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Feynman Physics Lectures V2 Ch27 1963-01-21 Field Energy and Momentum

Feynman Physics Lectures V2 Ch27 1963-01-21 Field Energy and Momentum

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Unformatted text preview: 27 Field Energy and Field Momentum 27—1 Local conservation It is clear that the energy of matter is not conserved. When an object radiates light it loses energy. However, the energy lost is possibly describable in some other form, say in the light. Therefore the theory of the conservation of energy 15 incomplete without a consrderation of the energy which is associated with the light or, in general, with the electromagnetic field. We take up now the law of conserva- tion of energy and, also, of momentum for the fields. Certainly, we cannot treat one without the other, because in the relatiVity theory they are different aspects of the same four-vector. Very early in Volume I, we discussed the conservation of energy; we said then merely that the total energy in the world is constant. Now we want to extend the idea of the energy conservation law in an important way—in a way that says something in detail about how energy is conserved. The new law will say that if energy goes away from a region, it is because it flows away through the boundaries of that region. It is a somewhat stronger law than the conservation of energy without such a restriction. To see what the statement means, let’s look at how the law of the conservation of charge works. We described the conservation of charge by saying that there is a current density j and a charge density p, and that when the charge decreases at some place there must be a flow of charge away from that place. We call that the conservation of charge. The mathematical form of the conservation law is . 6p V-J———:9?- (27.1) This law has the consequence that the total charge in the world is always constant— there 15 never any net gain or loss of charge. However, the total charge in the world could be constant in another way. Suppose that there is some charge Q1 near some pomt (1) while there is no charge near some point (2) some distance away (Fig. 27—1). Now suppose that, as time goes on, the charge Q1 were to gradually fade away and that simultaneously with the decrease of Q1 some charge Q2 would appear near point (2), and in such a way that at every instant the sum of Q1 and Q2 was a constant. In other words, at any intermediate state the amount of charge lost by Q1 would be added to Q2. Then the total amount of charge in the world would be conserved. That’s a “world-wide” conservation, but not what we will call a “local” conservation, because in order for the charge to get from (1) t0 (2). it didn’t have to appear anywhere in the space between pomt (l) and point (2). Locally, the charge was just “lost.” There is a difficulty With such a “world-wide” conservation law in the theory of relativity. The concept of “simultaneous moments” at distant pomts is one which is not equivalent in different systems. Two events that are simultaneous in one system are not Simultaneous for another system moving past. For “world-wide" conservation of the kind described, it is necessary that the charge lost from Q1 should appear simultaneously in Q2. Otherwise there would be some moments when the charge was not conserved. There seems to be no way to make the law of charge conservation relativistically invariant without making it a “local” conservation law. As a matter of fact, the requirement of the Lorentz relativistic invariance seems to restrict the possible laws of nature in surprismg ways. In modern quantum field theory, for example, people have often wanted to alter the theory by allowing what we call a “nonlocal” interaction—where something here 27—1 27—1 Local conservation 27—2 Energy conservation and electromagnetism 27—3 Energy density and energy flow in the electromagnetic field 27—4 The ambiguity of the field energy 27—5 Examples of energy flow 27—6 Field momentum (I) (2’ A QI v 02 (b) Fig. 27—]. Two ways to conserve charge: (a) Q] + 02 is constant; (b) th/dt = fi-nda = —sz/dt. has a direct effect on something there—but we get in trouble With the relatiVity principle. “Local” conservation involves another idea. It says that a charge can get from one place to another only ifthere is something happening in the space between. To describe the law we need not only the density of charge, p, but also another kind of quantity, namely j, a vector giving the rate of flow of charge across a surface. Then the flow is related to the rate of change of the density by Eq. (27.1). This 18 the more extreme kind of a conservation law. It says that charge is con- served in a special way—conserved “locally.” It turns out that energy conservation is also a local process. There is not only an energy density in a given region of space but also a vector to represent the rate of flow of the energy through a surface. For example, when a light source radiates, we can find the light energy movmg out from the source. If we imagine some mathe- matical surface surrounding the light source, the energy lost from inside the surface is equal to the energy that flows out through the surface. 