Electromagnetism 7

(Eq. 5.31), and therefore the charge density is zero; any unbalanced charge resides on the surface. (We proved this long ago, for the case of stationary charges, using the fact that E = 0; evidently, it is still true when the charges are allowed to move.) It follows, in particular, that Laplace's equation holds within a homogeneous ohmic material carrying a steady current, so all the tools and tricks of Chapter 3 are available for computing the potential

(a) Calculate R this way. (b) Explain why this method is fundamentally flawed. [See J. D. Romano and R. H. Price, Am. J. Phys. 64, 1150 (1996).] (c) Suppose the ends are, instead, spherical surfaces, centered at the apex ofthe cone. Calculate the resistance in that case. (Let L be the distance between the centers ofthe circular perimeter' of the end caps.) [Answer: (pI2nab)(b - a)2 I(JL2 + (b - a)2 - L)] Problem 7.41 A rare case in which the electrostatic field E for a circuit can actually be calculated is the following [M. A. Heald, Am. J. Phys. 52,522 (1984)]: Imagine an infinitel: long cylindrical sheet, of uniform resistivity and radius a. A slot (corresponding to the batter: I is maintained at ± Va/2, at ¢ = ±n, and a steady current flows over the surface, as indicated in Fig. 7.51. According to Ohm's law, then, Va¢ V(a, ¢) =~' (-n < ¢ < +n). (a) Use separation of variables in cylindrical coordinates to determine V (s, ¢) inside and outside the cylinder. [Answer: (Vain) tan -1 [(s sin ¢)I (a +s cos ¢)], (s < a); (Vain) tan - I [(asin¢)/(s+acos¢)], (s > a)] (b) Find the surface charge density on the cylinder. [Answer: (EO Valna) tan(¢12)] Problem 7.42 In a perfect conductor, the conductivity is infinite, so E = 0 (Eq. 7.3), and an: net charge resides on the surface Gust as it does for an imperfect conductor, in electrostatics). (a) Show that the magnetic field is constant (aB/at = 0), inside a perfect conductor. (b) Show that the magnetic flux through a perfectly conducting loop is constant

.2.1 Faraday's Law Figure 7.19 301 In 1831 Michael Faraday reported on a series of experiments, including three that (with some violence to history) can be characterized as follows: Experiment 1. He pulled a loop of wire to the right through a magnetic field (Fig. 7.20a). A current flowed in the loop. Experiment 2. He moved the magnet to the left, holding the loop still (Fig. 7.20b). Again, a current flowed in the loop. Experiment 3. With both the loop and the magnet at rest (Fig. 7.20c), he changed the strength of the field (he used an electromagnet, and varied the current in the coil). Once again, current flowed in the loop.

7.1.1 Ohm's Law To make a current flow, you have to push on the charges. How fast they move, in response to a given push, depends on the nature of the material. For most substances, the current density J is proportional to the force per unit charge, f: J = af. (7.1) The proportionality factor a (not to be confused with surface charge) is an empirical constant that varies from one material to another; it's called the conductivity of the medium. Actually, the handbooks usually list the reciprocal of a, called the resistivity: p = 1/a (not to be confused with charge density-I'm sorry, but we're running out of Greek letters, and this is the standard notation). Some typical values are listed in Table 7.1. Notice that even insulators conduct slightly, though the conductivity of a metal is astronomically greater-by a factor of 1022 or so. In fact, for most purposes metals can be regarded as perfect conductors, with a = 00. In principle, the force that drives the charges to produce the current could be anythingchemical, gravitational, or trained ants with tiny harnesses. For our purposes, though, it's usually an electromagnetic force that does the job. In this case Eq. 7.1 becomes J = aCE +v x B). (7.2) Ordinarily, the velocity of the charges is sufficiently small that the second term can be ignored: (7.3) (However, in plasmas, for instance, the magnetic contribution to f can be significant.) Equation 7.3 is called Ohm's law, though the physics behind it is really contained in Eq. 7.1, of which 7.3 is just a special case.

7.1.3 Motional emf Figure 7.9 R In the last section Ilisted several possible sources ofelectromotive force in a circuit, batterie\ being the most familiar. But I did not mention the most common one of all: the generator. Generators exploit motional emf's, which arise when you move a wire through a magnetic field. Figure 7.10 shows a primitive model for a generator. In the shaded region there is a uniform magnetic field B, pointing into the page, and the resistor R represents whatever it is (maybe a light bulb or a toaster) we're trying to drive current through. If the entire loop is pulled to the right with speed v, the charges in segment ab experience a magnetic force whose vertical component qvB drives current around the loop, in the clockwise direction. The emf is [; = ffmag . dl = vBh, (7.111 where h is the width of the loop. (The horizontal segments be and ad contribute nothing. since the force here is perpendicular to the wire.) Notice that the integral you perform to calculate [; (Eq. 7.9 or 7.11) is carried out at 0111' instant oftime-take a "snapshot" of the loop, if you like, and work from that. Thus dL for the segment ab in Fig. 7.10, points straight up, even though the loop is moving to the right. You can't quarrel with this-it's simply the way emf is defined-but it is important to be clear about it.

