20
Chapter 3 Conservation Laws In orer to check the physical consistency of the above set of equations governing Maxwell-Lorentz electroynamics [(2.10) an (2.12) or (1.65) an (1.68)], we examine the action of, an reaction on, the sources of the electromagnetic fiels. To be precise, we ask whether there is a correct balance in the exchange of energy, momentum, an angular momentum between the charge particles an the electromagnetic fiels. As we shall see, the Maxwell-Lorentz system as it stans implies the conservation of these mechanical properties, no matter how rapily the charges are moving. 3.1 Conservation of Energy We start with a consieration of the rate at which work is one on the particles, that is, the rate of energy transfer, or the power absorbe by the particles. For one particle, we know that the rate at which work is one on it is F v = ev E + gv B = (r)(j e E + j m B), (3.1) where we have use the Lorentz force law, (2.12), an the expressions for the currents, (1.44) an (2.7), for a point particle. We interpret this equation as meaning, even for general current istributions, that j e E+j m B is the rate of energy transfer from the fiel to the particles, per unit volume. Then through elimination of the currents by use of Maxwell s equations, (2.10), this rate can be rewritten as j e E + j m B = c = t ( B 1 ) c t E E + c ( E 2 + B 2 8π ( E 1 ) c t B B ) ( c E B ). (3.2) The general form of any local conservation law, (1.45) or (1.46), suggests the following interpretations: 21
22 CHAPTER 3. CONSERVATION LAWS 1. In the absence of charges (j e = j m = 0), this is the local energy conservation law E 2 + B 2 + t 8π c E B = 0. (3.3) We label the two objects appearing here as energy ensity = U = E2 + B 2, (3.4) 8π energy flux vector = S = c E B. (3.5) [The latter is usually calle the Poynting vector, after John Henry Poynting (1852 1914).] 2. In the presence of charges, the relation (3.2) is t U + S + j e E + j m B = 0, (3.6) which, if we integrate over an arbitrary volume V, boune by a surface S, becomes (r)u + S S + (r)(j e E + j m B) = 0. (3.7) t S V V The three terms here are ientifie, respectively, as the rate of change of the electromagnetic fiel energy within the volume, the rate of flow of electromagnetic energy out of the volume, an the rate of transfer of electromagnetic energy to the charge particles. Thus, (3.6) gives a complete escription of energy conservation. 3.2 Conservation of Momentum Next we consier the force on a particle, (2.12), as the rate of change of momentum, F = e (E + v ) c B + g (B v ) c E ( = (r) ρ e E + 1 c j e B + ρ m B 1 ) c j m E (r)f, (3.8) where f is the force ensity. Removing reference to the (generalize) charge an current ensities by use of Maxwell s equations, (2.10), we rewrite the force ensity f as f = 1 [E( E) + B( B)]
3.2. CONSERVATION OF MOMENTUM 23 + 1 = t [( 1 c ) ( t E + B B + E 1 )] c t B E E B c + 1 [ E ( E) + E( E) B ( B) + B( B)]. The quaratic structure in E occurring here is (3.9) E ( E) + E( E) = E2 2 + (E )E + E( E) = ( ) 1 E2 2 + EE, (3.10) which introuces yaic notation, incluing the unit yaic 1, with components 1 kl = δ kl = { 1, k = l, 0, k l, (3.11) where δ kl is the Kronecker δ symbol. (See Problem 3.1.) The analogous result hols for B. Accoringly, the force ensity is f = E B t c ( 1 E2 + B 2 ) EE + BB. (3.12) 8π We interpret this equation physically by ientifying an momentum ensity = G = E B c, (3.13) momentum flux (stress tensor) = T = 1 E2 + B 2 8π EE + BB. (3.14) When f = 0, we obtain the local statement of the conservation of momentum of the electromagnetic fiel. A full account of momentum balance is containe in t G + T + f = 0. (3.15) The volume integral of this equation for electromagnetic momentum is interprete analogously to the energy result, (3.7). The components of the stress tensor are given by T kl = δ kl U E ke l + B k B l. (3.16) Notice that the stress tensor is symmetrical, T kl = T lk, which, as we shall see in the next section, is require in orer to obtain a local conservation law for angular momentum. The trace of T, the sum of the iagonal elements T kk, is simply the energy ensity, (3.4), TrT = k T kk = U. (3.17)