27—2 Energy conservation and electromagnetism We want now to write quantitatively the conservation of energy for electro- magnetism. To do that, we have to describe how much energy there is in any volume element of space, and also the rate of energy flow. Suppose we think first only of the electromagnetic field energy. We Will let u represent the energy denszty in the field (that is, the amount of energy per unit volume in space) and let the vector S represent the energy flux of the field (that is, the flow of energy per unit time across a unit area perpendicular to the flow). Then, in perfect analogy with the conservation of charge, Eq (27 l), we can write the “local” law of energy conservation in the field as 611 71 — —V - S. (27.2) Of course, this law is not true in general; it is not true that the field energy is conserved. Suppose you are in a dark room and then turn on the light sw1tch. All of a sudden the room is full of light, so there is energy in the field, although there wasn’t any energy there before. Equation (27.2) is not the complete conservation law, because the field energy alone is not conserved, only the total energy in the world—there is also the energy of matter. The field energy Will change if there is some work being done by matter on the field or by the field on matter However, if there is matter inSide the volume of interest, we know how much energy it has: Each particle has the energy muc2/x/l ~— 7‘2/62. The total energy of the matter isjust the sum of all the particle energies, and the flow of this energy through a surface is just the sum of the energy carried by each particle that crosses the surface We want now to talk only about the energy of the electromagnetic field. So we must write an equation which says that the total field energy in a given volume decreases either because field energy flows out of the volume or because the field loses energy to matter (or gains energy, which is just a negative loss). The field energy inside a volume V is fudV, V and its rate of decrease is minus the time derivative of this integral. The flow of field energy out of the volume V is the integral of the normal component of S over the surface 2 that encloses V, / S ' n da. 2 7% / udV 2 f S - n da + (work done on matter inside V). (273) _ v x So 27—2 We have seen before that the field does work on each unit volume of matter at the rate E -j. [The force on a particle is F = q(E + v x B), and the rate of doing work is F~ v = qE' v. If there are N particles per unit volume, the rate of dOing work per unit volume is NqE- v, but qu = j] So the quantity E ‘j must be equal to the loss of energy per unit time and per unit volume by the field. Equation (27.3) then becomes _9_/ udV=/s.nda+/E-jdV. (27.4) at V 2 V This is our conservation law for energy in the field. We can convert it into a differential equation like Eq. (27.2) if we can change the second term to a volume integral That is easy to do with Gauss’ theorem. The surface integral of the normal component of S is the integral of its divergence over the volume inSide. So Eq. (27.3) is equivalent to — @dV=/ v-SdV+/E~jdV, V dt V V where we have put the time derivative of the first term inside the integral. Since this equation is true for any volume, we can take away the integrals and we have the energy equation for the electromagnetic fields: —6—u=v-S+E-j. (27.5) 6! Now this equation doesn’t do us a bit of good unless we know what u and S are. Perhaps we should just tell you what they are in terms of E and B, because all we really want is the result. However. we would rather show you the kind of argument that was used by Poynting in 1884 to obtain formulas for S and u, so you can see where they come from. (You won’t, however. need to learn this de- rivation for our later work.) 27—3 Energy density and energy flow in the electromagnetic field The Idea is to suppose that there is a field energy density u and a flux S that depend only upon the fields E and B. (For example, we know that in electrostatics, at least. the energy density can be written %60E - E.) Of course, the u and S might depend on the potentials or something else. but let’s see what we can work out We can try to rewrite the quantity E -j in such a way that it becomes the sum of two terms. one that is the time derivative of one quantity and another that is the divergence ofa second quantity. The first quantity would then be u and the second would be S (With suitable signs). Both quantities must be written in terms of the fields only; that is, we want to write our equality as E-j=—_—v.s. (27.6) The left-hand side must first be expressed in terms of the fields only. How can we do that" By using Maxwell’s equations. of course. From Maxwell’s equation for the curl of B, . _ 2 _ £3 — soc V X B 60 6t Substituting this in (27 6) we Will have only E’s and B’s: E-j = eoczE-(V x B) — 605%? (27.7) We are already partly finished. The last term is a time derivative—it is (a/at)(§eoE - E). So fieOE - E is at least one part of u. It’s the same thing we found in electrostatics. Now, all we have to do is to make the other term into the divergence of something. 