7.2.4 Energy in Magnetic Fields It takes a certain amount of energy to start a current flowing in a circuit. I'm not talking about the energy delivered to the resistors and converted into heat-that is irretrievably lost as far as the circuit is concerned and can be large or small, depending on how long you let the current run. What I am concerned with, rather, is the work you must do against the back emfto get the current going. This is afixed amount, and it is recoverable: you get it back when the current is turned off. In the meantime it represents energy latent in the circuit; as we'll see in a moment, it can be regarded as energy stored in the magnetic field. The work done on a unit charge, against the back emf, in one trip around the circuit is -E (the minus sign records the fact that this is the work done by you against the emf, not the work done by the emf). The amount of charge per unit time passing down the wire is I. So the total work done per unit time is dW dI - = -EI=LI-. dt dt If we start with zero current and build it up to a final value I, the work done (integrating the last equation over time) is IW = ~LI2·1 (7.29) It does not depend on how long we take to crank up the current, only on the geometry of the loop (in the form of L) and the final current I. There is a nicer way to write W, which has the advantage that it is readily generalized to surface and volume currents. Remember that the flux through the loop is equal to LI (Eq. 7.25). On the other hand, = LB. da = L(V x A) . da = £A . dl, where P is the perimeter of the loop and S is any surface bounded by P. Thus, LI = fA. dl,

7.3.5 Maxwell's Equations in Matter Maxwell's equations in the form 7.39 are complete and correct as they stand. However. when you are working with materials that are subject to electric and magnetic polarization there is a more convenient way to write them. For inside polarized matter there will be accumulations of "bound" charge and current over which you exert no direct control. It would be nice to reformulate Maxwell's equations in such a way as to make explicitreference only to those sources we control directly: the "free" charges and currents. We have already learned, from the static case, that an electric polarization P produces a bound charge density Ph = -V· P (7.46) (Eq. 4.12). Likewise, a magnetic polarization (or "magnetization") M results in a bound current Jb = V x M (7.47) (Eq. 6.13). There's just one new feature to consider in the nonstatic case: Any change in the electric polarization involves a flow of (bound) charge (call it Jp), which must be included in the total current. For suppose we examine a tiny chunk of polarized material (Fig. 7.45.) The polarization introduces a charge density Cfb = P at one end and -Cfb at the other (Eq. 4.11). If P now increases a bit, the charge on each end increases accordingly. giving a net current

A superconductor is a perfect conductor with the additional property that the (constant) B inside is in fact zero. (This "flux exclusion" is known as the Meissner effect. 18 ) (c) Show that the current in a superconductor is confined to the surface. (d) Superconductivity is lost above a certain critical temperature (Td, which varies from one material to another. Suppose you had a sphere (radius a) above its critical temperature, and you held it in a uniform magnetic field Boz while cooling it below Tc . Find the induced surface current density K, as a function of the polar angle e. Problem 7.43 A familiar demonstration of superconductivity (Prob. 7.42) is the levitation of a magnet over a piece of superconducting material. This phenomenon can be analyzed using the method of images. I 9 Treat the magnet as a perfect dipole m, a height z above the origin (and constrained to point in the z direction), and pretend that the superconductor occupies the entire half-space below the xy plane. Because of the Meissner effect, B = 0 for z .:s 0, and since B is divergenceless, the normal (z) component is continuous, so Bz = 0 just above the surface. This boundary condition is met by the image configuration in which an identical dipole is placed at -z, as a stand-in for the superconductor; the two arrangements therefore produce the same magnetic field in the region z > O. (a) Which way should the image dipole point (+z or -z)? (b) Find the force on the magnet due to the induced currents in the superconductor (which is to say, the force due to the image dipole). Set it equal to Mg (where M is the mass of the magnet) to determine the height h at which the magnet will "float." [Hint: refer to Prob. 6.3.]

As the loop moves, the flux decreases: (b) Integration path for calculating work done (follow the charge around the loop). Figure 7.12 det> dx - = Bh- = -Bhv. dt dt (The minus sign accounts for the fact that dx / dt is negative.) But this is precisely the emf (Eq. 7.11); evidently the emf generated in the loOp is minus the rate of change of flux through the loop: (7.13) This is the flux rule for motional emf. Apart from its delightful simplicity, it has the virtue of applying to nonrectangular loops moving in arbitrary directions through nonuniform magnetic fields; in fact, the loop need not even maintain a fixed shape. Proof: Figure 7.13 shows a loop of wire at tinw t and also a short time dt later. Suppose we compute the flux at time t, using surface S, and the flux at time t + dt, using the surface consisting of S plus the "ribbon" that connects the new position of the loop to the old. The change in flux, then, is det> = et>(t +dt) - et>(t) = et>ribbon = [ B . da. Jribbon Focus your attention on point P: in time dt it moves to pl. Let v be the velocitY of the wire, and u the velocity of a charge down the wire; w = v + u is the resultant velocity of a charge at P. The infinitesimal element of area on the ribbon can be written as da = (v x dl) dt

As these examples illustrate, the total current flowing from one electrode to the other is proportional to the potential difference between them: (7.4) This, of course, is the more familiar version of Ohm's law. The constant of proportionality R is called the resistance; it's a function of the geometry of the arrangement and the conductivity of the medium between the electrodes. (In Ex. 7.1, R = (LI C5 A); in Ex. 7.2, R = In (bla)/2nC5 L.) Resistance is measured in ohms (Q): an ohm is a volt per ampere. Notice that the proportionality between V and I is a direct consequence of Eq. 7.3: if you want to double V, you simply double the charge everywhere-but that doubles E, which doubles J, which doubles I.

