Page 436 - Electromagnetics

We now state Love’s equivalence principle [39]. Consider the fields within a homoge-
c
neous, source-free region V with parameters (˜ , ˜µ) bounded by a surface S. We know
how to compute the fields using the Franz formula and the surface fields. Now consider
a second problem in which the same surface S exists in an unbounded medium with
identical parameters. If the surface carries the equivalent sources (6.41) then the electro-
magnetic fields within V calculated using the Hertzian potentials (6.39) and (6.40) are
identical to those of the first problem, while the fields calculated outside V are zero. We
see that this must be true since the Franz formulas and the field/potential formulas are
identical, and the Franz formula (since it was derived from the Stratton–Chu formula)
gives the null field outside V . The two problems are equivalent in the sense that they
produce identical fields within V .
The fields produced by the equivalent sources obey the appropriate boundary condi-
tions across S. From (2.194) and (2.195) we have the boundary conditions
˜
˜
˜
ˆ n × (H 1 − H 2 ) = J s ,
˜
˜
˜
ˆ n × (E 1 − E 2 ) =−J ms .
˜
˜
˜
˜
Here ˆ n points inward to V , (E 1 , H 1 ) are the fields within V , and (E 2 , H 2 ) are the fields
within the excluded region. If the fields produced by the equivalent sources within the
excluded region are zero, then the fields must obey
˜
˜ eq
ˆ n × H 1 = J ,
s
˜ eq
˜
ˆ n × E 1 =−J ,
ms
eq
eq
˜
˜
which is true by the definition of (J s , J sm ).
Note that we can extend the equivalence principle to the case where the media are
different internal to V than external to V . See Chen [29].
With the equivalent sources identified we may compute the electromagnetic field in
V using standard techniques. Specifically, we may use the Hertzian potentials as shown
above or, since the Hertzian potentials are a simple remapping of the vector potentials,
we may use (5.60) and (5.61) to write
1
˜
2 ˜
˜
˜

ω
E =− j ∇(∇· A e ) + k A e − ∇× A h ,
k 2 ˜ c
1
˜
˜
˜
2 ˜

ω
H =− j ∇(∇· A h ) + k A h + ∇× A e ,
k 2 ˜ µ
where

˜ ˜ eq
A e (r,ω) = ˜ µ(ω)J (r ,ω)G(r|r ; ω) dS (6.42)
s
S

˜

= ˜ µ(ω)[ˆ n × H(r ,ω)]G(r|r ; ω) dS , (6.43)
S

˜ c ˜ eq
A h (r,ω) = ˜ (ω)J (r ,ω)G(r|r ; ω) dS (6.44)
ms
S

˜
c

= ˜ (ω)[−ˆ n × E(r ,ω)]G(r|r ; ω) dS . (6.45)
S
At points where the source is zero we can write the fields in the alternative form
ω 1
˜
˜
˜
E =− j ∇× ∇× A e − ∇× A h , (6.46)
k 2 ˜ c
ω 1
˜
˜
˜
H =− j ∇× ∇× A h + ∇× A. (6.47)
k 2 ˜ µ
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