Page 425 - Electromagnetics

Since this gives the contribution to the field in V from the fields on the surface receding
to infinity, we expect that this term should be zero. If the medium has loss, then the
exponential term decays and drives the contribution to zero. For a lossless medium the
contributions are zero if
˜
lim rE(r,ω) < ∞, (6.9)
r→∞
˜ ˜
lim r ηˆ r × H(r,ω) + E(r,ω) = 0. (6.10)
r→∞
To accompany (6.8) we also have

˜
lim rH(r,ω) < ∞, (6.11)
r→∞
˜ ˜
lim r ηH(r,ω) − ˆ r × E(r,ω) = 0. (6.12)
r→∞
We refer to (6.9) and (6.11) as the finiteness conditions, and to (6.10) and (6.12) as the
Sommerfeld radiation condition, for the electromagnetic field. They show that far from
the sources the fields must behave as a wave TEM to the r-direction. We shall see in
§ 6.2 that the waves are in fact spherical TEM waves.

6.1.3 Fields in the excluded region: the extinction theorem
The Stratton–Chu formula provides a solution for the field within the region V , external
to the excluded regions. An interesting consequence of this formula, and one that helps
˜
˜
us identify the equivalence principle, is that it gives the null result H = E = 0 when
evaluated at points within the excluded regions.
We can show this by considering two cases. In the first case we do not exclude the
particular region V m , but do exclude the remaining regions V n , n = m. Then the electric
field everywhere outside the remaining excluded regions (including at points within V m )
is, by (6.7),

˜ ρ

i
˜ ˜ i ˜ i
E(r,ω) = −J ×∇ G + ∇ G − jω ˜µJ G dV +
m c
V +V m ˜

˜ ˜
˜
+ (ˆ n × E) ×∇ G + (ˆ n · E)∇ G − jω ˜µ(ˆ n × H)G dS −
n =m S n
˜
1

− (dl · H)∇ G, r ∈ V + V m .
jω˜ c
n =m
na + nb
In the second case we apply the Stratton–Chu formula only to V m , and exclude all other
regions. We incur a sign change on the surface and line integrals compared to the first
case because the normal is now directed oppositely. By (6.7) we have

˜ ρ i
˜ ˜ i ˜ i
E(r,ω) = −J ×∇ G + ∇ G − jω ˜µJ G dV −
m c
V m ˜

˜
˜ ˜
− (ˆ n × E) ×∇ G + (ˆ n · E)∇ G − jω ˜µ(ˆ n × H)G dS +
S m
1
˜

+ (dl · H)∇ G, r ∈ V m .
jω˜ c
na + nb
© 2001 by CRC Press LLC