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when we measure the weight of something. The most interesting and illustrative normal force question, that is often asked, has to do with a scale in a lift. Using Newton’s third law we can solve these problems quite easily. When you stand on a scale to measure your weight you are pulled down by gravity. There is no acceleration downwards because there is a reaction force we call the normal force acting upwards on you. This is the force that the scale would measure. If the gravitational force were less then the reading on the scale would be less. Worked Example 54 Normal Forces 1 Question: A man weighing 100kg stands on a scale (measuring newtons). What is the reading on the scale? Answer: Step 1 : Decide what information is supplied We are given the mass of the man. We know the gravitational acceleration that acts on him - g = 10m=s2. Step 2 : Decide what equation to use to solve the problem The scale measures the normal force on the man. This is the force that balances gravity. We can use Newton’s laws to solve the problem: Fr = Fg + FN (10.13) where Fr is the resultant force on the man. 1Newton’s third law! 178 Step 3 : Firstly we determine the net force acting downwards on the man due to gravity Fg = mg m s2 = 100kg 9:8 = 980 £ kgm s2 = 980N downwards Step 4 : Now determine the normal force acting upwards on the man We now know the gravitational force downwards. We know that the sum of all the forces must equal the resultant acceleration times the mass. The overall resultant acceleration of the man on the scale is 0 - so Fr = 0. Fr = Fg + FN 0 = FN = 980N upwards 980N + FN ¡ Step 5 : Quote the ﬂnal answer The normal force is then 980N upwards. It exactly balances the gravitational force downwards so there is no net force and no acceleration on the man. Now we are going to add things to exactly the same problem to show how things change slightly. We will now move to a lift moving at constant velocity. Remember if velocity is constant then acceleration is zero. Worked Example 55 Normal Forces 2 s. What is the reading on the scale? Question: A man weighing 100kg stands on a scale (measuring newtons) inside a lift moving downwards at 2 m Answer: Step 1 : Decide what information is supplied We are given
the mass of the man and the acceleration of the lift. We know the gravitational acceleration that acts on him. Step 2 : Decide which equation to use to solve the problem Once again we can use Newton’s laws. We know that the sum of all the forces must equal the resultant acceleration times the mass (This is the resultant force, Fr). Fr = Fg + FN (10.14) 179 Step 3 : Firstly we determine the net force acting downwards on the man due to gravity Fg = mg m s2 = 100kg 9:8 = 980 £ kgm s2 = 980N downwards Step 4 : Now determine the normal force acting upwards on the man The scale measures this normal force, so once we’ve determined it we will know the reading on the scale. Because the lift is moving at constant velocity the overall resultant acceleration of the man on the scale is 0. If we write out the equation: Fr = Fg + FN 0 = FN = 980N upwards 980N + FN ¡ Step 5 : Quote the ﬂnal answer The normal force is then 980N upwards. It exactly balances the gravitational force downwards so there is no net force and no acceleration on the man. In this second example we get exactly the same result because the net acceleration on the man was zero! If the lift is accelerating downwards things are slightly diﬁerent and now we will get a more interesting answer! Worked Example 56 Normal Forces 3 s2. What is the reading on the scale? Question: A man weighing 100kg stands on a scale (measuring newtons) inside a lift accelerating downwards at 2 m Answer: Step 1 : Decide what information is supplied We are given the mass of the man and his resultant acceleration - this is just the acceleration of the lift. We know the gravitational acceleration that acts on him. Step 2 : Decide which equation to use to solve the problem Once again we can use Newton’s laws. We know that the sum of all the forces must equal the resultant acceleration times the mass (This is the resultant force, Fr). Fr = Fg + FN (10.15) 180 Step 3 : Firstly we determine the net force acting downwards on the man due to gravity, Fg Fg = mg m s2 = 100kg 9:8 = 980 £ kgm s2 = 980N downwards Step 4 : Now determine the normal force acting upwards on the man, FN We know that the sum of all the forces must equal
the resultant acceleration times the mass. The overall resultant acceleration of the man on the scale is 2 m s2 downwards. If we write out the equation: 100kg 2) ( ¡ £ 200 ¡ kgm s2 Fr = Fg + FN m s2 = = ¡ 980N + FN 980N + FN ¡ 200N = ¡ FN = 780N upwards 980N + FN ¡ Step 5 : Quote the ﬂnal answer The normal force is then 780N upwards. It balances the gravitational force downwards just enough so that the man only accelerates downwards at 2 m s2. Worked Example 57 Normal Forces 4 s2. What is the reading on the scale? Question: A man weighing 100kg stands on a scale (measuring newtons) inside a lift accelerating upwards at 4 m Answer: Step 1 : Decide what information is supplied We are given the mass of the man and his resultant acceleration - this is just the acceleration of the lift. We know the gravitational acceleration that acts on him. Step 2 : Decide which equation to use to solve the problem Once again we can use Newton’s laws. We know that the sum of all the forces must equal the resultant acceleration times the mass (This is the resultant force, Fr). Fr = Fg + FN (10.16) 181 Step 3 : Firstly we determine the net force acting downwards on the man due to gravity, Fg Fg = mg m s2 = 100kg 9:8 = 980 £ kgm s2 = 980N downwards Step 4 : Now determine the normal force upwards, FN We know that the sum of all the forces must equal the resultant acceleration times the mass. The overall resultant acceleration of the man on the scale is 2 m s2 downwards. if we write out the equation: 100kg (4) £ Fr = Fg + FN m s2 = = ¡ 980N + FN 980N + FN ¡ 400 kgm s2 400N = FN = 1380N upwards 980N + FN ¡ Step 5 : Quote the ﬂnal answer The normal force is then 1380N upwards. then in addition applies su–cient force to accelerate the man upwards at 4 m s2. It balances the gravitational force and 10.4 Comparative problems Here always work with multiplicative factors to ﬂnd something new in terms of something old. Worked Example 58 Comparative Problem 1 Question:On Earth a man weighs 70kg
. Now if the same man was instantaneously beamed to the planet Zirgon, which has the same size as the Earth but twice the mass, what would he weigh? (NOTE TO SELF: Vanessa: isn’t this confusing weight and mass?) Answer: 182 Step 1 : We start with the situation on Earth mEm r2 Step 2 : Now we consider the situation on Zirgon W = mg = G WZ = mgZ = G mZm r2 Z (10.17) (10.18) Step 3 : Relation between conditions on Earth and Zirgon but we know that mZ = 2mE and we know that rZ = r so we could write the equation again and substitute these relationships in: Step 4 : Substitute WZ = mgZ = G (2mE)m (r)2 WZ = 2(G (mE)m (r)2 ) Step 5 : Relation between weight on Zirgon and Earth Step 6 : Quote the ﬂnal answer so on Zirgon he weighs 140kg. WZ = 2(W ) (10.19) (10.20) (10.21) 10.4.1 Principles Write out ﬂrst case Write out all relationships between variable from ﬂrst and second case Write out second case Substitute all ﬂrst case variables into second case Write second case in terms of ﬂrst case † † † † † Interesting Fact: The acceleration due to gravity at the Earth’s surface is, by convention, equal to 9.80665 ms¡2. (The actual value varies slightly over the surface of the Earth). This quantity is known as g. The following is a list of the gravitational accelerations (in multiples of g) at the surfaces of each of the planets in our solar system: Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Pluto 0.376 0.903 1 0.38 2.34 1.16 1.15 1.19 0.066 183 Note: The "surface" is taken to mean the cloud tops of the gas giants (Jupiter, Saturn, Uranus and Neptune) in the above table. Worked Example 59 Comparative Problem 2 Question: On Earth a man weighs 70kg. On the planet Beeble how much will he weigh if Beeble has mass half of that of the Earth and a radius
one quarter that of the earth. Answer: Step 1 : Start with the situation on Earth Step 2 : Now consider the situation on Beeble W = mg = G mEm r2 WB = mgB = G mBm r2 B (10.22) (10.23) Step 3 : Relation between conditions on Earth and on Beeble We know that mB = 1 again and substitute these relationships in: Step 4 : Substitute 2 mE and we know that rB = 1 4 r so we could write the equation WB = mgB = G = mgB = G (mB)m (rB)2 ( 1 2 mE)m ( 1 4 r)2 = 8(G (mE)m (r)2 ) Step 5 : Relation between weight on Earth and weight on Beeble WB = 8(W ) (10.24) Step 6 : Quote the ﬂnal answer So the man weighs 560kg on Beeble! 184 Interesting Fact: Did you know that the largest telescope in the Southern Hemisphere is the South African Large Telescope (SALT) which came online in 2005 outside Sutherland in the Karoo. 10.5 Falling bodies Objects on the earth fall because there is a gravitation force between them and the earth - which results in an acceleration - as we saw above. So if you hold something in front of you and let it go - it will fall. It falls because of an acceleration toward the centre of the earth which results from the gravitational force between the two. These bodies move in a straight line from the point where they start to the centre for the earth. This means we can reuse everything we learnt in rectilinear motion. the only thing that needs thinking about is the directions we are talking about. We need to choose either up or down as positive just like we had to choose a positive direction in standard rectilinear motion problems. this is the hardest part. If you can do rectilinear motion you can do falling body problems. Just remember the acceleration they feel is constant and because of gravity - but once you have chosen your directions you can forget that gravity has anything to do with the problem - all you have is a rectilinear motion problem with a constant acceleration!! 10.6 Terminal velocity Physics is all about being simple - all we do is look at the world around us and notice how it really works. It is the one thing everyone is qualiﬂed to do - we spend
most of our time when we are really young experimenting to ﬂnd out how things work. Take a book - wave it in the air - change the angle and direction. what happens of course there is resistance. diﬁerent angles make it greater - the faster the book moves the greater it is. The bigger the area of the book moving in the direction of motion the greater the force. So we know that air resistance exists! it is a force. So what happens when an object falls? of course there is air resistance - or drag as it is normally called. There is an approximate formula for the drag force as well. The important thing to realise is that when the drag force and the gravitational force are equal for a falling body there is no net force acting on it - which means no net acceleration. That does not mean it does not move - but it means that its speed does not change. It falls at a constant velocity! This velocity is called terminal velocity. 10.7 Drag force The actual force of air resistance is quite complicated. Experiment by moving a book through the air with the face of the book and then the side of the book forward, you will agree that the area of the book makes a diﬁerence as to how much you must work in order to move the book at the same speed in both cases. This is why racing cars are slim-lined in design, and not shaped like a big box! 185 Get a plastic container lid (or anything waterproof) swing it around in air and then try to swing it around under water. The density of the water is much larger than the air, making you have to work harder at swinging the lid in water. This is why boats and submarines are a lot slower than aeroplanes! So we know that density, area and speed all play a role in the drag force. The expression we use for drag force is 1 2 where C is a constant which depends on the object and uid interactions, ‰ is the density, A is the area and v is the velocity. C‰Av2 (10.25) D = 10.8 Important Equations and Quantities Quantity mass velocity force energy Symbol Unit \| \| N J m ¡!v ¡!F E Units kg m s kg:m s2 kg:m s2 2 S.I. Units or \| or m:s¡1 or kg:m:s¡2 or kg:m2:s¡2 Direction \|
X X \| Table 10.1: Units used in Newtonian Gravitation 186 Chapter 11 Pressure 11.1 Important Equations and Quantities Quantity Symbol Unit S.I. Units Direction Units or Table 11.1: Units used in Pressure 187 Essay 3 : Pressure and Forces Author: Asogan Moodaly Asogan Moodaly received his Bachelor of Science degree (with honours) in Mechanical Engineering from the University of Natal, Durban in South Africa. For his ﬂnal year design project he worked on a 3-axis ﬂlament winding machine for composite (Glass re-enforced plastic in this case) piping. He worked in Vereeniging, Gauteng at Mine Support Products (a subsidiary of Dorbyl Heavy Engineering) as the design engineer once he graduated. He currently lives in the Vaal Triangle area and is working for Sasol Technology Engineering as a mechanical engineer, ensuring the safety and integrity of equipment installed during projects. Pressure and Forces In the mining industry, the roof (hangingwall) tends to drop as the face of the tunnel (stope) is excavated for rock containing gold. As one can imagine, a roof falling on one’s head is not a nice prospect! Therefore the roof needs to be supported. Roof Face The roof is not one big uniform chunk of rock. Rather it is broken up into smaller chunks. It is assumed that the biggest chunk of rock in the roof has a mass of less than 20 000 kgs therefore each support has to be designed to resist a force related to that mass. The strength of the material (either wood or steel) making up the support is taken into account when working out the minimum required size and thickness of the parts to withstand the force of the roof. Roof Face Supports 188 Sometimes the design of the support is such that the support needs to withstand the rock mass without the force breaking the roof.. Therefore hydraulic supports (hydro = water) use the principles of force and pressure such that as a force is exerted on the support, the water pressure increases. A pressure relief valve then squirts out water when the pressure (and thus the force) gets too large. Imagine a very large, modiﬂed doctor’s syringe. Hydraulic support Doctors syringe Force Force Seal to trap water in tube Water filled tube Pressure relief valve Rubber seal to trap medicine in syringe Pressure Pressure In the petrochemical industry, there are many vessels and pipes that are under high pressures.
A vessel is a containment unit (Imagine a pot without handles, that has the lid welded to the pot that would be a small vessel) where chemicals mix and react to form other chemicals, amongst other uses. End Product Chemical The end product chemicals are sold to companies that use these chemicals to make shampoo, dishwashing liquid, plastic containers, fertilizer, etc. Anyway, some of these chemical reactions require high temperatures and pressures in order to work. These pressures result in forces being applied to the insides of the vessels and pipes. Therefore the minimum thickness of the pipe and vessels walls must be determined using calculations, to withstand these forces. These calculations take into account the strength of the material (typically steel, plastic or composite), the diameter and of course the pressure inside the equipment. Let examine the concepts of force and pressure in further detail. 189 Chapter 12 Heat and Properties of Matter 12.1 Phases of matter 12.1.1 Density Matter is a substance which has mass and occupies space. The density of matter refers to how much mass is in a given volume. Said diﬁerently, you can imagine the density to be the amount of mass packed into a given volume. density = M ass V olume If we consider a bar of soap and a bar of steel with the same volume, the steel will have more mass because it has a greater density. The density is greater in steal because more atoms are closely packed in comparison to the soap. Although they are both the same size, the bar of steel will be "heavier" because it has more mass. Worked Example 60 Density of objects A bar of aluminum (Al) has dimensions 2cm x 3cm x 5cm with a mass of 81g. A bar of lead (Pb) has dimensions 3cm x 3cm x 5cm and a mass of 510.3g. Calculate the density of the aluminum and lead. Solution: First we calculate the volume of Al and Pb: For Aluminum: volume = 2cm ⁄ 3cm For Lead: volume = 3cm ⁄ ⁄ volume = Length W idth Height ⁄ ⁄ 5cm = 30cm3 3cm 5cm = 45cm3 ⁄ We can now calculate the densities using the mass and volume of each material. 30cm3 = 2:7g=cm3 For Aluminum: density = 81g For Lead: density = 510:3g 45cm3 = 11:34g
=cm3 190 2cm 5cm 3cm 5cm 3cm 3cm Now that you know the density of aluminum and lead, which object would be bigger (larger volume): 1kg of Lead or 1kg of Aluminum. Solution: 1kg of aluminum will be much larger in volume than 1kg of lead. Aluminum has a smaller density so it will take a lot more of it to have a weight of 1kg. Lead is much more dense, so it will take less for it to weigh 1kg. The density of liquids and gases can be calculated the same way as in solids. If the mass and volume of a liquid is known, the density can be calculated. We can often determine which liquid has a greater density by mixing two liquids and seeing how they settle. The more dense liquid will fall towards the bottom, or ’sink’. If you have ever added olive oil to water, you have seen it sits on the surface, or ’oats’. This is because olive oil is less dense than water. Fog occurs when water vapor becomes more dense than air("a cloud that sinks in air"). This principle can be used with solids and liquids. In fact, it is the density of an object that determines if it will oat or sink in water. Objects with densities greater than water will sink. Worked Example 61 Objects oating in water Ivory soap is famous for "soap that oats". If a 5cm x 3cm x 10cm bar of ivory soap weighs 1.35 Newtons, show that its density is less than water. Solution: First calculate the bars volume: volume = 3cm Now we must determine the mass of the bar based on its weight. We will use Newton’s Second law (F = ma): 10cm = 150cm3 5cm ⁄ ⁄ W eight = mass gravity = ) ⁄ W eight = 9:8m=s2 M ass ⁄ 191 M ass = 1:35N 9:8m=s2 = :138kg Using the mass and the volume we determine the density of the soap: density = 138g 150cm3 = :92g=cm3 Water has a density of 1g=cm3, therefore the soap is less dense than water, allowing it to oat. 12.2 Phases of matter Although phases are conceptually simple, they are hard to deﬂne precisely. A good deﬂnition
of a phase of a system is a region in the parameter space of the system’s thermodynamic variables in which the free energy is analytic. Equivalently, two states of a system are in the same phase if they can be transformed into each other without abrupt changes in any of their thermodynamic properties. All the thermodynamic properties of a system { the entropy, heat capacity, magnetization, compressibility, and so forth { may be expressed in terms of the free energy and its derivatives. For example, the entropy is simply the ﬂrst derivative of the free energy with temperature. As long as the free energy remains analytic, all the thermodynamic properties will be well-behaved. When a system goes from one phase to another, there will generally be a stage where the free energy is non-analytic. This is known as a phase transition. Familiar examples of phase transitions are melting (solid to liquid), freezing (liquid to solid), boiling (liquid to gas), and condensation (gas to liquid). Due to this non-analyticity, the free energies on either side of the transition are two diﬁerent functions, so one or more thermodynamic properties will behave very diﬁerently after the transition. The property most commonly examined in this context is the heat capacity. During a transition, the heat capacity may become inﬂnite, jump abruptly to a diﬁerent value, or exhibit a "kink" or discontinuity in its derivative. In practice, each type of phase is distinguished by a handful of relevant thermodynamic properties. For example, the distinguishing feature of a solid is its rigidity; unlike a liquid or a gas, a solid does not easily change its shape. Liquids are distinct from gases because they have much lower compressibility: a gas in a large container ﬂlls the container, whereas a liquid forms a puddle in the bottom. Many of the properties of solids, liquids, and gases are not distinct; for instance, it is not useful to compare their magnetic properties. On the other hand, the ferromagnetic phase of a magnetic material is distinguished from the paramagnetic phase by the presence of bulk magnetization without an applied magnetic ﬂeld. To take another example, many substances can exist in a variety of solid phases each corresponding to a unique crystal structure. These varying crystal phases of the same substance are called polymorphs. Diamond and graphite are examples of polymorphs of
carbon. Graphite is composed of layers of hexagonally arranged carbon atoms, in which each carbon atom is strongly bound to three neighboring atoms in the same layer and is weakly bound to atoms in the neighboring layers. By contrast in diamond each carbon atom is strongly bound to four neighboring carbon atoms in a cubic array. The unique crystal structures of graphite and diamond are responsible for the vastly diﬁerent properties of these two materials. Metastable phases 192 Metastable states may sometimes be considered as phases, although strictly speaking they aren’t because they are unstable. For example, each polymorph of a given substance is usually only stable over a speciﬂc range of conditions. For example, diamond is only stable at extremely high pressures. Graphite is the stable form of carbon at normal atmospheric pressures. Although diamond is not stable at atmospheric pressures and should transform to graphite, we know that diamonds exist at these pressures. This is because at normal temperatures the transformation If we were to heat the diamond, the rate of from diamond to graphite is extremely slow. transformation would increase and the diamond would become graphite. However, at normal temperatures the diamond can persist for a very long time. Another important example of metastable polymorphs occurs in the processing of steel. Steels are often subjected to a variety of thermal treatments designed to produce various combinations of stable and metastable iron phases. In this way the steel properties, such as hardness and strength can be adjusted by controlling the relative amounts and crystal sizes of the various phases that form. Phase diagrams The diﬁerent phases of a system may be represented using a phase diagram. The axes of the diagrams are the relevant thermodynamic variables. For simple mechanical systems, we generally use the pressure and temperature. The following ﬂgure shows a phase diagram for a typical material exhibiting solid, liquid and gaseous phases. The markings on the phase diagram show the points where the free energy is non-analytic. The open spaces, where the free energy is analytic, correspond to the phases. The phases are separated by lines of non-analyticity, where phase transitions occur, which are called phase boundaries. In the above diagram, the phase boundary between liquid and gas does not continue indefinitely. Instead, it terminates at a point on the phase diagram called the critical point. This reects the fact that, at extremely high temperatures and pressures, the liquid and gaseous phases become indistinguishable. In water, the critical point occurs at
around 647 K (374 C or 705 F) and 22.064 MPa. The existence of the liquid-gas critical point reveals a slight ambiguity in our above deﬂnitions. When going from the liquid to the gaseous phase, one usually crosses the phase boundary, but it is possible to choose a path that never crosses the boundary by going to the right of the critical point. Thus, phases can sometimes blend continuously into each other. We should note, however, that this does not always happen. For example, it is impossible for the solid-liquid phase boundary to end in a critical point in the same way as the liquid-gas boundary, because the solid and liquid phases have diﬁerent symmetry. An interesting thing to note is that the solid-liquid phase boundary in the phase diagram of most substances, such as the one shown above, has a positive slope. This is due to the solid phase having a higher density than the liquid, so that increasing the pressure increases the melting temperature. However, in the phase diagram for water the solid-liquid phase boundary has a negative slope. This reects the fact that ice has a lower density than water, which is an unusual property for a material. 193 12.2.1 Solids, liquids, gasses 12.2.2 Pressure in uids 12.2.3 change of phase 12.3 Deformation of solids 12.3.1 strain, stress Stress () and strain (†) is one of the most fundamental concepts used in the mechanics of materials. The concept can be easily illustrated by considering a solid, straight bar with a constant cross section throughout its length where a force is distributed evenly at the ends of the bar. This force puts a stress upon the bar. Like pressure, the stress is the force per unit area. In this case the area is the cross sectional area of the bar. stress = F orce Areacrosssection = ) = F A (A) Bar under compression (B) Bar under tension Figure 12.1: Illustration of Bar The bar in ﬂgure 1a is said to be under compression. If the direction of the force (¡!F ), were reversed, stretching the bar, it would be under tension (ﬂg. 1b). Using intuition, you can imagine how the bar might change in shape under compression and tension. Under a compressive load, the bar will shorten and thicken. In contrast
