Page 104 - Electromagnetics

tain metals may obey the diffusion equation rather than the wave equation, and must
thereby exhibit different behavior. In the study of quasistatic fields we often ignore the
displacement current term in Maxwell’s equations, producing solutions that are most
important near the sources of the fields and having little associated radiation. When the
displacement term is significant we produce solutions with the properties of waves.

2.10.2 Wave equation for bianisotropic materials
In deriving electromagnetic wave equations we transform the first-order coupled par-
tial differential equations we know as Maxwell’s equations into uncoupled second-order
equations. That is, we perform a set of operations (and make appropriate assumptions)
to reduce the set of four differential equations in the four unknown fields E, D, B, and
H, into a set of differential equations each involving a single unknown (usually E or
H). It is possible to derive wave equations for E and H even for the most general cases
of inhomogeneous, bianisotropic media, as long as the constitutive parameters ¯µ and
¯
ξ are constant with time. Substituting the constitutive relations (2.19)–(2.20) into the
Maxwell–Minkowski curl equations (2.169)–(2.170) we get
∂
¯
∇× E =− (ζ · E + ¯µ · H) − J m , (2.306)
∂t
∂
¯
∇× H = (¯ · E + ξ · H) + J. (2.307)
∂t
¯
Separate equations for E and H are facilitated by introducing a new dyadic operator ∇,
which when dotted with a vector field V gives the curl:
¯
∇· V =∇ × V. (2.308)
¯
It is easy to verify that in rectangular coordinates ∇ is
 
0 −∂/∂z ∂/∂y
¯  ∂/∂z
[∇] = 0 −∂/∂x  .
−∂/∂y ∂/∂x 0
With this notation, Maxwell’s curl equations (2.306)–(2.307) become simply

∂ ∂
¯ ¯
∇+ ζ · E =− ¯ µ · H − J m , (2.309)
∂t ∂t
∂ ∂

¯ ¯
∇− ξ · H = ¯ · E + J. (2.310)
∂t ∂t
Obtaining separate equations for E and H is straightforward. Defining the inverse
dyadic ¯µ −1 through
¯
¯ µ · ¯µ −1 = ¯µ −1 · ¯µ = I,
we can write (2.309) as
∂ −1 ∂ −1
¯
¯
H =− ¯µ · ∇+ ζ · E − ¯µ · J m (2.311)
∂t ∂t
¯
where we have assumed that ¯µ is independent of time. Assuming that ξ is also indepen-
dent of time, we can differentiate (2.310) with respect to time to obtain
2
∂ ∂H ∂ ∂J
¯ ¯
∇− ξ · = (¯ · E) + .
∂t ∂t ∂t 2 ∂t

© 2001 by CRC Press LLC