Page 218 - Electromagnetics

By integration
1 ∞ dω ∞ dω
˜
˜

w e (t) = F(ω − ω 0 )F(ω − ω 0 ) ·
8 −∞ 2π −∞ 2π
j(ω−ω )t j(ω −ω)t

ˇ

ˇ
ˇ ∗
ˇ ∗
· E · E g (ω 0 )e + E · E g (ω 0 )e +

ˇ
ˇ
ˇ ∗ ˇ ∗
+ E · E˜ (ω 0 )e j(ω+ω )t + E · E ˜ (ω 0 )e − j(ω+ω )t .
Our last step is to compute the time-average value of w e and let α → 0. Applying
(4.56) we find
1 ∞ dω ∞ dω
˜
˜

w e = F(ω − ω 0 )F(ω − ω 0 ) ·
8 −∞ 2π −∞ 2π
π π

ˇ
ˇ
ˇ
ˇ ∗ ˇ ∗

ˇ ∗
· 2E · E g (ω 0 ) sinc [ω − ω ] + E · E + E · E ˜ (ω 0 ) sinc [ω + ω ]

ω 0 ω 0
where sinc(x) is defined in (A.9) and we note that sinc(−x) = sinc(x). Finally we let
˜
α → 0 and use (4.53) to replace F(ω) by a δ-function. Upon integration these δ-functions
set ω = ω 0 and ω = ω 0 . Since sinc(0) = 1 and sinc(2π) = 0, the time-average stored

electric energy density becomes simply

1 ∂[ω˜ ]
ˇ 2
w e = |E| . (4.59)
4 ∂ω
ω=ω 0
Similarly,

1 ∂[ω ˜µ]
ˇ 2
w m = |H| .
4 ∂ω
ω=ω 0
This approach can also be applied to anisotropic materials to give

1 ∂[ω ˜ ¯ ]
ˇ
ˇ ∗
w e = E · · E, (4.60)
4 ∂ω
ω=ω 0
1 ∂[ω ˜ ¯µ]
ˇ
ˇ ∗
w m = H · · H. (4.61)
4 ∂ω
ω=ω 0
See Collin [39] for details. For the case of a lossless, nondispersive material where the
constitutive parameters are frequency independent, we can use (4.49) and (A.76) to
simplify this and obtain
1 1
ˇ
ˇ
ˇ ∗
ˇ ∗
w e = E · ¯ · E = E · D , (4.62)
4 4
1 1
ˇ
ˇ
ˇ ∗
ˇ ∗
w m = H · ¯µ · H = H · B , (4.63)
4 4
in the anisotropic case and
1 1
ˇ
ˇ 2
ˇ ∗
w e = |E| = E · D , (4.64)
4 4
1 1
ˇ
ˇ 2
ˇ ∗
w m = µ|H| = H · B , (4.65)
4 4
ˇ ˇ ˇ
ˇ
in the isotropic case. Here E, D, B, H are all phasor fields as defined by (4.54).
© 2001 by CRC Press LLC