, a tensile load will lengthen the bar and make it thinner. Figure 12.2: Bar changes length under tensile stress For a bar with an original length L, the addition of a stress will result in change of length L and L we can now deﬂne strain as the ratio between the two. That is, strain is L. With 4 deﬂned as the fractional change in length of the bar: 4 Strain L 4 L · 12.3.2 Elastic and plastic behavior Material properties are often characterized by a stress versus strain graph (ﬂgure x.xx). One way in which these graphs can be determined is by tensile testing. In this process, a machine 194 L L 4 Figure 12.3: Left end of bar is ﬂxed as length changes \| \| + Figure 12.4: dashed line represents plastic recovery incomplete stretches a the material by constant amounts and the corresponding stress is measured and plotted. Typical solid metal bars will show a result like that of ﬂgure x.xx. This is called a Type II response. Other materials may exibit diﬁerent responses. We will only concern ourself with Type II materials. The linear region of the graph is called the elastic region. By obtaining the slope of the linear region, it is easy to ﬂnd the strain for a given stress, or vice-versa. This slope shows itself to be very useful in characterizing materials, so it is called the Modulus of Elasticity, or Young’s Modulus: E = stress strain = F=A ¢L=L The elastic region has the unique property that allows the material to return to its original shape when the stress is removed. As the stress is removed it will follow line back to zero. One may think of stretching a spring and then letting it return to its original length. When a stress is applied in the linear region, the material is said to undergo elastic deformation. When a stress is applied that is in the non-linear region, the material will no longer return to its original shape. This is referred to as plastic deformation. If you have overstretched a spring you have seen that it no longer returns to its initial length; it has been plastically deformed. The stress where plastic behavior begins is called the yield strength (point A, ﬂg x). When a material has plastically deformed it will still recover
some of its shape (like an overstretched spring). When a stress in the non-linear region is removed, the stress strain graph will follow a line with a slope equal to the modulus of elasticity (see the dashed line in ﬂgure x.xx). The plastically deformed material will now have a linear region that follows the dashed line. Greater stresses in the plastic region will eventually lead to fracture (the material breaks). The maximum stress the material can undergo before fracture is the ultimate strength. 195 \| \| + Figure 12.5: dashed line represents plastic recovery incomplete 12.4 Ideal gasses Author: G¶erald Wigger G¶erald Wigger started his Physics studies at ETH in Zuerich, Switzerland. He moved to Cape Town, South Africa, for his Bachelor of Science degree (with honours) in Physics from the University of Cape Town in 1998. Returned to Switzerland, he ﬂnished his Diploma at ETH in 2000 and followed up with a PhD in the Solid State Physics group of Prof. Hans-Ruedi Ott at ETH. He graduated in the year 2004. Being awarded a Swiss fellowship, he moved to Stanford University where he is currently continuing his Physics research in the ﬂeld of Materials with novel electronic properties. Any liquid or solid material, heated up above its boiling point, undergoes a transition into a gaseous state. For some materials such as aluminium, one has to heat up to three thousand degrees Celsius (–C), whereas Helium is a gas already at -269 –C. For more examples see Table 12.1. As we ﬂnd very strong bonding between the atoms in a solid material, a gas consists of molecules which do interact very poorly. If one forgets about any electrostatic or intermolecular attractive forces between the molecules, one can assume that all collisions are perfectly elastic. One can visualize the gas as a collection of perfectly hard spheres which collide but which otherwise do not interact with each other. In such a gas, all the internal energy is in the form of kinetic energy and any change in internal energy is accompanied by a change in temperature. Such a gas is called an ideal gas. In order for a gas to be described as an ideal gas, the temperature should be raised far enough above the melting point. A few examples of ideal gases at room temperature are Helium, Argon and hydrogen. Despite the fact that there are only a few gases which can be accurately described
as an ideal gas, the underlying theory is widely used in Physics because of its beauty and simplicity. A thermodynamic system may have a certain substance or material whose quantity can be expressed in mass or mols in an overall volume. These are extensive properties of the system. In the following we will be considering often intensive versus extensive quantities. A material’s intensive property, is a quantity which does not depend on the size of the material, such as temperature, pressure or density. Extensive properties like volume, mass or number of atoms on 196 Material Aluminium Water Ethyl alcohol Methyl ether Nitrogen Helium Temperature in Celsius Temperature in Kelvin 2467 –C 100 –C 78.5 –C -25 –C -195.8 –C -268.9 –C 2740 K 373.15 K 351.6 K 248 K 77.3 K 4.2 K Table 12.1: Boiling points for various materials in degrees Celsius and in Kelvin quantity pressure p volume V unit Pa m3 molar volume vmol m3/mol temperature T mass M density ‰ internal energy E K kg kg/m3 J intensive or extensive intensive extensive intensive intensive extensive intensive extensive Table 12.2: Intensive versus extensive properties of matter the other hand gets bigger the bigger the material is (see Table 12.2 for various intensive/extensive properties). If the substance is evenly distributed throughout the volume in question, then a value of volume per amount of substance may be used as an intensive property. For an example, for an amount called a mol, volume per mol is typically called molar volume. Also, a volume per mass for a speciﬂc substance may be called speciﬂc volume. In the case of an ideas gas, a simple equation of state relates the three intensive properties, temperature, pressure, and molar or speciﬂc volume. Hence, for a closed system containing an ideal gas, the state can be speciﬂed by giving the values of any two of pressure, temperature, and molar volume. 12.4.1 Equation of state The ideal gas can be described with a single equation. However, in order to arrive there, we will be introducing three diﬁerent equations of state, which lead to the ideal gas law. The combination of these three laws leads to a complete picture of the ideal gas. 1661 - Robert Boyle used a U-tube and Mercury to develop a mathematical relationship between pressure
and volume. To a good approximation, the pressure and volume of a ﬂxed amount of gas at a constant temperature were related by V = constant p ¢ p V : pressure (P a) : Volume (m3) In other words, if we compress a given quantity of gas, the pressure will increase. And if we put it under pressure, the volume of the gas will decrease proportionally. 197 Figure 12.6: Pressure-Volume diagram for the ideal gas at constant temperature. Worked Example 62 compressed Helium gas A sample of Helium gas at 25–C is compressed from 200 cm3 to 0.240 cm3. pressure is now 3.00 cm Hg. What was the original pressure of the Helium? Solution: It’s always a good idea to write down the values of all known variables, indicating whether the values are for initial or ﬂnal states. Boyle’s Law problems are essentially special cases of the Ideal Gas Law: Initial: p1 =?; V1 = 200 cm3; Final: p2 = 3.00 cm Hg; V2 = 0.240 cm3; Since the number of molecules stays constant and the temperature is not changed along the process, so Its V1 = p2 p1 ¢ V2 ¢ hence p1 = p2 ¢ V2=V1 = 3:00cmHg 0:240cm3=200cm3 ¢ Setting in the values yields p1 = 3.60 Did you notice that the units for the pressure are in cm Hg? You may wish to convert this to a more common unit, such as millimeters of mercury, atmospheres, or pascals. 3.60 10mm/1 cm = 3.60 10¡2 mm Hg 10¡3 Hg ¢ 10¡3 cm Hg. ¢ ¢ ¢ 198 3.60 ¢ 10¡3 Hg ¢ 1 atm/76.0 cm Hg = 4.74 10¡5 atm ¢ One way to experience this is to dive under water. There is air in your middle ear, which is normally at one atmosphere of pressure to balance the air outside your ear drum. The water will put pressure on the ear drum, thereby compressing the air in your middle ear. Divers must push air into the ear through their Eustacean tubes to equalize this pressure. Work
ed Example 63 pressure in the ear of a diver How deep would you have to dive before the air in your middle ear would be compressed to 75% of its initial volume? Assume for the beginning that the temperature of the sea is constant as you dive. Solution: First we write down the pressure as a function of height h: where we take for p0 the atmospheric pressure at height h = 0, ‰ is the density of water at 20 degrees Celsius 998.23 kg/m3, g = 9.81 ms¡2. p = p0 + ‰ h g ¢ ¢ As the temperature is constant, it holds for both heights h Now solving for h using the fact that V0 = (p0 + ‰gh) p0 ¢ Ve ¢ yields Ve=V0 = 0:75 h = (0:75 p0 ⁄ ¡ p0)=(‰g) Now, how far can the diver dive down before the membranes of his ear brake. Solution: As the result is negative, h determines the way he can dive down. h is given as roughly 2.6 m. In 1809, the French chemist Joseph-Louis Gay-Lussac investigated the relationship between the Pressure of a gas and its temperature. Keeping a constant volume, the pressure of a gas sample is directly proportional to the temperature. Attention, the temperature is measured in Kelvin! The mathematical statement is as follows: 199 p1=T1 = p2=T2 = constant p1;2 T1;2 : pressures (P a) : Temperatures (K) That means, that pressure divided by temperature is a constant. On the other hand, if we plot pressure versus temperature, the graph crosses 0 pressure for T = 0 K = -273.15 –C as shown in the following ﬂgure. That point is called the absolute Zero. That is where any motion of molecules, electrons or other particles stops. Figure 12.7: Pressure-temperature diagram for the ideal gas at constant volume. Worked Example 64 Gay-Lussac Suppose we have the following problem: A gas cylinder containing explosive hydrogen gas has a pressure of 50 atm at a temperature of 300 K. The cylinder can withstand a pressure of 500 atm before it bursts, causing a building-attening explosion. What is the maximum temperature the cylinder can withstand before bursting? Solution: Let’s rewrite this, identifying the variables: A gas
cylinder containing explosive hydrogen gas has a pressure of 50 atm (p1) at a temperature of 300 K (T1). The cylinder can withstand a pressure of 500 atm 200 (p2) before it bursts, causing a building-attening explosion. What is the maximum temperature the cylinder can withstand before bursting? Plugging in the known variables into the expression for the Gay-Lussac law yields we ﬂnd the answer to be 3000 K. T2 = p2=p1 T1 = 500atm=50atm ⁄ 300K = 3000K ⁄ The law of combining volumes was interpreted by the Italian chemist Amedeo Avogadro in 1811, using what was then known as the Avogadro hypothesis. We would now properly refer to it as Avogadro’s law: Equal volumes of gases under the same conditions of temperature and pressure contain equal numbers of molecules. This can be understood in the following. As in an ideal gas, all molecules are considered to be tiny particles with no spatial extension which collide elastically with each other. So, the kind of gas is irrelevant. Avogadro found that at room temperature, in atmospheric pressure the 1023 molecules or atoms, occupies the volume of 22.4 volume of a mol of a substance, i.e. 6.022 ¢ l. Figure 12.8: Two diﬁerent gases occupying the same volume under the same circumstances. Combination of the three empirical gas laws, described in the preceding three sections leads to the Ideal Gas Law which is usually written as: p ¢ V = n R T ¢ ¢ : pressure (P a) : Volume (m3) : number of mols (mol) p V n R : gas konstant (J=molK) : temperature (K) T 201 where p = pressure, V = volume, n = number of mols, T = kelvin temperature and R the ideal gas constant. The ideal gas constant R in this equation is known as the universal gas constant. It arises from a combination of the proportionality constants in the three empirical gas laws. The universal gas constant has a value which depends only upon the units in which the pressure and volume are measured. The best available value of the universal gas constant is: 8.3143510 Another value which is sometimes convenient is 0.08206 dm3 atm/mol K. R is related to the
molK or 8.3143510 kP adm molK J 3 Boltzmann-constant as: R = N0 kB ¢ where N0 is the number of molecules in a mol of a substance, i.e. 6.022 ¢ 1.308 10¡23 J/K is valid for one single particle. ¢ This ideal gas equation is one of the most used equations in daily life, which we show in the (12.1) 1023 and kB is following problem set: Worked Example 65 ideal gas 1 A sample of 1.00 mol of oxygen at 50 –C and 98.6 kPa occupies what volume? Solution: We solve the ideal gas equation for the volume V = nRT p and plug in the values n = 1, T = 273.15 + 50 K = 323.15 K and p = 98.6 ¢ yielding for the volume V = 0.0272 m3 = 27.2 dm3. 103 Pa, This equation is often used to determine the molecular masses from gas data. Worked Example 66 ideal gas 2 A liquid can be decomposed by electricity into two gases. In one experiment, one of the gases was collected. The sample had a mass of 1.090 g, a volume of 850 ml, a pressure of 746 torr, and a temperature of 25 –C. Calculate its molecular mass. Solution: To calculate the molecular mass we need the number of grams and the number of mols. We can get the number of grams directly from the information in the question. We can calculate the mols from the rest of the information and the ideal gas equation. 202 V = 850mL = 0:850L = 0:850dm3 P = 746torr=760torr = 0:982atm T = 25:0–C + 273:15 = 298:15K pV = nRT (0:982atm)(0:850L) = (n)(0:0821Latmmol 1K ¡ ¡ 1)(298:15K) n = 0:0341mol molecular mass = g/mol = 1.090 g/ 0.0341 mol = 31.96 g/mol. The gas is oxygen. Or the equation can be comfortably used to design a gas temperature controller: Worked Example 67 ideal gas 3 In a gas thermometer, the pressure needed to ﬂx the volume of 0
.20 g of Helium at 0.50 L is 113.3 kPa. What is the temperature? Solution: We transform ﬂrst need to ﬂnd the number of mols for Helium. Helium consists of 2 protons and 2 neutrons in the core (see later) and therefore has a molar volume of 4 g/mol. Therefore, we ﬂnd plugging this into the ideal gas equation and solving for the temperature T we ﬂnd: n = 0:20g=4g=mol = 0:05mol T = pV nR = 113:3 103P a 0:5 10¡3m3 ¢ 0:05mol ¢ ¢ 8:314J=molK = 136:3K ¢ The temperature is 136 Kelvin. 12.4.2 Kinetic theory of gasses The results of several experiments can lead to a scientiﬂc law, which describes then all experiments performed. This is an empirical, that is based on experience only, approach to Physics. A law, however, only describes results; it does not explain why they have been obtained. Significantly stronger, a theory is a formulation which explains the results of experiments. A theory usually bases on postulates, that is a proposition that is accepted as true in order to provide a basis for logical reasoning. The most famous postulate in Physics is probably the one formulated by Walter Nernst which states that if one could reach absolute zero, all bodies would have the same entropy. 203 The kinetic-molecular theory of gases is a theory of great explanatory power. We shall see how it explains the ideal gas law, which includes the laws of Boyle and of Charles; Dalton’s law of partial pressures; and the law of combining volumes. The kinetic-molecular theory of gases can be stated as four postulates: † † † † A gas consists of particles (atoms or molecules) in continuous, random motion. Gas molecules inuence each other only by collision; they exert no other forces on each other. All collisions between gas molecules are perfectly elastic; all kinetic energy is conserved. The average energy of translational motion of a gas particle is directly proportional to temperature. In addition to the postulates above, it is assumed that the volumes of the particles are negligible as compared to container volume. These postulates, which correspond to a physical model of a gas much like
a group of billiard balls moving around on a billiard table, describe the behavior of an ideal gas. At room temperatures and pressures at or below normal atmospheric pressure, real gases seem to be accurately described by these postulates, and the consequences of this model correspond to the empirical gas laws in a quantitative way. We deﬂne the average kinetic energy of translation Et of a particle in a gas as Et = 1=2 mv2 (12.2) ¢ where m is the mass of the particle with average velocity v. The forth postulate states that the average kinetic energy is a constant deﬂning the temperature, i.e. we can formulate Et = 1=2 ¢ mv2 = c T ¢ (12.3) where the temperature T is given in Kelvin and c is a constant, which has the same value for all gases. As we have 3 diﬁerent directions of motion and each possible movement gives kBT, we ﬂnd for the energy of a particle in a gas as Et = 1=2 ¢ mv2 = 3=2kBT = 3=2 R NA T (12.4) Hence, we can ﬂnd an individual gas particle’s speed rms = root mean square, which is the average square root of the speed of the individual particles (ﬂnd u) r where Mmol is the molar mass, i.e. the mass of the particle m times the Avogadro number vrms = 3RT Mmol (12.5) NA. Worked Example 68 kinetic theory 1 204 Calculate the root-mean-square velocity of oxygen molecules at room temperature, 25 –C. Solution: Using vrms = 3RT =Mmol ; the molar mass of molecular oxygen is 31.9998 g/mol; the molar gas constant has the value 8.3143 J/mol K, and the temperature is 298.15 K. Since the joule is the s¡2, the molar mass must be expressed as 0.0319998 kg/mol. The root-meankg ¢ square velocity is then given by: m2 ¢ p vrms = 3(8:3143)(298:15)=(0:0319998) = 482:1m=s A speed of 482.1 m/s is 1726 km/h, much faster than
a jetliner can y and faster than most rie bullets. p The very high speed of gas molecules under normal room conditions would indicate that a gas molecule would travel across a room almost instantly. In fact, gas molecules do not do so. If a small sample of the very odorous (and poisonous!) gas hydrogen sulﬂde is released in one corner of a room, our noses will not detect it in another corner of the room for several minutes unless the air is vigorously stirred by a mechanical fan. The slow diﬁusion of gas molecules which are moving very quickly occurs because the gas molecules travel only short distances in straight lines before they are deected in a new direction by collision with other gas molecules. The distance any single molecule travels between collisions will vary from very short to very long distances, but the average distance that a molecule travels between collisions in a gas can be calculated. This distance is called the mean free path l of the gas molecules. If the rootmean-square velocity is divided by the mean free path of the gas molecules, the result will be the number of collisions one molecule undergoes per second. This number is called the collision frequency Z1 of the gas molecules. The postulates of the kinetic-molecular theory of gases permit the calculation of the mean free path of gas molecules. The gas molecules are visualized as small hard spheres. A sphere (d=2)2 and length vrms each of diameter d sweeps through a cylinder of cross-sectional area … second, colliding with all molecules in the cylinder. ¢ The radius of the end of the cylinder is d because two molecules will collide if their diameters overlap at all. This description of collisions with stationary gas molecules is not quite accurate, however, because the gas molecules are all moving relative to each other. Those relative velocities range between zero for two molecules moving in the same direction and 2vrms for a head-on collision. The average relative velocity is that of a collision at right angles, which is p2vrms. The total number of collisions per second per unit volume, Z1, is Z1 = …d2p2vrms (12.6) This total number of collisions must now be divided by the number of molecules which are present per unit volume. The number of gas molecules present per unit volume is found by rearrangement of the ideal gas law to n=V = p=RT and use of Avogadro’s number, n
= N=NA; thus N=V = pNA=RT. This gives the mean free path of the gas molecules, l, as (urms=Z1)=(N=V ) = l = RT =…d2pNAp2 (12.7) 205 According to this expression, the mean free path of the molecules should get longer as the temperature increases; as the pressure decreases; and as the size of the molecules decreases. Worked Example 69 mean free path Calculate the length of the mean free path of oxygen molecules at room temperature, 25 –C, taking the molecular diameter of an oxygen molecule as 370 pm. Solution: Using the formula for mean free path given above and the value of the root-meansquare velocity urms, l = …(370 ¢ (8:3143kgm2s¡2=Kmol)(298:15K) 10¡12m)2(101325kg=ms2)(6:0225 1023mol¡1)p2 ¢ ; 10¡8 m = 67 nm. so l = 6.7 ¢ The apparently slow diﬁusion of gas molecules takes place because the molecules travel only a very short distance before colliding. At room temperature and atmospheric pressure, oxygen 10¡12 m) = 180 molecular diameters between collisions. 10¡8 m)/(370 molecules travel only (6.7 ¢ ¢ The same thing can be pointed out using the collision frequency for a single molecule Z1, which is the root-mean-square velocity divided by the mean free path: Z1 = …d2pNAp2 =RT = vrms=l (12.8) For oxygen at room temperature, each gas molecule collides with another every 0.13 nanosec10+9 collisions per second 10¡9 s), since the collision frequency is 7.2 onds (one nanosecond is 1.0 ¢ ¢ per molecule. For an ideal gas, the number of molecules per unit volume is given using pV = nRT and n = N=NA as N=V = NAp=RT (12.9) which for oxygen at 25 –C would be (6.022 1023 mol¡1)(101325 kg/m s2) / (8.3143 kg m2/s2 ¢ 1025 molecules/m3. The number of collisions between two molecules
K mol)(298.15 K) or 2.46 ¢ in a volume, Z11, would then be the product of the number of collisions each molecule makes times the number of molecules there are, Z1N=V, except that this would count each collision twice (since two molecules are involved in each one collision). The correct equation must be Z11 = …d2p2N 2 Ap2vrms 2R2T 2 (12.10) If the molecules present in the gas had diﬁerent masses they would also have diﬁerent speeds, so an average value of vrms would be using a weighted average of the molar masses; the partial pressures of the diﬁerent gases in the mixture would also be required. Although such calculations involve no new principles, they are beyond our scope. 206 12.4.3 Pressure of a gas In the kinetic-molecular theory of gases, pressure is the force exerted against the wall of a container by the continual collision of molecules against it. From Newton’s second law of motion, the force exerted on a wall by a single gas molecule of mass m and velocity v colliding with it is: F = m a = m ¢ ¢v ¢t (12.11) In the above equation, the change in a quantity is indicated by the symbol ¢, that means by changing the time t by a fraction, we change the velocity v by some other minimal amount. It is assumed that the molecule rebounds elastically and no kinetic energy is lost in a perpendicular collision, so ¢v = v - (-v) = 2v (see ﬂgure below). If the molecule is moving perpendicular to the wall it will strike the opposite parallel wall, rebound, and return to strike the original wall again. If the length of the container or distance between the two walls is the path length l, then the time between two successive collisions on the same wall is ¢t = 2l/v. The continuous force which the molecule moving perpendicular to the wall exerts is therefore Figure 12.9: Change in momentum as a particle hits a wall. F = m 2v 2l=v = mv2 l (12.12) The molecules in a sample of gas are not, of course, all moving perpendicularly to a wall, but the components of their actual movement can be considered to be along the three mutually perpendicular x, y, and z axes.
