot be determined 36. Find the force exerted due to a particle having a charge of 3.2×10−19 C on another identical particle 5 cm away. 37. Suppose that the force exerted on an electron is 5.6×10−17 N, directed to the east. a. Find the magnitude of the electric field that exerts the force. b. What will be the direction of the electric field? c. If the electron is replaced by a proton, what will be the magnitude of force exerted? d. What will be the direction of force on the proton? 18.6 Electric Field Lines: Multiple Charges 38. Figure 18.65 An electric dipole (with +2q and –2q as the two charges) is shown in the figure above. A
third charge, −q is Figure 18.64 Millikan oil drop experiment. The classic Millikan oil drop experiment setup is shown above. In this experiment oil drops are suspended in a vertical electric field against the gravitational force to measure their charge. If the mass of a negatively charged drop suspended in an electric field of 1.18×10−4 N/C strength is 3.85×10−21 g, find the number of excess electrons in the drop. 18.4 Coulomb’s Law 25. For questions 25–27, suppose that the electrostatics force between two charges is F. What will be the force if the distance between them is halved? a. 4F b. 2F c. F/4 d. F/2 26. Which of the following is false? a. b. If the charge of one of the particles is doubled and that of the second is unchanged, the force will become 2F. If the charge of one of the particles is doubled and that of the second is halved, the force will remain F. If the charge of both the particles is doubled, the force will become 4F. d. None of the above. c. 27. Which of the following is true about the gravitational force between the particles? a. b. c. d. It will be 3.25×10−38 F. It will be 3.25×1038 F. It will be equal to F. It is not possible to determine the gravitational force as the masses of the particles are not given. 28. Two massive, positively charged particles are initially held a fixed distance apart. When they are moved farther apart, the magnitude of their mutual gravitational force changes by a factor of n. Which of the following indicates the factor by which the magnitude of their mutual electrostatic force changes? a. 1/n2 b. 1/n c. n d. n2 29. a. What is the electrostatic force between two charges of 1 C each, separated by a distance of 0.5 m? b. How will this force change if the distance is increased to 1 m? 30. a. Find the ratio of the electrostatic force to the gravitational force between two electrons. 820 Chapter 18 \| Electric Charge and Electric Field placed equidistant from the dipole charges. What will be the direction of the net force on the third charge? negative and that the sign of the charge of object S is positive. a. → b. ← c
. ↓ d. ↑ 39. ii) Briefly describe the characteristics of the field diagram that indicate that the magnitudes of the charges of objects R and T are equal and that the magnitude of the charge of object S is about twice that of objects R and T. For the following parts, an electric field directed to the right is defined to be positive. (b) On the axes below, sketch a graph of the electric field E along the x-axis as a function of position x. Figure 18.68 An Electric field (E) axis and Position (x) axis. (c) Write an expression for the electric field E along the x-axis as a function of position x in the region between objects S and T in terms of q, d, and fundamental constants, as appropriate. (d) Your classmate tells you there is a point between S and T where the electric field is zero. Determine whether this statement is true, and explain your reasoning using two of the representations from parts (a), (b), or (c). Figure 18.66 Four objects, each with charge +q, are held fixed on a square with sides of length d, as shown in the figure. Objects X and Z are at the midpoints of the sides of the square. The electrostatic force exerted by object W on object X is F. Use this information to answer questions 39–40. What is the magnitude of force exerted by object W on Z? a. F/7 b. F/5 c. F/3 d. F/2 40. What is the magnitude of the net force exerted on object X by objects W, Y, and Z? a. F/4 b. F/2 c. 9F/4 d. 3F 41. Figure 18.67 Electric field with three charged objects. The figure above represents the electric field in the vicinity of three small charged objects, R, S, and T. The objects have charges −q, +2q, and −q, respectively, and are located on the x-axis at −d, 0, and d. Field vectors of very large magnitude are omitted for clarity. (a) i) Briefly describe the characteristics of the field diagram that indicate that the sign of the charges of objects R and T is This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 20 \| Electric Current, Resistance, and Ohm's Law 867 20 ELECTRIC
