Electricity and Magnetism For 50 years, Edward M Purcell’s classic textbook has introduced students to the world of electricity and magnetism This third edition has been brought up to date and is now in SI units It features hundreds of new examples, problems, and figures, and contains discussions of real-life applications The textbook covers all the standard introductory topics, such as electrostatics, magnetism, circuits, electromagnetic waves, and electric and magnetic fields in matter Taking a nontraditional approach, magnetism is derived as a relativistic effect Mathematical concepts are introduced in parallel with the physical topics at hand, making the motivations clear Macroscopic phenomena are derived rigorously from the underlying microscopic physics With worked examples, hundreds of illustrations, and nearly 600 end-of-chapter problems and exercises, this textbook is ideal for electricity and magnetism courses Solutions to the exercises are available for instructors at www.cambridge.org/Purcell-Morin EDWARD M PURCELL (1912–1997) was the recipient of many awards for his scientific, educational, and civic work In 1952 he shared the Nobel Prize for Physics for the discovery of nuclear magnetic resonance in liquids and solids, an elegant and precise method of determining the chemical structure of materials that serves as the basis for numerous applications, including magnetic resonance imaging (MRI) During his career he served as science adviser to Presidents Dwight D Eisenhower, John F Kennedy, and Lyndon B Johnson DAVID J MORIN is a Lecturer and the Associate Director of Undergraduate Studies in the Department of Physics, Harvard University He is the author of the textbook Introduction to Classical Mechanics (Cambridge University Press, 2008) THIRD EDITION ELECTRICITY AND MAGNETISM EDWARD M PURCELL DAVID J MORIN Harvard University, Massachusetts CA M B R I D G E U N I V E R S I T Y P R E S S Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi, Mexico City Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/Purcell-Morin © D Purcell, F Purcell, and D Morin 2013 This edition is not for sale in India This publication is in copyright Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press Previously published by Mc-Graw Hill, Inc., 1985 First edition published by Education Development Center, Inc., 1963, 1964, 1965 First published by Cambridge University Press 2013 Printed in the United States by Sheridan Inc A catalog record for this publication is available from the British Library Library of Congress cataloging-in-publication data Purcell, Edward M Electricity and magnetism / Edward M Purcell, David J Morin, Harvard University, Massachusetts – Third edition pages cm ISBN 978-1-107-01402-2 (Hardback) Electricity Magnetism I Title QC522.P85 2012 537–dc23 2012034622 ISBN 978-1-107-01402-2 Hardback Additional resources for this publication at www.cambridge.org/Purcell-Morin Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate Preface to the third edition of Volume xiii Preface to the second edition of Volume xvii Preface to the first edition of Volume xxi CHAPTER ELECTROSTATICS: CHARGES AND FIELDS 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 1.12 1.13 1.14 1.15 1.16 Electric charge Conservation of charge Quantization of charge Coulomb’s law Energy of a system of charges Electrical energy in a crystal lattice The electric field Charge distributions Flux Gauss’s law Field of a spherical charge distribution Field of a line charge Field of an infinite flat sheet of charge The force on a layer of charge Energy associated with the electric field Applications 1 11 14 16 20 22 23 26 28 29 30 33 35 CONTENTS vi CONTENTS Chapter summary Problems Exercises CHAPTER THE ELECTRIC POTENTIAL 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13 2.14 2.15 2.16 2.17 2.18 Line integral of the electric field Potential difference and the potential function Gradient of a scalar function Derivation of the field from the potential Potential of a charge distribution Uniformly charged disk Dipoles Divergence of a vector function Gauss’s theorem and the differential form of Gauss’s law The divergence in Cartesian coordinates The Laplacian Laplace’s equation Distinguishing the physics from the mathematics The curl of a vector function Stokes’ theorem The curl in Cartesian coordinates The physical meaning of the curl Applications Chapter summary Problems Exercises CHAPTER ELECTRIC FIELDS AROUND CONDUCTORS 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 Conductors and insulators Conductors in the electrostatic field The general electrostatic problem and the uniqueness theorem Image charges Capacitance and capacitors Potentials and charges on several conductors Energy stored in a capacitor Other views of the boundary-value problem Applications Chapter summary 38 39 47 58 59 61 63 65 65 68 73 78 79 81 85 86 88 90 92 93 95 100 103 105 112 124 125 126 132 136 141 147 149 151 153 155 CONTENTS Problems Exercises 155 163 CHAPTER ELECTRIC CURRENTS 177 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 Electric current and current density Steady currents and charge conservation Electrical conductivity and Ohm’s law The physics of electrical conduction Conduction in metals Semiconductors Circuits and circuit elements Energy dissipation