Page 250 - Electromagnetics

transported across the surface S, is given by the constant terms in (4.139):

3 3
1
i J i E 1
E D

− |J ||E i | cos(ξ i − ξ ) dV = ˇ ω|E i ||D i | sin(ξ − ξ )+
i
i
i
i
2 V i=1 2 V i=1
c
E
B

+ ˇω|B i ||H i | sin(ξ i H − ξ ) +|J ||E i | cos(ξ i J c − ξ ) dV +
i
i
i
3

1
E H
ˆ
ˆ
+ |E i ||H j |(i i × i j ) · ˆ n cos(ξ − ξ ) dS. (4.140)
i
j
2 S i, j=1
We associate one mechanism for time-average power loss with the phase lag between
applied field and resulting polarization or magnetization. We can see this more clearly
if we use the alternative form of the Poynting theorem (2.302) written in terms of the
polarization and magnetization vectors. Writing
3 3

P
M
P(r, t) = |P i (r)| cos[ ˇωt + ξ (r)], M(r, t) = |M i (r)| cos[ ˇωt + ξ (r)],
i
i
i=1 i=1
and substituting the time-harmonic fields, we see that
3 3
1
JE ˇ ω
PE MH

− |J i ||E i |C ii (t) dV + |P i ||E i |S ii (t) + µ 0 |M i ||H i |S ii (t) dV
2 V i=1 2 V i=1
3

ˇ ω
2 EE 2 HH
=− 0 |E i | S ii (t) + µ 0 |H i | S ii (t) dV +
2 V i=1
3

1
EH
ˆ
ˆ
+ |E i ||H j |(i i × i j ) · ˆ nC ij (t) dS. (4.141)
2 S i, j=1
Selection of the constant part gives the balance of time-average power:
3
1
J E
− |J i ||E i | cos(ξ − ξ ) dV
i
i
2 V i=1
3

ˇ ω
E P H M
= |E i ||P i | sin(ξ − ξ ) + µ 0 |H i ||M i | sin(ξ i − ξ ) dV +
i
i
i
2 V i=1
3
1
E H
ˆ
ˆ
+ |E i ||H j |(i i × i j ) · ˆ n cos(ξ − ξ ) dS. (4.142)
j
i
2 S i, j=1
Here the power loss associated with the lag in alignment of the electric and magnetic
dipoles is easily identified as the volume term on the right-hand side, and is seen to arise
through the interaction of the fields with the equivalent sources as described through the
phase difference between E and P and between H and M. If these pairs are in phase, then
the time-average power balance reduces to that for a dispersionless material, equation
(4.146).
4.8.2 Poynting’s theorem for nondispersive materials
For nondispersive materials (2.299) is appropriate. We shall carry out the details here
so that we may examine the power-balance implications of nondispersive media. We
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