CURRENT, RESISTANCE, AND OHM'S LAW Figure 20.1 Electric energy in massive quantities is transmitted from this hydroelectric facility, the Srisailam power station located along the Krishna River in India (http://en.wikipedia.org/wiki/Srisailam_Dam), by the movement of charge—that is, by electric current. (credit: Chintohere, Wikimedia Commons) Chapter Outline 20.1. Current 20.2. Ohm’s Law: Resistance and Simple Circuits 20.3. Resistance and Resistivity 20.4. Electric Power and Energy 20.5. Alternating Current versus Direct Current 20.6. Electric Hazards and the Human Body 20.7. Nerve Conduction–Electrocardiograms Connection for AP® Courses In our daily lives, we see and experience many examples of electricity which involve electric current, the movement of charge. These include the flicker of numbers on a handheld calculator, nerve impulses carrying signals of vision to the brain, an ultrasound device sending a signal to a computer screen, the brain sending a message for a baby to twitch its toes, an electric train pulling its load over a mountain pass, and a hydroelectric plant sending energy to metropolitan and rural users. Humankind has indeed harnessed electricity, the basis of technology, to improve the quality of life. While the previous two chapters concentrated on static electricity and the fundamental force underlying its behavior, the next few chapters will be 868 Chapter 20 \| Electric Current, Resistance, and Ohm's Law devoted to electric and magnetic phenomena involving electric current. In addition to exploring applications of electricity, we shall gain new insights into its nature – in particular, the fact that all magnetism results from electric current. This chapter supports learning objectives covered under Big Ideas 1, 4, and 5 of the AP Physics Curriculum Framework. Electric charge is a property of a system (Big Idea 1) that affects its interaction with other charged systems (Enduring Understanding 1.B), whereas electric current is fundamentally the movement of charge through a conductor and is based on the fact that electric charge is conserved within a system (Essential Knowledge 1.B.1). The conservation of charge also leads to the concept of an electric circuit as a closed loop of electrical current. In addition, this chapter discusses examples showing that the current in a circuit is resisted by the elements of the circuit and the strength of the resistance depends on the material of the elements. The macroscopic
properties of materials, including resistivity, depend on their molecular and atomic structure (Enduring Understanding 1.E). In addition, resistivity depends on the temperature of the material (Essential Knowledge 1.E.2). The chapter also describes how the interaction of systems of objects can result in changes in those systems (Big Idea 4). For example, electric properties of a system of charged objects can change in response to the presence of, or changes in, other charged objects or systems (Enduring Understanding 4.E). A simple circuit with a resistor and an energy source is an example of such a system. The current through the resistor in the circuit is equal to the difference of potentials across the resistor divided by its resistance (Essential Knowledge 4.E.4). The unifying theme of the physics curriculum is that any changes in the systems due to interactions are governed by laws of conservation (Big Idea 5). This chapter applies the idea of energy conservation (Enduring Understanding 5.B) to electric circuits and connects concepts of electric energy and electric power as rates of energy use (Essential Knowledge 5.B.5). While the laws of conservation of energy in electric circuits are fully described by Kirchoff's rules, which are introduced in the next chapter (Essential Knowledge 5.B.9), the specific definition of power (based on Essential Knowledge 5.B.9) is that it is the rate at which energy is transferred from a resistor as the product of the electric potential difference across the resistor and the current through the resistor. Big Idea 1 Objects and systems have properties such as mass and charge. Systems may have internal structure. Enduring Understanding 1.B Electric charge is a property of an object or system that affects its interactions with other objects or systems containing charge. Essential Knowledge 1.B.1 Electric charge is conserved. The net charge of a system is equal to the sum of the charges of all the objects in the system. Enduring Understanding 1.E Materials have many macroscopic properties that result from the arrangement and interactions of the atoms and molecules that make up the material. Essential Knowledge 1.E.2 Matter has a property called resistivity. Big Idea 4 Interactions between systems can result in changes in those systems. Enduring Understanding 4.E The electric and magnetic properties of a system can change in response to the presence of, or changes in, other objects or systems. Essential Knowledge 4.E.4 The resistance of a resistor, and the capacitance of a capacitor, can
be understood from the basic properties of electric fields and forces, as well as the properties of materials and their geometry. Big Idea 5: Changes that occur as a result of interactions are constrained by conservation laws. Enduring Understanding 5.B The energy of a system is conserved. Essential Knowledge 5.B.5 Energy can be transferred by an external force exerted on an object or system that moves the object or system through a distance; this energy transfer is called work. Energy transfer in mechanical or electrical systems may occur at different rates. Power is deﬁned as the rate of energy transfer into, out of, or within a system. [A piston ﬁlled with gas getting compressed or expanded is treated in Physics 2 as a part of thermodynamics.] Essential Knowledge 5.B.9 Kirchhoff's loop rule describes conservation of energy in electrical circuits. [The application of Kirchhoff's laws to circuits is introduced in Physics 1 and further developed in Physics 2 in the context of more complex circuits, including those with capacitors.] 