If the number of molecules moving randomly, N, is large, then on the average one-third of them can be considered as exerting their force along each of the three perpendicular axes. The square of the average velocity along each axis, v2(x), v2(y), or v2(z), will be one-third of the square of the average total velocity v2: v2(x) = v2(y) = v2(z) = v2=3 (12.13) The average or mean of the square of the total velocity can replace the square of the perpen- dicular velocity, and so for a large number of molecules N, Since pressure is force per unit area, and the area of one side of a cubic container must be l2, the pressure p will be given by F=l2 as: F = (N=3) mv2 l (12.14) 207 This equation rearranges to p = (N=3) mv2 l3 (12.15) pV = N mv2=3 (12.16) ¢ because volume V is the cube of the length l. The form of the ideal gas law given above shows the pressure-volume product is directly proportional to the mean-square velocity of the gas molecules. If the velocity of the molecules is a function only of the temperature, and we shall see in the next section that this is so, the kinetic-molecular theory gives a quantitative explanation of Boyle’s law. Worked Example 70 gas pressure A square box contains He (Helium) at 25 –C. If the atoms are colliding with the walls 1022 times per second, calculate the force perpendicularly (at 90–) at the rate of 4.0 ¢ (in Newtons) and the pressure exerted on the wall per mol of He given that the area of the wall is 100 cm2 and the speed of the atoms is 600 ms¡1. Solution: We use the equation 12.14 to calculate the force. mv2 l The fraction v=l is the collision frequency Z1 = 0.6679 s¡1. The product of N Z1 is the number of molecules impinging on the wall per second. This induces for the force: = (N=3)mv F = (N=3) v l ¢ F = (N=3)mv¿ = 6:022 1023
=3 ¢ ¢ 0:004g=mol 6:022 1023 ¢ ¢ 600m=s ¢ 0:6679s¡1 yielding for the force F = 0.534 N. The pressure is the force per area: p = F=A = 0:534N=0:01m2 = 53:4P a: The calculated force is 0.534 N and the resulting pressure is 53.4 Pa. 12.4.4 Kinetic energy of molecules In the following, we will make the connection between the kinetic theory and the ideal gas laws. We will ﬂnd that the temperature is an important quantity which is the only intrinsic parameter entering in the kinetic energy of a gas. We will consider an ensemble of molecules in a gas, where the molecules will be regarded as rigid large particles. We therefore neglect any vibrations or rotations in the molecule. Hence, making this assumption, Physics for a molecular gas is the same as for a single atom gas. 208 The square of the velocity is sometimes di–cult to conceive, but an alternative statement can be given in terms of kinetic energy. The kinetic energy Ek of a single particle of mass m moving at velocity v is mv2=2. For a large number of molecules N, the total kinetic energy Ek will depend on the mean-square velocity in the same way: mv2=2 = n The second form is on a molar basis, since n = N=NA and the molar mass M = mNA where M v2=2 Ek = N (12.17) ¢ ¢ NA is Avogadro’s number 6.022 1023. The ideal gas law then appears in the form: ¢ (12.18) Compare pV = nM v2=2. This statement that the pressure-volume product of an ideal gas is directly proportional to the total kinetic energy of the gas is also a statement of Boyle’s law, since the total kinetic energy of an ideal gas depends only upon the temperature. pV = 2Ek=3 Comparison of the ideal gas law, pV = nRT, with the kinetic-molecular theory expression pV = 2Ek=3 derived in the previous section shows that the total kinetic energy of a collection of gas molecules is directly proportional to the absolute temperature of the gas. Equating the pV term of both equations gives which rearranges to an explicit expression
for temperature, Ek = 3=2nRT ; T = 2 3R Ek n = M v2 3R (12.19) (12.20) We see that temperature is a function only of the mean kinetic energy Ek, the mean molecular velocity v, and the mean molar mass M. Worked Example 71 mean velocity 1 Calculate the kinetic energy of 1 mol of nitrogen molecules at 300 K? Solution: Assume nitrogen behaves as an ideal gas, then Ek = 3=2 ¢ RT = (3=2)8:3145J=(molK) ¢ 300K = 3742J=mol(or3:74kJ=mol) At 300 K, any gas that behaves like an ideal gas has the same energy per mol. As the absolute temperature decreases, the kinetic energy must decrease and thus the mean velocity of the molecules must decrease also. At T = 0, the absolute zero of temperature, all motion of gas molecules would cease and the pressure would then also be zero. No molecules would be moving. Experimentally, the absolute zero of temperature has never been attained, although modern experiments have extended to temperatures as low as 1 „K. However, at low temperatures, the interactions between the particles becomes important and we enter a new regime of Quantum Mechanics, which considers molecules, single atoms or protons and electrons simultaneously as waves and as rigid particles. However, this would go too far. 209 Worked Example 72 mean velocity 2 If the translational rms. speed of the water vapor molecules (H2O) in air is 648 m/s, what is the translational rms speed of the carbon dioxide molecules (CO2) in the same air? Both gases are at the same temperature. And what is the temperature we measure? Solution: The molar mass of H2O is ¢ As the temperature is constant we can write ¢ MH2O = 2 1g=mol + 1 16g=mol = 18g=mol T = M v2 3R = 0:018kg=mol (648m=s)2 K ¢ ¢ 3 8:314J=mol ¢ Now we calculate the molar mass of CO2 = 303:0K = 29:9–C MCO2 = 2 ¢ 16g=mol + 1 ¢ 12g=mol = 44g=mol The rms velocity is again calculated with eq. 12.20 vCO2 = 3 R T
¢ MCO2 s 3 ¢ = s 8:314J=molK 303:0K 0:044kg=mol ¢ = 414:5m=s The experiment was performed at 29.9 –C and the speed of the CO2-molecules is 414.5 m/s, that is much slower than the water molecules as they are much heavier. 12.5 Temperature Let us look back to the equation for the temperature of an ideal gas, T = 2 3R We can see that temperature is proportional to the average kinetic energy of a molecule in the gas. In other words temperature is a measure of how much energy is contained in an object { in hot things the atoms have a lot of kinetic energy, in cold things they have less. It may be surprising that ‘hot’ and ‘cold’ are really just words for how fast molecules or atoms are moving around, but it is true. (12.21) Ek n Deﬂnition: Temperature is a measure of the average kinetic energy of the particles in a body. It should now be clear that heat is nothing more than energy on the move. It can be carried 210 ice water Figure 12.10: A heat ow diagram showing the heat owing from the warmer water into the cooler ice cube. by atoms, molecules or electromagnetic radiation but it is always just transport of energy. This is very important when we describe movement of heat as we will do in the following sections. ‘Cold’ is not a physical thing. It does not move from place to place, it is just the word for a lack of heat, just like dark is the word for an absence of light. 12.5.1 Thermal equilibrium Now that we have deﬂned the temperature of an isolated object (usually referred to as a body) we need to consider how heat will move between bodies at diﬁerent temperatures. Let us take two bodies; A which has a ﬂxed temperature and B whose temperature is allowed to change. If we allow heat to move between the two bodies we say they are in thermal contact. First let us consider what happens if B is cooler than A. Remember { we have ﬂxed the temperture of A so we need only worry about the temperature of B changing. An example of such a situation is an ice cube being dropped into a large pan of boiling water on a ﬂre. The water temperature is �
�xed i.e. does not change, because the ﬂre keeps it constant. It should be obvious that the ice cube will heat up and melt. In physical terms we say that the heat is owing out of the (warmer) boiling water, into the (cooler) ice cube. This ow of heat into the ice cube causes it to warm up and melt. In fact the temperature of any cooler object in thermal contact with a warmer one will increase as heat from the warmer object ows into it. The reverse would be true if B were warmer than A. We can now picture putting a small amount of warm water in to a freezer. If we come back in an hour or so the water will have cooled down and possibly frozen. In physical terms we say that the heat is owing out of the (warmer) water, into the (cooler) air in the freezer. This ow of heat out of the ice cube in to the aircauses it to cool down and (eventually) freeze. Again, any warm object in thermal contact with a cooler one will cool down due to heat owing out of it. There is one special case which we have not yet discussed { what happens if A and B are at the same temperature? In this case B will neither warm up nor cool down, in fact, its temperature will remain constant. When two bodies are at the same temperature we say that they are in thermal equilibrium. Another way to express this is to say that two bodies are in thermal equilibrium if the particles within those bodies have the same average kinetic energies. 211 water air Figure 12.11: A heat ow diagram showing the heat owing from the warmer water into the cooler air in the fridge. Convince yourself that the last three paragraghs are correct before you continue. You should notice that heat always ows from the warmer object to the cooler object, never the other way around. Also, we never talk about coldness moving as it is not a real physical thing, only a lack of heat. Most importantly, it should be clear that the ow of heat between the two objects always attempts to bring them to the same temperature (or in other words, into thermal equilibrium). The logical conclusion of all this is that if two bodies are in thermal contact heat will ow from the hotter object to the cooler one until they are in thermal equilibrium (i.e. at the same temperature). We will see how to deal with this if the temperature of object A is not �
��xed in the section on heat capacities. 12.5.2 Temperature scales Temperature scales are often confusing and even university level students can be tricked into using the wrong one. For most purposes in physics we do not use the familiar celcius (often innaccurately called centigrade) scale but the closely related absolute (or kelvin) scale { why? Let us think about the celcius scale now that we have deﬂned temperature as a measure of the average kinteic energy of the atoms or molecules in a body. A scale is a way of assigning a number to a physical quantity. Consider distance { using a ruler we can measure a distance and ﬂnd its legnth. This legnth could be measured in metres, inches, or miles. The same is true of temperatures in that many diﬁerent scales exist to measure them. Table 12.3 shows a few of these scales. Just like a ruler the scales have two deﬂned points which ﬂx the scale (consider the values at the beginning and end of the ruler e.g. 0cm and 15cm). This is usually achieved by deﬂning the temperature of some physical process, e.g. the freezing point of water. Armed with our knowledge of temperture we can see that Celcius’s scale has a big problem { it allows us to have a negative temperature. 212 Scale Fahrenheit Symbol Deﬂnition –F Temperature at which an equal mixture of ice and salt melts = 0–F Temperature of blood = 96–F Celcius –C Temperature at which water freezes = 0–C Temperature at which water boils = 100–C Kelvin K Absolute zero is 0 K Triple point of water is 273:16 K Table 12.3: The most important temperature scales. We found that temperature is a measure of the average kinetic energy of the particles in a body. Therefore, a negative temperature suggests that that the particles have negative kinetic energy. This can not be true as kinetic energy can only be positive. Kelvin addressed this problem by redeﬂning the zero of the scale. He realised that the coldest temperature you could achieve would be when the particles in a body were not moving at all. There is no way to cool something further than this as there is no more kinetic energy to remove from the body. This temperature is called absolute zero. Kelvin chose his scale so that 0K was the same as absolute
zero and chose the size of his degree to be the same as one degree in the celcius scale. Interesting Fact: Rankine did a similar thing to Kelvin but set his degree to be the same size as one degree fahrenheit. Unfortunately for him, almost everyone preferred Kelvin’s absolute scale and the rankine scale is now hardly ever used! It turns out that the freezing point of water, 0–C, is equal to 273:15 K. So, in order to convert from celcius to kelvin need to subtact 273.15. Deﬂnition: T (K) = T (–C) 273:15 ¡ 12.5.3 Practical thermometers It is often important to be able to determine an object’s temperature precisely. This can be a challenge at very high or low tempertures or in inaccesible places. Consider a scientist who wishes to know how hot the magma in a volcano is. They are not going to be able to just lower a thermometer in to the magma as it will just melt as it reaches the superheated rock. We will now look at some less extreme situations and show how a variety of thermometry techniques can be developed. Consider ﬂrst the gas cylinder which we tried to explode in worked example 7 by heating it while sealed. We decided that we would need to heat it to around 3000K before it explodes. How can we check this experimentally? In a sealed gas cylinder the volume of the gas and the number of moles of gas remain constant as we heat, this is why we could use the Gay-Lussac law in example 7. The Gay-Lussac law tells us that pressure is directly proportional 213 to temperature for ﬂxed volume and amount of gas. Therefore by mesuring the pressure in the cylinder (which can be done by ﬂtting a pressure gauge to the top of it) we can indirectly work out the temperature. This is similar to familiar alcohol or mercury thermometers. In these we use the fact that expansion of a liquid as it is heated is approximately proportional to temperature so we can use this expansion to as a measure of temperature. In fact, any thermometer you can imagine uses some physical property which varies with temperature to measure it indirectly. 12.5.4 Speciﬂc heat capacity Conversion of macroscopic energy to microscopic kinetic energy thus tends to raise the temperature, while the reverse conversion
lowers it. It is easy to show experimentally that the amount of heating needed to change the temperature of a body by some amount is proportional to the amount of matter in the body. Thus, it is natural to write ¢Q = M C¢T (23.4) where M is the mass of material, ¢Q is the amount of energy transferred to the material, and ¢T is the change of the material’s temperature. The quantity C is called the speciﬂc heat of the material in question and is the amount of energy needed to raise the temperature of a unit mass of material one degree in temperature. C varies with the type of material. Values for common materials are given in table 22.2. Table 22.2: Speciﬂc heats of common materials. Material C (J kg¡1 K¡1) brass 385 glass 669 ice 2092 steel 448 methyl alcohol 2510 glycerine 2427 water 4184 12.5.5 Speciﬂc latent heat It can be seen that the speciﬂc heat as deﬂned above will be inﬂnitely large for a phase change, where heat is transferred without any change in temperature. Thus, it is much more useful to deﬂne a quantity called latent heat, which is the amount of energy required to change the phase of a unit mass of a substance at the phase change temperature. 12.5.6 Internal energy In thermodynamics, the internal energy is the energy of a system due to its temperature. The statement of ﬂrst law refers to thermodynamic cycles. Using the concept of internal energy it is possible to state the ﬂrst law for a non-cyclic process. Since the ﬂrst law is another way of stating the conservation of energy, the energy of the system is the sum of the heat and work input, i.e., E = Q + W. Here E represents the heat energy of the system along with the kinetic energy and the potential energy (E = U + K.E. + P.E.) and is called the total internal energy of the system. This is the statement of the ﬂrst law for non-cyclic processes. For gases, the value of K.E. and P.E. is quite small, so the important internal energy function is U. In particular, since for an
ideal gas the state can be speciﬂed using two variables, the state variable u is given by, where v is the speciﬂc volume and t is the temperature. Thus, by deﬂnition,, where cv is the speciﬂc heat at constant volume. 214 Internal energy of an Ideal gas In the previous section, the internal energy of an ideal gas was shown to be a function of both the volume and temperature. Joule performed an experiment where a gas at high pressure inside a bath at the same temperature was allowed to expand into a larger volume. picture required In the above image, two vessels, labeled A and B, are immersed in an insulated tank containing water. A thermometer is used to measure the temperature of the water in the tank. The two vessels A and B are connected by a tube, the ow through which is controlled by a stop. Initially, A contains gas at high pressure, while B is nearly empty. The stop is removed so that the vessels are connected and the ﬂnal temperature of the bath is noted. The temperature of the bath was unchanged at the and of the process, showing that the internal energy of an ideal gas was the function of temperature alone. Thus Joule’s law is stated as = 0. 12.5.7 First law of thermodynamics We now address some questions of terminology. The use of the terms \heat" and \quantity of heat" to indicate the amount of microscopic kinetic energy inhabiting a body has long been out of favor due to their association with the discredited \caloric" theory of heat. Instead, we use the term internal energy to describe the amount of microscopic energy in a body. The word heat is most correctly used only as a verb, e. g., \to heat the house". Heat thus represents the transfer of internal energy from one body to another or conversion of some other form of energy to internal energy. Taking into account these deﬂnitions, we can express the idea of energy conservation in some material body by the equation ¢E = ¢Q ¡ ¢W (ﬂrst law of thermodynamics) where ¢E is the change in internal energy resulting from the addition of heat ¢Q to the body and the work ¢W done by the body on the outside world. This equation expresses the ﬂrst law of thermodynamics. Note that the sign conventions are inconsistent as
to the direction of energy ow. However, these conventions result from thinking about heat engines, i. e., machines which take in heat and put out macroscopic work. Examples of heat engines are steam engines, coal and nuclear power plants, the engine in your automobile, and the engines on jet aircraft. 12.6 Important Equations and Quantities Quantity Symbol Unit S.I. Units Direction Units or Table 12.4: Units used in Electricity and Magnetism 215 Chapter 13 Electrostatics 13.1 What is Electrostatics? Electrostatics is the study of electric charge which is not moving i.e. is static. 13.2 Charge All objects surrounding us (including people!) contain large amounts of electric charge. Charge can be negative or positive and is measured in units called coulombs (C). Usually, objects contain the same amount of positive and negative charge so its eﬁect is not noticeable and the object is called electrically neutral. However, if a small imbalance is created (i.e. there is a little bit more of one type of charge than the other on the object) then the object is said to be electrically charged. Some rather amusing examples of what happens when a person becomes charged are for example when you charge your hair by combing it with a plastic comb and it stands right up on end! Another example is when you walk fast over a nylon carpet and then touch a metal doorknob and give yourself a small shock (alternatively you can touch your friend and shock them!) Charge has 3 further important properties: † † † Charge is always conserved. Charge, just like energy, cannot be created or destroyed. Charge comes in discrete packets. The smallest unit of charge is that carried by one electron called the elementary charge, e, and by convention, it has a negative sign (e = 10¡19C). 1:6 ¡ £ Charged objects exert electrostatic forces on each other. Like charges repel and unlike charges attract each other. Interesting Fact: The word ‘electron’ comes from the Greek word for amber! The ancient Greeks observed that if you rubbed a piece of amber, you could use it to pick up bits of straw. (Attractive electrostatic force!) 216 You can easily test the fact that like charges repel and unlike charges attract by doing a very simple experiment. Take a glass rod and rub it with a piece of silk, then hang it from its middle with a piece string so that
it is free to move. If you then bring another glass rod which you have also charged in the same way next to it, you will see the rod on the string turn away from the rod in your hand i.e. it is repelled. If, however, you take a plastic rod, rub it with a piece of fur and then bring it close to the rod on the string, you will see the rod on the string turn towards the rod in your hand i.e. it is attracted! ////////// ////////// + + + + + + - - - - What actually happens is that when you rub the glass with silk, tiny amounts of negative charge are transferred from the glass onto the silk, which causes the glass to have less negative charge than positive charge, making it positively charged. When you rub the plastic rod with the fur, you transfer tiny amounts of negative charge onto the rod and so it has more negative charge than positive charge on it, making it negatively charged. Conductors and Insulators Some materials allow charge carriers to move relatively freely through them (e.g. most metals, tap water, the human body) and these materials are called conductors. Other materials, which do not allow the charge carriers to move through them (e.g. plastic, glass), are called nonconductors or insulators. Aside: As mentioned above, the basic unit of charge, namely the elementary charge, e, is carried by the electron. In a conducting material (e.g. copper), when the atoms bond to form the material, some of the outermost, loosely bound electrons become detached from the individual atoms and so become free to move around. The charge carried by these electrons can move around in the material! In insulators, there are very few, if any, free electrons and so the charge cannot move around in the material. 217 If an excess of charge is put onto an insulator, it will stay where it is put and there will be a concentration of charge in that area on the object. However, if an excess of charge is put onto a conductor, the charges of like sign will repel each other and spread out over the surface of the object. When two conductors are made to touch, the total charge on them is shared between the two. If the two conductors are identical, then each conductor will be left with half of the total charge. 13.3 Electrostatic Force As we now know, charged objects exert a force on one another. If the charges are at
rest then this force between them is known as the electrostatic force. An interesting characteristic of the electrostatic force is that it can be either attractive or repulsive, unlike the gravitational force which is only ever attractive. The relative charges on the two objects is what determines whether the force between the charged objects is attractive or repulsive. If the objects have opposite charges they attract each other, while if their charges are similar they repel each other (e.g. two metal balls which are negatively charged will repel each other, while a positively charged ball and negatively charged ball will attract one another). F F - - F F - + It is this force that determines the arrangement of charge on the surface of conductors. When we place a charge on a spherical conductor the repulsive forces between the individual like charges cause them to spread uniformly over the surface of the sphere. However, for conductors with non-regular shape there is a concentration of charge near the point or points of the object. +++++ + + + + + + +++++ ----- - - - - - - - - ---- This collection of charge can actually allow charge to leak oﬁ the conductor if the point is sharp enough. It is for this reason that buildings often have a lightning rod on the roof to remove any charge the building has collected. This minimises the possibility of the building being struck by lightning. This \spreading out" of charge would not occur if we were to place the charge on an insulator since charge cannot move in insulators. 13.3.1 Coulomb’s Law The behaviour of the electrostatic force was studied in detail by Charles Coulomb around 1784. Through his observations he was able to show that the electrostatic force between two point-like 218 charges is inversely proportional to the square of the distance between the objects. He also discovered that the force is proportional to the product of the charges on the two objects. F / Q1Q2 r2 ; where Q1 is the charge on the one point-like object, Q2 is the charge on the second, and r is the distance between the two. The magnitude of the electrostatic force between two point-like charges is given by Coulomb’s Law: F = k Q1Q2 r2 (13.1) and the proportionality constant k is called the electrostatic constant. We will use the value k = 8:99 109N ¢ £ m2=C2: The value of the electro
static constant is known to a very high precision (9 decimal places). Not many physical constants are known to as high a degree of accuracy as k. Aside: Notice how similar Coulomb’s Law is to the form of Newton’s Universal Law of Gravitation between two point-like particles: FG = G m1m2 r2 ; where m1 and m2 are the masses of the two particles, r is the distance between them, and G is the gravitational constant. It is very interesting that Coulomb’s Law has been shown to be correct no matter how small the distance, nor how large the charge: for example it still applies inside the atom (over distances smaller than 10¡10m). Let’s run through a simple example of electrostatic forces. Worked Example 73 Coulomb’s Law I 10¡9C Question: Two point-like charges carrying charges of +3 are 2m apart. Determine the magnitude of the force between them and state whether it is attractive or repulsive. Answer: Step 1 : (NOTE TO SELF: step is deprecated, use westep instead.) First draw the situation: +3 10¡9C and 10¡9C 10¡9C 5 ¡ £ £ 5 ¡ £ £ ¢ ¢ 2m Step 2 : (NOTE TO SELF: step is deprecated, use westep instead.) Is everything in the correct units? Yes, charges are in coulombs [C] and distances in meters [m]. 219 Step 3 : (NOTE TO SELF: step is deprecated, use westep instead.) Determine the magnitude of the force: Using Coulomb’s Law we have F = k Q1Q2 r2 = (8:99 £ = 3:37 ¡ £ 109N m2=C2) ¢ 10¡8N (+3 £ 10¡9C)( 5 ¡ (2m)2 £ 10¡9C) Thus the magnitude of the force is 3:37 two point charges having opposite signs. Step 4 : (NOTE TO SELF: step is deprecated, use westep instead.) Is the force attractive or repulsive? Well, since the two charges are oppositely charged, the force is attractive. We can also conclude this from the fact that Coulomb’s Law gives a negative value for the force. 10¡8N. The minus sign is a result of the £