27—3 Notice that the first term on the right-hand side of (27.7) is the same as (V X B) ' E. (27.8) And, as you know from vector algebra, (a X b) c is the same as a- (b X c); so our term is also the same as v . (B x E) (27.9) and we have the divergence of “something,” just as we wanted. Only that‘s wrong' We warned you before that V is “like” a vector, but not “exactly" the same. The reason it is not is because there is an additional convention from cal- culus: when a derivative operator is in front of a product, it works on everything to the right. In Eq (27.7). the V operates only on B, not on E But in the form (27 9), the normal convention would say that V operates on both B and E So it’s not the same thing In fact, if we work out the components of V - (B X E) we can see that it is equal to E~ (V X B) plus some other terms. It’s like what happens when we take a derivative ofa product in algebra For instance, d __ dj‘ dg g;(fg)—Eg+/g; Rather than working out all the components of V - (B X E), we would like to show you a trick that is very useful for this kind of problem. It is a trick that allows you to use all the rules of vector algebra on expressions With the V operator. without getting into trouble The trick is to throw out—for a while at least-the rule of the calculus notation about what the derivative operator works on You see. ordinarily, the order of terms is used for two separate purposes. One is for calculus: f(d/dx)g is not the same as g(a’/dx)f; and the other is for vectors: a X b is difTerent from b X a. We can, ifwe want.chooseto abandon momentarily the calculus rule. Instead of saying that a derivative operates on everything to the right, we make a new rule that doesn’t depend on the order in which terms are writ- ten down Then we can Juggle terms around Without worrying Here is our new convention. we show. by a subscript. what a difl‘erential op— erator works on; the order has no meaning. Suppose we let the operator D stand for 0/6X. Then Df means that only the derivative of the variable quantity 1‘ IS taken. Then (if M = a; fog = (3%) g. But notice now that according to our new rule,fD,g means the same thing We can write the same thing any which way. Dxfg = gD/f = fog = fg D!- You see, the Df can even come after everything. (It‘s surprismg that such a handy notation is never taught in books on mathematics or physics.) You may wonder: What if I want to write the derivative of fig? I want the derivative of both terms. That’s easy, you Just say so; you write D/(fg) + Dg(fg). That is Just g(6f/6X) —l— f(6g/6x), which is what you mean in the old notation by a(fg)/6x. You Will see that it is now going to be very easy to work out a new expreSSion for V - (B X E). We start by changing to the new notation; we write But if we have D/ g. it means V-(BXE)=V,;-(BXE)+VE~(B><E). (27.10) The moment we do that we don’t have to keep the order straight any more We always know that VE operates on E only. and V]; operates on B only In these circumstances, we can use V as though it were an ordinary vector. (Of course, 27—4 when we are finished, we will want to return to the “standard” notation that everybody usually uses ) So now we can do the various things like interchanging dots and crosses and making other kinds of rearrangements of the terms. For instance, the middle term of Eq. (27.10) can be rewritten as [5- VB X B. (You remember that a - b X c = b - c X a.) And the last term is the same as B - E X VE. It looks freakish, but it is all right. Now if we try to go back to the ordinary convention, we have to arrange that the V operates only on its “own” variable. The first one is already that way, so we canjust leave off the subscript. The second one needs some rearranging to put the V in front of the E, which we can do by reversing the cross product and changing sign: B-(EX VE) = —B-(VE x E). Now it is in a conventional order, so we can return to the usual notation. Equation (27.10) is equivalent to V-(BXE)=E-(VXB)—B-(VXE). (27.11) (A quicker way would have been to use components in this special case, but it was worth taking the time to show you the mathematical trick. You probably won’t see it anywhere else, and it is very good for unlocking vector algebra from the rules about the order of terms with derivatives.) We now return to our energy conservation discussion and use our new result, Eq. (27.11), to transform the V X B term of Eq. (27.7). That energy equatlon becomes E-j = Eoc2V - (B x E) + 60623- (V X E) — .331 @6055) (27.12) Now you see we’re almost finished. We have one term which is a nice derivative with respect to t to use for u and another that is a beautiful divergence to represent S. Unfortunately, there is the center term left over, which is neither a divergence nor a derivative with respect to t. So we almost made it, but not quite. After some thought, we look back at the differential equations of Maxwell and discover that V X E is, fortunately, equal to —6B/6t, which means that we can turn the extra term into something that is a pure time derivative: B‘(VXE)=B-(—%>= —%<!