I don't suppose there is any formula in physics more widely known than Ohm's law, and yet it's not really a true law, in tbe sense of Gauss's law or Ampere's law; rather, it is a "rule of thumb" that applies pretty well to many substances. You're not going to win a Nobel prize for finding an exception. In fact, when you stop to think about it, it's a little surprising that Ohm's law ever holds. After all, a given field E produces a force qE (on a charge q), and according to Newton's second law the charge will accelerate. But if the charges are accelerating, why doesn't the current increase with time, growing larger and larger the longer you leave the field on? Ohm's law implies, on the contrary, that a constant field produces a constant current, which suggests a constant velocity. Isn't that a contradiction of Newton's law? No, for we are forgetting the frequent collisions electrons make as they pass down the wire. It's a little like this: Suppose you're driving down a street with a stop sign at every intersection, so that, although you accelerate constantly in between, you are obliged to start all over again with each new block. Your average speed is then a constant, in spite of the fact that (save for the periodic abrupt stops) you are always accelerating. Ifthe length of a block is A and your acceleration is a, the time it takes to go a block is t=/¥, and hence the average velocity is Vave = ~at = ~. But wait! That's no good either! It says that the velocity is proportional to the square root of the acceleration, and therefore that the current should be proportional to the square root of the field! There's another twist to the story: The charges in practice are already moving quite fast because of their thermal energy. But the thermal velocities have random directions, and average to zero. The net drift velocity we're concerned with is a tiny extra bit (Prob. 5.19). So the time between collisions is actually much shorter than we supposed; in fact, A t=-- Vthermal' and therefore I aA vave = -at = --- 2 2Vthennal Ifthere are n molecules per unit volume and f free electrons per molecule, each with charge q and mass m, the current density is J = nfqvave = nfqA !. = ( nfAq 2 ) E.

I know: you're confused because I said E = 0 inside a conductor (Sect. 2.5.1). But that's for stationary charges (J = 0). Moreover, for perfect conductors E = J/a = 0 even if current is flowing. In practice, metals are such good conductors that the electric field required to drive current in them is negligible. Thus we routinely treat the connecting wire~ in electric circuits (for example) as equipotentials. Resistors, by contrast, are made from poorly conducting materials

I must warn you, now, of a small fraud that tarnishes many applications ofFaraday,slaw: Electromagnetic induction, of course, occurs only when the magnetic fields are changing. and yet we would like to use the apparatus ofmagnetostatics (Ampere's law, the Biot-Savart law, and the rest) to calculate those magnetic fields. Technically, any result derived in this way is only approximately correct. But in practice the error is usually negligible unless the field fluctuates extremely rapidly, or you are interested in points very far from the source. Even the case of a wire snipped by a pair of scissors (Prob. 7.18) is static enough for Ampere's law to apply. This regime, in which magnetostatic rules can be used to calculate the magnetic field on the right hand side of Faraday's law, is called quasistatic. General!) speaking, it is only when we come to electromagnetic waves and radiation that we must worry seriously about the breakdown of magnetostatics itself

Identical reasoning, applied to equation (ii), yields IBt - Bt = 0·1 Turning to (iii), a very thin Amperian loop straddling the surface (Fig. 7.47) gives El·I-E2·1=-~ f B . da . dt 1s (7.60) But in the limit as the width of the loop goes to zero, the flux vanishes. (I have already dropped the contribution of the two ends to f E . dl, on the same grounds.) Therefore, (7.61 ) That is, the components of E parallel to the interface are continuous across the boundary. By the same token, (iv) implies where Ilene is the free current passing through the Amperian loop. No volume current density will contribute (in the limit of infinitesimal width) but a surface current can. In fact. ifnis a unit vector perpendicular to the interface (pointing from 2 toward I), so that (n x I) is normal to the Amperian loop, then Ilene = K I . (n x I) = (KI x n) . I, and hence IH~ - H~ = K I x n·1 (7.62) So the parallel components of H are discontinuous by an amount proportional to the free surface current density

If you think about a typical electric circuit (Fig. 7.7)-a battery hooked up to a light bulb, say-there arises a perplexing question: In practice, the current is the same all the way around the loop, at any given moment; why is this the case, when the only obvious driving force is inside the battery? Off hand, you might expect this to produce a large current in the battery and none at all in the lamp. Who's doing the pushing in the rest of the circuit, and how does it happen that this push is exactly right to produce the same current in each segment? What's more, given that the charges in a typical wire move (literally) at a snail's pace (see Prob. 5.19), why doesn't it take half an hour for the news to reach the light bulb? How do all the charges know to start moving at the same instant?

In particular, although the magnetic force is responsible for establishing the emf, it is certainly not doing any work-magnetic forces never do work. Who, then, is supplying the energy that heats the resistor? Answer: The person who's pulling on the loop! With the current flowing, charges in segment ab have a vertical velocity (call it u) in addition to the horizontal velocity v they inherit from the motion of the loop. Accordingly, the magnetic force has a component qu B to the left. To counteract this, the person pulling on the wire must exert a force per unit charge fpull = uB to the right (Fig. 7.11). This force is transmitted to the charge by the structure of the wire. Meanwhile, the particle is actually moving in the direction of the resultant velocity w, and the distance it goes is (hi cos e). The work done per unit charge is therefore f Cpull . dl = (uB) (_h_) sin e = vBh = E cose (sin e coming from the dot product). As it turns out, then, the work done per unit charge is exactly equal to the emf, though the integrals are taken along entirely different paths (Fig. 7.12) and completely different forces are involved. To calculate the emf you integrate around the loop at one instant, but to calculate the work done you follow a charge in its motion around the loop; Cpull contributes nothing to the emf, because it is perpendicular to the wire, whereas Cmag contributes nothing to work because it is perpendicular to the motion of the charge.4 There is a particularly nice way of expressing the emf generated in a moving loop. Let be the flux of B through the loop: == f B· da. For the rectangular loop in Fig. 7.10,

