Page 212 - Electromagnetics

when the constitutive parameters have the form (2.29)–(2.31). Physically, this term
describes both the energy stored in the electromagnetic field and the energy dissipated by
the material because of time lags between the application of E and H and the polarization
or magnetization of the atoms (and thus the response fields D and B). In principle this
term can also be used to describe active media that transfer mechanical or chemical
energy of the material into field energy.
Instead of attempting to interpret (4.40), we concentrate on the physical meaning of

−∇ · S(r, t) =−∇ · [E(r, t) × H(r, t)].

We shall postulate that this term describes the net flow of electromagnetic energy into the
point r at time t. Then (4.39) shows that in the absence of impressed sources the energy
flow must act to (1) increase or decrease the stored energy density at r, (2) dissipate
c
energy in ohmic losses through the term involving J , or (3) dissipate (or provide) energy
through the term (40). Assuming linearity we may write
∂ ∂ ∂
−∇ · S(r, t) = w e (r, t) + w m (r, t) + w Q (r, t), (4.41)
∂t ∂t ∂t
where the terms on the right-hand side represent the time rates of change of, respectively,
stored electric, stored magnetic, and dissipated energies.

4.5.1 Dissipation in a dispersive material

Although we may, in general, be unable to separate the individual terms in (4.41), we
can examine these terms under certain conditions. For example, consider a field that
builds from zero starting from time t =−∞ and then decays back to zero at t =∞.
Then by direct integration 1

∞

− ∇· S(t) dt = w em (t =∞) − w em (t =−∞) + w Q (t =∞) − w Q (t =−∞)
−∞
where w em = w e +w m is the volume density of stored electromagnetic energy. This stored
energy is zero at t =±∞ since the fields are zero at those times. Thus,

∞
w Q =− ∇· S(t) dt = w Q (t =∞) − w Q (t =−∞)
−∞
represents the volume density of the net energy dissipated by a lossy medium (or supplied
by an active medium). We may thus classify materials according to the scheme

w Q = 0, lossless,
w Q > 0, lossy,
w Q ≥ 0, passive,
w Q < 0, active.
For an anisotropic material with the constitutive relations

˜
˜
˜
˜
J = ˜ ¯σ · E,
D = ˜ ¯ · E, B = ˜ ¯µ · H, ˜ c ˜
1 Note that in this section we suppress the r-dependence of most quantities for clarity of presentation.

© 2001 by CRC Press LLC