Next is another example that demonstrates the diﬁerence in magnitude between the gravita- tional force and the electrostatic force. Worked Example 74 Coulomb’s Law II Question: Determine the electrostatic force and gravitational force between two electrons 1”Aapart (i.e. the forces felt inside an atom) Answer: Step 1 : (NOTE TO SELF: step is deprecated, use westep instead.) £ First draw the situation: e 10¡19C 10¡19C 1:60 1:60 £ e ¡ ¡ 1”A Step 2 : (NOTE TO SELF: step is deprecated, use westep instead.) Get everything into S.I. units: The charge on an electron is 10¡31kg, and 1”A=1 of an electron is 9:11 Step 3 : (NOTE TO SELF: step is deprecated, use westep instead.) Calculate the electrostatic force using Coulomb’s Law: 10¡10m 1:60 £ £ ¡ £ 10¡19C, the mass FE = k e e ¢ 1”A2 Q1Q2 r2 = k 109N £ ¢ 10¡8N = (8:99 = 2:30 £ m2=C2) 1:60 ( ¡ £ 10¡19C)( 1:60 ¡ (10¡10m)2 10¡19C) £ 10¡8N. Hence the magnitude of the electrostatic force between the electrons is 2:30 (Note that the electrons carry like charge and from this we know the force must be repulsive. Another way to see this is that the force is positive and thus repulsive.) £ 220 Step 4 : (NOTE TO SELF: step is deprecated, use westep instead.) Calculate the gravitational force: FE = G m1m2 r2 = G me ¢ me (1”A)2 = (6:67 10¡11N ¢ £ m2=kg2) (9:11 £ 10¡31C)(9:11 (10¡10m)2 £ 10¡31kg) = 5:54 10¡51N £ The magnitude of the gravitational force between the electrons is 5:54 10¡51N £ Notice that the gravitational force between the electrons is much smaller than the electrostatic force. For this reason
, the gravitational force is usually neglected when determining the force between two charged objects. We mentioned above that charge placed on a spherical conductor spreads evenly along the surface. As a result, if we are far enough from the charged sphere, electrostatically, it behaves as a point-like charge. Thus we can treat spherical conductors (e.g. metallic balls) as point-like charges, with all the charge acting at the centre. Worked Example 75 Coulomb’s Law: Challenge Question Question: In the picture below, X is a small negatively charged sphere with a mass of 10kg. It is suspended from the roof by an insulating rope which makes an angle of 60o with the roof. Y is a small positively charged sphere which has the same magnitude of charge as X. Y is ﬂxed to the wall by means of an insulating bracket. Assuming the system is in equilibrium, what is the magnitude of the charge on X? /////////// 60o 10kg X { Y + 50cm n n n n n Answer: How are we going to determine the charge on X? Well, if we know the force between X and Y we can use Coulomb’s Law to determine their charges as we know the distance between them. So, ﬂrstly, we need to determine the magnitude of the electrostatic force between X and Y. Step 1 : (NOTE TO SELF: step is deprecated, use westep instead.) Is everything in S.I. units? The distance between X and Y is 50cm = 0:5m, and the mass of X is 10kg. 221 Step 2 : (NOTE TO SELF: step is deprecated, use westep instead.) Draw the forces on X (with directions) and label. T : tension from the thread 60o FE: electrostatic force X Fg: gravitational force Step 3 : (NOTE TO SELF: step is deprecated, use westep instead.) Determine the magnitude of the electrostatic force (FE). Since nothing is moving (system is in equlibrium) the vertical and horizonal components of the forces must cancel. Thus FE = T sin(60o); Fg = T sin(60o): The only force we know is the gravitational force Fg = mg. Now we can calculate the magnitude of T from above: T = Fg sin(60o) = (10kg)(10m=s2) sin(60o) = 115
5N: Which means that FE is: FE = T cos(60o) = 1154N cos(60o) = 577:5N ¢ Step 4 : (NOTE TO SELF: step is deprecated, use westep instead.) Now that we know the magnitude of the electrostatic force between X and Y, we can calculate their charges using Coulomb’s Law. Don’t forget that the magnitudes of the charges on X and Y are the same:. The magnitude of the electrostatic = j force is QX j QY j j FE = k j QX j j = = j = k Q2 X r2 QXQY r2 FEr2 k r 8:99 s (577:5N)(0:5m)2 109N £ 10¡4C ¢ m2=C2 = 1:27 Thus the charge on X is 1:27 ¡ £ 10¡4C £ 222 13.4 Electric Fields We have learnt that objects that carry charge feel forces from all other charged objects. It is useful to determine what the eﬁect of a charge would be at every point surrounding it. To do this we need some sort of reference. We know that the force that one charge feels due to another depends on both charges (Q1 and Q2). How then can we talk about forces if we only have one charge? The solution to this dilemna is to introduce a test charge. We then determine the force that would be exerted on it if we placed it at a certain location. If we do this for every point surrounding a charge we know what would happen if we put a test charge at any location. This map of what would happen at any point we call a ﬂeld map. It is a map of the electric It tells us how large the force on a test charge would be and in what ﬂeld due to a charge. direction the force would be. Our map consists of the lines that tell us how the test charge would move if it were placed there. 13.4.1 Test Charge This is the key to mapping out an electric ﬂeld. The equation for the force between two electric charges has been shown earlier and is: F = k Q1Q2 r2 : (13.2) If we want to map the ﬂeld for Q1 then we need to know exactly what would happen if we put Q
2 at every point around Q1. But this obviously depends on the value of Q2. This is a time when we need to agree on a convention. What should Q2 be when we make the map? By convention we choose Q2 = +1C. This means that if we want to work out the eﬁects on any other charge we only have to multiply the result for the test charge by the magnitude of the new charge. The electric ﬂeld strength is then just the force per unit of charge and has the same magnitude and direction as the force on our test charge but has diﬁerent units: E = k Q1 r2 (13.3) The electric ﬂeld is the force per unit of charge and hence has units of newtons per coulomb [N/C]. So to get the force the electric ﬂeld exerts we use: F = EQ (13.4) Notice we are just multiplying the electric ﬂeld magnitude by the magnitude of the charge it is acting on. 13.4.2 What do ﬂeld maps look like? The maps depend very much on the charge or charges that the map is being made for. We will start oﬁ with the simplest possible case. Take a single positive charge with no other charges around it. First, we will look at what eﬁects it would have on a test charge at a number of points. 223 Positive Charge Acting on Test Charge At each point we calculate the force on a test charge, q, and represent this force by a vector. +Q We can see that at every point the positive test charge, q, would experience a force pushing it away from the charge, Q. This is because both charges are positive and so they repel. Also notice that at points further away the vectors are shorter. That is because the force is smaller if you are further away. If the charge were negative we would have the following result. Negative Charge Acting on Test Charge -Q Notice that it is almost identical to the positive charge case. This is important { the arrows are the same length because the magnitude of the charge is the same and so is the magnitude of the test charge. Thus the magnitude of the force is the same. The arrows point in the opposite direction because the charges now have opposite sign and so the test charge is attracted to the charge. Now, to make things simpler, we draw
continuous lines showing the path that the test charge would travel. This means we don’t have to work out the magnitude of the force at many diﬁerent points. 224 Electric Field Map due to a Positive Charge +Q Some important points to remember about electric ﬂelds: There is an electric ﬂeld at every point in space surrounding a charge. Field lines are merely a representation { they are not real. When we draw them, we just pick convenient places to indicate the ﬂeld in space. Field lines always start at a right-angle (90o) to the charged object causing the ﬂeld. Field lines never cross! † † † † 13.4.3 Combined Charge Distributions We look at the ﬂeld of a positive charge and a negative charge placed next to each other. The net resulting ﬂeld would be the addition of the ﬂelds from each of the charges. To start oﬁ with let us sketch the ﬂeld maps for each of the charges as though it were in isolation. Electric Field of Negative and Positive Charge in Isolation +Q -Q 225 Notice that a test charge starting oﬁ directly between the two would be pushed away from the positive charge and pulled towards the negative charge in a straight line. The path it would follow would be a straight line between the charges. +Q -Q Now let’s consider a test charge starting oﬁ a bit higher than directly between the charges. If it starts closer to the positive charge the force it feels from the positive charge is greater, but the negative charge does attract it, so it would move away from the positive charge with a tiny force attracting it towards the negative charge. As it gets further from the positive charge the force from the negative and positive charges change and they are equal in magnitude at equal distances from the charges. After that point the negative charge starts to exert a stronger force on the test charge. This means that the test charge moves towards the negative charge with only a small force away from the positive charge. +Q -Q Now we can ﬂll in the other lines quite easily using the same ideas. The resulting ﬂeld map is: +Q -Q Two Like Charges I: The Positive Case For the case of two positive charges things look a little diﬁerent. We can’t just turn
the arrows around the way we did before. In this case the test charge is repelled by both charges. This tells us that a test charge will never cross half way because the force of repulsion from both charges will be equal in magnitude. 226 +Q +Q The ﬂeld directly between the charges cancels out in the middle. The force has equal magnitude and opposite direction. Interesting things happen when we look at test charges that are not on a line directly between the two. +Q +Q We know that a charge the same distance below the middle will experience a force along a reected line, because the problem is symmetric (i.e. if we ipped vertically it would look the same). This is also true in the horizontal direction. So we use this fact to easily draw in the next four lines. +Q +Q 227 Working through a number of possible starting points for the test charge we can show the electric ﬂeld map to be: +Q +Q Two Like Charges II: The Negative Case We can use the fact that the direction of the force is reversed for a test charge if you change the sign of the charge that is inuencing it. If we change to the case where both charges are negative we get the following result: -Q -Q 13.4.4 Parallel plates One very important example of electric ﬂelds which is used extensively is the electric ﬂeld between two charged parallel plates. In this situation the electric ﬂeld is constant. This is used for many practical purposes and later we will explain how Millikan used it to measure the charge on the electron. 228 Field Map for Oppositely Charged Parallel Plates + + + + + + + + + - - - - - - - - - This means that the force that a test charge would feel at any point between the plates would be identical in magnitude and direction. The ﬂelds on the edges exhibit fringe eﬁects, i.e. they bulge outwards. This is because a test charge placed here would feel the eﬁects of charges only on one side (either left or right depending on which side it is placed). Test charges placed in the middle experience the eﬁects of charges on both sides so they balance the components in the horizontal direction. This isn’t the case on the edges. The Force on a Test Charge between Oppositely
Charged Parallel Plates + + + + + + + + + - - - - - - - - - 13.4.5 What about the Strength of the Electric Field? When we started making ﬂeld maps we drew arrows to indicate the strength of the ﬂeld and the direction. When we moved to lines you might have asked \Did we forget about the ﬂeld strength?". We did not. Consider the case for a single positive charge again: 229 A m mg h B Figure 13.1: A mass under the inuence of a gravitational ﬂeld. +Q Notice that as you move further away from the charge the ﬂeld lines become more spread out. In ﬂeld map diagrams the closer ﬂeld lines are together the stronger the ﬂeld. This brings us to an interesting case. What is the electric ﬂeld like if the object that is charged has an irregular shape. 13.5 Electrical Potential 13.5.1 Work Done and Energy Transfer in a Field When a charged particle moves in an electric ﬂeld work is done and energy transfers take place. This is exactly analogous to the case when a mass moves in a gravitational ﬂeld such as that set up by any massive object. Work done by a ﬂeld Gravitational Case A mass held at a height h above the ground has gravitational potential energy since, if released, it will fall under the action of the gravitational ﬂeld. Once released, in the absence of friction, only the force of gravity acts on the mass and the mass accelerates in the direction of the force (towards the earth’s centre). In this way, work is done by the ﬂeld. When the mass 230 + + + + + + + A +Q QE s B - - - - - - - Figure 13.2: A charged particle under the inuence of an electric ﬂeld. falls a distance h (from point A to B), the work done is, W = F s = mgh In falling, the mass loses gravitational potential energy and gains kinetic energy. The work done by the ﬂeld is equal to the energy transferred, Energy is conserved! W = Gain in Ek = Loss in Ep (a falling mass) Electrical Case A charge in an electric ﬂeld has
electrical potential energy since, if released, it will move under the action of the electric ﬂeld. When released, in the absence of friction, only the electric force acts on the charge and the charge accelerates in the direction of the force (for positive charges the force and acceleration are in the direction of the electric ﬂeld, while negative charges experience a force and acceleration in the opposite direction to the electric ﬂeld.) Consider a positive charge +Q placed in the uniform electric ﬂeld between oppositely charged parallel plates. The positive charge will be repelled by the positive plate and attracted by the negative plate (i.e. it will move in the direction of the electric ﬂeld lines). In this way, work is done by the ﬂeld. In moving the charge a distance s in the electric ﬂeld, the work done is, W = F s = QEs since E = F Q : In the process of moving, the charge loses electrical potential energy and gains kinetic energy. The work done by the ﬂeld is equal to the energy transferred, W = Gain in Ek = Loss in Ep (charge moving under the inuence of an electric ﬂeld) Work done by us Gravitational Case In order to return the mass m in Fig.13.1 to its original position (i.e. lift it a distance h from B back to A) we have to apply a force mg to balance the force of gravity. An amount of work mgh is done by the lifter. In the process, the mass gains gravitational potential energy, mgh = Gain in Ep (lifting a mass) 231 Electrical Case In order to return the charge in Fig.13.2 to its original position (i.e. from B back to A) we have to exert a force QE on the charge to balance the force exerted on it by the electric ﬂeld. An amount of work QEs is done by us. In the process, the charge gains electrical potential energy, Energy is conserved! QEs = Gain in Ep (charge moved against an electric ﬂeld) In summary, when an object moves under the inuence of a ﬂeld, the ﬂeld does work and potential energy is transferred into kinetic energy. Potential energy is lost, while kinetic energy is gained. When an object is moved against a �
started at rest, the gain in kinetic energy is the ﬂnal kinetic energy, Eat A k = 400 J 13.5.2 Electrical Potential Diﬁerence Consider a positive test charge +Q placed at A in the electric ﬂeld of another positive point charge. + +Q A B The test charge moves towards B under the inuence of the electric ﬂeld of the other charge. In the process the test charge loses electrical potential energy and gains kinetic energy. Thus, at A, the test charge has more potential energy than at B { A is said to have a higher electrical potential than B. The potential energy of a charge at a point in a ﬂeld is deﬂned as the work required to move that charge from inﬂnity to that point. The potential diﬁerence between two points in an electric ﬂeld is deﬂned as the work required to move a unit positive test charge from the point of lower potential to that of higher potential. If an amount of work W is required to move a charge Q from one point to another, then the potential diﬁerence between the two points is given by, V = W Q unit : J:C¡1 or V (the volt) 233 From this equation it follows that one volt is the potential diﬁerence between two points in an electric ﬂeld if one joule of work is done in moving one coulomb of charge from the one point to the other. Worked Example 77 Potential diﬁerence Question: A positively charged object Q is placed as shown in the sketch. The potential diﬁerence between two points A and B is 4 10¡4 V. £ +Q A B (a) Calculate the change in electrical potential energy of a +2nC charge when it moves from A to B. (b) Which point, A or B, is at the higher electrical potential? Explain. (c) If this charge were replaced with another of charge -2nC, in what way would its change in energy be aﬁected? Answer: (a) The electrical potential energy of the positive charge decreases as it moves from A to B since it is moving in the direction of the electric ﬂeld produced by the object Q. This loss in potential energy is equal to the work done by the �
��eld, Loss in Electrical Potential Energy = W = V Q (Since V = W Q ) = (4 = 8 £ 10¡4)(2 10¡13 J £ 10¡9) £ (b) Point A is at the higher electrical potential since work is required by us to move a positive test charge from B to A. (c) If the charge is replaced by one of negative charge, the electrical potential energy of the charge will increase in moving from A to B (in this case we would have to do work on the charge). As an example consider the electric ﬂeld between two oppositely charged parallel plates a distance d apart maintained at a potential diﬁerence V. 234 + + + + + + + P d O +Q - - - - - - - This electric ﬂeld is uniform so that a charge placed anywhere between the plates will experience the same force. Consider a positive test charge Q placed at point O just oﬁ the surface of the negative plate. In order to move it towards the positive plate we have to apply a force QE. The work done in moving the charge from the negative to the positive plate is, W = F s = QEd; but from the deﬂnition of electrical potential, Equating these two expressions for the work done, W = V Q: QEd = V Q; E = V d : and so, rearranging, Worked Example 78 Parallel plates Question: Two charged parallel plates are at a distance of 180 mm from each other. The potential diﬁerence between them is 3600 V as shown in the diagram. + + + + + + + d = 180mm X Y V = 3 600V - - - - - - - 235 + + + + + + + d -Q Fup=QE Fdown=mg - - - - - - - Figure 13.3: An oil drop suspended between oppositely charged parallel plates. 10¡9 C, is (a) If a small oil drop of negligible mass, carrying a charge of +6:8 placed between the plates at point X, calculate the magnitude and direction of the electrostatic force exerted on the droplet. (b) If the droplet is now moved to point Y, would the force exerted on it be bigger, smaller or the same as in (a)? Answer: (a) Step 1 : (NOTE TO SELF: step
is deprecated, use westep instead.) First ﬂnd the electric ﬂeld strength between the plates, £ E = = V d 3600 0:180 = 20000 N:C¡1 from the positive to the negative plate Step 2 : (NOTE TO SELF: step is deprecated, use westep instead.) Now the force exerted on the charge at X is, F = QE = (6:8 = 1:36 £ £ 10¡9)(20000) 10¡4 down (b) Step 3 : (NOTE TO SELF: step is deprecated, use westep instead.) The same. Since the electric ﬂeld strength is uniform, the force exerted on a charge is the same at all points between the plates. 13.5.3 Millikan’s Oil-drop Experiment Robert Millikan measured the charge on an electron by studying the motion of charged oil drops between oppositely charged parallel plates. Consider one such negative drop between the plates in Fig.13.3. Since this drop is negative, the electric ﬂeld exerts an upward force on the drop. In addition to this upward force, gravity exerts a downward force on the drop. Millikan adjusted 236 the electric ﬂeld strength between the plates by varying the potential diﬁerence applied across the plates. In this way, Millikan was able to bring the drops to rest. At equlibrum, Since E = V d, and, therefore, Fup = Fdown QE = mg Q V d = mg; Q = mgd V Millikan found that all drops had charges which were multiples of 1:6 become charged by gaining or losing electrons, the charge on an electron must be The magnitude of the electron’s charge is denoted by e, £ 10¡19 C. Since objects 10¡19 C. 1:6 ¡ £ e = 1:6 10¡19 C £ Worked Example 79 Charge Question: A metal sphere carries a charge of +3:2 to lose to attain its charge? Answer: Since the sphere is positive it lost electrons in the process of charging (when an object loses negative charges it is left positive). In fact, it lost, 10¡8 C. How many electrons did it have £ 10¡8 10¡19 = 2 £ 3:2 1:6 £ £ 1011 electrons Worked Example 80 Millikan oil
-drop experiment Question: In a Millikan-type experiment a positively charged oil drop is placed between two horizontal plates, 20 mm apart, as shown. 237 - - - - - - - + + + + + + + The potential diﬁerence across the plates is 4000V. The drop has a mass of 1:2 10¡14kg and a charge of 8 (a) Draw the electric ﬂeld pattern between the two plates. (b) Calculate: 10¡19C. £ £ 1. the electric ﬂeld intensity between the two plates. 2. the magnitude of the gravitational force acting on the drop. 3. the magnitude of the Coulomb force acting on the drop. (c) The drop is observed through a microscope. What will the drop be seen to do? Explain. (d) Without any further calculations, give two methods that could be used to make the drop remain in a ﬂxed position. Answer: (a 4000 0:02 = 2 £ 105 V:m¡1 up (b) 1. 2. Fgrav = mg = (1:2 = 1:2 10¡14)(10) 10¡13N £ £ 238 3. FCoulomb = QE = (8 = 1:6 £ £ 10¡19)(s 10¡13N 105) £ (c) Since Fup > Fdown, the drop accelerates upwards. (d) The Coulomb force can be decreased by decreasing the electric ﬂeld strength between the plates. Since E = V d, this can be done either by increasing d or decreasing V. 13.6 Important Equations and Quantities Units Quantity charge force mass acceleration radial distance electric ﬂeld work potential diﬁerence Unit Symbol q or Q C (Coulomb) N (Newton) \| \| \| A:s kg:m s2 kg m s2 m N=C or V =m kg:m A:s3 2 kg:m s2 2 kg:m A:s3 ¡!F m ¡!a r ¡!E W V J V (Volt) S.I. Units or \| or kg:m:s¡2 or \| or m:s¡2 or \| or kg:m:A¡1:s¡3 or kg:m2:s¡2 or
kg:m2:A¡1:s¡3 Direction \| X \| X \| X \| \| Table 13.1: Units used in Electrostatics 239 Chapter 14 Electricity Warning: We believe in experimenting and learning about physics at every opportunity, BUT playing with electricity can be EXTREMELY DANGEROUS! Do not try to build home made circuits without someone who knows if what you are doing is safe. Normal electrical outlets are dangerous. Treat electricity with respect in your everyday life. You will encounter electricity everyday for the rest of your life and to make sure you are able to make wise decisions we have included an entire chapter on electrical safety. Please read it - not only will it make you safer but it will show the applications of many of the ideas you will learn in this chapter. 14.1 Flow of Charge The normal motion of "free" electrons in a conductor has no particular direction or speed. However, electrons can be inuenced to move in a coordinated fashion through a conductive material. This motion of electrons is what we call electricity, or electric current. This is in contrast to static electricity, which is an unmoving accumulation of electric charge. Just like water owing through the emptiness of a pipe, electrons are able to move between the atoms of a conductor. The conductor may appear to be solid to our eyes, but any material composed of atoms is mostly empty space! The liquid-ow analogy is so ﬂtting that the motion of electrons through a conductor is often referred to as a "ow." As each electron moves through a conductor, it pushes on the one ahead of it. This push of one electron on another makes all of the electrons move together as a group. The motion of the each electron in a conductor may be very slow. However, the starting and stopping of electron ow through a conductor is virtually instantaneous from one end of it to the other. As an analogy consider a tube ﬂlled end-to-end with marbles: Tube Marble Marble The tube is full of marbles, just as a conductor is full of free electrons. If a single marble is suddenly inserted into this full tube on the left-hand side, another marble will immediately try to exit the tube on the right. Even though each marble only traveled a short distance, the 240 transfer of motion through the tube is virtually instantaneous from the left end to the right end. The nearly instantaneous transfer of motion through the tube occurs no matter how long the tube is. With electricity, the overall e�
��ect from one end of a conductor to the other is eﬁectively instantaneous. Each individual electron, though, travels through the conductor at a much slower pace. If we want electrons to ow in a certain direction to a certain place, we must provide the proper path for them to move. A path for electrons must be provided just as a plumber must install piping to get water to ow where he or she wants it to ow. Wires made of highly conductive metals such as copper or aluminum are used to form this path. This means that there can be electric current only where there exists a continuous path of conductive material (wire) providing a path for electrons. In the marble analogy, marbles can ow into the left-hand side of the tube only if the tube is open on the right-hand side for marbles to ow out. If the tube is blocked on the right-hand side, the marbles will just "pile up" inside the tube. Marble "ow" will not occur if the tube is blocked. The same holds true for electric current: the continuous ow of electrons requires there be an unbroken path. Let’s look at a diagram to illustrate how this works: A thin, solid line (as shown above) is the conventional symbol for a continuous piece of wire. The wire is made of a conductive material, such as copper or aluminum. The wire’s constituent atoms have many free electrons which can easily move through the wire. However, there will never be a continuous ow of electrons within this wire unless they have a place to come from and a place to go. Let’s add an hypothetical electron "Source" and "Destination:" Electron Source Electron Destination Now, with the Electron Source pushing new electrons into the wire on the left-hand side, electron ow through the wire can occur (as indicated by the arrows pointing from left to right). However, the ow will be interrupted if the conductive path formed by the wire is broken: Electron Source no flow! no flow! (break) Electron Destination Air is an insulator that impedes the ow of electrons. An air gap separates the two pieces of wire, the path has now been broken, and electrons cannot ow from Source to Destination. This is like cutting a water pipe in two and capping oﬁ the broken ends of the pipe: water can’t ow if there’s no exit out of the pipe
