Page 426 - Electromagnetics

˜
Each of the expressions for E is equally valid for points within V m . Upon subtraction we
get

˜ ρ i
˜ i ˜ i
0 = −J ×∇ G + ∇ G − jω ˜µJ G dV +
m c
V ˜
N

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

− (dl · H)∇ G, r ∈ V m .
jω˜ c
n=1
na + nb
This expression is exactly the Stratton–Chu formula (6.7) evaluated at points within the
˜
excluded region V m . The treatment of H is analogous and is left as an exercise. Since we
may repeat this for any excluded region, we find that the Stratton–Chu formula returns
the null field when evaluated at points outside V . This is sometimes referred to as the
vector Ewald–Oseen extinction theorem [90]. We must emphasize that the fields within
the excluded regions are not generally equal to zero; the Stratton–Chu formula merely
returns this result when evaluated there.

6.2 Fields in an unbounded medium
Two special cases of the Stratton–Chu formula are important because of their applica-
tion to antenna theory. The first is that of sources radiating into an unbounded region.
The second involves a bounded region with all sources excluded. We shall consider the
former here and the latter in § 6.3.
Assuming that there are no bounding surfaces in (6.7) and (6.8), except for one surface
that has been allowed to recede to infinity and therefore provides no surface contribution,
we find that the electromagnetic fields in unbounded space are given by

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

i
˜ ˜ i ˜ ρ m c ˜ i
H = J ×∇ G + ∇ G − jω˜ J G dV .
m
V ˜ µ
We can view the right-hand sides as superpositions of the fields present in the cases
where (1) electric sources are present exclusively, and (2) magnetic sources are present
i
˜ i
exclusively. With ˜ρ = 0 and J = 0 we find that
m m

˜ ρ i
˜ ˜ i
E = c ∇ G − jω ˜µJ G dV , (6.13)
V ˜

˜ ˜ i
H = J ×∇ GdV . (6.14)
V

Using ∇ G =−∇G we can write
˜ ρ (r ,ω)
i
˜ ˜ i
E(r,ω) =−∇ c G(r|r ; ω) dV − jω ˜ µ(ω)J (r ,ω)G(r|r ; ω) dV
V ˜ (ω) V
˜
˜
=−∇φ e (r,ω) − jωA e (r,ω),
© 2001 by CRC Press LLC