Page 256 - Electromagnetics

4.9.1 Boundary condition for the time-average Poynting vector
In § 2.9.5 we developed a boundary condition for the normal component of the time-
domain Poynting vector. For time-harmonic fields we can derive a similar boundary
condition using the time-average Poynting vector. Consider a surface S across which
the electromagnetic sources and constitutive parameters are discontinuous, as shown in
Figure 2.6. Let ˆ n 12 be the unit normal to the surface pointing into region 1 from region
2. If we apply the large-scale form of the complex Poynting theorem (4.155) to the two
separate surfaces shown in Figure 2.6, we obtain

1 1 1
1 c
ˇ
ˇ
ˇ ∗
ˇ ˇ ∗
ˇ ∗
E · J − 2 j ˇω E · D − B · H dV + S · ˆ n dS
2 V 4 4 2 S
1 c c
= ˆ n 12 · (S − S ) dS (4.159)
2
1
2
S 10
c
ˇ
ˇ ∗
where S = E × H is the complex Poynting vector. If, on the other hand, we apply the
large-scale form of Poynting’s theorem to the entire volume region including the surface
of discontinuity, and include the surface current contribution, we have
1 1 1
1 c
ˇ
ˇ
ˇ ∗
ˇ ∗
ˇ ˇ ∗
E · J − 2 j ˇω E · D − B · H dV + S · ˆ n dS
2 V V 4 4 2 S
1
ˇ ∗ ˇ
=− J · E dS. (4.160)
s
2
S 10
If we wish to have the integrals over V and S in (4.159) and (4.160) produce identical
results, then we must postulate the two conditions
ˇ
ˇ
ˆ n 12 × (E 1 − E 2 ) = 0
and
ˇ ∗ ˇ
c c
ˆ n 12 · (S − S ) =−J · E. (4.161)
1 2 s
The first condition is merely the continuity of tangential electric field; it allows us to be
nonspecific as to which value of E we use in the second condition. If we take the real
part of the second condition we have
ˆ n 12 · (S av,1 − S av,2 ) = p J , (4.162)
s
1 ˇ ˇ ∗
s
where S av = 2 Re{E × H } is the time-average Poynting power flow density and p J =
ˇ ∗ ˇ
1
− Re{J · E} is the time-average density of power delivered by the surface sources. This
2 s
is the desired boundary condition on the time-average power flow density.
4.10 Fundamental theorems for time-harmonic fields
4.10.1 Uniqueness
If we think of a sinusoidal electromagnetic field as the steady-state culmination of a
transient event that has an identifiable starting time, then the conditions for uniqueness
established in § 2.2.1 are applicable. However, a true time-harmonic wave, which has
existed since t =−∞ and thus has infinite energy, must be interpreted differently.

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