In the last section we put the finishing touches on Maxwell's equations: I (i) V ·E=-p (Gauss's law), EO (ii) V ·B=O (no name), (iii) aB VxE=-- (Faraday's law), at (7.391 (iv) aE V x B = fLoJ + fLOEoat (Ampere's law with Maxwell's correction). Together with the force law, F = q(E + v x B), (7.40) they summarize the entire theoretical content of classical electrodynamicsI5 (save for some special properties ofmatter, which we encountered in Chapters 4 and 6). Even the continuit~ equation, (7.41 ) which is the mathematical expression of conservation of charge, can be derived from Maxwell's equations by applying the divergence to number (iv). I have written Maxwell's equations in the traditional way, which emphasizes that the~ specify the divergence and curl ofE and B. In this form they reinforce the notion that electric fields can be produced either by charges (p) or by changing magnetic fields (aB/at), and magnetic fields can be produced either by currents (J) or by changing electric fields (aE/ at l. Actually, this is somewhat misleading, because when you come right down to it aB/at and aE/at are themselves due to charges and currents. I think it is logically preferable to write 1 aB I (i) V·E = -p, (iii) V xE+- =0, EO at (7.421 (iv) aE (ii) V ·B=O, V x B - fLoEo- = fLoJ, at with the fields (E and B) on the left and the sources (p and J) on the right. This notation emphasizes that all electromagnetic fields are ultimately attributable to charges and currents. Maxwell's equations tell you how charges produce fields; reciprocally, the force law tells you how fields affect charges

Inductance (like capacitance) is an intrinsically positive quantity. Lenz's law, which is enforced by the minus sign in Eq. 7.26, dictates that the emf is in such a direction as to oppose any change in current. For this reason, it is called a back emf. Whenever you try to alter the current in a wire, you must fight against this back emf. Thus inductance plays somewhat the same role in electric circuits that mass plays in mechanical systems: The greater L is, the harder it is to change the current, just as the larger the mass, the harder it is to change an object's velocity.

Maxwell's Equations 7.3.1 Electrodynamics Before Maxwell So far, we have encountered the following laws, specifying the divergence and curl of electric and magnetic fields: (i) (ii) (iii) (iv) 1 V·E= -p EO V ·B=O aB VxE=--at v x B = fLoJ (Gauss's law), (no name), (Faraday's law), (Ampere's law). These equations represent the state of electromagnetic theory over a century ago, when Maxwell began his work. They were not written in so compact a form in those days, but their physical content was familiar. Now, it happens there is a fatal inconsistency in these 322 CHAPTER 7. ELECTRODYNAMICS formulas. It has to do with the old rule that divergence of curl is always zero. If you appl) the divergence to number (iii), everything works out: ( aB) a V . (V x E) = V· -- = --(V·B). at at The left side is zero because divergence of curl is zero; the right side is zero by virtue of equation (ii). But when you do the same thing to number (iv), you get into trouble: V· (V x B) = fLO(V .J); (7.351 the left side must be zero, but the right side, in general, is not. For steady currents, the divergence of J is zero, but evidently when we go beyond magnetostatics Ampere's la\\ cannot be right. There's another way to see that Ampere's law is bound to fail for nonsteady currenh Suppose we're in the process of charging up a capacitor (Fig. 7.42). In integral form. Ampere's law reads f B . dl = fLOIene. I want to apply it to the Amperian loop shown in the diagram. How do I determine Ienc " Well, it's the total current passing thtough the loop, or, more precisely, the current piercing a surface that has the loop for its boundary. In this case, the simplest surface lies in the plane of the loop-the wire punctures this surface, so I ene = I. Fine-but what if I dra\\ instead the balloon-shaped surfate in Fig. 7.42? No current passes through this surface, and I conclude that I ene = O! We never had this problem in magnetostatics because the conflict arises only when charge is piling up somewhere (in this case, on the capacitor plates). But for nonsteady currents (such as this one) "the current enclosed by a loop" is an ill-defined notion, since it depends entirely oh what surface you use. (If this seems pedantic to you- "obviously one should use the planar surface"-remember that the Amperian loop could be some contorted shape that doesn't even lie in a plane.)

Meanwhile, Ampere's law (with Maxwell's term) becomes ( ap) aE V x B = flo Jt + V x M + - + fLoEo -, . at at or where, as before, aD VxH=Jr+-, . at (7.53) I H == -B - M. (7.54) flo Faraday's law and V . B = 0 are not affected by our separation of charge and current into free and bound parts, since they do not involve p or J. In terms ofjree charges and currents, then, Maxwell's equations read (i) V· D = Pt, (ii) V· B = 0, aB (iii) V x E = --at ' aD (iv) V xH=Jt+-. at (7.55) Some people regard these as the "true" Maxwell's equations, but please understand that they are in no way more "general" than 7.39; they simply reflect a convenient division of charge and current into free and nonfree parts. And they have the disadvantage of hybrid notation, since they contain both E and D, both Band H. They must be supplemented, therefore, by appropriate constitutive relations, giving D and H in terms of E and B. These depend on the nature of the material; for linear media so p = EOXeE, and M = XmH, (7.56) 1 D = EE, and H = -B, (7.57) fL where E == Eo(l + Xe) and fL == fLo(l + Xm). Incidentally, you'll remember that D is called the electric "displacement"; that's why the second term in the AmperelMaxwell equation (iv) is called the displacement current, generalizing Eq. 7.37: aD Jd=-'