. If we were to take another piece of wire leading to the Destination and connect it with the wire leading to the Source, we would once again have a continuous path for electrons to ow. The two dots in the diagram indicate physical (metal-to-metal) contact between the wire pieces: Electron Source no flow! (break) Electron Destination 241 Now, we have continuity from the Source, to the newly-made connection, down, to the right, and up to the Destination. Please take note that the broken segment of wire on the right hand side has no electrons owing through it. This is because it is no longer part of a complete path from Source to Destination. It is interesting to note that no "wear" occurs within wires due to this electric current. This is in contrast to water-carrying pipes which are eventually corroded and worn by prolonged ows. Electrons do encounter some degree of friction as they move and this friction can generate heat in a conductor. This is a topic we’ll discuss later. 14.2 Circuits In order for the Source-and-Destination scheme to work, both would have to have a huge reservoir of electrons in order to sustain a continuous ow! Using the marble-and-tube analogy, the source and destination buckets would have to be large reservoirs to contain enough marble capacity for a "ow" of marbles to be sustained. The answer to this paradox is found in the concept of a circuit: a never-ending looped pathway for electrons. If we take a wire, and loop it around so that it forms a continuous pathway, we have the means to support a uniform ow of electrons without having to resort to huge reservoirs. electrons can flow in a path without beginning or end, continuing forever! A marble-andhula-hoop "circuit" Each electron advancing clockwise in this circuit pushes on the one in front of it, which pushes on the one in front of it, and so on. A circuit is just like a hula-hoop ﬂlled with marbles. Now, we have the capability of supporting a continuous ow of electrons indeﬂnitely without the need for reservoirs. All we need to maintain this ow is a continuous means of motivation for those electrons. This topic will be addressed in the next section of this chapter. It must be realized that continuity is just as important in a circuit as it is in a straight piece of wire. Just as in the example with the straight piece
of wire between the electron Source and Destination, any break in this circuit will prevent electrons from owing through it: 242 no flow! continuous electron flow cannot occur anywhere in a "broken" circuit! (break) no flow! no flow! An important principle to realize here is that it doesn’t matter where the break occurs. Any discontinuity in the circuit will prevent electron ow throughout the entire circuit. no flow! continuous electron flow cannot occur anywhere in a "broken" circuit! no flow! (break) no flow! A combination of batteries and conductors with other components is called an electric circuit or circuit. The word circuit implies that you return to your starting point and this is an important property of electric circuits. They must contain a closed loop before charge can ow. The simplest possible circuit is a battery with a single conductor. Now how do we form a closed loop with these two components? The battery has two terminals (connection points). One is called the positive terminal and one the negative. When we describe charge owing we consider charges moving from the positive terminal of the battery around the conductor and back into the battery at the negative terminal. As much charge ows out of the positive terminal as ows into the negative terminal. Because of this, there is no build up of charge in the battery. The battery does work on the charges causing them to move round the circuit. We have covered the topics of batteries and circuits but we need to draw these things to help keep the ideas clear in our minds. To do this we need to agree on how to draw things so that other people can understand what we are doing. We need a convention. 243 14.3 Voltage and current As was previously mentioned, we need more than just a continuous path (circuit) before a continuous ow of electrons will occur. We also need some means to push these electrons around the circuit. Just like marbles in a tube or water in a pipe, it takes some kind of inuencing force to initiate ow. With electrons, this force is the same force at work in static electricity: the force produced by an imbalance of electric charge. The electric charge diﬁerence serves to store a certain amount of energy. This energy is not unlike the energy stored in a high reservoir of water that has been pumped from a lower-level pond: Reservoir Energy stored Water flow Pump Pond The inuence of gravity on the water in the reservoir creates a force that attempts to move the water down to the lower level again. If a suitable
pipe is run from the reservoir back to the pond, water will ow under the inuence of gravity down from the reservoir, through the pipe: 244 Reservoir Energy released Pond It takes energy to pump that water from the low-level pond to the high-level reservoir. The movement of water through the piping back down to its original level constitutes a releasing of energy stored from previous pumping. If the water is pumped to an even higher level, it will take even more energy to do so, and so more energy will be stored. The more energy is stored the more energy is released if the water is allowed to ow through a pipe back down again: 245 Energy stored Reservoir Pump Pond Reservoir Energy released More energy stored More energy released Pump Pond Electrons are not much diﬁerent. If we "pump" electrons away from their normal "levels," we create a condition where a force exists as the electrons seek to re-establish their former positions. The force attracting electrons back to their original positions is analogous to the force gravity exerts on water in the reservoir. Just as gravity tries to draw water down to its former level, the force exerted on the electrons attracts them back to their former posisions. Just as the pumping of water to a higher level results in energy being stored, "pumping" electrons to create an electric charge imbalance results in a certain amount of energy being stored in that imbalance. Providing a way for water to ow back down from the heights of the 246 reservoir results in a release of that stored energy. Similarly, providing a way for electrons to ow back to their original "levels" results in a release of stored energy. When the electrons are poised in that static condition (just like water sitting still, high in a reservoir), the energy stored there is called potential energy. It is given that name because it has the possibility (potential) of release that has not been fully realized yet. This potential energy, stored in the form of an electric charge imbalance, can be expressed as a term called voltage. Technically, voltage is a measure of potential energy per unit charge of electrons, or something a physicist would call speciﬂc potential energy. Deﬂned in the context of static electricity, voltage is the measure of work required to move a unit charge from one location to another. This work is against the force which tries to keep electric charges balanced. In the context of electrical power sources, voltage is the amount of potential energy available (work to be done)
per unit charge, to move electrons through a conductor. Voltage is an expression of potential energy. As such, it represents the possibility or potential for energy release as the electrons move from one "level" to another. Because of this, voltage is always referenced between two points. Consider the water reservoir analogy: Reservoir Drop Drop Location #1 Location #2 Because of the diﬁerence in the height of the drop, there’s potential for much more energy to be released from the reservoir through the piping to location 2 than to location 1. The principle can be intuitively understood in dropping a rock: which results in a more violent impact, a rock dropped from a height of one foot, or the same rock dropped from a height of one kilometre? Obviously, the drop of greater height results in greater energy released (a more violent impact). We cannot assess the amount of stored energy in a water reservoir simply by measuring the volume of water. Similarly, we cannot predict the severity of a falling rock’s impact simply from knowing the weight of the rock: in both cases we must also consider how far these masses will drop from their initial height. The amount of energy released by allowing a mass to drop is relative to the distance between its starting and ending points. Likewise, the potential energy available for moving electrons from one point to another is relative to those two points. Therefore, voltage is always expressed as a quantity between two points. Interestingly enough, the analogy of a mass potentially "dropping" from one height to another is such an apt model that voltage between two points is sometimes called a voltage drop. 247 Voltage can be generated in many ways. Chemical reactions, radiant energy, and the inuence of magnetism on conductors are a few ways in which voltage may be produced. Respective examples of these three sources of voltage are batteries, solar cells, and generators. For now, we won’t go into detail as to how each of these voltage sources works. The important thing is that we understand how voltage sources can be applied to create electron ow in a circuit. Let’s take the symbol for a chemical battery and build a circuit step by step: 1 - + 2 Battery Any source of voltage, including batteries, have two points for electrical contact. In this case, we have point 1 and point 2 in the above diagram. The horizontal lines of varying length indicate that this is a battery. The horizontal lines further indicate the direction which this battery’s voltage will try to push electrons
through a circuit. The horizontal lines in the battery symbol appear separated, and so make it appear as if the battery is unable to serve as a path for electrons to move. This is no cause for concern: in real life, those horizontal lines represent metallic plates immersed in a liquid or semi-solid material that not only conducts electrons, but also generates the voltage to push them along by interacting with the plates. Notice the little "+" and "-" signs to the immediate left of the battery symbol. The negative (-) end of the battery is always the end with the shortest dash, and the positive (+) end of the battery is always the end with the longest dash. By convention, electrons are said to be "negatively" charged, so the negative end of a battery is that end which tries to push electrons out of it. Likewise, the positive end is that end which tries to attract electrons. With the "+" and "-" ends of the battery not connected to anything, there will be voltage between those two points. However, there will be no ow of electrons through the battery, because there is no continuous path for the electrons to move. 248 Water analogy Reservoir Electric Battery No flow 1 - + 2 Battery No flow (once the reservoir has been completely filled) Pump Pond The same principle holds true for the water reservoir and pump analogy: without a return pipe back to the pond, stored energy in the reservoir cannot be released in the form of water ow. Once the reservoir is completely ﬂlled up, no ow can occur, no matter how much pressure the pump may generate. There needs to be a complete path (circuit) for water to ow from the pond, to the reservoir, and back to the pond in order for continuous ow to occur. We can provide such a path for the battery by connecting a piece of wire from one end of the battery to the other. Forming a circuit with a loop of wire, we will initiate a continuous ow of electrons in a clockwise direction: 249 Electric Circuit Battery 1 - + 2 electron flow! Water analogy Reservoir water flow! water flow! Pump Pond So long as the battery continues to produce voltage and the continuity of the electrical path isn’t broken, electrons will continue to ow in the circuit. Following the metaphor of water moving through a pipe, this continuous, uniform ow of electrons through the circuit is called a current. So long as the voltage source keeps "pushing" in the same direction, the electron ow will continue to move in the
same direction in the circuit. This single-direction ow of electrons is called a Direct Current, or DC. Because electric current is composed of individual electrons owing in unison through a conductor, just like marbles through a tube or water through a pipe, the amount of ow throughout a single circuit will be the same at any point. If we were to monitor a cross-section of the wire in a single circuit, counting the electrons owing by, we would notice the exact same quantity per unit of time as in any other part of the circuit. The same quantity would be observed regardless 250 of conductor length or conductor diameter. If we break the circuit’s continuity at any point, the electric current will cease in the entire loop. Futhermore, the full voltage produced by the battery will be manifested across the break, between the wire ends that used to be connected: 1 - + 2 no flow! Battery no flow! - (break) + voltage drop Notice the "+" and "-" signs drawn at the ends of the break in the circuit, and how they correspond to the "+" and "-" signs next to the battery’s terminals. These markers indicate the direction that the voltage attempts to push electron ow. Remember that voltage is always relative between two points. Whether a point in a circuit gets labeled with a "+" or a "-" depends on the other point to which it is referenced. Take a look at the following circuit, where each corner of the loop is marked with a number for reference: 1 - + 4 no flow! 2 - Battery (break) + 3 no flow! With the circuit’s continuity broken between points 2 and 3, voltage dropped between points 2 and 3 is "-" for point 2 and "+" for point 3. Now let’s see what happens if we connect points 2 and 3 back together again, but place a break in the circuit between points 3 and 4: 251 1 - + 4 no flow! 2 Battery no flow! + - (break) 3 With the break between 3 and 4, the voltage drop between those two points is "+" for 4 and "-" for 3. Take special note of the fact that point 3’s "sign" is opposite of that in the ﬂrst example, where the break was between points 2 and 3 (where point 3 was labeled "+"). It is impossible for us to say that point 3 in this circuit will always be either "+" or "-", because the sign is not speci�
�c to a single point, but is always relative between two points! 14.4 Resistance The circuit in the previous section is not a very practical one. In fact, it can be quite dangerous to directly connect the poles of a voltage source together with a single piece of wire. This is because the magnitude of electric current may be very large in such a short circuit, and the release of energy very dramatic (usually in the form of heat). Usually, electric circuits are constructed in such a way as to make practical use of that released energy, in as safe a manner as possible. One practical and popular use of electric current is for the operation of electric lighting. The simplest form of electric lamp is a tiny metal "ﬂlament" inside of a clear glass bulb, which glows white-hot ("incandesces") with heat energy when su–cient electric current passes through it. Like the battery, it has two conductive connection points, one for electrons to enter and the other for electrons to exit. Connected to a source of voltage, an electric lamp circuit looks something like this: Battery - + electron flow electron flow Electric lamp (glowing) As the electrons work their way through the thin metal ﬂlament of the lamp, they encounter more opposition to motion than they typically would in a thick piece of wire. This opposition to 252 electric current depends on the type of material, its cross-sectional area, and its temperature. It is technically known as resistance. It serves to limit the amount of current through the circuit with a given amount of voltage supplied by the battery. The "short circuit" where we had nothing but a wire joining one end of the voltage source (battery) to the other had not such a limiting resistance. Interesting Fact: Materials known as conductors have a low resistance, while insulators have a very high one. When electrons move against the opposition of resistance, "friction" is generated. Just like mechanical friction, the friction produced by electrons owing against a resistance manifests itself in the form of heat. The concentrated resistance of a lamp’s ﬂlament results in a relatively large amount of heat energy dissipated at that ﬂlament. This heat energy is enough to cause the ﬂlament to glow white-hot, producing light. The wires connecting the lamp to the battery hardly even get warm while conducting the same amount of current. This is because of their much lower resistance due to their larger cross-section. As
in the case of the short circuit, if the continuity of the circuit is broken at any point, electron ow stops throughout the entire circuit. With a lamp in place, this means that it will stop glowing: no flow! - Battery - + no flow! + (break) voltage drop Electric lamp (not glowing) no flow! As before, with no ow of electrons the entire potential (voltage) of the battery is available across the break, waiting for the opportunity of a connection to bridge across that break and permit electron ow again. This condition is known as an open circuit, where a break in the continuity of the circuit prevents current throughout. All it takes is a single break in continuity to "open" a circuit. Once any breaks have been connected once again and the continuity of the circuit re-established, it is known as a closed circuit. What we see here is the basis for switching lamps on and oﬁ by switches. Because any break in a circuit’s continuity results in current stopping throughout the entire circuit, we can use a device designed to intentionally break that continuity (called a switch). This switch can be mounted at any convenient location that we can run wires to. It controls the ow of electrons in the hole circuit: 253 switch Battery - + It doesn’t matter how twisted or convoluted a route the wires take conducting current, so long as they form a complete, uninterrupted loop (circuit). This is how a switch mounted on the wall of a house can control a lamp that is mounted down a long hallway, or even in another room, far away from the switch. The switch itself is constructed of a pair of conductive contacts (usually made of some kind of metal) forced together by a mechanical lever actuator or pushbutton. When the contacts touch each other, electrons are able to ow from one to the other and the circuit’s continuity is established; when the contacts are separated, electron ow from one to the other is prevented by the insulation of the air between, and the circuit’s continuity is broken. In keeping with the "open" and "closed" terminology of circuits, a switch that is making contact from one connection terminal to the other provides continuity for electrons to ow through, and is called a closed switch. Conversely, a switch that is breaking continuity won’t allow electrons to pass through and is called an open switch. This terminology is often confusing to the new student of electronics, because the words "open" and "closed" are commonly
understood in the context of a door, where "open" is equated with free passage and "closed" with blockage. With electrical switches, these terms have opposite meaning: "open" means no ow while "closed" means free passage of electrons. 14.5 Voltage and current in a practical circuit Because it takes energy to force electrons to ow against the opposition of a resistance, there will be voltage manifested (or "dropped") between any points in a circuit with resistance between them. It is important to note that although the amount of current is uniform in a simple circuit, the amount of voltage between diﬁerent sets of points in a single circuit may vary considerably: 254 same rate of current... Battery 1 - + 4 2 3... at all points in this circuit (NOTE TO SELF: How do you actually measure currents and voltages? In the next two paragraphs there’s a lot of ado what we ’see’ at diﬁerent points of a circuit, but I don’t see anything!) Take this circuit as an example. We labelled four points with the numbers 1, 2, 3, and 4. The amount of current conducted through the wire between points 1 and 2 is exactly the same as the amount of current conducted through the lamp (between points 2 and 3). This same quantity of current passes through the wire between points 3 and 4, and through the battery (between points 1 and 4). However, we will ﬂnd the voltage appearing between any two of these points to be directly proportional to the resistance within the conductive path between those two points. In a normal lamp circuit, the resistance of a lamp will be much greater than the resistance of the connecting wires. So we should expect to see a substantial amount of voltage drop between points 2 and 3, and only a very small one between points 1 and 2, or between 3 and 4. The voltage drop between points 1 and 4, of course, will be the full voltage oﬁered by the battery. This will be only slightly higher than the voltage drop across the lamp (between points 2 and 3). This, again, is analogous to the water reservoir system: 255 2 Reservoir 1 Waterwheel (energy released) 3 (energy stored) Pump 4 Pond Between points 2 and 3, where the falling water is releasing energy at the water-wheel, there is a diﬁerence of pressure between the two points. This reects the opposition
to the ow of water through the water-wheel. From point 1 to point 2, or from point 3 to point 4, where water is owing freely through reservoirs with little opposition, there is little or no diﬁerence of pressure (no potential energy). However, the rate of water ow in this continuous system is the same everywhere (assuming the water levels in both pond and reservoir are unchanging): through the pump, through the water-wheel, and through all the pipes. So it is with simple electric circuits: the rate of electron ow is the same at every point in the circuit, although voltages may diﬁer between diﬁerent sets of points. 14.6 Direction of current ow in a circuit We know now that the moving charges in an electrical ciruit are the negatively chargend electrons. These electrons naturally ow from the negative pole of a battery to the positive pole. This form of symbology became known as electron ow notation: Electron flow notation + - Electric charge moves from the negative (surplus) side of the battery to the positive (deficiency) side. 256 However, for historical reasons the current ow in a circuit is conventionally denoted in the opposite direction. That is, it ows from the positive pole to the negative one. This became known as conventional ow notation: Conventional flow notation + - Electric charge moves from the positive (surplus) side of the battery to the negative (deficiency) side. In conventional ow notation, we show the motion of charge according to the (technically. This way the labels make sense, but the direction of charge ow is incorrect) labels of + and incorrect. ¡ Does it matter, really, how we designate charge ow in a circuit? Not really, so long as we’re consistent in the use of our symbols. You may follow an imagined direction of current (conventional ow) or the actual (electron ow) with equal success insofar as circuit analysis is concerned. Concepts of voltage, current, resistance, continuity, and even mathematical treatments such as Ohm’s Law (section 2 (NOTE TO SELF: make this reference dynamic)) and Kirchhoﬁ’s Laws (section 6 (NOTE TO SELF: make this reference dynamic)) remain just as valid with either style of notation. Aside: Benjamin Franklin made a conjecture regarding the direction of charge ow when rubbing smooth wax with rough wool. By assuming that the observed charges ow from the wax to the
wool, he set the precedent for electrical notation that exists to this day. Because Franklin assumed electric charge moved in the opposite direction that it actually does, electrons are said to have a negative charge, and so objects he called "negative" (representing a deﬂciency of charge) actually have a surplus of electrons. By the time the true direction of electron ow was discovered, the nomenclature of "positive" and "negative" had already been so well established in the scientiﬂc community that no eﬁort was made to change it. It would have made more sense to call electrons "positive" in referring to "excess" charge. You see, the terms "positive" and "negative" are human inventions, and as such have no absolute meaning beyond our own conventions of language and scientiﬂc description. Franklin could have just as easily referred to a surplus of charge as "black" and a deﬂciency as "white", in which case scientists would speak of electrons having a "white" charge. However, because we tend to associate the word "positive" with "surplus" and "negative" with "deﬂciency," the standard label for electron charge does seem backward. As discussed above, many engineers decided to retain the old concept of electricity with "positive" referring to a surplus of charge, and label charge ow (current) accordingly. 257 14.7 How voltage, current, and resistance relate First lets recap some of the ideas we have learnt so far. We will need these to understand how voltage, current and resistance relate. An electric circuit is formed when a conductive path is created to allow free electrons to continuously move. This continuous movement of free electrons through the conductors of a circuit is called a current. It is often referred to in terms of "ow," just like the ow of a liquid through a hollow pipe. The force motivating electrons to "ow" in a circuit is called voltage. Voltage is a speciﬂc measure of potential energy that is always relative between two points. When we speak of a certain amount of voltage being present in a circuit, we are referring to the measurement of how much potential energy exists to move electrons from one particular point in that circuit to another particular point. Without reference to two particular points, the term "voltage" has no meaning. Free electrons tend to move through conductors with some degree of friction, or opposition to motion. This