in current flow Electromotive force and the voltaic cell Networks with voltage sources Variable currents in capacitors and resistors Applications Chapter summary Problems Exercises CHAPTER THE FIELDS OF MOVING CHARGES 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 From Oersted to Einstein Magnetic forces Measurement of charge in motion Invariance of charge Electric field measured in different frames of reference Field of a point charge moving with constant velocity Field of a charge that starts or stops Force on a moving charge Interaction between a moving charge and other moving charges Chapter summary Problems Exercises CHAPTER THE MAGNETIC FIELD 6.1 6.2 Definition of the magnetic field Some properties of the magnetic field 177 180 181 189 198 200 204 207 209 212 215 217 221 222 226 235 236 237 239 241 243 247 251 255 259 267 268 270 277 278 286 vii viii CONTENTS Vector potential Field of any current-carrying wire Fields of rings and coils Change in B at a current sheet How the fields transform Rowland’s experiment Electrical conduction in a magnetic field: the Hall effect 6.10 Applications Chapter summary Problems Exercises 6.3 6.4 6.5 6.6 6.7 6.8 6.9 CHAPTER ELECTROMAGNETIC INDUCTION 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10 7.11 Faraday’s discovery Conducting rod moving through a uniform magnetic field Loop moving through a nonuniform magnetic field Stationary loop with the field source moving Universal law of induction Mutual inductance A reciprocity theorem Self-inductance Circuit containing self-inductance Energy stored in the magnetic field Applications Chapter summary Problems Exercises CHAPTER ALTERNATING-CURRENT CIRCUITS 8.1 8.2 8.3 8.4 8.5 8.6 8.7 A resonant circuit Alternating current Complex exponential solutions Alternating-current networks Admittance and impedance Power and energy in alternating-current circuits Applications Chapter summary Problems Exercises 293 296 299 303 306 314 314 317 322 323 331 342 343 345 346 352 355 359 362 364 366 368 369 373 374 380 388 388 394 402 405 408 415 418 420 421 424 Chapter summary 1.5 volt battery The main idea behind the converter is the fact that inductors resist sudden changes in current Consider a circuit where current flows from a battery through an inductor If a switch is opened downstream from the inductor, and if an alternative path is available through a capacitor, then current will still flow for a brief time through the inductor onto the capacitor Charge will therefore build up on the capacitor This process is repeated at a high frequency, perhaps 50 kHz Even if the back voltage from the capacitor is higher than the forward voltage from the battery, a positive current will still flow briefly onto the capacitor each time the switch is opened (Backward current can be prevented with a diode.) The capacitor then serves as a higher-voltage effective battery for powering the LED The magnetic field of the earth cannot be caused by a permanent magnet, because the interior temperature is far too hot to allow the iron core to exist in a state of permanent magnetization Instead, the field is caused by the dynamo effect (see Problem 7.19 and Exercise 7.47) A source of energy is needed to drive the dynamo, otherwise the field would decay on a time scale of 20,000 years or so This source isn’t completely understood; possibilities include tidal forces, gravitational setting, radioactivity, and the buoyancy of lighter elements The dynamo mechanism requires a fluid region inside the earth (this region is the outer core) and also a means of charge separation (perhaps friction between layers) so that currents can exist It also requires that the earth be rotating, so that the Coriolis force can act on the fluid Computer models indicate that the motion of the fluid is extremely complicated, and also that the reversal of the field (which happens every 200,000 years, on average) is likewise complicated The poles don’t simply rotate into each other Rather, all sorts of secondary poles appear on the surface of the earth during the process, which probably takes a few thousand years Who knows where all the famous explorers and their compasses would have ended up if a reversal had been taking place during the last thousand years! In recent years, the magnetic north pole has been moving at the brisk rate of about 50 km per year This speed isn’t terribly unusual, though, so it doesn’t necessarily imply that a reversal is imminent CHAPTER SUMMARY • Faraday discovered that a current in one circuit can be induced by a changing current in another circuit • If a loop moves through a magnetic field, the induced emf equals E = vw(B1 − B2 ), where w is the length of the transverse sides, and the B’s are the fields at these sides This emf can be viewed as a consequence of the Lorentz force acting on the charges in the transverse sides 373 374 Electromagnetic induction • More generally, the emf can be written in terms of the magnetic flux as d (7.81) E =− dt This is known as Faraday’s law of induction, and it holds in all cases: the loop can be moving, or the source of the magnetic field can be moving, or the flux can be changed by some other arbitrary means The sign of the induced emf is determined by Lenz’s law: the induced current flows in the direction that produces