S. GLASGOW. M. WARE AND J PEATROSS PHYSICAL REVIEW E 64 046610 E0= u, (x, dx Wappror(x, t)=5E(x, t).D(x, )+5B(x, t).H(x,t) (x,)d2x;,t=≤t≤t Some texts on classical electrodynamics originally identified (63) Eq.(70)as the total energy density, i.e., as the object con- served in Poynting's theorem(48). Subsequent editions have The boundary term (the second integral)appears since the clarified that the quantity (70)is valid only for the time av- the boundary term appears since the ball's radius decreases strated that the correct object to be considered sad.demon- dimensions of the ball depend on time t. The -c multiplying erage of a single frequency. However, they have not demon- in size as time proceeds forward, and does so at the rate c. total energy density (i.e, a positive definite form indicating The boundary ab[apex, c(tapex-n)] is the surface of the the closed nature of the dynamics) ball embedded at time t(dimension 2) To make a comparison with Eq (50), we eliminate D(x, t) Using the conservation law(48)to eliminate u,(x, t)in and B(x, t) from the expression by way of Eq(22),(39) the first integral, and then using the divergence theorem to (without the time derivatives), and (34)(and the"magnetic exchange the volume integral for a surface integral, one gets analogs of these relations). Writing the result as closely as at Et is determined by the values of certain quantities possible to the form of Eq. (50), we get only on the balls boundary, IS(x, n)n(x E(x,)|2+;|H(x,t) +l(x,)dx,t;≤t≤ taper de(x, o)cE(x, o) Here n(x) is the unit outward normal to the boundary of the Xt)e ball at position xe db(xaper, c(tapert)). In Sec. Ill we show that u(x, t)>S(x, t)l. Since u is positive definite, this establishes the more useful fact that ExT l(x,1)=|a(x,D)=|S(x,t)≥S(x,1)m(x)≥-S(x1)·n(x) (65) +H(x,/e" ior &h(x, o)aH(x,) S(x,D)n(x)+u(x,1)≥0. dreH(x,T)+cc (71) Thus the integrand in Eq.(64) is non-negative and so the energy does not increase, Clearly densities(50)and(71) constitute different quadratic forms in the fields. In particular, whereas the dynamic total E认(1)≤0,t1≤t≤ taper (67) energy density (50) is manifestly positive definite for any field history, the approximate total energy density(71)can Equation(67)together with the initial data(62)demands that be shown to alternate sign for certain physically relevant E认1)≤Et1)=0;t≤t≤ taper (68) examples To illustrate this effect, we here consider the simple case which contradicts the non-negativity of Eu(t) Eq(61)unless of monochromatic electric fields given by E1)=Et)=0,1≤t≤1aper E(x,)=E0(x)e-1+c Since u is positive definite in the fields E(x, n) and (x,n), Eu(t vanishes for time in the indicated interval (In this example we examine only the electric contribution. only if those fields vanish in the cone YXaper, taper).This Also. it is useful to recall the distributional identities together with the causal relationship of the other two fields to hese fields then demands that all four fields vanish in the cone, thereby establishing luminal front velocity e -i(o-w)t drei(o-o)T=lim 0++(a-’) 3. The relationship between the dynamical energy and the From definition(50)it is clear that the dynamical energy density differs from the approximate energy density 046610-8E˙V~t!5 E B~xapex ,c(tapex2t)… ut~x,t!d3x 2c E ]B~xapex ,c(tapex2t)… u~x,t!d2x; ti<t<tapex . ~63! The boundary term ~the second integral! appears since the dimensions of the ball depend on time t. The 2c multiplying the boundary term appears since the ball’s radius decreases in size as time proceeds forward, and does so at the rate c. The boundary ]B@xapex ,c(tapex2t)# is the surface of the ball embedded at time t ~dimension 2!. Using the conservation law ~48! to eliminate ut(x,t) in the first integral, and then using the divergence theorem to exchange the volume integral for a surface integral, one gets that E˙V(t) is determined by the values of certain quantities only on the ball’s boundary, E˙V~t!52c E ]B„xapex ,c(tapex2t)… @S~x,t!•n~x! 1u~x,t!#d2x; ti<t<tapex . ~64! Here n(x) is the unit outward normal to the boundary of the ball at position xP]B„xapex ,c(tapex2t)…. In Sec. III we show that uu(x,t)u>iS(x,t)i. Since u is positive definite, this establishes the more useful fact that u~x,t!5uu~x,t!u>iS~x,t!i>uS~x,t!•n~x!u>2S~x,t!•n~x!, ~65! i.e., S~x,t!•n~x!1u~x,t!>0. ~66! Thus the integrand in Eq. ~64! is non-negative and so the energy does not increase, E˙V~t!<0; ti<t<tapex . ~67! Equation ~67! together with the initial data ~62! demands that EV~t!<EV~ti!50; ti<t<tapex , ~68! which contradicts the non-negativity of EV(t) Eq. ~61! unless EV~t!5EV~ti!50; ti<t<tapex . ~69! Since u is positive definite in the fields E(x,t) and H(x,t), EV(t) vanishes for time in the indicated interval only if those fields vanish in the cone V(xapex ,tapex). This together with the causal relationship of the other two fields to these fields then demands that all four fields vanish in the cone, thereby establishing luminal front velocity. 3. The relationship between the dynamical energy and the traditional approximate kinematic energy From definition ~50! it is clear that the dynamical energy density differs from the approximate energy density, uapprox~x,t!ª1 2 E~x,t!•D~x,t!1 1 2 B~x,t!•H~x,t!. ~70! Some texts on classical electrodynamics originally identified Eq. ~70! as the total energy density, i.e., as the object con￾served in Poynting’s theorem ~48!. Subsequent editions have clarified that the quantity ~70! is valid only for the time av￾erage of a single frequency. However, they have not demon￾strated that the correct object to be considered is a dynamical total energy density ~i.e., a positive definite form indicating the closed nature of the dynamics!. To make a comparison with Eq. ~50!, we eliminate D(x,t) and B(x,t) from the expression by way of Eq. ~22!, ~39! ~without the time derivatives!, and ~34! ~and the ‘‘magnetic’’ analogs of these relations!. Writing the result as closely as possible to the form of Eq. ~50!, we get uapprox~x,t! 5 1 2 iE~x,t!i 21 1 2 iH~x,t!i 2 1 i 4pE 2` 1` dvF E† ~x,t!e2ivt aˆ E † ~x,v!aˆ E~x,v! v 3 E 2` t dt eivtE~x,t! 1H† ~x,t!e2ivt aˆ H † ~x,v!aˆ H~x,v! v 3 E 2` t dt eivt H~x,t!G 1c.c. ~71! Clearly densities ~50! and ~71! constitute different quadratic forms in the fields. In particular, whereas the dynamic total energy density ~50! is manifestly positive definite for any field history, the approximate total energy density ~71! can be shown to alternate sign for certain physically relevant examples. To illustrate this effect, we here consider the simple case of monochromatic electric fields given by E~x,t!5E0~x!e2iVt 1c.c. ~72! ~In this example we examine only the electric contribution.! Also, it is useful to recall the distributional identities e2i(v2v8)t E 2` t dt ei(v2v8)t 5 lim e→01 1 e1i~v2v8! 5pd~v2v8!2iPS 1 v2v8 D . ~73! S. GLASGOW, M. WARE, AND J. PEATROSS PHYSICAL REVIEW E 64 046610 046610-8