Page 435 - Electromagnetics

c ˜
˜
Finally, using Ampere’s law ∇× H = jω˜ E in the source free region V , and taking the
curl in the first term outside the integral, we have
N N
˜ 1 ˜ ˜
E(r,ω) = ∇× ∇× (ˆ n × H)GdS + ∇× (ˆ n × E)GdS . (6.37)
jω˜ c
n=1 S n n=1 S n
Similarly
N N
˜ 1 ˜ ˜
H(r,ω) =− ∇× ∇× (ˆ n × E)GdS + ∇× (ˆ n × H)GdS . (6.38)
jω ˜µ
n=1 S n n=1 S n
These expressions together constitute the Franz formula for the vector Huygens principle
[192].

6.3.3 Love’s equivalence principle

Love’s equivalence principle allows us to identify the equivalent Huygens sources for the
fields within a bounded, source-free region V . It then allows us to replace a problem in
the bounded region with an “equivalent” problem in unbounded space where the source-
excluding surfaces are replaced by equivalent sources. The field produced by both the
real and the equivalent sources gives a field in V identical to that of the original problem.
This is particularly useful since we know how to compute the fields within an unbounded
region by employing potential functions.
We identify the equivalent sources by considering the electric and magnetic Hertzian
potentials produced by electric and magnetic current sources. Consider an impressed
eq
˜
eq
˜
electric surface current J s and a magnetic surface current J ms flowing on the closed
surface S in a homogeneous, isotropic medium with permeability ˜µ(ω) and complex
c
permittivity ˜ (ω). These sources produce
eq
˜

J s (r ,ω)
˜
Π e (r,ω) = G(r|r ; ω) dS , (6.39)
c
S jω˜ (ω)
˜ eq
J ms (r ,ω)
˜
Π h (r,ω) = G(r|r ; ω) dS , (6.40)
S jω ˜µ(ω)
which in turn can be used to find
˜
˜
˜
E =∇ × (∇× Π e ) − jω ˜µ∇× Π h ,
˜
˜
˜
c
H = jω˜ ∇× Π e +∇ × (∇× Π h ).
Upon substitution we find that

˜ 1 ˜ eq ˜ eq
E(r,ω) =∇ × ∇ × J G dS +∇ × [−J ]GdS ,
ms
s
S jω˜ c S

˜ 1 ˜ eq ˜ eq
H(r,ω) =−∇ × ∇ × −J G dS +∇ × J GdS .
ms
s
S jω ˜µ S
These are identical to the Franz equations (6.37) and (6.38) if we identify
˜
˜
J ˜ eq = ˆ n × H, J ˜ eq =−ˆ n × E. (6.41)
s
ms
These are the equivalent source densities for the Huygens principle.
© 2001 by CRC Press LLC