a magnetic field that opposes the change in flux The differential form of Faraday’s law is ∇ ×E=− ∂B ∂t (7.82) This is one of Maxwell’s equations • If we have two circuits C1 and C2 , a current I1 in one circuit will produce a flux 21 through the other The mutual inductance M21 is defined by M21 = 21 /I1 It then follows that a changing I1 produces an emf in C2 equal to E21 = −M21 dI1 /dt The two coefficients of mutual inductance are symmetric: M12 = M21 • The self-inductance L is defined analogously A current I in a circuit will produce a flux through the circuit, and the self-inductance is defined by L = /I The emf is then E = −L dI/dt • If a circuit contains an inductor, and if a switch is opened (or closed), the current can’t change discontinuously, because that would create an infinite value of E = −L dI/dt The current must therefore gradually change If a switch is closed in an RL circuit, the current takes the form E0 − e−(R/L)t (7.83) R The quantity L/R is the time constant of the circuit • The energy stored in an inductor equals U = LI /2 It can be shown that this is equivalent to the statement that a magnetic field contains an energy density of B2 /2μ0 (just as an electric field contains an energy density of E2 /2) I= Problems 7.1 Current in a bottle ** An ocean current flows at a speed of knots (approximately m/s) in a region where the vertical component of the earth’s magnetic field is 0.35 gauss The conductivity of seawater in that region is (ohm-m)−1 On the assumption that there is no other horizontal component of E than the motional term v × B, find the density J of the horizontal electric current If you were to carry a bottle of seawater through the earth’s field at this speed, would such a current be flowing in it? Problems 7.2 7.3 What’s doing work? *** In Fig 7.27 a conducting rod is pulled to the right at speed v while maintaining contact with two rails A magnetic field points into the page From the reasoning in Section 7.3, we know that an induced emf will cause a current to flow in the counterclockwise direction around the loop Now, the magnetic force qu × B is perpendicular to the velocity u of the moving charges, so it can’t work on them However, the magnetic force f in Eq (7.5) certainly looks like it is doing work What’s going on here? Is the magnetic force doing work or not? If not, then what is? There is definitely something doing work because the wire will heat up 7.5 (B into page) v Rod Figure 7.27 Pulling a square frame ** A square wire frame with side length has total resistance R It is being pulled with speed v out of a region where there is a uniform B field pointing out of the page (the shaded area in Fig 7.28) Consider the moment when the left corner is a distance x inside the shaded area (a) What force you need to apply to the square so that it moves with constant speed v? (b) Verify that the work you √ from x = x0 (which you can assume is less than / 2) down to x = equals the energy dissipated in the resistor 7.4 375 Loops around a solenoid ** We can think of a voltmeter as a device that registers the line integral E · ds along a path C from the clip at the end of its (+) lead, through the voltmeter, to the clip at the end of its (−) lead Note that part of C lies inside the voltmeter itself Path C may also be part of a loop that is completed by some external path from the (−) clip to the (+) clip With that in mind, consider the arrangement in Fig 7.29 The solenoid is so long that its external magnetic field is negligible Its cross-sectional area is 20 cm2 , and the field inside is toward the right and increasing at the rate of 100 gauss/s Two identical voltmeters are connected to points on a loop that encloses the solenoid and contains two 50 ohm resistors, as shown The voltmeters are capable of reading microvolts and have high internal resistance What will each voltmeter read? Make sure your answer is consistent, from every point of view, with Eq (7.26) Total charge ** A circular coil of wire, with N turns of radius a, is located in the field of an electromagnet The magnetic field is perpendicular to the coil (that is, parallel to the axis of the coil), and its strength has the constant value B0 over that area The coil is connected by a pair of twisted leads to an external resistance The total resistance of v x B out of page Figure 7.28 Resistance R 376 Electromagnetic induction – + 50 Ω dB dt 50 Ω – Figure 7.29 + this closed circuit, including that of the coil itself, is R Suppose the electromagnet is turned off, its field dropping more or less rapidly to zero The induced electromotive force causes current to flow around the circuit Derive a formula for the total charge Q = I dt that passes through the resistor, and explain why it does not depend on the rapidity with which the field drops to zero 7.6 Growing current in a solenoid ** An infinite solenoid has radius R and n turns per unit length The current grows linearly with time, according to I(t) = Ct Use the integral form of Faraday’s law to find the electric field at radius r, both inside and outside the solenoid Then verify that your answers satisfy the differential form of the law 7.7 Maximum emf for a thin loop *** A long straight stationary wire is parallel to the y axis and passes through the point z = h on the z axis A current