Page 43 - Electromagnetics

¯ ¯ ¯
¯
where each of the quantities P, L, M, Q may be dyadics in the usual sense, or dyadic
operators containing space or time derivatives or integrals, or some nonlinear operations
on the fields. We may write these expressions as a single matrix equation

cD E
¯
= [C] (2.18)
H cB
where the 6 × 6 matrix
¯ ¯

P L
¯
[C] = .
¯ ¯
M Q
This most general relationship between fields is the property of a bianisotropic material.
We may wonder why D is not related to (E, B, H), E to (D, B), etc. The reason is
that since the field pairs (E, B) and (D, H) convert identically under a Lorentz transfor-
mation, a constitutive relation that maps fields as in (2.18) is form invariant, as are the
Maxwell–Minkowski equations. That is, although the constitutive parameters may vary
numerically between observers moving at different velocities, the form of the relationship
given by (2.18) is maintained.
Many authors choose to relate (D, B) to (E, H), often because the expressions are
simpler and can be more easily applied to specific problems. For instance, in a linear,
isotropic material (as shown below) D is directly proportional to E and B is directly
proportional to H. To provide the appropriate expression for the constitutive relations,
we need only remap (2.18). This gives
¯
D = ¯ · E + ξ · H, (2.19)
¯
B = ζ · E + ¯µ · H, (2.20)
or

D ¯
E
= C EH , (2.21)
B H
¯ ¯
where the new constitutive parameters ¯ , ξ, ζ, ¯µ can be easily found from the original
¯ ¯ ¯
¯
constitutive parameters P, L, M, Q. We do note, however, that in the form (2.19)–(2.20)
the Lorentz invariance of the constitutive equations is not obvious.
In the following paragraphs we shall characterize some of the most common materials
according to these classifications. With this approach effects such as temporal or spatial
dispersion are not part of the classification process, but arise from the nature of the
constitutive parameters. Hence we shall not dwell on the particulars of the constitutive
parameters, but shall concentrate on the form of the constitutive relations.

Constitutive relations for fields in free space. In a vacuum the fields are related
by the simple constitutive equations
D =
0 E, (2.22)
1
H = B. (2.23)
µ 0
The quantities µ 0 and
0 are, respectively, the free-space permeability and permittivity
constants. It is convenient to use three numerical quantities to describe the electromag-
netic properties of free space — µ 0 ,
0 , and the speed of light c — and interrelate them
through the equation
c = 1/(µ 0
0 ) 1/2 .

© 2001 by CRC Press LLC