20.1 Current Learning Objectives By the end of this section, you will be able to: • Define electric current, ampere, and drift velocity. • Describe the direction of charge flow in conventional current. • Use drift velocity to calculate current and vice versa. The information presented in this section supports the following AP® learning objectives and science practices: • 1.B.1.1 The student is able to make claims about natural phenomena based on conservation of electric charge. (S.P. 6.4) • 1.B.1.2 The student is able to make predictions, using the conservation of electric charge, about the sign and relative quantity of net charge of objects or systems after various charging processes, including conservation of charge in simple circuits. (S.P. 6.4, 7.2) This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 20 \| Electric Current, Resistance, and Ohm's Law 869 Electric Current Electric current is defined to be the rate at which charge flows. A large current, such as that used to start a truck engine, moves a large amount of charge in a small time, whereas a small current, such as that used to operate a hand-held calculator, moves a small amount of charge over a long period of time. In equation form, electric current is defined to be = Δ Δ, (20.1) where Δ
is the amount of charge passing through a given area in time Δ. (As in previous chapters, initial time is often taken to be zero, in which case Δ =.) (See Figure 20.2.) The SI unit for current is the ampere (A), named for the French physicist André-Marie Ampère (1775–1836). Since = Δ / Δ, we see that an ampere is one coulomb per second: 1 A = 1 C/s (20.2) Not only are fuses and circuit breakers rated in amperes (or amps), so are many electrical appliances. Figure 20.2 The rate of flow of charge is current. An ampere is the flow of one coulomb through an area in one second. Example 20.1 Calculating Currents: Current in a Truck Battery and a Handheld Calculator (a) What is the current involved when a truck battery sets in motion 720 C of charge in 4.00 s while starting an engine? (b) How long does it take 1.00 C of charge to flow through a handheld calculator if a 0.300-mA current is flowing? Strategy We can use the definition of current in the equation = Δ / Δ to find the current in part (a), since charge and time are given. In part (b), we rearrange the definition of current and use the given values of charge and current to find the time required. Solution for (a) Entering the given values for charge and time into the definition of current gives = Δ Δ = 180 A. = 720 C 4.00 s = 180 C/s (20.3) Discussion for (a) This large value for current illustrates the fact that a large charge is moved in a small amount of time. The currents in these “starter motors” are fairly large because large frictional forces need to be overcome when setting something in motion. Solution for (b) Solving the relationship = Δ / Δ for time Δ, and entering the known values for charge and current gives Δ = Δ = 1.00 C 0.30010-3 C/s = 3.33103 s. (20.4) Discussion for (b) This time is slightly less than an hour. The small current used by the hand-held calculator takes a much longer time to move a smaller charge than the large current of the truck starter. So why can we operate our calculators only seconds after turning them on?
It's because calculators require very little energy. Such small current and energy demands allow handheld calculators to operate from solar cells or to get many hours of use out of small batteries. Remember, calculators do not have moving parts in the same way that a truck engine has with cylinders and pistons, so the technology requires smaller currents. 870 Chapter 20 \| Electric Current, Resistance, and Ohm's Law Figure 20.3 shows a simple circuit and the standard schematic representation of a battery, conducting path, and load (a resistor). Schematics are very useful in visualizing the main features of a circuit. A single schematic can represent a wide variety of situations. The schematic in Figure 20.3 (b), for example, can represent anything from a truck battery connected to a headlight lighting the street in front of the truck to a small battery connected to a penlight lighting a keyhole in a door. Such schematics are useful because the analysis is the same for a wide variety of situations. We need to understand a few schematics to apply the concepts and analysis to many more situations. Figure 20.3 (a) A simple electric circuit. A closed path for current to flow through is supplied by conducting wires connecting a load to the terminals of a battery. (b) In this schematic, the battery is represented by the two parallel red lines, conducting wires are shown as straight lines, and the zigzag represents the load. The schematic represents a wide variety of similar circuits. Note that the direction of current in Figure 20.3 is from positive to negative. The direction of conventional current is the direction that positive charge would flow. In a single loop circuit (as shown in Figure 20.3), the value for current at all points of the circuit should be the same if there are no losses. This is because current is the flow of charge and charge is conserved, i.e., the charge flowing out from the battery will be the same as the charge flowing into the battery. Depending on the situation, positive charges, negative charges, or both may move. In metal wires, for example, current is carried by electrons—that is, negative charges move. In ionic solutions, such as salt water, both positive and negative charges move. This is also true in nerve cells. A Van de Graaff generator used for nuclear research can produce a current of pure positive charges, such as protons. Figure 20.4 illustrates the movement of charged particles that compose a current. The fact
