Page 46 - Electromagnetics

the null field:
E = D = B = H = 0.
By Ampere’s and Faraday’s laws we must also have J = J m = 0; hence, by the continuity
equation, ρ = ρ m = 0.
In addition to the null field, we have the condition that the tangential electric field
on the surface of a PEC must be zero. Similarly, the tangential magnetic field on the
surface of a PMC must be zero. This implies (§ 2.8.3) that an electric surface current
may exist on the surface of a PEC but not on the surface of a PMC, while a magnetic
surface current may exist on the surface of a PMC but not on the surface of a PEC.
A PEC may be regarded as the limit of a conducting material as σ →∞. In many
practical cases, good conductors such as gold and copper can be assumed to be perfect
electric conductors, which greatly simplifies the application of boundary conditions. No
physical material is known to behave as a PMC, but the concept is mathematically
useful for applying symmetry conditions (in which a PMC is sometimes referred to as a
“magnetic wall”) and for use in developing equivalence theorems.

Constitutive relations in a linear anisotropic material. In a linear anisotropic
material there are relationships between B and H and between D and E, but the field
vectors are not aligned as in the isotropic case. We can thus write

D = ¯ · E, B = ¯µ · H, J = ¯σ · E,
where ¯ is called the permittivity dyadic, ¯µ is the permeability dyadic, and ¯σ is the
conductivity dyadic. In terms of the general constitutive relation (2.18) we have
¯ µ −1
¯
¯
¯
¯
P = c¯ , Q = , L = M = 0.
c
Many different types of materials demonstrate anisotropic behavior, including opti-
cal crystals, magnetized plasmas, and ferrites. Plasmas and ferrites are examples of
gyrotropic media. With the proper choice of coordinate system, the frequency-domain
permittivity or permeability can be written in matrix form as
   

11
12 0 µ 11 µ 12 0
[ ˜ ¯ ] =  −
12
11 0  , [ ˜ ¯µ] =  −µ 12 µ 11 0  . (2.32)
0 0
33 0 0 µ 33
Each of the matrix entries may be complex. For the special case of a lossless gyrotropic
material, the matrices become hermitian:
   

− jδ 0 µ − jκ 0
[ ˜ ¯ ] =  jδ
0  , [ ˜ ¯µ] =  jκµ 0  , (2.33)
0 0
3 0 0 µ 3
where
,
3 , δ, µ, µ 3 , and κ are real numbers.
Crystals have received particular attention because of their birefringent properties. A
birefringent crystal can be characterized by a symmetric permittivity dyadic that has real
permittivity parameters in the frequency domain; equivalently, the constitutive relations
do not involve constitutive operators. A coordinate system called the principal system,
with axes called the principal axes, can always be found so that the permittivity dyadic
in that system is diagonal:
 

x 00
[ ˜ ¯ ] =  0
y 0  .
00
z

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