I flows in this wire, returning by a remote conductor whose field we may neglect Lying in the xy plane is a thin rectangular loop with two of its sides, of length , parallel to the long wire The length b of the other two sides is very small The loop slides with constant speed v in the xˆ direction Find the magnitude of the electromotive force induced in the loop at the moment the center of the loop has position x For what values of x does this emf have a local maximum or minimum? (Work in the approximation where b x, so that you can approximate the relevant difference in B fields by a derivative.) Problems 7.8 Faraday’s law for a moving tilted sheet **** Recall the “tilted sheet” example in Section 5.5, in which a charged sheet was tilted at 45◦ in the lab frame We calculated the electric field in the frame F moving to the right with speed v (which was 0.6c in the example) The goal of this problem is to demonstrate that Faraday’s law holds in this setup (a) For a general speed v, find the component of the electric field that is parallel to the sheet in frame F (in which the sheet moves to the left with speed v) If you solved Exercise 5.12, you’ve already done most of the work (b) Use the Lorentz transformations to find the magnetic field in F (c) In F , verify that E · ds = −d /dt holds for the rectangle shown in Fig 7.30 (this rectangle is fixed in F ) 7.9 377 zЈ xЈ Figure 7.30 Mutual inductance for two solenoids ** Figure 7.31 shows a solenoid of radius a1 and length b1 located inside a longer solenoid of radius a2 and length b2 The total number of turns is N1 on the inner coil, N2 on the outer Work out an approximate formula for the mutual inductance M 7.10 Mutual-inductance symmetry ** In Section 7.7 we made use of the vector potential to prove that M12 = M21 We can give a second proof, this time in the spirit of Exercise 3.64 Imagine increasing the currents in two circuits gradually from zero to the final values of I1f and I2f (“f” for “final”) Due to the induced emfs, some external agency has to supply power to increase (or maintain) the currents The final currents can be brought about in many different ways Two possible ways are of particular interest (a) Keep I2 at zero while raising I1 gradually from zero to I1f Then raise I2 from zero to I2f while holding I1 constant at I1f (b) Carry out a similar program with the roles of and exchanged, that is, raise I2 from zero to I2f first, and so on N2 turns N1 turns 2a2 2a1 b1 b2 v Figure 7.31 378 Electromagnetic induction Compute the total work done by external agencies, for each of the two programs Then complete the argument See Crawford (1992) for further discussion 7.11 L for a solenoid * Find the self-inductance of a long solenoid with radius r, length , and N turns 7.12 Doubling a solenoid * (a) Two identical solenoids are connected end-to-end to make a solenoid of twice the length By what factor is the selfinductance increased? The answer quickly follows from the formula for a solenoid’s L, but you should also explain in words why the factor is what it is (b) Same question, but now with the two solenoids placed right on top of one another (Imagine that one solenoid is slightly wider and surrounds the other.) They are connected so that the current flows in the same direction in each 7.13 Adding inductors * (a) Two inductors, L1 and L2 , are connected in series, as shown in Fig 7.32(a) Show that the effective inductance L of the system is given by L = L1 + L2 (a) L2 L1 Check the L1 → and L1 → ∞ limits (b) If the inductors are instead connected in parallel, as shown in Fig 7.32(b), show that the effective inductance is given by 1 + = L L1 L2 (b) L1 (7.84) (7.85) Again check the L1 → and L1 → ∞ limits 7.14 Current in an RL circuit ** Show that the expression for the current in an RL circuit given in Eq (7.69) follows from Eq (7.65) L2 Figure 7.32 7.15 Energy in an RL circuit * Consider the RL circuit discussed in Section 7.9 Show that the energy delivered by the battery up to an arbitrary time t equals the energy stored in the magnetic field plus the energy dissipated in the resistor To this, multiply Eq (7.65) by I to obtain I R = I(E0 − L dI/dt), and then integrate this equation 7.16 Energy in a superconducting solenoid * A superconducting solenoid designed for whole-body imaging by nuclear magnetic resonance is 0.9 meters in diameter and 2.2 meters long The field at its center is tesla Estimate roughly the energy stored in the field of this coil Problems 7.17 Two expressions for the energy * Two different expressions for the energy stored in a long solenoid are LI /2 and (B2 /2μ0 )(volume) Show that these expressions are consistent 7.18 Two expressions for the energy (general) *** The task of Problem 2.24 was to demonstrate that two different expressions for the electrostatic energy, ( E2 /2) dv and (ρφ/2) dv, are equivalent (as they must be, if they are both valid) The latter expression can quickly be converted to Cφ /2 in the case of oppositely charged conductors in a capacitor (see Exercise 3.65) The task of this problem is to demonstrate the analogous relation for the magnetic energy, that is, to show that if a circuit (of finite extent) with self-inductance L contains current I, then (B2 /2μ0 ) dv equals LI /2 This is a bit trickier than the electrostatic