Page 262 - Electromagnetics

if the materials within are reciprocal and the boundary fields obey (4.176), or if the
region is source-free. In each of these cases

ˇ
ˇ
ˇ
ˇ

E a × H b − E b × H a · dS = 0 (4.178)
S
and
ˇ
ˇ
f a , ˇ g b − f b , ˇ g a = 0. (4.179)
Rayleigh–Carson reciprocity theorem. The physical meaning behind reciprocity
can be made clear with a simple example. Consider two electric Hertzian dipoles, each
oscillating with frequency ˇω and located within an empty box consisting of PEC walls.
These dipoles can be described in terms of volume current density as
ˇ J a (r) = I a δ(r − r ),
ˇ

a
ˇ
ˇ

J b (r) = I b δ(r − r ).
b
ˇ
Since the fields on the surface obey (4.176) (specifically, ˆ n×E = 0), and since the medium
within the box is empty space (a reciprocal medium), the fields produced by the sources
must obey (4.179). We have

ˇ ˇ ˇ ˇ
E b (r) · I a δ(r − r ) dV = E a (r) · I b δ(r − r ) dV,
a b
V V
hence
ˇ
ˇ I a · E b (r ) = I b · E a (r ). (4.180)
ˇ
ˇ

a b
This is the Rayleigh–Carson reciprocity theorem. It also holds for two Hertzian dipoles
located in unbounded free space, because in that case the Sommerfeld radiation condition
satisfies (4.176).
As an important application of this principle, consider a closed PEC body located in
ˇ
free space. Reciprocity holds in the region external to the body since we have ˆ n × E = 0
at the boundary of the perfect conductor and the Sommerfeld radiation condition on the
boundary at infinity. Now let us place dipole a somewhere external to the body, and
dipole b adjacent and tangential to the perfectly conducting body. We regard dipole a
as the source of an electromagnetic field and dipole b as “sampling” that field. Since the
tangential electric field is zero at the surface of the conductor, the reaction between the
two dipoles is zero. Now let us switch the roles of the dipoles so that b is regarded as
the source and a is regarded as the sampler. By reciprocity the reaction is again zero
and thus there is no field produced by b at the position of a. Now the position and
orientation of a are arbitrary, so we conclude that an impressed electric source current
placed tangentially to a perfectly conducting body produces no field external to the body.
This result is used in Chapter 6 to develop a field equivalence principle useful in the study
of antennas and scattering.

4.10.3 Duality
A duality principle analogous to that found for time-domain fields in § 2.9.2 may be
established for frequency-domain and time-harmonic fields. Consider a closed surface S
˜
enclosing a region of space that includes a frequency-domain electric source current J

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