that conventional current is taken to be in the direction that positive charge would flow can be traced back to American politician and scientist Benjamin Franklin in the 1700s. He named the type of charge associated with electrons negative, long before they were known to carry current in so many situations. Franklin, in fact, was totally unaware of the small-scale structure of electricity. It is important to realize that there is an electric field in conductors responsible for producing the current, as illustrated in Figure 20.4. Unlike static electricity, where a conductor in equilibrium cannot have an electric field in it, conductors carrying a current have an electric field and are not in static equilibrium. An electric field is needed to supply energy to move the charges. Making Connections: Take-Home Investigation—Electric Current Illustration Find a straw and little peas that can move freely in the straw. Place the straw flat on a table and fill the straw with peas. When you pop one pea in at one end, a different pea should pop out the other end. This demonstration is an analogy for an electric current. Identify what compares to the electrons and what compares to the supply of energy. What other analogies can you find for an electric current? Note that the flow of peas is based on the peas physically bumping into each other; electrons flow due to mutually repulsive electrostatic forces. This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 20 \| Electric Current, Resistance, and Ohm's Law 871 Figure 20.4 Current is the rate at which charge moves through an area, such as the cross-section of a wire. Conventional current is defined to move in the direction of the electric field. (a) Positive charges move in the direction of the electric field and the same direction as conventional current. (b) Negative charges move in the direction opposite to the electric field. Conventional current is in the direction opposite to the movement of negative charge. The flow of electrons is sometimes referred to as electronic flow. Example 20.2 Calculating the Number of Electrons that Move through a Calculator If the 0.300-mA current through the calculator mentioned in the Example 20.1 example is carried by electrons, how many electrons per second pass through it? Strategy The current calculated in the previous example was defined for the flow of positive charge. For electrons, the magnitude is the same, but the sign is opposite, electrons = −0.300×10−3 C/
s.Since each electron (−) has a charge of –1.60×10−19 C, we can convert the current in coulombs per second to electrons per second. Solution Starting with the definition of current, we have electrons = Δelectrons Δ = –0.300×10−3 C s. We divide this by the charge per electron, so that s = –0.30010 – 3 C – = 1.881015 – s. s 1 – –1.6010−19 C (20.5) (20.6) Discussion There are so many charged particles moving, even in small currents, that individual charges are not noticed, just as individual water molecules are not noticed in water flow. Even more amazing is that they do not always keep moving forward like soldiers in a parade. Rather they are like a crowd of people with movement in different directions but a general trend to move forward. There are lots of collisions with atoms in the metal wire and, of course, with other electrons. Drift Velocity Electrical signals are known to move very rapidly. Telephone conversations carried by currents in wires cover large distances without noticeable delays. Lights come on as soon as a switch is flicked. Most electrical signals carried by currents travel at speeds on the order of 108 m/s, a significant fraction of the speed of light. Interestingly, the individual charges that make up the current move much more slowly on average, typically drifting at speeds on the order of 10−4 m/s. How do we reconcile these two speeds, and what does it tell us about standard conductors? The high speed of electrical signals results from the fact that the force between charges acts rapidly at a distance. Thus, when a free charge is forced into a wire, as in Figure 20.5, the incoming charge pushes other charges ahead of it, which in turn push on charges farther down the line. The density of charge in a system cannot easily be increased, and so the signal is passed on 872 Chapter 20 \| Electric Current, Resistance, and Ohm's Law rapidly. The resulting electrical shock wave moves through the system at nearly the speed of light. To be precise, this rapidly moving signal or shock wave is a rapidly propagating change in electric field. Figure 20.5 When charged particles are forced into this volume of a conductor, an equal number are quickly forced to leave. The repulsion between like charges makes it difficult to increase the number of charges in a volume. Thus, as one charge enters, another leaves