case, so here are some hints: (1) a useful vector identity is ∇ · (A × B) = B · (∇ × A) − A · (∇ × B), (2) the vector potential and magnetic field satisfy ∇ × A = B, (3) ∇ × B = μ0 J, (4) = A · dl from Eq (7.52), and (5) L is defined by = LI 7.19 Critical frequency of a dynamo *** A dynamo like the one in Exercise 7.47 has a certain critical speed ω0 If the disk revolves with an angular velocity less than ω0 , nothing happens Only when that speed is attained is the induced E large enough to make the current large enough to make the magnetic field large enough to induce an E of that magnitude The critical speed can depend only on the size and shape of the conductors, the conductivity σ , and the constant μ0 Let d be some characteristic dimension expressing the size of the dynamo, such as the radius of the disk in our example (a) Show by a dimensional argument that ω0 must be given by a relation of this form: ω0 = K/μ0 σ d2 , where K is some dimensionless numerical factor that depends only on the arrangement and relative size of the various parts of the dynamo (b) Demonstrate this result again by using physical reasoning that relates the various quantities in the problem (R, E, E, I, B, etc.) You can ignore all numerical factors in your calculations and absorb them into the constant K Additional comments: for a dynamo of modest size made wholly of copper, the critical speed ω0 would be practically unattainable It is ferromagnetism that makes possible the ordinary dc generator by providing a magnetic field much stronger than the current in the coils, unaided, could produce For an earth-sized dynamo, however, with d measured in hundreds of kilometers rather than meters, the critical speed is very much smaller The earth’s magnetic field is almost certainly produced by a nonferromagnetic 379 380 Electromagnetic induction dynamo involving motions in the fluid metallic core That fluid happens to be molten iron, but it is not even slightly ferromagnetic because it is too hot (That will be explained in Chapter 11.) We don’t know how the conducting fluid moves, or what configuration of electric currents and magnetic fields its motion generates in the core The magnetic field we observe at the earth’s surface is the external field of the dynamo in the core The direction of the earth’s field a million years ago is preserved in the magnetization of rocks that solidified at that time That magnetic record shows that the field has reversed its direction nearly 200 times in the last 100 million years Although a reversal cannot have been instantaneous (see Exercise 7.46), it was a relatively sudden event on the geological time scale The immense value of paleomagnetism as an indelible record of our planet’s history is well explained in Chapter 18 of Press and Siever (1978) Exercises 7.20 Induced voltage from the tides * Faraday describes in the following words an unsuccessful attempt to detect a current induced when part of a circuit consists of water moving through the earth’s magnetic field (Faraday, 1839, p 55): I made experiments therefore (by favour) at Waterloo Bridge, extending a copper wire nine hundred and sixty feet in length upon the parapet of the bridge, and dropping from its extremities other wires with extensive plates of metal attached to them to complete contact with the water Thus the wire and the water made one conducting circuit; and as the water ebbed or flowed with the tide, I hoped to obtain currents analogous to those of the brass ball I constantly obtained deflections at the galvanometer, but they were irregular, and were, in succession, referred to other causes than that sought for The different condition of the water as to purity on the two sides of the river; the difference in temperature; slight differences in the plates, in the solder used, in the more or less perfect contact made by twisting or otherwise; all produced effects in turn: and though I experimented on the water passing through the middle arches only; used platina plates instead of copper; and took every other precaution, I could not after three days obtain any satisfactory results Assume the vertical component of the field was 0.5 gauss, make a reasonable guess about the velocity of tidal currents in the Thames, and estimate the magnitude of the induced voltage Faraday was trying to detect 7.21 Maximum emf * What is the maximum electromotive force induced in a coil of 4000 turns, average radius 12 cm, rotating at 30 revolutions per second in the earth’s magnetic field where the field intensity is 0.5 gauss? Exercises 381 7.22 Oscillating E and B * In the central region of a solenoid that is connected to a radiofrequency power source, the magnetic field oscillates at 2.5 · 106 cycles per second with an amplitude of gauss What is the amplitude of the oscillating electric field at a point cm from the axis? (This point lies within the region where the magnetic field is nearly uniform.) 