Of course, we had no right to expect Ampere's law to hold outside of magnetostatics; after all, we derived it from the Biot-Savart law. However, in Maxwell's time there was no experimental reason to doubt that Ampere's law was of wider validity. The flaw was a purely theoretical one, and Maxwell fixed it by purely theoretical arguments. 7.3.2 How Maxwell Fixed Ampere's Law The problem is on the right side of Eq. 7.35, which should be zero, but isn't. Applying the continuity equation (5.29) and Gauss's law, the offending term can be rewritten: V.J=_ap =-~(EOV'E)=-V'(EOaE). at at at It might occur to you that if we were to combine Eo(aEjat) with J, in Ampere's law, it would be just right to kill off the extra divergence: IV x B = fLoJ + fLo EO ·1 (7.36) (Maxwell himself had other reasons for wanting to add this quantity to Ampere's law. To him the rescue ofthe continuity equation was a happy dividend rather than a primary motive. But today we recognize this argument as a far more compelling one than Maxwell's, which was based on a now-discredited model of the ether.) 13 Such a modification changes nothing, as far as magnetostatics is concerned: when E is constant, we still have V x B = fLoJ. In fact, Maxwell's term is hard to detect in ordinary electromagnetic experiments, where it must compete for recognition with J; that's why Faraday and the others never discovered it in the laboratory. However, it plays a crucial role in the propagation of electromagnetic waves, as we'll see in Chapter 9. Apart from curing the defect in Ampere's law, Maxwell's term has a certain aesthetic appeal: Just as a changing magnetic field induces an electric field (Faraday's law), so IA changing electric field induces a magnetic field. I Of course, theoretical convenience and aesthetic consistency are only suggestive-there might, after all, be other ways to doctor up Ampere's law. The real confirmation of Maxwell's theory came in 1888 with Hertz's experiments on electromagnetic waves. Maxwell called his extra term the displacement Current: aE Jd == 1'0-· at (7.37) It's a misleading name, since Eo(aEjat) has nothing to do with current, except that it adds to J in Ampere's law. Let's see now how the displacement current resolves the paradox of the charging capacitor (Fig. 7.42). If the capacitor plates are very close together (I didn't

Problem 7.49 An atomic electron (charge q) circles about the nucleus (charge Q) in an orbit of radius r; the centripetal acceleration is provided, of course, by the Coulomb attraction of opposite charges Now a small magnetic field dB is slowly turned on, perpendicular to the plane of the orbit. Show that the increase in kinetic energy, dT, imparted by the induced electric field, is just right to sustain circular motion at the same radius r. (That's why, in my discussion of diamagnetism, I assumed the radius is fixed. See Sect. 6.1.3 and the references cited there.) Problem 7.50 The current in a long solenoid is increasing linearly with time, so that the flux is proportional to t:

Suppose now that you vary the current in loop 1. The flux through loop 2 will vary accordingly, and Faraday's law says this changing flux will induce an emf in loop 2: d¢>2 dh [2=--=-M-. dt dt (In quoting Eq. 7.2I-which was based on the Biot-Savart law-I am tacitly assuming that the currents change slowly enough for the configuration to be considered quasistatic.) What

Suppose you have two loops of wire, at rest (Fig. 7.29). If you run a steady current II around loop 1, it produces a magnetic field B]. Some of the field lines pass through loop 2; let <1>2 be the flux ofB] through 2. You might have a tough time actually calculating BI• but a glance at the Biot-Savart law, reveals one significant fact about this field: It is proportional to the current h. Therefore. so too is the flux through loop 2:

The current density, therefore, is 329 ap Jp = at. (7.48) This polarization current has nothing whatever to do with the bound current Jb. The latter is associated with magnetization of the material and involves the spin and orbital motion of electrons; Jp, by contrast, is the result of the linear motion of charge when the electric polarization changes. If P points to the right and is increasing, then each plus charge moves a bit to the right and each minus charge to the left; the cumulative effect is the polarization current Jp . In this connection, we ought to check that Eq. 7.48 is consistent with the continuity equation: ap a apb v .J = v . - = - (V . P) = - -. p at at at Yes: The continuity equation is satisfied; in fact, Jp is essential to account for the conservation of bound charge. (Incidentally, a changing magnetization does not lead to any anaiogous accumulation of charge or current. The bound current Jb = V x M varies in response to changes in M, to be sure, but that's about it.) In view of all this, the total charge density can be separated into two parts: P = PI + Pb = PI - V . P, and the current density into three parts: Gauss's law can now be written as I V . E = - (pI - V . P), EO or where D, as in the static case, is given by D == EoE+P