almost immediately, carrying the signal rapidly forward. Good conductors have large numbers of free charges in them. In metals, the free charges are free electrons. Figure 20.6 shows how free electrons move through an ordinary conductor. The distance that an individual electron can move between collisions with atoms or other electrons is quite small. The electron paths thus appear nearly random, like the motion of atoms in a gas. But there is an electric field in the conductor that causes the electrons to drift in the direction shown (opposite to the field, since they are negative). The drift velocity d is the average velocity of the free charges. Drift velocity is quite small, since there are so many free charges. If we have an estimate of the density of free electrons in a conductor, we can calculate the drift velocity for a given current. The larger the density, the lower the velocity required for a given current. Figure 20.6 Free electrons moving in a conductor make many collisions with other electrons and atoms. The path of one electron is shown. The average velocity of the free charges is called the drift velocity, d, and it is in the direction opposite to the electric field for electrons. The collisions normally transfer energy to the conductor, requiring a constant supply of energy to maintain a steady current. Conduction of Electricity and Heat Good electrical conductors are often good heat conductors, too. This is because large numbers of free electrons can carry electrical current and can transport thermal energy. The free-electron collisions transfer energy to the atoms of the conductor. The electric field does work in moving the electrons through a distance, but that work does not increase the kinetic energy (nor speed, therefore) of the electrons. The work is transferred to the conductor's atoms, possibly increasing temperature. Thus a continuous power input is required to maintain current. An exception, of course, is found in superconductors, for reasons we shall explore in a later chapter. Superconductors can have a steady current without a continual supply of energy—a great energy savings. In contrast, the supply of energy can be useful, such as in a lightbulb filament. The supply of energy is necessary to increase the temperature of the tungsten filament, so that the filament glows. Making Connections: Take-Home Investigation—Filament Observations Find a lightbulb with a filament. Look carefully at the filament and describe its structure. To what points is the filament connected? We can obtain an expression for the relationship between current and drift velocity by considering the number of
free charges in a segment of wire, as illustrated in Figure 20.7. The number of free charges per unit volume is given the symbol and depends on the material. The shaded segment has a volume, so that the number of free charges in it is. The charge Δ in this segment is thus, where is the amount of charge on each carrier. (Recall that for electrons, is This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 20 \| Electric Current, Resistance, and Ohm's Law 873 −1.60×10−19 C.) Current is charge moved per unit time; thus, if all the original charges move out of this segment in time Δ, the current is = Δ Δ = Δ. Note that / Δ is the magnitude of the drift velocity, d, since the charges move an average distance in a time Δ. Rearranging terms gives = d, (20.7) (20.8) where is the current through a wire of cross-sectional area made of a material with a free charge density. The carriers of the current each have charge and move with a drift velocity of magnitude d. Figure 20.7 All the charges in the shaded volume of this wire move out in a time, having a drift velocity of magnitude d = /. See text for further discussion. Note that simple drift velocity is not the entire story. The speed of an electron is much greater than its drift velocity. In addition, not all of the electrons in a conductor can move freely, and those that do might move somewhat faster or slower than the drift velocity. So what do we mean by free electrons? Atoms in a metallic conductor are packed in the form of a lattice structure. Some electrons are far enough away from the atomic nuclei that they do not experience the attraction of the nuclei as much as the inner electrons do. These are the free electrons. They are not bound to a single atom but can instead move freely among the atoms in a “sea” of electrons. These free electrons respond by accelerating when an electric field is applied. Of course as they move they collide with the atoms in the lattice and other electrons, generating thermal energy, and the conductor gets warmer. In an insulator, the organization of the atoms and the structure do not allow for such free electrons. Example 20.3 Calculating Drift Velocity in a Common Wire Calculate the drift velocity of electrons in a 12-gau
ge copper wire (which has a diameter of 2.053 mm) carrying a 20.0-A current, given that there is one free electron per copper atom. (Household wiring often contains 12-gauge copper wire, and the maximum current allowed in such wire is usually 20 A.) The density of copper is 8.80×103 kg/m3. Strategy We can calculate the drift velocity using the equation = d. The current = 20.0 A is given, and = – 1.60×10 – 19 C is the charge of an electron. We can calculate the area of a cross-section of the wire using the formula = 2, where is one-half the given diameter, 2.053 mm. We are given the density of copper, 8.80×103 kg/m3, and the periodic table shows that the atomic mass of copper is 63.54 g/mol. We can use these two quantities along with Avogadro's number, 6.02×1023 atoms/mol, cubic meter. the number of free electrons per to determine, Solution First, calculate the density of free electrons in copper. There is one free electron per copper atom. Therefore, is the same as the number of copper atoms per m3. We can now find as follows: = 1 − atom× 6.02×1023 atoms mol × 1 mol 63.54 g × 1000 g kg × 8.80×103 kg 1 m3 (20.9) = 8.342×1028 − /m3. The cross-sectional area of the wire is 874 Chapter 20 \| Electric Current, Resistance, and Ohm's Law = 2 = 2.053×10−3 m 2 2 = 3.310×10–6 m2. Rearranging = d to isolate drift velocity gives = d = 20.0 A (8.342×1028/m3)(–1.60×10–19 C)(3.310×10–6 m2) = –4.53×10–4 m/s. (20.10) (20.11) Discussion The minus sign indicates that the negative charges are moving in the direction opposite to conventional current. The small value for drift velocity (on the order of 10−4 m/s ) confirms that the signal moves on the order of 1012 times faster (about 108 m/s ) than the charges that carry it. 20.2 Ohm’s
