Page 98 - Electromagnetics Handbook

hence
∂D
w em =−∇ · (E × H) + H · (∇× E) − · E
∂t
by (B.44). Substituting for ∇× E from (2.1) we have

∂D ∂B
w em =−∇ · (E × H) − E · + H · .
∂t ∂t
This is not quite of the form (2.285) since a single term representing the time rate of
change of energy density is not present. However, for a linear isotropic medium in which

and µ do not depend on time (i.e., a nondispersive medium) we have

∂D ∂E 1 ∂ 1 ∂
E · =
E · =
(E · E) = (D · E), (2.295)
∂t ∂t 2 ∂t 2 ∂t
∂B ∂H 1 ∂ 1 ∂
H · = µH · = µ (H · H) = (H · B). (2.296)
∂t ∂t 2 ∂t 2 ∂t
Using this we obtain
∂
∇· S em + W em =−f em · v =−J · E (2.297)
∂t
where
1
W em = (D · E + B · H)
2
and

S em = E × H. (2.298)

Equation (2.297) is the point form of the energy conservation theorem, also called Poynt-
ing’s theorem after J.H. Poynting who first proposed it. The quantity S em given in
(2.298) is known as the Poynting vector. Integrating (2.297) over a volume and using the
divergence theorem, we obtain the large-scale form

1 ∂

− J · E dV = (D · E + B · H) dV + (E × H) · dS. (2.299)
V V 2 ∂t S
This also holds for a nondispersive, linear, bianisotropic medium with a symmetric con-
stitutive matrix [101, 185].
We see that the electromagnetic energy conservation theorem (2.297) is identical in
form to the mechanical energy conservation theorem (2.285). Thus, if the system is com-
posed of just the kinetic and electromagnetic subsystems, the mechanical force exactly
balances the Lorentz force, and (2.297) and (2.285) add to give

∂
∇· (S em + S k ) + (W em + W k ) = 0, (2.300)
∂t
showing that energy is conserved for the entire system.
As in the mechanical system, we identify W em as the volume electromagnetic energy
density in V , and S em as the density of electromagnetic energy flowing across the bound-
ary of V . This interpretation is somewhat controversial, as discussed below.

© 2001 by CRC Press LLC