The first experiment, of course, is an example of motional emf, conveniently expressed by the flux rule: det> E=--. dt I don't think it will surprise you to learn that exactly the same emf arises in Experiment 2- all that really matters is the relative motion of the magnet and the loop. Indeed, in the light of special relativity is has to be so. But Faraday knew nothing of relativity, and in classical electrodynamics this simple reciprocity is a coincidence, with remarkable implications. For if the loop moves, it's a magnetic force that sets up the emf, but if the loop is stationarY. the force cannot be magnetic-stationary charges experience no magnetic forces. In that case, what is responsible? What sort of field exerts a force on charges at rest? Well, electric fields do, of course, but in this case there doesn't seem to be any electric field in sight. Faraday had an ingenious inspiration: A changing magnetic field induces an electric field. It is this "induced" electric field that accounts for the emf in Experiment 2.6 Indeed, if (a, Faraday found empirically) the emf is again equal to the rate of change of the flux, f det> E= E·dl=--dt ' then E is related to the change in B by the equation 1E . dl = - f ~~ .da. (7.141 (7.151 (7.161 This is Faraday's law, in integral form. We can convert it to differential form by applying Stokes' theorem: IVXE=-~·I Note that Faraday's law reduces to the old rule #E . dl 0 (or, in differential form. V x E =0) in the static case (constant B) as, of course, it should. In Experiment 3 the magnetic field changes for entirely different reasons, but according to Faraday's law an electric field will again be induced, giving rise to an emf -det>/ dt. Indeed, one can subsume all three cases (and for that matter any combination of them) into a kind of universal flux rule: Whenever (and for whatever reason) the magnetic flux through a loop changes, an emf will appear in the loop. det> E=-- dt (7.17) 6you might argue that the magnetic field in Experiment 2 is not really changing-just moving. What I mean j, that if you sit at ajixed location, the field does change, as the magnet passes by. 7.2. ELECTROMAGNETIC INDUCTION 303 Many people call this "Faraday's law." Maybe I'm overly fastidious, but I find this confusing. There are really two totally different mechanisms underlying Eq. 7.17, and to identify them both as "Faraday's law" is a little like saying that because identical twins look alike we ought to call them by the same name. In Faraday's first experiment it's the Lorentz force law at work; the emf is magnetic. But in the other two it's an electric field (induced by the changing magnetic field) that does th~ job. Viewed in this light, it is quite astonishing that all three processes yield the same formula for the emf. In fact, it was precisely this "coincidence" that led Einstein to the special theory of relativity-he sought a deeper understanding of what is, in classical electrodynamics, a peculiar accident. But that's a story for Chapter 12. In the meantime I shall reserve the term "Faraday's law" for electric fields induced by changing magnetic fields, and I do nat regard Experiment 1 as an instance of Faraday's law.

The induced current will flow in such a direction that the flux it produces tends to cancel the change. (As the front end of the magnet in Ex. 7.5 enters the ring, the flux increases, so the current in the ring must generate a field to the right-it therefore flows clockwise.) Notice that it is the change in flux, not the flux itself, that nature abhors (when the tail end of the magnet exits the ring, the flux drops, so the induced current flows counterclockwise, in an effort to restore it). Faraday induction is a kind of "inertial" phenomenon: A conducting loop "likes" to maintain a constant flux through it; if you try to change the flux, the loop responds by sending a current around in such a direction as to frustrate your efforts. (It doesn't succeed completely; the flux produced by the induced current is typically only a tiny fraction of the original. All Lenz's law tells you is the direction of the flow.)

There is a pleasing symmetry about Maxwell's equations; it is particularly striking in free space, where P and J vanish: aB ) V ·E=O, VxE=-- at ' V·B =0, aE V x B = /LoEo-. at If you replace E by Band B by -/LOEOE, the first pair of equations turns into the second, and vice versa. This symmetry16 between E and B is spoiled, though, by the charge term in Gauss's law and the current term in Ampere's law. You can't help wondering why the corresponding quantities are "missing" from V .B = ° and V x E = -aBjat. What if we had (7.43) (ii) V· B = /LoPm, aD ) (iii) V x E = -/LoJm - at' (iv) V x B = /LoJe + /LoEo aE. at Then Pm would represent the density of magnetic "charge," and Pe the density of electric charge; Jm would be the current of maghetic charge, and Je the current of electric charge. Both charges would be conserved: apm V ·Jm = ---at ' ape and V ·Je = --. at (7.44) The former follows by application of the divergence to (iii), the latter by taking the divergence of (iv). In a sense, Maxwell's equations beg for magnetic charge to exist-it would fit in so nicely. And yet, in spite of a diligent search, no one has ever found any.17 As far as we know, Pm is zero everywhere, and so is Jm; B is not on equal footing with E: there exist

There is a sign ambiguity in the definition of emf CEq. 7.9): Which way around the loop are you supposed to integrate? There is a compensatory ambiguity in the definition offlux CEq. 7.12): Which is the positive direction for da? In applying the flux rule, sign consistency is governed Cas alwaYs) by your right hand: If your fingers define the positive direction around the loop, then your thumb indicates the direction of da. Should the emf come out negative, it means the current will flow in the negative direction around the circuit. The flux rule is a nifty short-cut for calculating motional emf's. It does not contain any new physics. Occasionally you will run across problems that cannot be handled by the flux rule; for these one must go back to the Lorentz force law itself.

This is an astonishing conclusion: Whatever the shapes and positions of the loops, thefiux through 2 when we run a current I around I is identical to thefiux through 1 when we send the same current 1 around 2. We may as well drop the subscripts and call them both M.