opposition to motion is more properly called resistance. The amount of current in a circuit depends on the amount of voltage available to motivate the electrons (NOTE TO SELF: Motivated electrons?), and also the amount of resistance in the circuit to oppose electron ow. Just like voltage, resistance is a quantity relative between two points. For this reason, the quantities of voltage and resistance are often stated as being "between" or "across" two points in a circuit. To be able to make meaningful statements about these quantities in circuits, we need to be able to describe their quantities in the same way that we might quantify mass, temperature, volume, length, or any other kind of physical quantity. For mass we might use the units of "pound" or "gram". For temperature we might use degrees Fahrenheit or degrees Celsius. Here are the standard units of measurement for electrical current, voltage, and resistance: Quantity Current Voltage Resistance Charge Symbol I V (or E) R Q Unit of Measurement Abbreviation of Unit Ampere volt Ohm coulomb A V › C The "symbol" given for each quantity is the standard alphabetical letter used to represent that quantity in an algebraic equation. Standardized letters like these are common in the disciplines of physics and engineering, and are internationally recognized. The "unit abbreviation" for each quantity represents the alphabetical symbol used as a shorthand notation for its particular unit of measurement. And, yes, that strange-looking "horseshoe" symbol is the capital Greek letter › (called omega), just a character in a foreign alphabet (apologies to any Greek readers here). Aside: Each unit of measurement is named after a famous experimenter in electricity: The amp after the Frenchman Andre M. Ampere, the volt after the Italian Alessandro Volta, and the ohm after the German Georg Simon Ohm. The mathematical symbol for each quantity is meaningful as well. The "R" for resistance and the "V " for voltage are both self-explanatory. The "I" is thought to have been meant to represent "Intensity" (of electron ow). The other symbol for voltage, "E", stands for "Electromotive force". The symbols "E" and "V " are interchangeable for the most part, although some texts reserve "E" to represent voltage across a source (such as a battery or generator) and "V " to represent voltage across anything else. 258 One foundational unit of electrical measurement is the unit of the
coulomb. It is a measure of electric charge proportional to the number of electrons in an imbalanced state. One coulomb of charge is rougly equal to the charge of 6,250,000,000,000,000,000 electrons. The symbol for electric charge quantity is the capital letter "Q", with the unit of coulombs abbreviated by the capital letter "C". It so happens that the unit for electron ow, the ampere, is equal to 1 coulomb of electrons passing by a given point in a circuit in 1 second of time. Cast in these terms, current is the rate of electric charge motion through a conductor. As stated before, voltage is the measure of potential energy per unit charge available to motivate electrons from one point to another. Before we can precisely deﬂne what a "volt" is, we must understand how to measure this quantity we call "potential energy". The general metric unit for energy of any kind is the joule, equal to the amount of work performed by a force of 1 newton exerted through a motion of 1 meter (in the same direction). (NOTE TO SELF: Make a reference to the Mechanics chapter.) Deﬂned in these scientiﬂc terms, 1 volt is equal to 1 joule of electric potential energy per (divided by) 1 coulomb of charge. Thus, a 9 volt battery releases 9 joules of energy for every coulomb of electrons moved through a circuit. These units and symbols for electrical quantities will become very important to know as we begin to explore the relationships between them in circuits. The ﬂrst, and perhaps most important, relationship between current, voltage, and resistance is called Ohm’s Law. It states that the amount of electric current through a metal conductor in a circuit is directly proportional to the voltage impressed across it, for any given temperature. It can be expressed in the form of a simple equation, describing how voltage, current, and resistance interrelate: ¢ In this algebraic expression, voltage (V ) is equal to current (I) multiplied by resistance (R). V = I R : (14.1) Aside: Georg Simon Ohm published his law in his 1827 paper, The Galvanic Circuit Investigated Mathematically. Using algebra techniques, we can manipulate this equation into two variations, solving for I and for R, respectively: Let’s see how these equations
might work to help us analyze simple circuits14.2) electron flow Battery + - Electric lamp (glowing) electron flow 259 (NOTE TO SELF: replace E by V) In the above circuit, there is only one source of voltage (the battery, on the left) and only one source of resistance to current (the lamp, on the right). This makes it very easy to apply Ohm’s Law. If we know the values of any two of the three quantities (voltage, current, and resistance) in this circuit, we can use Ohm’s Law to determine the third. In this ﬂrst example, we will calculate the amount of current (I) in a circuit, given values of voltage (V ) and resistance (R): Worked Example 81 Question: What is the amount of current (I) in this circuit? Battery E = 12 V + - I =??? I =??? Lamp R = 3 W (NOTE TO SELF: replace E by V) Answer: I = V R = 12 V 3 › = 4 A : (14.3) In the second example, we will calculate the amount of resistance (R) in a circuit, given values of voltage (V ) and current (I): Worked Example 82 Question: What is the amount of resistance (R) oﬁered by the lamp? 260 Battery E = 36 Lamp R =??? (NOTE TO SELF: replace E by V) Answer: R = V I = 36 V 4 A = 9 › : (14.4) In the last example, we will calculate the amount of voltage supplied by a battery, given values of current (I) and resistance (R): Worked Example 83 Question: What is the amount of voltage provided by the battery? Battery E =??? + - I = 2 A I = 2 A Lamp R = 7 W (NOTE TO SELF: replace E by V) Answer: V = I ¢ R = (2 A) ¢ (7 ›) = 14 V : (14.5) 261 Ohm’s Law is a very simple and useful tool for analyzing electric circuits. It is used so often in the study of electricity and electronics that it needs to be committed to memory by the serious student. All you need to do is commit V = I R to memory and derive the other two formulae from that when you need them! ¢ 14.8 Voltmeters, ammet
ers, and ohmmeters As we have seen in previous sections, an electric circuit is made up of a number of diﬁerent components such as batteries and resistors. In electronics, there are many types of meters used to measure the properties of the individual components of an electric circuit. For example, one may be interested in measuring the amount of current owing through a circuit, or measure the voltage provided by a battery. In this section we will discuss the practical usage of voltmeters, ammeters, and ohmmeters. A voltmeter is an instrument for measuring the voltage between two points in an electric circuit. In analogy with a water circuit, a voltmeter is like a meter designed to measure pressure diﬁerence. Since one is interested in measuring the voltage between two points in a circuit, a voltmeter must be connected in parallel with the portion of the circuit on which the measurement is made: V The above illustration shows a voltmeter connected in parallel with a battery. One lead of the voltmeter is connected to one end of the battery and the other lead is connected to the opposite end. The voltmeter may also be used to measure the voltage across a resistor or any other component of a circuit that has a voltage drop. An ammeter is an instrument used to measure the ow of electric current in a circuit. Since one is interested in measuring the current owing through a circuit component, the ammeter must be connected in series with the measured circuit component: A An ohmmeter is an instrument for measuring electrical resistance. The basic ohmmeter can function much like an ammeter. The ohmmeter works by suppling a constant voltage to the resistor and measuring the current owing through it. The measured current is then converted into a corresponding resistance reading through Ohm’s law. One cautionary detail needs to be 262 mentioned with regard to ohmmeters: they only function correctly when measuring resistance that is not being powered by a voltage or current source. In other words, you cannot measure the resistance of a component that is already connected to a circuit. The reason for this is simple: the ohmmeter’s accurate indication depends only on its own source of voltage. The presence of any other voltage across the measured circuit component interferes with the ohmmeter’s operation. The circuit diagram below shows an ohmmeter solely connected with a resistor: › The table below summarizes the use of each measuring instrument that we discussed and the way it should be connected to a circuit
component. Instrument Measured Quantity Proper Connection Voltmeter Ammeter Ohmmeter In Parallel In Series Only with Resistor Voltage Current Resistance 14.9 An analogy for Ohm’s Law In our water-and-pipe analogy, Ohm’s Law also exists. Think of a water pump that exerts pressure (voltage) to push water around a "circuit" (current) through a restriction (resistance). If the resistance to water ow stays the same and the pump pressure increases, the ow rate must also increase. " V = " I R If the pressure stays the same and the resistance increases (making it more di–cult for the water to ow), then the ow rate must decrease: V = # I " R If the ow rate stays the same while the resistance to ow decreases, the required pressure from the pump decreases: # V = I # R As odd as it may seem, the actual mathematical relationship between pressure, ow, and resistance is actually more complex for uids like water than it is for electrons. If you pursue further studies in physics, you will discover this for yourself. Thankfully for the electronics student, the mathematics of Ohm’s Law is very simple. 263 14.10 Power in electric circuits In addition to voltage and current, there is another measure of free electron activity in a circuit: power. The concept of power was introduced in Chapter 8. Basically, it is a measure of how rapidly a standard amount of work is done. In electric circuits, power is a function of both voltage and current: P = IV: So power (P ) is exactly equal to current (I) multiplied by voltage (V ) and there is no extra constant of proportionality. The unit of measurement for power is the Watt (abbreviated W). Aside: You can verify for yourself that the eqution for power in an electric cicuit makes sense. Remember that voltage is the speciﬂc work (or potential energy) per unit charge, while current is the amount electric charge that ow though a conductor per time unit. So the product of those two qunatities is the oumount of work per time unit, which is exactly the power. It is important to realise that only the combination of a voltage drop and the ow of current corresponds to power. So, a circuit with high voltage and low current may be dissipating the same amount of power as a circuit with low voltage and high current. In an open circuit, where
voltage is present between the terminals of the source and there is zero current, there is zero power dissipated, no matter how great that voltage may be. Since P = IV and I = 0, the power dissipated in any open circuit must be zero. 14.11 Calculating electric power We’ve seen the formula for determining the power in an electric circuit: by multiplying the voltage in volts by the current in Amperes we arrive at an answer in watts." Let’s apply this to a circuit example: Battery E = 18 V + - I =??? I =??? Lamp R = 3 W In the above circuit, we know we have a battery voltage of 18 Volts and a lamp resistance of 3 ›. Using Ohm’s Law to determine current, we get: I = V R = 18V 3› = 6A: 264 Now that we know the current, we can take that value and multiply it by the voltage to determine power: P = IV = (6A)(18V = 108W: Answer: the lamp is dissipating (releasing) 108 W of power, most likely in the form of both light and heat. Let’s try taking that same circuit and increasing the battery voltage to see what happens: Battery E = 36 V + - I =??? I =??? Lamp R = 3 W Since the resistance stays the same, the current will increase when we increase the voltage: 36V 3› Note that Ohm’s Law is linear, so the current exactly doubles when we double the voltage. = 12A: V R I = = Now, let’s calculate the power: P = IV = (12A)(36V) = 432W: Notice that the power has increased just as we might have suspected, but it increased quite a bit more than the current. Why is this? Because power is a function of voltage multiplied by current, and both voltage and current doubled from their previous values, the power will increase by a factor of 2 x 2, or 4: the ratio of the new power 432 W and the old power 108 W, is exactly 4. We could in fact have arrived at this result without the intermediate step of calculating the current. From we can expres power directly as a function of voltage: I = V R and P = IV The analogous relation between power and current is P = IIR = I 2R: 265 Interesting Fact: It was James Prescott Joule, not Georg
Simon Ohm, who ﬂrst discovered the mathematical relationship between power dissipation and current through a resistance. This discovery, published in 1841, followed the form of the last equation (P = I2R), and is properly known as Joule’s Law. However, these power equations are so commonly associated with the Ohm’s Law equations relating voltage, current, and resistance (V = IR ; I = V =R ; and R = V =I) that they are frequently credited to Ohm. 14.12 Resistors Because the relationship between voltage, current, and resistance in any circuit is so regular, we can reliably control any variable in a circuit simply by controlling the other two. Perhaps the easiest variable in any circuit to control is its resistance. This can be done by changing the material, size, and shape of its conductive components (remember how the thin metal ﬂlament of a lamp created more electrical resistance than a thick wire?). Special components called resistors are made for the express purpose of creating a precise quantity of resistance for insertion into a circuit. They are typically constructed of metal wire or carbon, and engineered to maintain a stable resistance value over a wide range of environmental conditions. Unlike lamps, they do not produce light, but they do produce heat as electric power is dissipated by them in a working circuit. Typically, though, the purpose of a resistor is not to produce usable heat, but simply to provide a precise quantity of electrical resistance. The most common schematic symbol for a resistor is a zig-zag line: Resistor values in ohms are usually shown as an adjacent number, and if several resistors are present in a circuit, they will be labeled with a unique identiﬂer number such as R1, R2, R3, etc. As you can see, resistor symbols can be shown either horizontally or vertically: R1 150 This is resistor "R1" with a resistance value of 150 ohms. R2 25 This is resistor "R2" with a resistance value of 25 ohms. In keeping more with their physical appearance, an alternative schematic symbol for a resistor looks like a small, rectangular box: Resistors can also be shown to have varying rather than ﬂxed resistances. This might be for the purpose of describing an actual physical device designed for the purpose of providing an adjustable resistance, or it could be to show some component that just happens to have an unstable resistance: 266 variable resistance
... or... In fact, any time you see a component symbol drawn with a diagonal arrow through it, that component has a variable rather than a ﬂxed value. This symbol "modiﬂer" (the diagonal arrow) is standard electronic symbol convention. In practice, resistors are not only rated in terms of their resistance in Aside: ohms, but also in term the amount of power they can dissipate in watts. Resistors dissipate heat as the electric currents through them overcome the "friction" of their resistance and can in fact become quite hot in actual applications. Most resistors found in small electronic devices such as portable radios are rated at 1/4 (0.25) watt or less. The power rating of any resistor is roughly proportional to its physical size. Also note how resistances (in ohms) have nothing to do with size! 14.13 Nonlinear conduction Ohm’s Law is a powerful tool for analyzing electric circuits, but it has a practical limitation. In the application of Ohm’s Law, we alwasy assume that the restistance does not change as a function of voltage and current. For most conductors, this is a reasonable approximation as long ads the temperature does not change too much. In a normal lightbulb, the resistance of the ﬂlament wire will increase dramatically as it warms from room temperature to operating temperature. If we increase the supply voltage in a real lamp circuit, the resulting increase in current causes the ﬂlament to increase in temperature, which increases its resistance. This eﬁectively limits the increase in current. Consequently, voltage and current do not follow the simple equation I = V =R, with a constant R (of 3 › ion our example). The lamp’s ﬂlament resistance does not remain stable for diﬁerent currents. The phenomenon of resistance changing with variations in temperature is one shared by almost all metals, of which most wires are made. For most applications, these changes in resistance are small enough to be ignored. In the application of metal lamp ﬂlaments, which increase a lot in temperature (up to about 1000oC, and starting from room temperature) the change is quite large. A more realistic analysis of a lamp circuit over several diﬁerent values of battery voltage would generate a plot of this shape: 267 I (current) E (voltage) The
plot is no longer a straight line. It rises sharply on the left, as voltage increases from zero to a low level. As it progresses to the right we see the line attening out, the circuit requiring greater and greater increases in voltage to achieve equal increases in current. If we apply Ohm’s Law to ﬂnd the resistance of this lamp circuit with the voltage and current values plotted above, the calculated values will change with voltage or curreny. We could say that the resistance here is nonlinear, increasing with increasing current and voltage. The nonlinearity is caused by the eﬁects of high temperature on the metal wire of the lamp ﬂlament. 14.14 Circuit wiring So far, we’ve been analyzing single-battery, single-resistor circuits with no regard for the connecting wires between the components, so long as a complete circuit is formed. Does the wire length or circuit "shape" matter to our calculations? Let’s look at a couple of circuit conﬂgurations and ﬂnd out: Battery 10 V 1 4 2 Resistor 5 W 3 Battery 10 V 1 4 2 Resistor 5 W 3 268 When we draw wires connecting points in a circuit, we usually assume those wires have negligible resistance. As such, they contribute no appreciable eﬁect to the overall resistance of the circuit, and so the only resistance we have to contend with is the resistance in the components. In the above circuits, the only resistance comes from the 5 › resistors, so that is all we will consider in our calculations. In real life, metal wires actually do have resistance (and so do power sources!), but those resistances are generally so much smaller than the resistance present in the other circuit components that they can be safely ignored. If connecting wire resistance is very little or none, we can regard the connected points in a circuit as being electrically common. That is, points 1 and 2 in the above circuits may be physically joined close together or far apart, and it doesn’t matter for any voltage or resistance measurements relative to those points. The same goes for points 3 and 4. It is as if the ends of the resistor were attached directly across the terminals of the battery, so far as our Ohm’s Law calculations and voltage measurements are concerned. This is useful to know, because it means you can re-draw a circuit diagram or re-wire a circuit, shortening or leng
thening the wires as desired without appreciably impacting the circuit’s function. All that matters is that the components attach to each other in the same sequence. It also means that voltage measurements between sets of "electrically common" points will be the same. That is, the voltage between points 1 and 4 (directly across the battery) will be the same as the voltage between points 2 and 3 (directly across the resistor). Take a close look at the following circuit, and try to determine which points are common to each other: 1 Battery 10 V 4 2 3 Resistor 5 W 6 5 Here, we only have 2 components excluding the wires: the battery and the resistor. Though the connecting wires take a convoluted path in forming a complete circuit, there are several electrically common points in the electrons’ path. Points 1, 2, and 3 are all common to each other, because they’re directly connected together by wire. The same goes for points 4, 5, and 6. The voltage between points 1 and 6 is 10 volts, coming straight from the battery. However, since points 5 and 4 are common to 6, and points 2 and 3 common to 1, that same 10 volts also exists between these other pairs of points: Between points 1 and 4 = 10 volts Between points 2 and 4 = 10 volts Between points 3 and 4 = 10 volts (directly across the resistor) Between points 1 and 5 = 10 volts Between points 2 and 5 = 10 volts Between points 3 and 5 = 10 volts Between points 1 and 6 = 10 volts (directly across the battery) Between points 2 and 6 = 10 volts Between points 3 and 6 = 10 volts Since electrically common points are connected together by (zero resistance) wire, there is no signiﬂcant voltage drop between them regardless of the amount of current conducted from one 269 to the next through that connecting wire. Thus, if we were to read voltages between common points, we should show (practically) zero: Between points 1 and 2 = 0 volts Between points 2 and 3 = 0 volts Between points 1 and 3 = 0 volts Between points 4 and 5 = 0 volts Between points 5 and 6 = 0 volts Between points 4 and 6 = 0 volts Points 1, 2, and 3 are electrically common Points 4, 5, and 6 are electrically common This makes sense mathematically, too. With a 10 volt battery and a 5 › resistor, the circuit current will be 2 amps. With wire resistance
being zero, the voltage drop across any continuous stretch of wire can be determined through Ohm’s Law as such: E = I R E = (2 A)(0 W) E = 0 V It should be obvious that the calculated voltage drop across any uninterrupted length of wire in a circuit where wire is assumed to have zero resistance will always be zero, no matter what the magnitude of current, since zero multiplied by anything equals zero. Because common points in a circuit will exhibit the same relative voltage and resistance measurements, wires connecting common points are often labeled with the same designation. This is not to say that the terminal connection points are labeled the same, just the connecting wires. Take this circuit as an example: 1 wire #2 2 wire #2 Battery 10 V 4 wire #1 3 Resistor 5 W 6 5 wire #1 wire #1 Points 1, 2, and 3 are all common to each other, so the wire connecting point 1 to 2 is labeled the same (wire 2) as the wire connecting point 2 to 3 (wire 2). In a real circuit, the wire stretching from point 1 to 2 may not even be the same color or size as the wire connecting point 2 to 3, but they should bear the exact same label. The same goes for the wires connecting points 6, 5, and 4. Knowing that electrically common points have zero voltage drop between them is a valuable troubleshooting principle. If I measure for voltage between points in a circuit that are supposed to be common to each other, I should read zero. If, however, I read substantial voltage between those two points, then I know with certainty that they cannot be directly connected together. 270 If those points are supposed to be electrically common but they register otherwise, then I know that there is an "open failure" between those points. One ﬂnal note: for most practical purposes, wire conductors can be assumed to possess zero resistance from end to end. In reality, however, there will always be some small amount of resistance encountered along the length of a wire, unless it’s a superconducting wire. Knowing this, we need to bear in mind that the principles learned here about electrically common points are all valid to a large degree, but not to an absolute degree. That is, the rule that electrically common points are guaranteed to have zero voltage between them is more accurately stated as such: electrically common points will have very little voltage dropped between them. That small, virtually unavoidable trace of resistance
found in any piece of connecting wire is bound to create a small voltage across the length of it as current is conducted through. So long as you understand that these rules are based upon ideal conditions, you won’t be perplexed when you come across some condition appearing to be an exception to the rule. 14.15 Polarity of voltage drops We can trace the direction that electrons will ow in the same circuit by starting at the negative (-) terminal and following through to the positive (+) terminal of the battery, the only source of voltage in the circuit. From this we can see that the electrons are moving counter-clockwise, from point 6 to 5 to 4 to 3 to 2 to 1 and back to 6 again. As the current encounters the 5 › resistance, voltage is dropped across the resistor’s ends. The signs of this voltage drop is negative (-) at point 4 with respect to positive (+) at point 3. We can mark the polarity of the resistor’s voltage drop with these negative and positive symbols, in accordance with the direction of current (whichever end of the resistor the current is entering is negative with respect to the end of the resistor it is exiting: Battery 10 V 1 + - 6 2 3 current current - + Resistor 5 W 4 5 We could make our table of voltages a little more complete by marking the polarity of the voltage for each pair of points in this circuit: Between points 1 (+) and 4 (-) = 10 volts Between points 2 (+) and 4 (-) = 10 volts Between points 3 (+) and 4 (-) = 10 volts Between points 1 (+) and 5 (-) = 10 volts Between points 2 (+) and 5 (-) = 10 volts Between points 3 (+) and 5 (-) = 10 volts Between points 1 (+) and 6 (-) = 10 volts Between points 2 (+) and 6 (-) = 10 volts Between points 3 (+) and 6 (-) = 10 volts 271 While it might seem a little silly to document polarity of voltage drop in this circuit, it is an important concept to master. It will be critically important in the analysis of more complex circuits involving multiple resistors and/or batteries. It should be understood that polarity has nothing to do with Ohm’s Law: there will never be negative voltages, currents, or resistance entered into any Ohm’s Law equations! There are other mathematical principles of electricity that do take polarity into account through the use of signs (+ or -
), but not Ohm’s Law. 14.16 What are "series" and "parallel" circuits? Circuits consisting of just one battery and one load resistance are very simple to analyze, but they are not often found in practical applications. Usually, we ﬂnd circuits where more than two components are connected together. There are two basic ways in which to connect more than two circuit components: series and parallel. First, an example of a series circuit: Series R1 R3 1 + - 4 2 R2 3 Here, we have three resistors (labeled R1, R2, and R3), connected in a long chain from one terminal of the battery to the other. (It should be noted that the subscript labeling { those little numbers to the lower-right of the letter "R" { are unrelated to the resistor values in ohms. They serve only to identify one resistor from another.) The deﬂning characteristic of a series circuit is that there is only one path for electrons to ow. In this circuit the electrons ow in a counter-clockwise direction, from point 4 to point 3 to point 2 to point 1 and back around to 4. Now, let’s look at the other type of circuit, a parallel conﬂguration: 1 + - 8 Parallel 2 3 4 R1 R2 R3 7 6 5 Again, we have three resistors, but this time they form more than one continuous path for electrons to ow. There’s one path from 8 to 7 to 2 to 1 and back to 8 again. There’s another from 8 to 7 to 6 to 3 to 2 to 1 and back to 8 again. And then there’s a third path from 8 to 7 to 272 6 to 5 to 4 to 3 to 2 to 1 and back to 8 again. Each individual path (through R1, R2, and R3) is called a branch. The deﬂning characteristic of a parallel circuit is that all components are connected between the same set of electrically common points. Looking at the schematic diagram, we see that points 1, 2, 3, and 4 are all electrically common. So are points 8, 7, 6, and 5. Note that all resistors as well as the battery are connected between these two sets of points. And, of course, the complexity doesn’t stop at simple series and parallel either! We can have circuits that are a