24 CHAPTER 3. CONSERVATION LAWS We also note that the Poynting vector, (3.5), is proportional to the momentum ensity, S = c 2 G, (3.18) which has the structure of energy ensity velocity = c 2 (mass ensity velocity). (3.19) This is the first inication of the relativistic connection between energy an mass, E = mc 2. 3.3 Conservation of Angular Momentum. Virial Theorem Having iscusse momentum, we now turn to angular momentum. We will use tensor notation to write (3.15) in component form, t G k + l T lk + f k = 0, (3.20) where we have also use the summation convention: Whenever an inex is repeate, a sum over all values of that inex is assume, a i b i 3 a i b i = a b. (3.21) i=1 The rate of change of angular momentum is the torque τ, which, for one particle, is τ = r F = (r) r f, (3.22) where the volume-integrate form is no longer restricte to a single particle. The torque ensity, the moment of the force ensity, can be written in component form as (r f) i = ǫ ijk x j f k, (3.23) where we have introuce the totally antisymmetric (Levi-Civita) symbol ǫ ijk, which changes sign uner any interchange of two inices, ǫ ijk = ǫ jik = ǫ kji = ǫ ikj = +ǫ kij = +ǫ jki, (3.24) an is normalize by ǫ 123 = 1. In particular, then, it vanishes if any two inices are equal, ǫ 112 = 0, for example. The torque ensity may be obtaine by first taking the moment of the force ensity equation (3.20), where we have note that t x jg k + l (x j T lk ) T jk + x j f k = 0, (3.25) l x j = δ lj. (3.26)
3.4. CONSERVATION LAWS AND THE SPEED OF LIGHT 25 When we now multiply (3.25) with ǫ ijk an sum over repeate inices, we fin that the terms involving spatial erivatives can be written as a ivergence: t (ǫ ijkx j G k ) + l (ǫ ijk x j T lk ) + ǫ ijk x j f k = 0. (3.27) This final step is justifie only because T kl is symmetrical (thus this symmetry is require for the existence of a local conservation law of angular momentum). We therefore ientify the following electromagnetic angular momentum quantities: angular momentum ensity = J = r G (3.28) angular momentum flux tensor = K, K ij = ǫ jkl x k T il. (3.29) The interpretation of (3.27) as a local account of angular momentum conservation for fiels an particles procees as before. (See Problem 3.5.) Another important application of (3.25) results if we set j = k an sum. With the ai of (3.17) this gives t (r G) + (T r) U + r f = 0, (3.30) which we call the electromagnetic virial theorem, in analogy with the mechanical virial theorem of Ruolf Clausius (1822 1888). (See Chapter 8.) 3.4 Conservation Laws an the Spee of Light In this section, we restrict our attention to electromagnetic fiels in omains free of charge particles, specifically, moving, finite regions occupie by electromagnetic fiels, which we will refer to as electromagnetic s. The total electromagnetic energy of such a is constant in time: t E = (r) t U = (r) S = 0, (3.31) inasmuch as the resulting surface integral, conucte over an enclosing surface on which all fiels vanish, equals zero. Similar consierations apply to the total electromagnetic linear an angular momentum, t P = t J = (r) t G = (r) t (r G) = (r) T = 0, (3.32) (r) ( T r) = 0. (3.33) With an eye towar relativity, we consier the space an time moments of (3.6) an (3.15), respectively, combine as a single vector statement: ( ) ( ) 0 = x k t U + S c 2 t t G k + l T lk, (3.34)