%5>. Now we have exactly what we want. Our energy equation reads . 2 6 60C2 60 E-J=V.(€OCBXE)—a_tTB.B+7EIE, (2713) which is exactly like Eq. (27.6), if we make the definitions 2 u=%E-E+E°TCB.B (27.14) and = eoc2E x B. (27.15) (Reversing the cross product makes the signs come out right.) Our program was successful. We have an expression for the energy densrty that IS the sum of an “electric” energy density and a “magnetic” energy density, whose forms are just like the ones we found in statics when we worked out [he energy in terms of the fields. Also, we have found a formula for the energy flow vector of the electromagnetic field. This new vector, S = eoczE X B, is called “Poynting’s vector,” after 1ts discoverer. It tells us the rate at which the field energy moves around in space. The energy which flows through a small area do per second is S ' n da, where n is the unit vector perpendicular to do. (Now that we have our formulas for u and S, you can forget the derivations if you want.) 27—5 S -‘4vv———A4—e» V MRECHON OF WAVE PROPAGKHON Fig. 27—2. The vectors E, B, and S for a light wave. 27—4 The ambiguity of the field energy Before we take up some applications of the Poynting formulas [Eqs. (27.14) and (27.15)], we would like to say that we have not really “proved” them. All we did was to find a possible “it” and a posszble “S.” How do we know that by juggling the terms around some more we couldn’t find another formula for “u” and another formula for “S”? The new S and the new u would be different, but they would still satisfy Eq. (27.6). It’s possrble. It can be done, but the forms that have been found always involve various derivatives of the field (and always with second-order terms like a second derivative or the square of a first derivative). There are, in fact, an infinite number of different possibilities for u and S, and so far no one has thought of an experimental way to tell which one is right! People have guessed that the simplest one is probably the correct one, but we must say that we do not know for certain what is the actual location in space of the electro- magnetic field energy. So we too will take the easy way out and say that the field energy is given by Eq. (27.14). Then the flow vector S must be given by Eq. (27.15). It is interesting that there seems to be no unique way to resolve the indefinite- ness in the location of the field energy. It is sometimes claimed that this problem can be resolved by using the theory of gravitation in the following argument. In the theory of gravity, all energy is the source of gravitational attraction. There- fore the energy density of electricity must be located properly if we are to know in which direction the gravity force acts. As yet, however, no one has done such a delicate experiment that the precise location of the gravitational influence of electromagnetic fields could be determined. That electromagnetic fields alone can be the source of gravitational force is an idea it is hard to do without. It has, in fact, been observed that light is deflected as it passes near the sun—we could say that the sun pulls the light down toward it. Do you not want to allow that the light pulls equally on the sun? Anyway, everyone always accepts the Simple expreSSions we have found for the location of electromagnetic energy and its flow. And although sometimes the results obtained from using them seem strange, noboby has ever found anything wrong With them—that is, no disagreement with experiment. So we will follow the rest of the world—besides, we believe that it is probably perfectly right. We should make one further remark about the energy formula. In the first place, the energy per unit volume in the field is very simple: It is the electrostatic energy plus the magnetic energy, ifwe write the electrostatic energy in terms of E2 and the magnetic energy as 82. We found two such expressions as possible expreSSions for the energy when we were dOing static problems. We also found a number of other formulas for the energy in the electrostatic field, such as p¢>, which is equal to the integral of E - E in the electrostatic case However, in an electrodynamic field the equality failed, and there was no obvrous chOice as to which was the right one. Now we know which is the right one. Similarly, we have fOund the formula for the magnetic energy that is correct in general The right formula for the energy dens1ty of dynarmc fields 15 Eq (27.14) 27—5 Examples of energy flow Our formula for the energy flow vector S is something quite new. We want now to see how it works in some specral cases and also to see whether it checks out with anything that we knew before. The first example we will take is light. In a light wave we have an E vector and a B vector at right angles to each other and to the direction of the wave p...
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