Page 252 - Electromagnetics

This can be written more compactly using phasor notation as

p J (r) dV = p σ (r) dV + S av (r) · ˆ n dS (4.147)
V V S
where
1
ˇ
ˇ i∗
p J (r) =− Re E(r) · J (r)
2
is the time-average density of power delivered by the sources to the fields in V ,
1
ˇ
ˇ c∗
p σ (r) = E(r) · J (r)
2
is the time-average density of power transferred to the conducting material as heat, and
1
ˇ
ˇ ∗
S av (r) · ˆ n = Re E(r) × H (r) · ˆ n
2
is the density of time-average power transferred across the boundary surface S. Here
c
ˇ
S = E(r) × H (r)
ˇ ∗
is called the complex Poynting vector and S av is called the time-average Poynting vector.
Comparison of (4.146) with (4.140) shows that nondispersive materials cannot manifest
the dissipative (or active) properties determined by the term
3
1
E D H B c J c E
ˇ ω|E i ||D i | sin(ξ − ξ ) + ˇω|B i ||H i | sin(ξ i − ξ ) +|J ||E i | cos(ξ i − ξ ) dV.
i
i
i
i
i
2 V i=1
This term can be used to classify materials as lossless, lossy, or active, as shown next.
4.8.3 Lossless, lossy, and active media
In § 4.5.1 we classified materials based on whether they dissipate (or provide) energy
over the period of a transient event. We can provide the same classification based on
their steady-state behavior.
We classify a material as lossless if the time-average flow of power entering a homoge-
neous body is zero when there are sources external to the body, but no sources internal
to the body. This implies that the mechanisms within the body either do not dissipate
power that enters, or that there is a mechanism that creates energy to exactly balance the
dissipation. If the time-average power entering is positive, then the material dissipates
power and is termed lossy. If the time-average power entering is negative, then power
must originate from within the body and the material is termed active. (Note that the
power associated with an active body is not described as arising from sources, but is
rather described through the constitutive relations.)
Since materials are generally inhomogeneous we may apply this concept to a vanish-
ingly small volume, thus invoking the point-form of Poynting’s theorem. From (4.140)
we see that the time-average influx of power density is given by
3
1
E D H B
−∇ · S av (r) = p in (r) = ˇ ω|E i ||D i | sin(ξ − ξ ) + ˇω|B i ||H i | sin(ξ i − ξ )+
i
i
i
2
i=1
c
E

+|J ||E i | cos(ξ J c − ξ ) .
i i i
© 2001 by CRC Press LLC