7.23 Vibrating wire * A taut wire passes through the gap of a small magnet (Fig 7.33), where the field strength is 5000 gauss The length of wire within the gap is 1.8 cm Calculate the amplitude of the induced alternating voltage when the wire is vibrating at its fundamental frequency of 2000 Hz with an amplitude of 0.03 cm, transverse to the magnetic field 7.24 Pulling a frame ** The shaded region in Fig 7.34 represents the pole of an electromagnet where there is a strong magnetic field perpendicular to the plane of the paper The rectangular frame is made of a mm diameter aluminum rod, bent and with its ends welded together Suppose that by applying a steady force of newton, starting at the position shown, the frame can be pulled out of the magnet in second Then, if the force is doubled, to newtons, the frame seconds Brass has about twice the will be pulled out in resistivity of aluminum If the frame had been made of a mm brass rod, the force needed to pull it out in second would be newtons If the frame had been made of a cm diameter aluminum rod, the force required to pull it out in second would newtons You may neglect in all cases the inertia of the be frame 7.25 Sliding loop ** A long straight stationary wire is parallel to the y axis and passes through the point z = h on the z axis A current I flows in this wire, returning by a remote conductor whose field we may neglect Lying in the xy plane is a square loop with two of its sides, of length b, parallel to the long wire This loop slides with constant speed v in the xˆ direction Find the magnitude of the electromotive force induced in the loop at the moment when the center of the loop crosses the y axis 7.26 Sliding bar ** A metal crossbar of mass m slides without friction on two long parallel conducting rails a distance b apart; see Fig 7.35 A resistor R is connected across the rails at one end; compared with R, the resistance of bar and rails is negligible There is a uniform field B perpendicular to the plane of the figure At time t = the crossbar is given a velocity v0 toward the right What happens afterward? Figure 7.33 Figure 7.34 382 Electromagnetic induction m R B 7.27 Ring in a solenoid ** An infinite solenoid with radius b has n turns per unit length The current varies in time according to I(t) = I0 cos ωt (with positive defined as shown in Fig 7.36) A ring with radius r < b and resistance R is centered on the solenoid’s axis, with its plane perpendicular to the axis Figure 7.35 (a) What is the induced current in the ring? (b) A given little piece of the ring will feel a magnetic force For what values of t is this force maximum? (c) What is the effect of the force on the ring? That is, does the force cause the ring to translate, spin, flip over, stretch/ shrink, etc.? Positive I 7.28 A loop with two surfaces ** Consider the loop of wire shown in Fig 7.37 Suppose we want to calculate the flux of B through this loop Two surfaces bounded by the loop are shown in parts (a) and (b) of the figure What is the essential difference between them? Which, if either, is the correct surface to use in performing the surface integral B · da to find the flux? Describe the corresponding surface for a three-turn coil Show that this is all consistent with our previous assertion that, for a compact coil of N turns, the electromotive force is just N times what it would be for a single loop of the same size and shape r b Figure 7.36 b (a) Does the rod ever stop moving? If so, when? (b) How far does it go? (c) How about conservation of energy? 7.29 Induced emf in a loop *** Calculate the electromotive force in the moving loop in Fig 7.38 at the instant when it is in the position shown Assume the resistance of the loop is so great that the effect of the current in the loop itself is negligible Estimate very roughly how large a resistance would be safe, in this respect Indicate the direction in which current would flow in the loop, at the instant shown 7.30 Work and dissipated energy ** Suppose the loop in Fig 7.6 has a resistance R Show that whoever is pulling the loop along at constant speed does an amount of work during the interval dt that agrees precisely with the energy dissipated in the resistance during this interval, assuming that the self-inductance of the loop can be neglected What is the source of the energy in Fig 7.14 where the loop is stationary? Exercises 7.31 Sinusoidal emf ** Does the prediction of a simple sinusoidal variation of electromotive force for the rotating planar loop in Fig 7.13 depend on the loop being rectangular, on the magnetic field being uniform, or on both? Explain Can you suggest an arrangement of rotating loop and stationary coils that will give a definitely nonsinusoidal emf? Sketch the voltage–time curve you would expect to see on the oscilloscope, with that arrangement 7.32 Emfs and voltmeters ** The circular wire in Fig 7.39(a) encircles a solenoid in which the magnetic flux d /dt is increasing at a constant rate E0 out of the page So the clockwise emf around the loop is E0 In Fig 7.39(b) the solenoid has been removed, and a capacitor has been inserted in the loop The upper plate is positive The voltage difference between the plates is E0 , and this voltage is maintained by someone physically dragging positive charges from the negative plate to the positive plate (or rather, dragging electrons the other way) So this person is the source of the emf In Fig 7.39(c) the above capacitor has been replaced by N little capacitors, each with a voltage difference of E0 /N The figure shows N = 12, but assume that N is large, essentially infinite As above, the emf is maintained by people dragging charges from one plate to the