What Faraday's discovery tells us is that there are really two distinct kinds of electric fields: those attributable directly to electric charges, and those associated with changing magnetic fields. 9 The former can be calculated (in the static case) using Coulomb's law; the latter can be found by exploiting the analogy between Faraday's law, aB VxE=--at '

a remarkable thing: Every time you change the current in loop 1, an induced current flows in loop 2-even though there are no wires connecting them! Come to think of it, a changing current not only induces an emf in any nearby loops, it also induces an emf in the source loop itself(Fig 7.32). Once again, the field (and therefore also the flux) is proportional to the current: = LI. (7.25) The constant of proportionality L is called the self-inductance (or simply the inductance) of the loop. As with M, it depends on the geometry (size and shape) of the loop. If the current changes, the emf induced in the loop is dI E=-L-. dt Inductance is measured in henries (H); a henry is a volt-second per ampere.

amount 1(A . J) per unit volume. The distinction is one of bookkeeping; the important quantity is the total energy W, and we shall not worry about where (if anywhere) the energy is "located." You might find it strange that is takes energy to set up a magnetic field-after all, magnetic fields themselves do no work. The point is that producing a magnetic field, where previously there was none, requires changing the field, and a changing B-field, according to Faraday, induces an electric field. The latter, of course, can do work. In the beginning there is no E, and at the end there is no E; but in between, while B is building up, there is an E, and it is against this that the work is done. (You see why I could not calculate the energy stored in a magnetostatic field back in Chapter 5.) In the light of this, it is extraordinary how similar the magnetic energy formulas are to their electrostatic counterparts: Example 7.13 1 f EO f 2 Welec =2 (Vp)dr=2 E dr, Wmag = -If (A· J) dr = - 1 f B2 dr. 2 2/-LO (2.43 and 2.45) (7.31 and 7.34)

and Ampere's law, CHAPTER 7. ELECTRODYNAMICS v x B = fLoJ. Of course, the curl alone is not enough to determine a field-you must also specify the divergence. But as long as E is a pure Faraday field, due exclusively to a changing B (with p = 0), Gauss's law says V ·E=O, while for magnetic fields, of course, V ·B=O always. So the parallel is complete, and I conclude that Faraday-induced electric fields are determined by -caB/at) in exactly the same way as magnetostatic fields are determined by fLoJ. In particular, if symmetry permits, we can use all the tricks associated with Ampere's law in integral form, f B . dl = fLO/ene, only this rime it's Faraday's law in integral form: J: E . dl = _ del> . r dt (7.181 The rate of change of (magnetic) flux through the Amperian loop plays the role formerl~ assigned to fLO/enc.

be true ... What's gone wrong? Answer: We have overstepped the limits of the quasistatic approximation. As we shall see in Chapter 9, electromagnetic "news" travels at the speed of light, and at large distances B depends not on the current now, but on the current as it was at some earlier time (indeed, a whole range of earlier times, since different points on the wire are different distances away). If r is the time it takes 1 to change substantially, then the quasistatic approximation should hold only for s « cr, and hence Eq. 7.19 simply does not apply, at extremely large

does indicate the basic ingredients, and it correctly predicts that conductivity is proportional to the density ofthe moving charges and (ordinarily) decreases with increasing temperature. As a result of all the collisions; the work done by the electricai force is converted into heat in the resistor. Since the work done per unit charge is V and the charge flowing per unit time is I, the power delivered is (7.71 This is the Joule heating law. With I in amperes and R in ohms, P comes out in wath (joules per second).

draw them that way, but the calculation is simpler if you assume this), then the electric field between them is I I Q E=-cr=-- EO EO A' where Q is the charge on the plate and A is its area. Thus, between the plates aE I dQ I -=--=-1. at EoA dt EoA Now, Eq. 7.36 reads, in integral form, f B . dl = {tolenc + {tOEO f (~~) .da. (7.38) If we choose the flat surface, then E = 0 and Ienc = I. If, on the other hand, we use the balloon-shaped surface, then Ienc = 0, but !(aElat). da = IIEo. So we get the same answer for either surface, though in the first case it comes from the genuine current and in the second from the displacement current.

electrical impulse; in a thermocouple it's a temperature gradient that does the job; in a photoelectric cell it's light; and in a Van de Graaff generator the electrons are literally loaded onto a conveyer belt and swept along. Whatever the mechanism, its net effect is determined by the line integral of f around the circuit: I [; == 1f . dl = 1fs . dl.l (7.9) (Because :p E . dl = 0 for electrostatic fields, it doesn't matter whether you use f or fs.) [; is called the electromotive force, or emf, of the circuit. It's a lousy term, since this is not aforce at all-it's the integral of a force per unit charge. Some people prefer the word electromotance, but emf is so ingrained that I think we'd better stick with it. Within an ideal source of emf (a resistanceless battery,3 for instance), the net force on the charges is zero (Eq. 7.1 with a = (0), so E = -fs . The potential difference between the terminals (a and b) is therefore v = -l h E· dl = I h fs . dl = f fs . dl = [; (7.10) (we can extend the integral to the entire loop because fs = 0 outside the source). The function of a battery, then, is to establish and maintain a voltage difference equal to the electromotive force (a 6 V battery, for example, holds the positive terminal 6 V above the negative terminal). The resulting electrostatic field drives current around the rest of the circuit (notice, however, that inside the battery fs drives current in the direction opposite to E). Because it's the line integral of(" [; can be interpreted as the work done, per unit charge, by the source-indeed, in some books electromotive force is defined this way. However, as you'll see in the next section, there is some subtlety involved in this interpretation, so I prefer Eq. 7.9.