Law: Resistance and Simple Circuits Learning Objectives By the end of this section, you will be able to: • Explain the origin of Ohm's law. • Calculate voltages, currents, and resistances with Ohm's law. • Explain the difference between ohmic and non-ohmic materials. • Describe a simple circuit. The information presented in this section supports the following AP® learning objectives and science practices: • 4.E.4.1 The student is able to make predictions about the properties of resistors and/or capacitors when placed in a simple circuit based on the geometry of the circuit element and supported by scientific theories and mathematical relationships. (S.P. 2.2, 6.4) What drives current? We can think of various devices—such as batteries, generators, wall outlets, and so on—which are necessary to maintain a current. All such devices create a potential difference and are loosely referred to as voltage sources. When a voltage source is connected to a conductor, it applies a potential difference that creates an electric field. The electric field in turn exerts force on charges, causing current. Ohm's Law The current that flows through most substances is directly proportional to the voltage applied to it. The German physicist Georg Simon Ohm (1787–1854) was the first to demonstrate experimentally that the current in a metal wire is directly proportional to the voltage applied: ∝. (20.12) This important relationship is known as Ohm's law. It can be viewed as a cause-and-effect relationship, with voltage the cause and current the effect. This is an empirical law like that for friction—an experimentally observed phenomenon. Such a linear relationship doesn't always occur. Resistance and Simple Circuits If voltage drives current, what impedes it? The electric property that impedes current (crudely similar to friction and air resistance) is called resistance. Collisions of moving charges with atoms and molecules in a substance transfer energy to the substance and limit current. Resistance is defined as inversely proportional to current, or ∝ 1. (20.13) Thus, for example, current is cut in half if resistance doubles. Combining the relationships of current to voltage and current to resistance gives This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 20 \| Electric Current, Resistance, and Ohm's Law =. 875 (20.14) This relationship is also called
Ohm's law. Ohm's law in this form really defines resistance for certain materials. Ohm's law (like Hooke's law) is not universally valid. The many substances for which Ohm's law holds are called ohmic. These include good conductors like copper and aluminum, and some poor conductors under certain circumstances. Ohmic materials have a resistance that is independent of voltage and current. An object that has simple resistance is called a resistor, even if its resistance is small. The unit for resistance is an ohm and is given the symbol Ω (upper case Greek omega). Rearranging = gives =, and so the units of resistance are 1 ohm = 1 volt per ampere: Figure 20.8 shows the schematic for a simple circuit. A simple circuit has a single voltage source and a single resistor. The wires connecting the voltage source to the resistor can be assumed to have negligible resistance, or their resistance can be included in. 1 Ω = 1. (20.15) Figure 20.8 A simple electric circuit in which a closed path for current to flow is supplied by conductors (usually metal wires) connecting a load to the terminals of a battery, represented by the red parallel lines. The zigzag symbol represents the single resistor and includes any resistance in the connections to the voltage source. Making Connections: Real World Connections Ohm's law ( = ) is a fundamental relationship that could be presented by a linear function with the slope of the line being the resistance. The resistance represents the voltage that needs to be applied to the resistor to create a current of 1 A through the circuit. The graph (in the figure below) shows this representation for two simple circuits with resistors that have different resistances and thus different slopes. Figure 20.9 The figure illustrates the relationship between current and voltage for two different resistors. The slope of the graph represents the resistance value, which is 2Ω and 4Ω for the two lines shown. 876 Chapter 20 \| Electric Current, Resistance, and Ohm's Law Making Connections: Real World Connections The materials which follow Ohm's law by having a linear relationship between voltage and current are known as ohmic materials. On the other hand, some materials exhibit a nonlinear voltage-current relationship and hence are known as nonohmic materials. The figure below shows current voltage relationships for the two types of materials. Figure 20.10 The relationship between voltage and current for ohmic and non-ohmic materials are