other in every capacitor This setup is similar to the setup in Fig 7.39(a), in that the emf is evenly distributed around the circuit By definition, the voltage difference between two points is b given by Vb − Va ≡ − a E · ds This is what a voltmeter measures For each of the above three setups, find the voltage difference Vb − Va along path (shown in part (a) of the figure), and also the voltage difference Va − Vb along path Comment on the similarities and differences in your results, and also on each of the total voltage drops in a complete round trip 7.33 Getting a ring to spin ** A nonconducting thin ring of radius a carries a static charge q This ring is in a magnetic field of strength B0 , parallel to the ring’s axis, and is supported so that it is free to rotate about that axis If the field is switched off, how much angular momentum will be added to the ring? Supposing the mass of the ring to be m, show that the ring, if initially at rest, will acquire an angular velocity ω = qB0 /2m Note that, as in Problem 7.5, the result depends only on the initial and final values of the field strength, and not on the rapidity of change 383 (a) (b) Figure 7.37 eres p 100 am v= 15 cm 5m /s 10 c m cm Figure 7.38 384 Electromagnetic induction (a) Path Wire a b Path (b) a + – b (c) + – a b _ dΦ dt 7.34 Faraday’s experiment *** The coils that first produced a slight but detectable kick in Faraday’s galvanometer he describes as made of 203 feet of copper wire each, wound around a large block of wood; see Fig 7.1(a) The turns of the second spiral (that is, single-layer coil) were interposed between those of the first, but separated from them by twine The diameter of the copper wire itself was 1/20 inch He does not give the dimensions of the wooden block or the number of turns in the coils In the experiment, one of these coils was connected to a “battery of 100 plates.” (Assume that one plate is roughly volt.) See if you can make a rough estimate of the duration in seconds (it will be small) and magnitude in amperes of the pulse of current that passed through his galvanometer 7.35 M for two rings ** Derive an approximate formula for the mutual inductance of two circular rings of the same radius a, arranged like wheels on the same axle with their centers a distance b apart Use an approximation good for b a 7.36 Connecting two circuits ** Part (a) of Fig 7.40 shows two coils with self-inductances L1 and L2 In the relative position shown, their mutual inductance is M The positive current direction and the positive electromotive force direction in each coil are defined by the arrows in the figure The equations relating currents and electromotive forces are E1 = −L1 Figure 7.39 dI1 dI2 ±M dt dt and E2 = −L2 dI2 dI1 ±M (7.86) dt dt (a) Given that M is always to be taken as a positive constant, how must the signs be chosen in these equations? What if we had chosen, as we might have, the other direction for positive current, and for positive electromotive force, in the lower coil? (b) Now connect the two coils together, as in part (b) of the figure, to form a single circuit What is the self-inductance L of this circuit, expressed in terms of L1 , L2 , and M? What is the selfinductance L of the circuit formed by connecting the coils as shown in (c)? Which circuit, (b) or (c), has the greater selfinductance? (c) Considering that the self-inductance of any circuit must be a positive quantity (why couldn’t it be negative?), see if you can draw a general conclusion, valid for any conceivable pair of coils, concerning the relative magnitude of L1 , L2 , and M 7.37 Flux through two rings ** Discuss the implications of the theorem 21 /I1 = 12 /I2 in the case of the large and small concentric rings in Fig 7.20 With Exercises fixed current I1 in the outer ring, obviously 21 , the flux through the inner ring, decreases if R1 is increased, simply because the field at the center gets weaker But with fixed current in the inner ring, why should 12 , the flux through the outer ring, decrease as R1 increases, holding R2 constant? It must so to satisfy our theorem 385 (a) I1 L1 E1 7.38 Using the mutual inductance for two rings *** Can you devise a way to use the theorem 21 /I1 = 12 /I2 to find the magnetic field strength due to a ring current at points in the plane of the ring at a distance from the ring much greater than the ring radius? (Hint: Consider the effect of a small change R1 in the radius of the outer ring in Fig 7.20; it must have the same effect on 12 /I2 as on 21 /I1 ) 7.39 Small L * How could we wind a resistance coil so that its self-inductance would be small? I2 L1 E2 (b) 7.40 L for a cylindrical solenoid ** Calculate the self-inductance of a cylindrical solenoid 10 cm in diameter and m long It has a single-layer winding containing a total of 1200 turns Assume that the magnetic field inside the solenoid is approximately uniform right out to the ends Estimate roughly the magnitude of the error you will thereby incur Is the true L larger or smaller than your approximate result? 