Page 88 - Electromagnetics

that for ηH. That is, the fields

E 2 = ηH 1 , (2.260)
H 2 =−E 1 /η, (2.261)
are also a solution to Maxwell’s equations, and thus the dual problem merely involves
replacing E by ηH and H by −E/η. However, the final forms of E and H will not be
identical after appropriate boundary values are imposed.
This form of duality is very important for the solution of fields within waveguides or
the fields scattered by objects where the sources are located outside the region where the
fields are evaluated.

2.9.3 Reciprocity
The reciprocity theorem, also called the Lorentz reciprocity theorem, describes a spe-
cific and often useful relationship between sources and the electromagnetic fields they
produce. Under certain special circumstances we find that an interaction between inde-
pendent source and mediating fields called “reaction” is a spatially symmetric quantity.
The reciprocity theorem is used in the study of guided waves to establish the orthogonal-
ity of guided wave modes, in microwave network theory to obtain relationships between
terminal characteristics, and in antenna theory to demonstrate the equivalence of trans-
mission and reception patterns.
Consider a closed surface S enclosing a volume V . Assume that the fields within and
on S are produced by two independent source fields. The source (J a , J ma ) produces the
field (E a , D a , B a , H a ) as described by Maxwell’s equations

∂B a
∇× E a =−J ma − , (2.262)
∂t
∂D a
∇× H a = J a + , (2.263)
∂t
while the source field (J b , J mb ) produces the field (E b , D b , B b , H b ) as described by

∂B b
∇× E b =−J mb − , (2.264)
∂t
∂D b
∇× H b = J b + . (2.265)
∂t
The sources may be distributed in any way relative to S: they may lie completely inside,
completely outside, or partially inside and partially outside. Material media may lie
within S, and their properties may depend on position.
Let us examine the quantity

R ≡∇ · (E a × H b − E b × H a ).

By (B.44) we have

R = H b ·∇ × E a − E a ·∇ × H b − H a ·∇ × E b + E b ·∇ × H a
so that by Maxwell’s curl equations

∂B b ∂B a ∂D b ∂D a
R = H a · − H b · − E a · − E b · +
∂t ∂t ∂t ∂t
+ [J a · E b − J b · E a − J ma · H b + J mb · H a ] .

© 2001 by CRC Press LLC