Page 251 - Electromagnetics

have, substituting the field expressions,

3 3
1
i J E 1
c J E

c
i
− |J ||E i |C ii (t) dV = |J ||E i |C ii (t) dV +
i
i
2 V i=1 2 V i=1
3 " #
∂
1 DE 1 BH

+ |D i ||E i |C ii (t) + |B i ||H i |C ii (t) dV +
∂t V i=1 4 4
3
1
EH

ˆ
ˆ
+ |E i ||H j |(i i × i j ) · ˆ nC ij (t) dS. (4.143)
2 S i, j=1
c
Here we remember that the conductivity relating E to J must also be nondispersive.
Note that the electric and magnetic energy densities w e (r, t) and w m (r, t) have the time-
average values w e (r, t) and w m (r, t) given by
3
1 T/2 1 1
E D
w e (r, t) = E(r, t) · D(r, t) dt = |E i ||D i | cos(ξ − ξ )
i
i
T −T/2 2 4 i=1
1
ˇ
ˇ ∗
= Re E(r) · D (r) (4.144)
4
and
3
1 T/2 1 1
H B
w m (r, t) = B(r, t) · H(r, t) dt = |B i ||H i | cos(ξ i − ξ )
i
T −T/2 2 4 i=1
1
ˇ
ˇ ∗
= Re H(r) · B (r) , (4.145)
4
where T = 2π/ ˇω. We have already identified the energy stored in a nondispersive material
ˇ
ˇ
E
D
(§ 4.5.2). If (4.144) is to match with (4.62), the phases of E and D must match: ξ = ξ .
i i
B
We must also have ξ H = ξ . Since in a dispersionless material σ must be independent
i i c
ˇ
E
ˇ c
of frequency, from J = σE we also see that ξ J = ξ .
i i
Upon differentiation the time-average stored energy terms in (4.143) disappear, giving
3 3
1
i J E 1
c EE

i
− |J ||E i |C ii (t) dV = |J ||E i |C ii (t) dV −
i
i
2 V i=1 2 V i=1
3 " #

1 EE 1 BB
−2 ˇω |D i ||E i |S ii (t) + |B i ||H i |S ii (t) dV +
V i=1 4 4
3
1
EH
ˆ
ˆ
+ |E i ||H j |(i i × i j ) · ˆ nC ij (t) dS.
2 S i, j=1
Equating the constant terms, we find the time-average power balance expression
3 3

1
i J i E 1
c
− |J ||E i | cos(ξ i − ξ ) dV = |J ||E i | dV +
i
i
i
2 V i=1 2 V i=1
3
1
E H
ˆ
ˆ
+ |E i ||H j |(i i × i j ) · ˆ n cos(ξ − ξ ) dS. (4.146)
i
j
2 S i, j=1
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