combination of series and parallel, too: Series-parallel R1 1 + - 6 2 5 R2 3 4 R3 In this circuit, we have two loops for electrons to ow through: one from 6 to 5 to 2 to 1 and back to 6 again, and another from 6 to 5 to 4 to 3 to 2 to 1 and back to 6 again. Notice how both current paths go through R1 (from point 2 to point 1). In this conﬂguration, we’d say that R2 and R3 are in parallel with each other, while R1 is in series with the parallel combination of R2 and R3. This is just a preview of things to come. Don’t worry! We’ll explore all these circuit conﬂg- urations in detail, one at a time! The basic idea of a "series" connection is that components are connected end-to-end in a line to form a single path for electrons to ow: Series connection R1 R2 R3 R4 only one path for electrons to flow! The basic idea of a "parallel" connection, on the other hand, is that all components are connected across each other’s leads. In a purely parallel circuit, there are never more than two sets of electrically common points, no matter how many components are connected. There are many paths for electrons to ow, but only one voltage across all components: 273 Parallel connection These points are electrically common R1 R2 R3 R4 These points are electrically common Series and parallel resistor conﬂgurations have very diﬁerent electrical properties. We’ll ex- plore the properties of each conﬂguration in the sections to come. 14.17 Simple series circuits Let’s start with a series circuit consisting of three resistors and a single battery: 9 V 1 + - 4 R1 3 kW 5 kW R3 2 10 kW R2 3 The ﬂrst principle to understand about series circuits is that the amount of current is the same through any component in the circuit. This is because there is only one path for electrons to ow in a series circuit, and because free electrons ow through conductors like marbles in a tube, the rate of ow (marble speed) at any point in the circuit (tube) at any speciﬂc point in time must be equal. From the way that the 9
volt battery is arranged, we can tell that the electrons in this circuit will ow in a counter-clockwise direction, from point 4 to 3 to 2 to 1 and back to 4. However, we have one source of voltage and three resistances. How do we use Ohm’s Law here? An important caveat to Ohm’s Law is that all quantities (voltage, current, resistance, and power) must relate to each other in terms of the same two points in a circuit. For instance, with a single-battery, single-resistor circuit, we could easily calculate any quantity because they all applied to the same two points in the circuit: 274 kW 3 =I 9 volts 3 kW = 3 mA Since points 1 and 2 are connected together with wire of negligible resistance, as are points 3 and 4, we can say that point 1 is electrically common to point 2, and that point 3 is electrically common to point 4. Since we know we have 9 volts of electromotive force between points 1 and 4 (directly across the battery), and since point 2 is common to point 1 and point 3 common to point 4, we must also have 9 volts between points 2 and 3 (directly across the resistor). Therefore, we can apply Ohm’s Law (I = E/R) to the current through the resistor, because we know the voltage (E) across the resistor and the resistance (R) of that resistor. All terms (E, I, R) apply to the same two points in the circuit, to that same resistor, so we can use the Ohm’s Law formula with no reservation. However, in circuits containing more than one resistor, we must be careful in how we apply Ohm’s Law. In the three-resistor example circuit below, we know that we have 9 volts between points 1 and 4, which is the amount of electromotive force trying to push electrons through the series combination of R1, R2, and R3. However, we cannot take the value of 9 volts and divide it by 3k, 10k or 5k › to try to ﬂnd a current value, because we don’t know how much voltage is across any one of those resistors, individually. 9 V 1 + - 4 R1 3 kW 5 kW R3 2 10 kW R2 3 The ﬂgure of 9 volts is a total quantity for the whole
circuit, whereas the ﬂgures of 3k, 10k, and 5k › are individual quantities for individual resistors. If we were to plug a ﬂgure for total voltage into an Ohm’s Law equation with a ﬂgure for individual resistance, the result would not relate accurately to any quantity in the real circuit. For R1, Ohm’s Law will relate the amount of voltage across R1 with the current through R1, given R1’s resistance, 3k›: 275 IR1 = ER1 3 kW ER1 = IR1(3 kW) But, since we don’t know the voltage across R1 (only the total voltage supplied by the battery across the three-resistor series combination) and we don’t know the current through R1, we can’t do any calculations with either formula. The same goes for R2 and R3: we can apply the Ohm’s Law equations if and only if all terms are representative of their respective quantities between the same two points in the circuit. So what can we do? We know the voltage of the source (9 volts) applied across the series combination of R1, R2, and R3, and we know the resistances of each resistor, but since those quantities aren’t in the same context, we can’t use Ohm’s Law to determine the circuit current. If only we knew what the total resistance was for the circuit: then we could calculate total current with our ﬂgure for total voltage (I=E/R). This brings us to the second principle of series circuits: the total resistance of any series circuit is equal to the sum of the individual resistances. This should make intuitive sense: the more resistors in series that the electrons must ow through, the more di–cult it will be for those electrons to ow. In the example problem, we had a 3 k›, 10 k›, and 5 k› resistor in series, giving us a total resistance of 18 k›: Rtotal = R1 + R2 + R3 + Rtotal = 3 kW 10 kW 5 kW + Rtotal = 18 kW In essence, we’ve calculated the equivalent resistance of R1, R2, and R3 combined. Knowing this, we could re-draw the circuit with a single equivalent resistor representing the series combination of
R1, R2, and R3: 9 V 1 + - 4 R1 + R2 + R3 = 18 kW Now we have all the necessary information to calculate circuit current, because we have the voltage between points 1 and 4 (9 volts) and the resistance between points 1 and 4 (18 k›): Itotal= Etotal Rtotal Itotal = 9 volts 18 kW = 500 mA Knowing that current is equal through all components of a series circuit (and we just determined the current through the battery), we can go back to our original circuit schematic and note the current through each component: 276 9 V 1 + - 4 R1 3 kW I = 500 mA I = 500 mA 2 R2 10 kW R3 5 kW 3 Now that we know the amount of current through each resistor, we can use Ohm’s Law to determine the voltage drop across each one (applying Ohm’s Law in its proper context): ER1 = IR1 R1 ER2 = IR2 R2 ER3 = IR3 R3 ER1 = (500 mA)(3 kW) = 1.5 V ER2 = (500 mA)(10 kW) = 5 V ER3 = (500 mA)(5 kW) = 2.5 V Notice the voltage drops across each resistor, and how the sum of the voltage drops (1.5 + 5 + 2.5) is equal to the battery (supply) voltage: 9 volts. This is the third principle of series circuits: that the supply voltage is equal to the sum of the individual voltage drops. However, the method we just used to analyze this simple series circuit can be streamlined for better understanding. By using a table to list all voltages, currents, and resistances in the circuit, it becomes very easy to see which of those quantities can be properly related in any Ohm’s Law equation: R1 R2 R3 Total E I R Volts Amps Ohms Ohm’s Law Ohm’s Law Ohm’s Law Ohm’s Law The rule with such a table is to apply Ohm’s Law only to the values within each vertical column. For instance, ER1 only with IR1 and R1; ER2 only with IR2 and R2; etc. You begin your analysis by ﬂlling in those elements of the table that are given to you from the beginning: R1 R2
R3 3k 10k 5k E I R Total 9 Volts Amps Ohms As you can see from the arrangement of the data, we can’t apply the 9 volts of ET (total 277 voltage) to any of the resistances (R1, R2, or R3) in any Ohm’s Law formula because they’re in diﬁerent columns. The 9 volts of battery voltage is not applied directly across R1, R2, or R3. However, we can use our "rules" of series circuits to ﬂll in blank spots on a horizontal row. In this case, we can use the series rule of resistances to determine a total resistance from the sum of individual resistances: R1 R2 R3 Total 9 3k 10k 5k 18k E I R Volts Amps Ohms Rule of series circuits RT = R1 + R2 + R3 Now, with a value for total resistance inserted into the rightmost ("Total") column, we can apply Ohm’s Law of I=E/R to total voltage and total resistance to arrive at a total current of 500 „A: R1 R2 R3 3k 10k 5k E I R Volts Amps Ohms Total 9 500m 18k Ohm’s Law Then, knowing that the current is shared equally by all components of a series circuit (another "rule" of series circuits), we can ﬂll in the currents for each resistor from the current ﬂgure just calculated: R1 R2 R3 E I R 500m 3k 500m 10k 500m 5k Total 9 500m 18k Volts Amps Ohms Rule of series circuits IT = I1 = I2 = I3 Finally, we can use Ohm’s Law to determine the voltage drop across each resistor, one column at a time: 278 R1 1.5 500m 3k E I R R2 5 500m 10k R3 2.5 500m 5k Total 9 500m 18k Volts Amps Ohms Ohm’s Law Ohm’s Law Ohm’s Law 14.18 Simple parallel circuits Let’s start with a parallel circuit consisting of three resistors and a single battery: 1 + - 9 V 2 3 4 R1 10 kW R2 2 kW R3 1 kW 8 7 6 5 The �
��rst principle to understand about parallel circuits is that the voltage is equal across all components in the circuit. This is because there are only two sets of electrically common points in a parallel circuit, and voltage measured between sets of common points must always be the same at any given time. Therefore, in the above circuit, the voltage across R1 is equal to the voltage across R2 which is equal to the voltage across R3 which is equal to the voltage across the battery. This equality of voltages can be represented in another table for our starting values: R1 9 10k E I R R2 9 2k R3 9 1k Total 9 Volts Amps Ohms Just as in the case of series circuits, the same caveat for Ohm’s Law applies: values for voltage, current, and resistance must be in the same context in order for the calculations to work correctly. However, in the above example circuit, we can immediately apply Ohm’s Law to each resistor to ﬂnd its current because we know the voltage across each resistor (9 volts) and the resistance of each resistor: 279 IR1 = ER1 R1 IR2 = ER2 R2 IR3 = ER3 R3 IR1 = 9 V 10 kW = 0.9 mA IR2 = IR3 = E I R 9 V 2 kW 9 V 1 kW R1 9 0.9m 10k = 4.5 mA = 9 mA R2 9 4.5m 2k R3 9 9m 1k Total 9 Volts Amps Ohms Ohm’s Law Ohm’s Law Ohm’s Law At this point we still don’t know what the total current or total resistance for this parallel circuit is, so we can’t apply Ohm’s Law to the rightmost ("Total") column. However, if we think carefully about what is happening it should become apparent that the total current must equal the sum of all individual resistor ("branch") currents: 1 + - 9 V IR1 IT IT 2 3 4 IR2 IR3 R1 10 kW 2 kW R2 R3 1 kW 8 6 7 5 As the total current exits the negative (-) battery terminal at point 8 and travels through the circuit, some of the ow splits oﬁ at point 7 to go up through R1, some more splits oﬁ at point 6 to go up through R2, and the
remainder goes up through R3. Like a river branching into several smaller streams, the combined ow rates of all streams must equal the ow rate of the whole river. The same thing is encountered where the currents through R1, R2, and R3 join to ow back to the positive terminal of the battery (+) toward point 1: the ow of electrons from point 2 to point 1 must equal the sum of the (branch) currents through R1, R2, and R3. This is the second principle of parallel circuits: the total circuit current is equal to the sum of the individual branch currents. Using this principle, we can ﬂll in the IT spot on our table with the sum of IR1, IR2, and IR3: 280 R1 9 0.9m 10k E I R R2 9 4.5m 2k R3 9 9m 1k Total 9 14.4m Volts Amps Ohms Rule of parallel circuits Itotal = I1 + I2 + I3 Finally, applying Ohm’s Law to the rightmost ("Total") column, we can calculate the total circuit resistance: R1 9 0.9m 10k E I R R2 9 4.5m 2k R3 9 9m 1k Total 9 Volts 14.4m Amps 625 Ohms Rtotal = Etotal Itotal = 9 V 14.4 mA = 625 W Ohm’s Law Please note something very important here. The total circuit resistance is only 625 ›: less than any one of the individual resistors. In the series circuit, where the total resistance was the sum of the individual resistances, the total was bound to be greater than any one of the resistors individually. Here in the parallel circuit, however, the opposite is true: we say that the individual resistances diminish rather than add to make the total. This principle completes our triad of "rules" for parallel circuits, just as series circuits were found to have three rules for voltage, current, and resistance. Mathematically, the relationship between total resistance and individual resistances in a parallel circuit looks like this: Rtotal = 1 1 R2 + 1 R3 1 R1 + The same basic form of equation works for any number of resistors connected together in parallel, just add as many 1/R terms on the denominator of the fraction as needed to accommodate all parallel resistors in the circuit. 14.19 Power calculations When calculating the
power dissipation of resistive components, use any one of the three power equations to derive and answer from values of voltage, current, and/or resistance pertaining to each component: 281 Power equations P = IE P = E2 E R P = I2R This is easily managed by adding another row to our familiar table of voltages, currents, and resistances: E I R P R1 R2 R3 Total Volts Amps Ohms Watts Power for any particular table column can be found by the appropriate Ohm’s Law equation (appropriate based on what ﬂgures are present for E, I, and R in that column). An interesting rule for total power versus individual power is that it is additive for any conﬂguration of circuit: series, parallel, series/parallel, or otherwise. Power is a measure of rate of work, and since power dissipated must equal the total power applied by the source(s) (as per the Law of Conservation of Energy in physics), circuit conﬂguration has no eﬁect on the mathematics. 14.20 Correct use of Ohm’s Law When working through worked examples it is important to try to ﬂgure out what you are doing correctly as well as what you are doing wrong. Make sure you don’t stop doing the good things and try to correct the mistakes. Circuit questions form a large part of high school and early university courses and it is important to understand the concepts properly. One common mistake which students make we’ll discuss here so you know to look out for it when you are working through examples and studying. When applying Ohm’s Laws students often mix up the contexts of voltage, current, and resistance. This means a student might mistakenly use a value for I through one resistor and the value for V across another resistor or a set of connected resistors. Remember this important rule: The variables used in Ohm’s Law equations must be common to the same two points in the circuit under consideration. This is especially important in series-parallel combination circuits where nearby components may have diﬁerent values for both voltage drop and current. When using Ohm’s Law to calculate a variable for a single component: be sure the voltage you’re using is solely across that single component and the current you’re referencing is solely through that single component and † † the resistance you’re referencing
is solely for that single component. † When calculating a variable for a set of components in a circuit, be sure that the voltage, current, and resistance values are speciﬂc to that complete set of components only! 282 A good way to remember this is to pay close attention to the two points on either side of the component or set of components. Making sure that the voltage in question is across those two points, that the current in question is the electron ow from one of those points all the way to the other point, that the resistance in question is the equivalent of a single resistor between those two points, and that the power in question is the total power dissipated by all components between those two points. The "table" method presented for both series and parallel circuits in this chapter is a way to keep the components correct when using Ohm’s Law. In a table like the one shown below, you are only allowed to apply an Ohm’s Law equation for the values of a single vertical column at a time: R1 R2 R3 Total E I R P Volts Amps Ohms Watts Ohm’s Law Ohm’s Law Ohm’s Law Ohm’s Law Deriving values horizontally across columns is allowable as per the principles of series and parallel circuits: For series circuits: R1 R2 R3 Total E I R P Add Volts Equal Amps Add Add Ohms Watts Etotal = E1 + E2 + E3 Itotal = I1 = I2 = I3 Rtotal = R1 + R2 + R3 Ptotal = P1 + P2 + P3 283 For parallel circuits: R1 R2 R3 Total E I R P Equal Volts Add Diminish Add Amps Ohms Watts Etotal = E1 = E2 = E3 Itotal = I1 + I2 + I3 Rtotal = 1 1 R2 + 1 R3 1 R1 + Ptotal = P1 + P2 + P3 The "table" method helps to keep track of all relevant quantities. It also facilitates crosschecking of answers by making it easy to solve for the original unknown variables through other methods, or by working backwards to solve for the initially given values from your solutions. For example, if you have just solved for all unknown voltages, currents, and resistances in a circuit, you can check your work by adding a row at the bottom for power calculations on
each resistor, seeing whether or not all the individual power values add up to the total power. If not, then you must have made a mistake somewhere! While this technique of "cross-checking" your work is nothing new, using the table to arrange all the data for the cross-check(s) results in a minimum of confusion. Aside: Although checking your work might not be fun when you have just worked hard on the problem the beneﬂts are great. Coming back to a problem after a small break (trying another problem) often helps to ﬂnd simple mistakes. If you have done all the work then ﬂnding a simple mistake will be quick to ﬂx because you know exactly what you need to do. Also if you start ﬂnding mistakes while checking you’ll build a mental list and ﬂnd that you’ll stop making them after a while. Do it and you’ll ﬂnd it will pay oﬁ! 14.21 Conductor size The width of a conductor aﬁects the ow of electrons through it. The broader the cross-sectional area (thickness or area of a sl) of the conductor, the more room for electrons to ow, and consequently, the easier it is for ow to occur (less resistance). Electrical wire is usually round in cross-section (although there are some unique exceptions to this rule), and comes in two basic varieties: solid and stranded. Solid copper wire is just as it sounds: a single, solid strand of copper the whole length of the wire. Stranded wire is composed of smaller strands of solid copper wire twisted together to form a single, larger conductor. The greatest beneﬂt of stranded wire is its mechanical exibility, being able to withstand repeated 284 bending and twisting much better than solid copper (which tends to fatigue and break after time). 14.22 Fuses Normally, the ampacity rating of a conductor is a circuit design limit never to be intentionally exceeded, but there is an application where ampacity exceedence is expected: in the case of fuses. A fuse is nothing more than a short length of wire designed to melt and separate in the event of excessive current. Fuses are always connected in series with the component(s) to be protected from overcurrent, so that when the fuse blows (opens) it will open the entire circuit and stop current through the component(s). A fuse connected
in one branch of a parallel circuit, of course, would not aﬁect current through any of the other branches. Normally, the thin piece of fuse wire is contained within a safety sheath to minimize hazards of arc blast if the wire burns open with violent force, as can happen in the case of severe overcurrents. In the case of small automotive fuses, the sheath is transparent so that the fusible element can be visually inspected. Residential wiring used to commonly employ screw-in fuses with glass bodies and a thin, narrow metal foil strip in the middle. 14.23 Important Equations and Quantities Quantity Symbol Unit S.I. Units Direction Units or Table 14.1: Units used in Electricity and Magnetism † † † † † † † † REVIEW: A circuit is an unbroken loop of conductive material that allows electrons to ow through continuously without beginning or end. If a circuit is "broken," that means it’s conductive elements no longer form a complete path, and continuous electron ow cannot occur in it. The location of a break in a circuit is irrelevant to its inability to sustain continuous electron ow. Any break, anywhere in a circuit prevents electron ow throughout the circuit. REVIEW: Electrons can be motivated to ow through a conductor by a the same force manifested in static electricity. Voltage is the measure of speciﬂc potential energy (potential energy per unit charge) between two locations. In layman’s terms, it is the measure of "push" available to motivate electrons. Voltage, as an expression of potential energy, is always relative between two locations, or points. Sometimes it is called a voltage "drop." 285 † † † † † † † † † † † † † † † † † † † † † When a voltage source is connected to a circuit, the voltage will cause a uniform ow of electrons through that circuit called a current. In a single (one loop) circuit, the amount current of current at any point is the same as the amount of current at any other point. If a circuit containing a voltage source is broken, the full voltage of that source will appear across the points of the break. The +/- orientation a voltage drop is called the polarity. It is also relative between two points. REVIEW: Resistance is the measure of opposition
to electric current. A short circuit is an electric circuit oﬁering little or no resistance to the ow of electrons. Short circuits are dangerous with high voltage power sources because the high currents encountered can cause large amounts of heat energy to be released. An open circuit is one where the continuity has been broken by an interruption in the path for electrons to ow. A closed circuit is one that is complete, with good continuity throughout. A device designed to open or close a circuit under controlled conditions is called a switch. The terms "open" and "closed" refer to switches as well as entire circuits. An open switch is one without continuity: electrons cannot ow through it. A closed switch is one that provides a direct (low resistance) path for electrons to ow through. REVIEW: Connecting wires in a circuit are assumed to have zero resistance unless otherwise stated. Wires in a circuit can be shortened or lengthened without impacting the circuit’s function { all that matters is that the components are attached to one another in the same sequence. Points directly connected together in a circuit by zero resistance (wire) are considered to be electrically common. Electrically common points, with zero resistance between them, will have zero voltage dropped between them, regardless of the magnitude of current (ideally). The voltage or resistance readings referenced between sets of electrically common points will be the same. These rules apply to ideal conditions, where connecting wires are assumed to possess absolutely zero resistance. In real life this will probably not be the case, but wire resistances should be low enough so that the general principles stated here still hold. REVIEW: Power is additive in any conﬂguration of resistive circuit: PT otal = P1 + P2 +... Pn REVIEW: 286 † † † † When electrons ow through a conductor, a magnetic ﬂeld will be produced around that conductor. The left-hand rule states that the magnetic ux lines produced by a current-carrying wire will be oriented the same direction as the curled ﬂngers of a person’s left hand (in the "hitchhiking" position), with the thumb pointing in the direction of electron ow. The magnetic ﬂeld force produced by a current-carrying wire can be greatly increased by shaping the wire into a coil instead of a straight line. If wound in a coil shape, the magnetic ﬂeld will be oriented along
the axis of the coil’s length. The magnetic ﬂeld force produced by an electromagnet (called the magnetomotive force, or mmf), is proportional to the product (multiplication) of the current through the electromagnet and the number of complete coil "turns" formed by the wire. 287 Chapter 15 Magnets and Electromagnetism sectionPermanent magnets Magnetism has been known to mankind for many thousands of years. Lodestone, a magnetized form of the iron oxide mineral magnetite which has the property of attracting iron objects, is referred to in old European and Asian historical records, around 800 BC in Europe and earlier in the East, around 2600 BC. The root of the English word magnet is the Greek word magnes, thought to be derived from Magnesia in Asia Minor, once an important source of lodestone. Lodestone was used as a navigational compass as it was found to orient itself in a north-south direction if left free to rotate by suspension on a string or on a oat in water. Interesting Fact: A compass is a navigational instrument for ﬂnding directions. It consists of a magnetised pointer free to align itself accurately with Earth’s magnetic ﬂeld. A compass provides a known reference direction which is of great assistance in navigation. The cardinal points are north, south, east and west. A compass can be used in conjunction with a clock and a sextant to provide a very accurate navigation capability. This device greatly improved maritime trade by making travel safer and more e–cient. A compass can be any magnetic device using a needle to indicate the direction of the magnetic north of a planet’s magnetosphere. Any instrument with a magnetized bar or needle turning freely upon a pivot and pointing in a northerly and southerly direction can be considered a compass. Aside: In 1269, Frenchmen Peter Peregrinus and Pierre de Maricourt, using a compass and a lodestone, found that the magnetic force of the lodestone was diﬁerent at the opposite ends, which they deﬂned to be the poles of the magnet. Like poles of magnets repel one another whilst unlike poles attract. These poles always occur in pairs. It is impossible to isolate a single pole. Breaking a piece of magnet in half results in two pieces, each with it’s own pair of poles. 288 N magnet S... after breaking in half.