7.41 Opening a switch ** In the circuit shown in Fig 7.41 the 10 volt battery has negligible internal resistance The switch S is closed for several seconds, then opened Make a graph with the abscissa time in milliseconds, showing the potential of point A with respect to ground, just before and then for milliseconds after the opening of switch S Show also the variation of the potential at point B in the same period of time 7.42 RL circuit ** A coil with resistance of 0.01 ohm and self-inductance 0.50 millihenry is connected across a large 12 volt battery of negligible internal resistance How long after the switch is closed will the current reach 90 percent of its final value? At that time, how much energy, in joules, is stored in the magnetic field? How much energy has been withdrawn from the battery up to that time? 7.43 Energy in an RL circuit ** Consider the RL circuit discussed in Section 7.9 Show that the energy delivered by the battery up to an arbitrary time t equals the energy stored in the magnetic field plus the energy dissipated in the resistor Do this by using the expression for I(t) in Eq (7.69) M LЈ = ? (c) LЉ = ? Figure 7.40 386 Electromagnetic induction S and explicitly calculating the relevant integrals This method is rather tedious, so feel free to use a computer to evaluate the integrals See Problem 7.15 for a much quicker method A 0.1 henry 10 volts B 150 ohms 50 ohms Figure 7.41 7.44 Magnetic energy in the galaxy * A magnetic field exists in most of the interstellar space in our galaxy There is evidence that its strength in most regions is between 10−6 and 10−5 gauss Adopting · 10−6 gauss as a typical value, find, in order of magnitude, the total energy stored in the magnetic field of the galaxy For this purpose you may assume the galaxy is a disk roughly 1021 m in diameter and 1019 m thick To see whether the magnetic energy amounts to much, on that scale, you might consider the fact that all the stars in the galaxy are radiating about 1037 joules/second How many years of starlight is the magnetic energy worth? 7.45 Magnetic energy near a neutron star * It has been estimated that the magnetic field strength at the surface of a neutron star, or pulsar, may be as high as 1010 tesla What is the energy density in such a field? Express it, using the mass–energy equivalence, in kilograms per m3 a a I Figure 7.42 7.46 Decay time for current in the earth ** Magnetic fields inside good conductors cannot change quickly We found that current in a simple inductive circuit decays exponentially with characteristic time L/R; see Eq (7.71) In a large conducting body such as the metallic core of the earth, the “circuit” is not easy to identify Nevertheless, we can find the order of magnitude of the decay time, and what it depends on, by making some reasonable approximations Consider a solid doughnut of square cross section, as shown in Fig 7.42, made of material with conductivity σ A current I flows around it Of course, I is spread out in some manner over the cross section, but we shall assume the resistance is that of a wire of area a2 and length π a, that is, R ≈ π/aσ For the field B we adopt the field at the center of a ring with current I and radius a/2 For the stored energy U, a reasonable estimate would be B2 /2μ0 times the volume of the doughnut Since dU/dt = −I R, the decay time of the energy U will be τ ≈ U/I R Show that, except for some numerical factor depending on our various approximations, τ ≈ μ0 a2 σ The radius of the earth’s core is 3000 km, and its conductivity is believed to be 106 (ohm-m)−1 , roughly one-tenth that of iron at room temperature Evaluate τ in centuries 7.47 A dynamo ** In this question the term dynamo will be used for a generator that works in the following way By some external agency – the shaft of a steam turbine, for instance – a conductor is driven through Exercises 387 a magnetic field, inducing an electromotive force in a circuit of which that conductor is part The source of the magnetic field is the current that is caused to flow in that circuit by that electromotive force An electrical engineer would call it a self-excited dc generator One of the simplest dynamos conceivable is sketched in Fig 7.43 It has only two essential parts One part is a solid metal disk and axle which can be driven in rotation The other is a twoturn “coil” which is stationary but is connected by sliding contacts, or “brushes,” to the axle and to the rim of the revolving disk One of the two devices pictured is, at least potentially, a dynamo The other is not Which is the dynamo? Note that the answer to this question cannot depend on any convention about handedness or current directions An intelligent extraterrestrial being inspecting the sketches could give the answer, provided only that it knows about arrows! What you think determines the direction of the current in such a dynamo? What will determine the magnitude of the current? Figure 7.43 ... 477 479 482 483 489 492 495 497 500 504 505 507 509 511 513 516 523 524 ix x CONTENTS 11 .2 11 .3 11 .4 11 .5 11 .6 11 .7 11 .8 11 .9 11 .10 11 .11 11 .12 The absence of magnetic “charge” The field of a current... CHAPTER 10 ELECTRIC FIELDS IN MATTER 10 .1 10.2 10 .3 10 .4 10 .5 10 .6 10 .7 10 .8 10 .9 10 .10 10 .11 10 .12 10 .13 10 .14 10 .15 10 .16 Dielectrics The moments of a charge distribution The potential and field... first edition of Volume xxi CHAPTER ELECTROSTATICS: CHARGES AND FIELDS 1. 1 1. 2 1. 3 1. 4 1. 5 1. 6 1. 7 1. 8 1. 9 1. 10 1. 11 1 .12 1. 13 1. 14 1. 15 1. 16 Electric charge Conservation of charge Quantization of