Page 86 - Electromagnetics

Comparing (2.244) with (2.242) and (2.245) with (2.243), we conclude that

¯
¯
¯
¯
2
ζ 2 =−ξ 1 , ξ 2 =−ζ 1 , ¯ µ 2 = η ¯ 1 , ¯ 2 = ¯ µ 1 .
0 2
η 0
As an important special case, we see that for a linear, isotropic medium specified by a
permittivity
and permeability µ, the dual problem is obtained by replacing
r with µ r
and µ r with
r . The solution to the dual problem is then given by
E 2 = η 0 H 1 , η 0 H 2 =−E 1 ,
as before. We thus see that the medium in the dual problem must have electric properties
numerically equal to the magnetic properties of the medium in the original problem, and
magnetic properties numerically equal to the electric properties of the medium in the
original problem. This is rather inconvenient for most applications. Alternatively, we
may divide Ampere’s law by η = (µ/
) 1/2 instead of η 0 . Then the dual problem has
J replaced by J m /η, and J m replaced by −ηJ, and the solution to the dual problem is
given by

E 2 = ηH 1 , ηH 2 =−E 1 .
In this case there is no need to swap
r and µ r , since information about these parameters
is incorporated into the replacement sources.
We must also remember that to obtain a unique solution we need to specify the bound-
ary values of the fields. In a true dual problem, the boundary values of the fields used
in the original problem are used on the swapped fields in the dual problem. A typical
example of this is when the condition of zero tangential electric field on a perfect electric
conductor is replaced by the condition of zero tangential magnetic field on the surface of
a perfect magnetic conductor. However, duality can also be used to obtain the mathe-
matical form of the field expressions, often in a homogeneous (source-free) situation, and
boundary values can be applied later to specify the solution appropriate to the problem
geometry. This approach is often used to compute waveguide modal fields and the elec-
tromagnetic fields scattered from objects. In these cases a TE/TM field decomposition
is employed, and duality is used to find one part of the decomposition once the other is
known.
Dualityof electric and magnetic point source fields. By duality, we can some-
times use the known solution to one problem to solve a related problem by merely sub-
stituting different variables into the known mathematical expression. An example of this
is the case in which we have solved for the fields produced by a certain distribution of
electric sources and wish to determine the fields when the same distribution is used to
describe magnetic sources.
Let us consider the case when the source distribution is that of a point current, or
Hertzian dipole, immersed in free space. As we shall see in Chapter 5, the fields for a
general source may be found by using the fields produced by these point sources. We
begin by finding the fields produced by an electric dipole source at the origin aligned
along the z-axis,
J = ˆ zI 0 δ(r),
then use duality to find the fields produced by a magnetic current source J m = ˆ zI m0 δ(r).
The fields produced by the electric source must obey
∂
∇× E e =− µ 0 H e , (2.246)
∂t

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