.. N magnet S N magnet S The Earth itself is a magnet. Its magnetic poles are approximately aligned along the Earth’s axis of rotation. The magnitude of forces between the poles of magnets follows an inverse square law; i. e. it varies inversely as the square of the distance of separation. Magnetic forces are a result of magnetic ﬂelds. By placing a magnet underneath a piece of paper and sprinkling iron ﬂlings on top one can map the magnetic ﬂeld. The ﬂlings align themselves parallel to the ﬂeld. Magnetic ﬂelds can be represented by magnetic ﬂeld lines which are parallel to the magnetic ﬂeld and whose spacing represents the relative strength of the magnetic ﬂeld. The strength of the magnetic ﬂeld is referred to as the magnetic ux. Magnetic ﬂeld lines form closed loops. In a bar magnet magnetic ﬂeld lines emerge at one pole and then curve around to the other pole with the rest of the loop being inside the magnet. magnetic field N magnet S As already said, opposite poles of a magnet attract each other and bringing them together results in their magnetic ﬂeld lines converging. Like poles of a magnet repel each other and bringing them together results in their magnetic ﬂeld lines diverging. 289 Ferromagnetism is a phenomenon exhibited by materials like iron, nickel or cobalt. These materials are known as permanent magnets. They always magnetize so as to be attracted to a magnet, regardless of which magnetic pole is brought toward the unmagnetized iron: N iron S N magnet S attraction Interesting Fact: The cause of Earth’s magnetic ﬂeld is not known for certain, but is possibly explained by the dynamo theory. The magnetic ﬂeld extends several tens of thousands of kilometers into space. The ﬂeld is approximately a magnetic dipole, with one pole near the geographic north pole and the other near the geographic south pole. An imaginary line joining the magnetic poles would be inclined by approximately 11.3 from the planet’s axis of rotation. The location of the magnetic poles is not static but wanders as much as several kilometers a year. The two poles wander independently of each other and are not at exact opposite positions on the globe. The Earth’s magnetic ﬂeld reverses
at intervals, ranging from tens of thousands to many millions of years, with an average interval of 250,000 years. It is believed that this last occurred some 780,000 years ago. The mechanism responsible for geomagnetic reversals is not well understood. When the North reappears in the opposite direction, we would interpret this as a reversal, whereas turning oﬁ and returning in the same direction is called a geomagnetic excursion. At present, the overall geomagnetic ﬂeld is becoming weaker at a rate which would, if it continues, cause the ﬂeld to disappear, albeit temporarily, by about around 3000-4000 AD. The deterioration began roughly 150 years ago and has accelerated in the past several years. So far the strength of the earth’s ﬂeld has decreased by 10 to 15 percent. The ability of a ferromagnetic material tends to retain its magnetization after an external ﬂeld is removed is called it’s retentivity. 290 Paramagnetic materials are materials like aluminum or platinum which become magnetized in an external magnetic ﬂeld in a similar way to ferromagnetic materials but lose their magnetism when the external magnetic ﬂeld is removed. Diamagnetism is exhibited by materials like copper or bismuth which become magnetized in a magnetic ﬂeld with a polarity opposite to the external magnetic ﬂeld. Unlike iron, they are slightly repelled by a magnet S diamagnetic material N N magnet S repulsion The cause of Earth’s magnetic ﬂeld is not known for certain, but is possibly explained by the dynamo theory. The magnetic ﬂeld extends several tens of thousands of kilometers into space. The ﬂeld is approximately a magnetic dipole, with one pole near the geographic north pole and the other near the geographic south pole. An imaginary line joining the magnetic poles would be inclined by approximately 11.3 from the planet’s axis of rotation. The location of the magnetic poles is not static but wanders as much as several kilometers a year. The two poles wander independently of each other and are not at exact opposite positions on the globe. Currently the south magnetic pole is further from the geographic south pole than than north magnetic pole is from the north geographic pole. The strength of the ﬂeld at the Earth’s surface at this time ranges from less
than 30 microtesla (0.3 gauss) in an area including most of South America and South Africa to over 60 microtesla (0.6 gauss) around the magnetic poles in northern Canada and south of Australia, and in part of Siberia. The ﬂeld is similar to that of a bar magnet, but this similarity is superﬂcial. The magnetic ﬂeld of a bar magnet, or any other type of permanent magnet, is created by the coordinated motions of electrons (negatively charged particles) within iron atoms. The Earth’s core, however, is hotter than 1043 K, the temperature at which the orientations of electron orbits within iron become randomized. Therefore the Earth’s magnetic ﬂeld is not caused by magnetised iron deposits, but mostly by electric currents (known as telluric currents). Another feature that distinguishes the Earth magnetically from a bar magnet is its magnetosphere. A magnetosphere is the region around an astronomical object, in which phenomena are dominated by its magnetic ﬂeld. Earth is surrounded by a magnetosphere, as are the magnetized planets Jupiter, Saturn, Uranus and Neptune. Mercury is magnetized, but too weakly to trap plasma. Mars has patchy surface magnetization. The distant ﬂeld of Earth is greatly modiﬂed by the solar wind, a hot outow from the sun, consisting of solar ions (mainly hydrogen) moving at about 400 km/s. Earth’s magnetic ﬂeld forms an obstacle to the solar wind. The Earth’s magnetic ﬂeld reverses at intervals, ranging from tens of thousands to many It is believed that this last oc- millions of years, with an average interval of 250,000 years. 291 curred some 780,000 years ago. The mechanism responsible for geomagnetic reversals is not well understood. When the North reappears in the opposite direction, we would interpret this as a reversal, whereas turning oﬁ and returning in the same direction is called a geomagnetic excursion. At present, the overall geomagnetic ﬂeld is becoming weaker at a rate which would, if it continues, cause the ﬂeld to disappear, albeit temporarily, by about around 3000-4000 AD. The deterioration began roughly 150 years ago and has accelerated in the past several years. So far the
strength of the earth’s ﬂeld has decreased by 10 to 15 percent. 15.1 Electromagnetism The discovery of the relationship between magnetism and electricity was, like so many other scientiﬂc discoveries, stumbled upon almost by accident. The Danish physicist Hans Christian Oersted was lecturing one day in 1820 on the possibility of electricity and magnetism being related to one another, and in the process demonstrated it conclusively by experiment in front of his whole class! By passing an electric current through a metal wire suspended above a magnetic compass, Oersted was able to produce a deﬂnite motion of the compass needle in response to the current. What began as conjecture at the start of the class session was conﬂrmed as fact at the end. Needless to say, Oersted had to revise his lecture notes for future classes! His serendipitous discovery paved the way for a whole new branch of science: electromagnetics. Detailed experiments showed that the magnetic ﬂeld produced by an electric current is always oriented perpendicular to the direction of ow. A simple method of showing this relationship is called the left-hand rule. Simply stated, the left-hand rule says that the magnetic ux lines produced by a current-carrying wire will be oriented the same direction as the curled ﬂngers of a person’s left hand (in the "hitchhiking" position), with the thumb pointing in the direction of electron ow: The "left-hand" rule I I I The magnetic ﬂeld encircles this straight piece of current-carrying wire, the magnetic ux I lines having no deﬂnite "north" or "south’ poles. (NOTE TO SELF: Need to add wires attracting or wires repelling) While the magnetic ﬂeld surrounding a current-carrying wire is indeed interesting, it is quite weak for common amounts of current, able to deect a compass needle and not much more. To 292 create a stronger magnetic ﬂeld force (and consequently, more ﬂeld ux) with the same amount of electric current, we can wrap the wire into a coil shape, where the circling magnetic ﬂelds around the wire will join to create a larger ﬂeld with a deﬂnite magnetic (north and south) polarity
: S N magnetic field The amount of magnetic ﬂeld force generated by a coiled wire is proportional to the current through the wire multiplied by the number of "turns" or "wraps" of wire in the coil. This ﬂeld force is called magnetomotive force (mmf), and is very much analogous to electromotive force (E) in an electric circuit. An electromagnet is a piece of wire intended to generate a magnetic ﬂeld with the passage of electric current through it. Though all current-carrying conductors produce magnetic ﬂelds, an electromagnet is usually constructed in such a way as to maximize the strength of the magnetic ﬂeld it produces for a special purpose. Electromagnets ﬂnd frequent application in research, industry, medical, and consumer products. As an electrically-controllable magnet, electromagnets ﬂnd application in a wide variety of "electromechanical" devices: machines that eﬁect mechanical force or motion through electrical power. Perhaps the most obvious example of such a machine is the electric motor. Relay Applying current through the coil causes the switch to close. Relays can be constructed to actuate multiple switch contacts, or operate them in "reverse" (energizing the coil will open the switch contact, and unpowering the coil will allow it to spring closed again). 293 Multiple-contact relay Relay with "normallyclosed" contact 15.2 Magnetic units of measurement If the burden of two systems of measurement for common quantities (English vs. metric) throws your mind into confusion, this is not the place for you! Due to an early lack of standardization in the science of magnetism, we have been plagued with no less than three complete systems of measurement for magnetic quantities. First, we need to become acquainted with the various quantities associated with magnetism. There are quite a few more quantities to be dealt with in magnetic systems than for electrical systems. With electricity, the basic quantities are Voltage (E), Current (I), Resistance (R), and Power (P). The ﬂrst three are related to one another by Ohm’s Law (E=IR ; I=E/R ; R=E/I), while Power is related to voltage, current, and resistance by Joule’s Law (P=IE ; P=I2R ; P=E2
/R). With magnetism, we have the following quantities to deal with: Magnetomotive Force { The quantity of magnetic ﬂeld force, or "push." Analogous to electric voltage (electromotive force). Field Flux { The quantity of total ﬂeld eﬁect, or "substance" of the ﬂeld. Analogous to electric current. Field Intensity { The amount of ﬂeld force (mmf) distributed over the length of the elec- tromagnet. Sometimes referred to as Magnetizing Force. Flux Density { The amount of magnetic ﬂeld ux concentrated in a given area. Reluctance { The opposition to magnetic ﬂeld ux through a given volume of space or material. Analogous to electrical resistance. Permeability { The speciﬂc measure of a material’s acceptance of magnetic ux, analogous to the speciﬂc resistance of a conductive material (‰), except inverse (greater permeability means easier passage of magnetic ux, whereas greater speciﬂc resistance means more di–cult passage of electric current)... But wait. the fun is just beginning! Not only do we have more quantities to keep track of with magnetism than with electricity, but we have several diﬁerent systems of unit measurement for each of these quantities. As with common quantities of length, weight, volume, and temperature, we have both English and metric systems. However, there is actually more than 294 one metric system of units, and multiple metric systems are used in magnetic ﬂeld measurements! One is called the cgs, which stands for Centimeter-Gram-Second, denoting the root measures upon which the whole system is based. The other was originally known as the mks system, which stood for Meter-Kilogram-Second, which was later revised into another system, called rmks, standing for Rationalized Meter-Kilogram-Second. This ended up being adopted as an international standard and renamed SI (Systeme International). Quantity Symbol Unit of Measurement and abbreviation Field Force mmf Gilbert (Gb) Amp-turn Amp-turn CGS SI English Field Flux Field Intensity Flux Density Reluctance F H B ´ Permeability m Maxwell (Mx) Weber (Wb) Line Oersted (
Oe) Amp-turns per meter Amp-turns per inch Gauss (G) Tesla (T) Lines per square inch Gilberts per Maxwell Amp-turns per Weber Amp-turns per line Gauss per Oersted Tesla-meters per Amp-turn Lines per inch-Ampturn And yes, the „ symbol is really the same as the metric preﬂx "micro." I ﬂnd this especially confusing, using the exact same alphabetical character to symbolize both a speciﬂc quantity and a general metric preﬂx! As you might have guessed already, the relationship between ﬂeld force, ﬂeld ux, and reluctance is much the same as that between the electrical quantities of electromotive force (E), current (I), and resistance (R). This provides something akin to an Ohm’s Law for magnetic circuits: A comparison of "Ohm’s Law" for electric and magnetic circuits: E = IR Electrical mmf = F´ Magnetic And, given that permeability is inversely analogous to speciﬂc resistance, the equation for ﬂnding the reluctance of a magnetic material is very similar to that for ﬂnding the resistance of a conductor: A comparison of electrical and magnetic opposition: R = r l A Electrical ´ = l mA Magnetic 295 In either case, a longer piece of material provides a greater opposition, all other factors being equal. Also, a larger cross-sectional area makes for less opposition, all other factors being equal. 15.3 Electromagnetic induction While Oersted’s surprising discovery of electromagnetism paved the way for more practical applications of electricity, it was Michael Faraday who gave us the key to the practical generation of electricity: electromagnetic induction. Faraday discovered that a voltage would be generated across a length of wire if that wire was exposed to a perpendicular magnetic ﬂeld ux of changing intensity. An easy way to create a magnetic ﬂeld of changing intensity is to move a permanent magnet next to a wire or coil of wire. Remember: the magnetic ﬂeld must increase or decrease in intensity perpendicular to the wire (so that the lines of ux "cut across" the conductor), or else no voltage will be induced: Electromagnetic induction current changes direction with change in magnet motion voltage changes polarity with change in magnet motion
- + V + - N S magnet moved back and forth Faraday was able to mathematically relate the rate of change of the magnetic ﬂeld ux with induced voltage (note the use of a lower-case letter "e" for voltage. This refers to instantaneous voltage, or voltage at a speciﬂc point in time, rather than a steady, stable voltage.): e = N dF dt Where, e = (Instantaneous) induced voltage in volts N = F = t = Number of turns in wire coil (straight wire = 1) Magnetic flux in Webers Time in seconds The "d" terms are standard calculus notation, representing rate-of-change of ux over time. 296 "N" stands for the number of turns, or wraps, in the wire coil (assuming that the wire is formed in the shape of a coil for maximum electromagnetic e–ciency). This phenomenon is put into obvious practical use in the construction of electrical generators, which use mechanical power to move a magnetic ﬂeld past coils of wire to generate voltage. However, this is by no means the only practical use for this principle. If we recall that the magnetic ﬂeld produced by a current-carrying wire was always perpendicular to that wire, and that the ux intensity of that magnetic ﬂeld varied with the amount of current through it, we can see that a wire is capable of inducing a voltage along its own length simply due to a change in current through it. This eﬁect is called self-induction: a changing magnetic ﬂeld produced by changes in current through a wire inducing voltage along the length of that same wire. If the magnetic ﬂeld ux is enhanced by bending the wire into the shape of a coil, and/or wrapping that coil around a material of high permeability, this eﬁect of self-induced voltage will be more intense. A device constructed to take advantage of this eﬁect is called an inductor, and will be discussed in greater detail in the next chapter. A device speciﬂcally designed to produce the eﬁect of mutual inductance between two or more coils is called a transformer. Because magnetically-induced voltage only happens when the magnetic ﬂeld ux is changing in strength relative to the wire, mutual inductance between two coils can only happen with alternating (changing { AC
) voltage, and not with direct (steady { DC) voltage. The only applications for mutual inductance in a DC system is where some means is available to switch power on and oﬁ to the coil (thus creating a pulsing DC voltage), the induced voltage peaking at every pulse. A very useful property of transformers is the ability to transform voltage and current levels If the according to a simple ratio, determined by the ratio of input and output coil turns. energized coil of a transformer is energized by an AC voltage, the amount of AC voltage induced in the unpowered coil will be equal to the input voltage multiplied by the ratio of output to input wire turns in the coils. Conversely, the current through the windings of the output coil compared to the input coil will follow the opposite ratio: if the voltage is increased from input coil to output coil, the current will be decreased by the same proportion. This action of the transformer is analogous to that of mechanical gear, belt sheave, or chain sprocket ratios: 297 Torque-reducing geartrain Large gear (many teeth) Small gear (few teeth) + + high torque, low speed low torque, high speed "Step-down" transformer high voltage AC voltage source many turns low voltage few turns Load high current low current A transformer designed to output more voltage than it takes in across the input coil is called a "step-up" transformer, while one designed to do the opposite is called a "step-down," in reference to the transformation of voltage that takes place. The current through each respective coil, of course, follows the exact opposite proportion. 15.4 AC Most students of electricity begin their study with what is known as direct current (DC), which is electricity owing in a constant direction, and/or possessing a voltage with constant polarity. DC is the kind of electricity made by a battery (with deﬂnite positive and negative terminals), or the kind of charge generated by rubbing certain types of materials against each other. As useful and as easy to understand as DC is, it is not the only "kind" of electricity in use. Certain sources of electricity (most notably, rotary electro-mechanical generators) naturally produce voltages alternating in polarity, reversing positive and negative over time. Either as a voltage switching polarity or as a current switching direction back and forth, this "kind" of electricity is known as Alternating Current (AC): 298 DIRECT CURRENT (DC) ALTERN
ATING CURRENT (AC) I I I I Whereas the familiar battery symbol is used as a generic symbol for any DC voltage source, the circle with the wavy line inside is the generic symbol for any AC voltage source. One might wonder why anyone would bother with such a thing as AC. It is true that in some cases AC holds no practical advantage over DC. In applications where electricity is used to dissipate energy in the form of heat, the polarity or direction of current is irrelevant, so long as there is enough voltage and current to the load to produce the desired heat (power dissipation). However, with AC it is possible to build electric generators, motors and power distribution systems that are far more e–cient than DC, and so we ﬂnd AC used predominately across the world in high power applications. To explain the details of why this is so, a bit of background knowledge about AC is necessary. If a machine is constructed to rotate a magnetic ﬂeld around a set of stationary wire coils with the turning of a shaft, AC voltage will be produced across the wire coils as that shaft is rotated, in accordance with Faraday’s Law of electromagnetic induction. This is the basic operating principle of an AC generator, also known as an alternator : Alternator operation Step #1 Step #2 S N no current! Load Step #3 N S no current! Load N S + I - I Load Step #4 S N - I I Load + Notice how the polarity of the voltage across the wire coils reverses as the opposite poles of 299 the rotating magnet pass by. Connected to a load, this reversing voltage polarity will create a reversing current direction in the circuit. The faster the alternator’s shaft is turned, the faster the magnet will spin, resulting in an alternating voltage and current that switches directions more often in a given amount of time. While DC generators work on the same general principle of electromagnetic induction, their construction is not as simple as their AC counterparts. With a DC generator, the coil of wire is mounted in the shaft where the magnet is on the AC alternator, and electrical connections are made to this spinning coil via stationary carbon "brushes" contacting copper strips on the rotating shaft. All this is necessary to switch the coil’s changing output polarity to the external circuit so the external circuit sees a constant polarity: (DC) Generator operation Step #1 Step #2 N S SN Load Step #3 N S SN -
+ N S + Load Step #4 - + SN + N S - I N S - I Load Load The generator shown above will produce two pulses of voltage per revolution of the shaft, both pulses in the same direction (polarity). In order for a DC generator to produce constant voltage, rather than brief pulses of voltage once every 1/2 revolution, there are multiple sets of coils making intermittent contact with the brushes. The diagram shown above is a bit more simpliﬂed than what you would see in real life. The problems involved with making and breaking electrical contact with a moving coil should be obvious (sparking and heat), especially if the shaft of the generator is revolving at high speed. If the atmosphere surrounding the machine contains ammable or explosive vapors, the practical problems of spark-producing brush contacts are even greater. An AC generator (alternator) does not require brushes and commutators to work, and so is immune to these problems experienced by DC generators. The beneﬂts of AC over DC with regard to generator design is also reected in electric motors. While DC motors require the use of brushes to make electrical contact with moving coils of wire, AC motors do not. In fact, AC and DC motor designs are very similar to their 300 generator counterparts (identical for the sake of this tutorial), the AC motor being dependent upon the reversing magnetic ﬂeld produced by alternating current through its stationary coils of wire to rotate the rotating magnet around on its shaft, and the DC motor being dependent on the brush contacts making and breaking connections to reverse current through the rotating coil every 1/2 rotation (180 degrees). So we know that AC generators and AC motors tend to be simpler than DC generators and DC motors. This relative simplicity translates into greater reliability and lower cost of manufacture. But what else is AC good for? Surely there must be more to it than design details of generators and motors! Indeed there is. There is an eﬁect of electromagnetism known as mutual induction, whereby two or more coils of wire placed so that the changing magnetic ﬂeld created by one induces a voltage in the other. If we have two mutually inductive coils and we energize one coil with AC, we will create an AC voltage in the other coil. When used as such, this device is known as a transformer : Transformer AC voltage source Induced AC voltage The fundamental signiﬂcance of a transformer is its ability to step
voltage up or down from the powered coil to the unpowered coil. The AC voltage induced in the unpowered ("secondary") coil is equal to the AC voltage across the powered ("primary") coil multiplied by the ratio of secondary coil turns to primary coil turns. If the secondary coil is powering a load, the current through the secondary coil is just the opposite: primary coil current multiplied by the ratio of primary to secondary turns. This relationship has a very close mechanical analogy, using torque and speed to represent voltage and current, respectively: 301 Speed multiplication geartrain Large gear (many teeth) Small gear (few teeth) high torque low speed + + low torque high speed "Step-down" transformer high voltage AC voltage source many turns low voltage few turns Load high current low current If the winding ratio is reversed so that the primary coil has less turns than the secondary coil, the transformer "steps up" the voltage from the source level to a higher level at the load: Speed reduction geartrain Large gear (many teeth) Small gear (few teeth) low torque high speed + + high torque low speed "Step-up" transformer high voltage AC voltage source low voltage few turns high current many turns Load low current The transformer’s ability to step AC voltage up or down with ease gives AC an advantage unmatched by DC in the realm of power distribution. When transmitting electrical power over long distances, it is far more e–cient to do so with stepped-up voltages and stepped-down currents 302 (smaller-diameter wire with less resistive power losses), then step the voltage back down and the current back up for industry, business, or consumer use use. high voltage Power Plant Step-up low voltage... to other customers Step-down Home or Business low voltage Transformer technology has made long-range electric power distribution practical. Without the ability to e–ciently step voltage up and down, it would be cost-prohibitive to construct power systems for anything but close-range (within a few miles at most) use. As useful as transformers are, they only work with AC, not DC. Because the phenomenon of mutual inductance relies on changing magnetic ﬂelds, and direct current (DC) can only produce steady magnetic ﬂelds, transformers simply will not work with direct current. Of course, direct current may be interrupted (pulsed) through the primary winding of a transformer to create a changing magnetic ﬂeld (as is

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