Page 249 - Electromagnetics

(2.301), which is valid for dispersive materials, or (2.299). For nondispersive materials
(2.299) allows us to interpret the volume integral term as the time rate of change of
stored energy. But if the operating frequency lies within the realm of material dispersion
and loss, then we can no longer identify an explicit stored energy term.

4.8.1 General form of Poynting’s theorem

We begin with (2.301). Substituting the time-harmonic representations we obtain the
term
3 3
∂D(r, t)
E ∂
D
E(r, t) · = ˆ i i |E i | cos[ ˇωt + ξ ] · ˆ i i |D i | cos[ ˇωt + ξ ]
i
i
∂t ∂t
i=1 i=1
3

E D
=− ˇω |E i ||D i | cos[ ˇωt + ξ ] sin[ ˇωt + ξ ].
i
i
i=1
Since 2 sin A cos B ≡ sin(A + B) + sin(A − B) we have
3
∂ 1
DE
E(r, t) · D(r, t) =− ˇ ω|E i ||D i |S ii (t),
∂t 2
i=1
where
E
D
E
D
S DE (t) = sin(2 ˇωt + ξ + ξ ) + sin(ξ − ξ )
ii i i i i
describes the temporal dependence of the field product. Separating the current into an
i
c
impressed term J and a secondary term J (assumed to be the conduction current) as
i
c
J = J + J and repeating the above steps with the other terms, we obtain
3 3
1
i J E 1
EH

ˆ
ˆ
i
− |J ||E i |C ii (t) dV = |E i ||H j |(i i × i j ) · ˆ nC ij (t) dS +
i
2 V i=1 2 S i, j=1
3
1
DE BH c J E

c
+ − ˇω|D i ||E i |S ii (t) − ˇω|B i ||H i |S ii (t) +|J ||E i |C ii (t) dV, (4.139)
i
2 V i=1
where
H
H
B
B
S ii BH (t) = sin(2 ˇωt + ξ + ξ ) + sin(ξ − ξ ),
i
i
i
i
E
H
E
H
C ij EH (t) = cos(2 ˇωt + ξ + ξ ) + cos(ξ − ξ ),
j
i
i
j
and so on.
We see that each power term has two temporal components: one oscillating at fre-
quency 2 ˇω, and one constant with time. The oscillating component describes power that
cycles through the various mechanisms of energy storage, dissipation, and transfer across
the boundary. Dissipation may be produced through conduction processes or through
polarization and magnetization phase lag, as described by the volume term on the right-
hand side of (4.139). Power may also be delivered to the fields either from the sources,
as described by the volume term on the left-hand side, or from an active medium, as
described by the volume term on the right-hand side. The time-average balance of power
supplied to the fields and extracted from the fields throughout each cycle, including that
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