lit second after the switch is closed), then charge is piling up somewhere, and-here's the crucial point-the electric field of this accumulating charge is in such a direction as to even out the flow. Suppose, for instance, that the current into the bend in Fig. 7.8 is greater than the current out. Then charge piles up at the "knee," and this produces a field aiming away from the kink. This field opposes the current flowing in (slowing it down) and promotes the current flowing out (speeding it up) until these currents are equal, at which point there is no further accumulation of charge, and equilibrium is established. It's a beautiful system, automatically self-correcting to keep the current uniform, and it does it all so quickly that, in practice, you can safely assume the current is the same all around the circuit even in systems that oscillate at radio frequencies. The upshot of all this is that there are really two forces involved in driving current around a circuit: the source, fs, which is ordinarily confined to one portion of the loop (a battery, say), and the electrostatic force, which serves to smooth out the flow and communicate the influence of the source to distant parts of the circuit: f=fs+E. (7.8) The physical agency responsible for fs can be anyone of many different things: in a battery it's a chemical force; in a piezoelectric crystal mechanical pressure is converted into an

nd therefore CHAPTER 7. ELECTRODYNAMICS w= 11 fA. dl. The vector sign might as well go on the I: w = f (A . I) dl. In this form, the generalization to volume currents is obvious: w = { (A. J) dr. 21v (7.30) (7.31 ) But we can do even better, and express W entirely in terms of the magnetic field: Ampere's law, V x B = fLoJ, lets us eliminate J: w = _1_ fA. (V x B) dr. 2fLO (7.32) Integration by parts enables us to move the derivative from B to A; specifically, product rule 6 states that V . (A x B) = B . (V x A) - A . (V x B), so A . (V x B) = B . B - V . (A x B). Consequently, w 2~O [I B 2 dr - 1V· (A x B)dr] _I [{ B2 dr _ J (A x B) . da] , 2fLO lv fs (7.33) where S is the surface bounding the volume V. Now, the integration in Eq. 7.31 is to be taken over the entire volume occupied by the current. But any region larger than this will do just as well, for J is zero out there anyway. In Eq. 7.33 the larger the region we pick the greater is the contribution from the volume integral, and therefore the smaller is that of the surface integral (this makes sense: as the surface gets farther from the current, both A and B decrease). In particular, if we agree to integrate over all space, then the surface integral goes to zero, and we are left with w= -11 B2 dr. 2fLo all space (7.34) In view of this result, we say the energy is "stored in the magnetic field," in the amount (B2 /2fLO) per unit volume. This is a nice way to think of it, though someone looking at Eq. 7.31 might prefer to say that the energy is stored in the current distribution, in the

stationary sources for E (electric charges) but none for B. (This is reflected in the fact that magnetic multipole expansions have no monopole term, and magnetic dipoles consist of current loops, not separated north and south "poles.") Apparently God just didn't make any magnetic charge. (In the quantum theory of electrodynamics, by the way, it's a more than merely aesthetic shame that magnetic charge does not seem to exist: Dirac showed that the existence of magnetic charge would explain why electric charge is quantized. See Prob.8.12.)

7.3.6 Boundary Conditions 331 In general, the fields E, B, D, and H will be discontinuous at a boundary between two different media, or at a surface that carries charge density (f or current density K. The explicit form of these discontinuities can be deduced from Maxwell's equations (7.55), in their integral form (i) (ii) Is D· da = Qfene ) is B·da=O over any closed surface S. (iii) (iv) J E.dl=-~ rB.da } !p dt 1s J H. dl = IF + rD. da !p Jene dt 1s for any surface S bounded by the closed loop P. Applying (i) to a tiny, wafer-thin Gaussian pillbox extending just slightly into the material on either side of the boundary, we obtain (Fig. 7.46): Dj • a - D2 . a = (ff a. (The positive direction for a isfrom 2 toward 1. The edge of the wafer contributes nothing in the limit as the thickness goes to zero, nor does any volume change density.) Thus, the component of D that is perpendicular to the interface is discontinuous in the amount

v7.3.6 Boundary Conditions 331 In general, the fields E, B, D, and H will be discontinuous at a boundary between two different media, or at a surface that carries charge density (f or current density K. The explicit form of these discontinuities can be deduced from Maxwell's equations (7.55), in their integral form (i) (ii) Is D· da = Qfene ) is B·da=O over any closed surface S. (iii) (iv) J E.dl=-~ rB.da } !p dt 1s J H. dl = IF + rD. da !p Jene dt 1s for any surface S bounded by the closed loop P. Applying (i) to a tiny, wafer-thin Gaussian pillbox extending just slightly into the material on either side of the boundary, we obtain (Fig. 7.46): Dj • a - D2 . a = (ff a. (The positive direction for a isfrom 2 toward 1. The edge of the wafer contributes nothing in the limit as the thickness goes to zero, nor does any volume change density.) Thus, the component of D that is perpendicular to the interface is discontinuous in the amount

where M21 is the constant of proportionality; it is known as the mutual inductance of the two loops. There is a cute formula for the mutual inductance, which you can derive by expressing the flux in terms of the vector potential and invoking Stokes' theorem: <1>2 = fBl . da2 = f (V x AI) . da2 = f Al . d12. Now, according to Eq. 5.63, and hence Evidently M21 = fLO 11 dll . d12 . (7.22) 4n r r This is the Neumann formula; it involves a double line integral-one integration around loop 1, the other around loop 2 (Fig. 7.30). It's not very useful for practical calculations, but it does reveal two important things about mutual inductance: 1. M21 is a purely geometrical quantity, having to do with the sizes, shapes, and relative positions of the two loops. 2. The integral in Eq. 7.22 is unchanged ifwe switch the roles ofloops 1 and 2; it follows tha

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