Page 261 - Electromagnetics

We see that if there are no sources within S then

ˇ ˇ ˇ ˇ
E a × H b − E b × H a · dS = 0. (4.175)
S
Whenever (4.175) holds we say that the “system” within S is reciprocal. Thus, for
instance, a region of empty space is a reciprocal system.
A system need not be source-free in order for (4.175) to hold. Suppose the relationship
ˇ
ˇ
between E and H on S is given by the impedance boundary condition
ˇ
ˇ
E t =−Z(ˆ n × H), (4.176)
ˇ
ˇ
where E t is the component of E tangential to S so that ˆ n × E = ˆ n × E t , and the complex
wall impedance Z may depend on position. By (4.176) we can write
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
(E a × H b − E b × H a ) · ˆ n = H b · (ˆ n × E a ) − H a · (ˆ n × E b )
ˇ
ˇ
ˇ
ˇ
=−ZH b · [ˆ n × (ˆ n × H a )] + ZH a · [ˆ n × (ˆ n × H b )].
ˇ
ˇ
ˇ
Since ˆ n × (ˆ n × H) = ˆ n(ˆ n · H) − H, the right-hand side vanishes. Hence (4.175) still holds
even though there are sources within S.
The reaction theorem. When sources lie within the surface S, and the fields on S
obey (4.176), we obtain an important corollary of the Lorentz reciprocity theorem. We
have from (4.174) the additional result
ˇ
ˇ
f a , ˇ g b − f b , ˇ g a = 0.
Hence a reciprocal system has
ˇ ˇ
f a , ˇ g b = f b , ˇ g a (4.177)
(which holds even if there are no sources within S, since then the reactions would be
identically zero). This condition for reciprocity is sometimes called the reaction theorem
and has an important physical meaning which we shall explore below in the form of
the Rayleigh–Carson reciprocity theorem. Note that in obtaining this relation we must
assume that the medium is reciprocal in order to eliminate the terms in (4.169). Thus,
in order for a system to be reciprocal, it must involve both a reciprocal medium and a
boundary over which (4.176) holds.
It is important to note that the impedance boundary condition (4.176) is widely appli-
ˇ
cable. If Z → 0, then the boundary condition is that for a PEC: ˆ n×E = 0.If Z →∞,a
ˇ
PMC is described: ˆ n×H = 0. Suppose S represents a sphere of infinite radius. We know
from (4.168) that if the sources and material media within S are spatially finite, the fields
far removed from these sources are described by the Sommerfeld radiation condition
ˇ
ˇ
ˆ r × E = η 0 H
where ˆ r is the radial unit vector of spherical coordinates. This condition is of the type
(4.176) since ˆ r = ˆ n on S, hence the unbounded region that results from S receding to
infinity is also reciprocal.

Summary of reciprocity for reciprocal systems. We can summarize reciprocity
as follows. Unbounded space containing sources and materials of finite size is a reciprocal
system if the media are reciprocal; a bounded region of space is a reciprocal system only

© 2001 by CRC Press LLC