Page 216 - Electromagnetics

-40 -20 0 20 40
ω t
0

ω/ω 0
-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0

Figure 4.2: Temporal (top) and spectral magnitude (bottom) dependences of E used to
compute energy stored in a dispersive material.

that we may treat as real. Then, since there is no dissipation, we conclude that the
term (4.40) represents the time rate of change of stored energy at time t, including the
effects of field build-up. Hence the interpretation 2
∂D ∂w e ∂B ∂w m
E · = , H · = .
∂t ∂t ∂t ∂t
We shall concentrate on the electric field term and later obtain the magnetic field term
by induction.
Since for periodic signals it is more convenient to deal with the time-averaged stored
energy than with the instantaneous stored energy, we compute the time average of w e (r, t)
over the period of the sinusoid centered at the time origin. That is, we compute

1 T/2
w e = w e (t) dt (4.56)
T −T/2

where T = 2π/ω 0 . With α → 0, this time-average value is accurate for all periods of the
sinusoidal wave.
Because the most expedient approach to the computation of (4.56) is to employ the
Fourier spectrum of E,weuse

1 ∞ jωt 1 ∞ − jω t
˜

˜ ∗
E(r, t) = E(r,ω)e dω = E (r,ω )e dω ,
2π 2π
−∞ −∞
∂D(r, t) 1 ∞ jωt 1 ∞ − jω t
˜

˜ ∗

= ( jω)D(r,ω)e dω = (− jω )D (r,ω )e dω .
∂t 2π 2π
−∞ −∞
2 Note that in this section we suppress the r-dependence of most quantities for clarity of presentation.
© 2001 by CRC Press LLC