shown. (a) (b) Clearly the resistance of an ohmic material (shown in (a)) remains constant and can be calculated by finding the slope of the graph but that is not true for a non-ohmic material (shown in (b)). Example 20.4 Calculating Resistance: An Automobile Headlight What is the resistance of an automobile headlight through which 2.50 A flows when 12.0 V is applied to it? Strategy We can rearrange Ohm's law as stated by = and use it to find the resistance. Solution Rearranging = and substituting known values gives = = 12.0 V 2.50 A = 4.80 Ω. (20.16) Discussion This is a relatively small resistance, but it is larger than the cold resistance of the headlight. As we shall see in Resistance and Resistivity, resistance usually increases with temperature, and so the bulb has a lower resistance when it is first switched on and will draw considerably more current during its brief warm-up period. Resistances range over many orders of magnitude. Some ceramic insulators, such as those used to support power lines, have resistances of 1012 Ω or more. A dry person may have a hand-to-foot resistance of 105 Ω, whereas the resistance of the human heart is about 103 Ω. A meter-long piece of large-diameter copper wire may have a resistance of 10−5 Ω, and superconductors have no resistance at all (they are non-ohmic). Resistance is related to the shape of an object and the material of which it is composed, as will be seen in Resistance and Resistivity. Additional insight is gained by solving = for, yielding = (20.17) This expression for can be interpreted as the voltage drop across a resistor produced by the current. The phrase drop is often used for this voltage. For instance, the headlight in Example 20.4 has an drop of 12.0 V. If voltage is measured at various points in a circuit, it will be seen to increase at the voltage source and decrease at the resistor. Voltage is similar to fluid pressure. The voltage source is like a pump, creating a pressure difference, causing current—the flow of charge. The resistor is like a pipe that reduces pressure and limits flow because of its resistance. Conservation of energy has important consequences here. The voltage source supplies energy (causing an electric field and a current), and the resistor converts it to another form
(such as thermal energy). In a simple circuit (one with a single simple resistor), the voltage supplied by the source equals the voltage drop across the resistor, since PE = Δ, and the same flows through each. Thus the energy supplied by the voltage source and the energy converted by the resistor are equal. (See Figure 20.11.) This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 20 \| Electric Current, Resistance, and Ohm's Law 877 Figure 20.11 The voltage drop across a resistor in a simple circuit equals the voltage output of the battery. Making Connections: Conservation of Energy In a simple electrical circuit, the sole resistor converts energy supplied by the source into another form. Conservation of energy is evidenced here by the fact that all of the energy supplied by the source is converted to another form by the resistor alone. We will find that conservation of energy has other important applications in circuits and is a powerful tool in circuit analysis. PhET Explorations: Ohm's Law See how the equation form of Ohm's law relates to a simple circuit. Adjust the voltage and resistance, and see the current change according to Ohm's law. The sizes of the symbols in the equation change to match the circuit diagram. Figure 20.12 Ohm's Law (http://cnx.org/content/m55356/1.2/ohms-law_en.jar) 20.3 Resistance and Resistivity Learning Objectives By the end of this section, you will be able to: • Explain the concept of resistivity. • Use resistivity to calculate the resistance of specified configurations of material. • Use the thermal coefficient of resistivity to calculate the change of resistance with temperature. The information presented in this section supports the following AP® learning objectives and science practices: • 1.E.2.1 The student is able to choose and justify the selection of data needed to determine resistivity for a given material. (S.P. 4.1) • 4.E.4.2 The student is able to design a plan for the collection of data to determine the effect of changing the geometry and/or materials on the resistance or capacitance of a circuit element and relate results to the basic properties of resistors and capacitors. (S.P. 4.1, 4.2) • 4.E.4.3 The student is able to analyze data to determine the effect of changing the geometry and
/or materials on the resistance or capacitance of a circuit element and relate results to the basic properties of resistors and capacitors. (S.P. 5.1) Material and Shape Dependence of Resistance The resistance of an object depends on its shape and the material of which it is composed. The cylindrical resistor in Figure 20.13 is easy to analyze, and, by so doing, we can gain insight into the resistance of more complicated shapes. As you might expect, the cylinder's electric resistance is directly proportional to its length, similar to the resistance of a pipe to fluid flow. The longer the cylinder, the more collisions charges will make with its atoms. The greater the diameter of the cylinder, the more current it can carry (again similar to the flow of fluid through a pipe). In fact, is inversely proportional to the cylinder's crosssectional