26 CHAPTER 3. CONSERVATION LAWS outsie the charge an current istributions. Exploiting the connection between S an G [(3.18)], we can rewrite (3.34) as a local conservation law, much as the equality of T jk an T kj lea to the conservation of angular momentum: t (ru c2 tg) + (Sr c 2 tt) = 0. (3.35) When (3.35) is integrate over a volume enclosing the electromagnetic, the surface term oes not contribute, an we fin (r)(ru c 2 tg) = 0. (3.36) t The volume integral of the momentum ensity is the total momentum P, (r)g = P, (3.37) which as note in (3.32) is constant in time. Consequently, we can rewrite (3.36) as (r)ru = c 2 P, (3.38) t where the integral here provies an energy weighting of the position vector, at each instant of time, (r)ru(r, t) = E r E (t), (3.39) where, as in (3.31), the energy E is E = (r) U. (3.40) Thus the motion of this energy-centroi vector is governe by E c 2 t r E(t) = P, (3.41) which is to say that the center of energy, r E (t), moves with constant velocity, t r E(t) = v E, (3.42) the total momentum being that velocity multiplie by a mass, m = E/c 2. (3.43) The application of the virial theorem, (3.30), to an electromagnetic supplies another velocity. We infer that (r)r G = E. (3.44) t
3.4. CONSERVATION LAWS AND THE SPEED OF LIGHT 27 By introucing a momentum weighting for the position vector, (r)r G(r, t) = r P (t) P, (3.45) we euce that the center of momentum moves with velocity which is constant in the irection of the momentum, We combine (3.47) with (3.41) to yiel t r P(t) = v P, (3.46) v P P = E. (3.47) v P v E = c 2. (3.48) If the flow of energy an momentum takes place in a single irection, it woul be reasonable to expect that these mechanical properties are being transporte with a common velocity, v E = v P = v, (3.49) which then has a efinite magnitue, v v = c 2, v = c, (3.50) which supplies the physical ientification of c as the spee of light. Of course, this ientification was an input to our inference of Maxwell s equations. We here recover it from a consieration of energy an momentum, thus inicating the consistency of Maxwell s equations. The relation between the momentum an the energy of this electromagnetic is then E = v P, P = E c2v, (3.51) so we learn that E = Pc, v = c P P, (3.52) which results express the mechanical properties of a localize electromagnetic carrying both energy an momentum at the spee of light, in the irection of the momentum. There is another, somewhat more irect, mechanical proof that electromagnetic s propagate at spee c. When no charges or currents are present, the local equation of energy conservation, (3.3), implies [ ] (r 2 c 2 t 2 ) t U + S = 0, (3.53) which can be rewritten, using (3.18), as t [(r2 c 2 t 2 )U] + [(r 2 c 2 t 2 )S] + 2c 2 [tu r G] = 0. (3.54)
28 CHAPTER 3. CONSERVATION LAWS Integrating this over all space an using the iea of energy an momentum weighting to efine averages, as before, we obtain [ r 2 E (t) c 2 t 2] E = 2c 2 [ r P (t) P te]. (3.55) t Accoring to (3.47), the combination appearing on the right is a constant of the motion, which we can put equal to zero by ientifying the coorinate origin with r P at t = 0. The time integral of this equation is then which implies, for large times, r 2 E = (ct) 2 + constant, (3.56) ( r 2 E (t) ) 1/2 ct; (3.57) the center of energy of the moves away from the origin at the spee of light. What are the fiels oing to enforce the conitions (3.51) of simple mechanical flow in a single irection? The relation between momentum an energy, (3.52), E = P c, can be expresse in terms of the fiels as (r) E2 + B 2 8π = (r) E B, (3.58) where the volume integrations are extene over the. Now, a sum of vectors of given magnitues is of maximum magnitue when all those vectors are parallel, which is to say here that (r) E2 + B 2 (r) E B, (3.59) 2 where equality hols only when E B everywhere points in the same irection, that of the s total momentum or velocity. On the other han, we note the inequality, (E B) 2 = E 2 B 2 (E B) 2 ( E 2 + B 2 ) 2 [ (E 2 B 2 ) 2 ] ( E = + (E B) 2 2 + B 2 ) 2, 2 2 2 (3.60) where the equality hols only if both E B = 0 an E 2 = B 2. So we euce the opposite inequality to (3.59), (r) E B (r) E2 + B 2. (3.61) 2 Comparing (3.59) an (3.61), we see that both equalities must hol, so that E B = 0, E 2 = B 2, (3.62)
3.5. PROBLEMS FOR CHAPTER 3 29 E B v Figure 3.1: Electric an magnetic fiels for an electromagnetic propagating with velocity v. an E B is uniirectional, pointing in the irection of propagation. Accoringly, the electric an magnetic fiels in a uniirectional are, everywhere within the, of equal magnitue, mutually perpenicular, an perpenicular to the irection of motion of the. (See Fig. 3.1.) These are the familiar properties of the electromagnetic fiels of a light wave, which are here erive without recourse to explicit solutions to Maxwell s equations. 3.5 Problems for Chapter 3 1. The unit yaic 1 is efine in terms of orthogonal unit vectors i, j, k by 1 = ii + jj + kk. Verify that (A is an arbitrary vector) A 1 = A, 1 A = A, 1 1 = 1. Repeat, using components, i.e., A B = A i B i. Expan the following proucts of vectors with yaics: A (BC), (AB) C, A (BC), (AB) C. 2. Let A(r) an B(r) be vector fiels. Show that (AB) = (A )B + B( A). Let λ(r) further be an arbitrary scalar function. Simplify (λab), (λa B). 3. An infinitesimal rotation is escribe by its effect on an arbitrary vector V by δv = δω V,
30 CHAPTER 3. CONSERVATION LAWS where the irection of δω points in the irection of the rotation, an has magnitue equal to the (infinitesimal) amount of the rotation. Check that δ(v 2 ) = 0. The statement that, if B an C are vectors, so is B C, is expresse by δ(b C) = δb C + B δc = δω (B C). Verify irectly the resulting relation among the arbitrary vectors. 4. Verify the following relations for the electromagnetic stress tensor: (a) (b) TrT = T kk = U, TrT 2 = T kl T lk = 3U 2 2(cG) 2 U 2, an (c) ett = U[U 2 (cg) 2 ]. Here the summation convention is employe, an the trace an eterminant refer to T thought of as a 3 3 matrix. 5. Show that the angular momentum conservation law erive in Section 3.3 can be written as t J + K + r f = 0, where the angular momentum ensity is J = r G, an the angular momentum flux tensor is K = T r, the cross prouct referring to the secon vector inex of T. 6. What if r P (0) 0 in (3.55)? Show that the integral of that equation can be interprete in analogy with a group of particles that, at time t = 0, are set off with various positions an velocities, thereafter to move with those constant velocities, r(t) = r(0) + vt. Square an average this position vector, an upon comparison with the solution of (3.55), ientify r(0) v an v.
3.5. PROBLEMS FOR CHAPTER 3 31 7. As in Problem 2.1, let F = E + ib, F = E ib. Ientify the scalar the vector an the yaic 1 8π F F, 1 8πi F F, 1 8π (FF + F F). What happens to these quantities if F is replace by e iφ F, φ being a constant? 8. Electric charge e is locate at the fixe point 1 2R. Magnetic charge g is statione at the fixe point 1 2R. What is the momentum ensity at the arbitrary point r? Verify that it is ivergenceless by writing it as a curl. Evaluate the electromagnetic angular momentum, the integrate moment of the momentum ensity. Recognize that it is a graient with respect to R. Continue the evaluation to iscover that it epens only on the irection of R, not its magnitue. This is the naive, semiclassical basis for the charge quantization conition of Dirac, eg = n 2 hc.