area. 878 Chapter 20 \| Electric Current, Resistance, and Ohm's Law Figure 20.13 A uniform cylinder of length and cross-sectional area. Its resistance to the flow of current is similar to the resistance posed by a pipe to fluid flow. The longer the cylinder, the greater its resistance. The larger its cross-sectional area, the smaller its resistance. For a given shape, the resistance depends on the material of which the object is composed. Different materials offer different resistance to the flow of charge. We define the resistivity of a substance so that the resistance of an object is directly proportional to. Resistivity is an intrinsic property of a material, independent of its shape or size. The resistance of a uniform cylinder of length, of cross-sectional area, and made of a material with resistivity, is =. (20.18) Table 20.1 gives representative values of. The materials listed in the table are separated into categories of conductors, semiconductors, and insulators, based on broad groupings of resistivities. Conductors have the smallest resistivities, and insulators have the largest; semiconductors have intermediate resistivities. Conductors have varying but large free charge densities, whereas most charges in insulators are bound to atoms and are not free to move. Semiconductors are intermediate, having far fewer free charges than conductors, but having properties that make the number of free charges depend strongly on the type and amount of impurities in the semiconductor. These unique properties of semiconductors are put to use in modern electronics, as will be explored in later chapters. This content is available for free at
http://cnx.org/content/col11844/1.13 Chapter 20 \| Electric Current, Resistance, and Ohm's Law 879 Table 20.1 Resistivities of Various materials at 20ºC Material Conductors Silver Copper Gold Aluminum Tungsten Iron Platinum Steel Lead Resistivity ρ ( Ω ⋅ m ) 1.59×10−8 1.72×10−8 2.44×10−8 2.65×10−8 5.6×10−8 9.71×10−8 10.6×10−8 20×10−8 22×10−8 Manganin (Cu, Mn, Ni alloy) 44×10−8 Constantan (Cu, Ni alloy) Mercury Nichrome (Ni, Fe, Cr alloy) 49×10−8 96×10−8 100×10−8 Semiconductors[1] Carbon (pure) Carbon Germanium (pure) Germanium Silicon (pure) Silicon Insulators Amber Glass Lucite Mica Quartz (fused) Rubber (hard) Sulfur Teflon Wood 3.5×105 (3.5 − 60)×105 600×10−3 (1 − 600)×10−3 2300 0.1–2300 5×1014 109 − 1014 >1013 1011 − 1015 75×1016 1013 − 1016 1015 >1013 108 − 1014 880 Chapter 20 \| Electric Current, Resistance, and Ohm's Law Example 20.5 Calculating Resistor Diameter: A Headlight Filament A car headlight filament is made of tungsten and has a cold resistance of 0.350 Ω. If the filament is a cylinder 4.00 cm long (it may be coiled to save space), what is its diameter? Strategy We can rearrange the equation = to find the cross-sectional area of the filament from the given information. Then its diameter can be found by assuming it has a circular cross-section. Solution The cross-sectional area, found by rearranging the expression for the resistance of a cylinder given in =, is Substituting the given values, and taking from Table 20.1, yields =. = (5.610–8 Ω ⋅ m)(4.0010–2 m) 0.350 Ω = 6.4010–9 m2. The area of a circle is related to its diameter by = 2 4. Solving for
the diameter, and substituting the value found for, gives 1 2 = 2 = 2 = 9.010–5 m. 6.4010–9 m2 3.14 1 2 (20.19) (20.20) (20.21) (20.22) Discussion The diameter is just under a tenth of a millimeter. It is quoted to only two digits, because is known to only two digits. Temperature Variation of Resistance The resistivity of all materials depends on temperature. Some even become superconductors (zero resistivity) at very low temperatures. (See Figure 20.14.) Conversely, the resistivity of conductors increases with increasing temperature. Since the atoms vibrate more rapidly and over larger distances at higher temperatures, the electrons moving through a metal make more collisions, effectively making the resistivity higher. Over relatively small temperature changes (about 100ºC or less), resistivity varies with temperature change Δ as expressed in the following equation = 0(1 + Δ), (20.23) where 0 is the original resistivity and is the temperature coefficient of resistivity. (See the values of in Table 20.2 below.) For larger temperature changes, may vary or a nonlinear equation may be needed to find. Note that is positive for metals, meaning their resistivity increases with temperature. Some alloys have been developed specifically to have a small temperature dependence. Manganin (which is made of copper, manganese and nickel), for example, has close to zero (to three digits on the scale in Table 20.2), and so its resistivity varies only slightly with temperature. This is useful for making a temperature-independent resistance standard, for example. 1. Values depend strongly on amounts and types of impurities This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 20 \| Electric Current, Resistance, and Ohm's Law 881 Figure 20.14 The resistance of a sample of mercury is zero at very low temperatures—it is a superconductor up to about 4.2 K. Above that critical temperature, its resistance makes a sudden jump and then increases nearly linearly with temperature. Table 20.2 Temperature Coefficients of Resistivity Material Conductors Silver Copper Gold Aluminum Tungsten Iron Platinum Lead Coefficient α (1/°C)[2] 3.8×10−3 3.9×10−3 3.4×10−3 3.9×10−3 4

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