Page 106 - Electromagnetics

Substituting for ∇× B from (2.317) we then obtain

2
∂ E ∂
∇× (∇× E) + µ 0
0 2 =−µ 0 (J + J M + J P ), (2.320)
∂t ∂t
which is the wave equation for E. Taking the curl of (2.317) and substituting from (2.316)
we obtain the wave equation

2
∂ B
∇× (∇× B) + µ 0
0 2 = µ 0 ∇× (J + J M + J P ) (2.321)
∂t
for B. Solution of the wave equations is often facilitated by writing the curl-curl operation
in terms of the vector Laplacian. Using (B.47), and substituting for the divergence from
(2.318) and (2.319), we can write the wave equations as

2
∂ E 1 ∂
2
∇ E − µ 0
0 2 = ∇(ρ + ρ P ) + µ 0 (J + J M + J P ), (2.322)
∂t
0 ∂t
2
∂ B
2
∇ B − µ 0
0 2 =−µ 0 ∇× (J + J M + J P ). (2.323)
∂t
The simplicity of these equations relative to (2.312) and (2.313) is misleading. We have
not considered the constitutive equations relating the polarization P and magnetization
M to the fields, nor have we considered interactions leading to secondary sources.
2.10.3 Wave equation in a conducting medium
As an example of the type of wave equation that arises when secondary sources are
included, consider a homogeneous isotropic conducting medium described by permittivity

, permeability µ, and conductivity σ. In a conducting medium we must separate the
i
source field into a causative impressed term J that is independent of the fields it sources,
s
and a secondary term J that is an effect of the sourced fields. In an isotropic conducting
s
medium the effect is described by Ohm’s law J = σE. Writing the total current as
s
i
J = J + J , and assuming that J m = 0, we write the wave equation (2.314) as
i
2
∂ E ∂(J + σE)
∇× (∇× E) + µ
=−µ . (2.324)
∂t 2 ∂t
Using (B.47) and substituting ∇· E = ρ/
, we can write the wave equation for E as
2
∂E ∂ E ∂J i 1
2
∇ E − µσ − µ
= µ + ∇ρ. (2.325)
∂t ∂t 2 ∂t
i
Substituting J = J + σE into (2.315) and using (B.47), we obtain
2
∂ H
2 i
∇(∇· H) −∇ H + µ
=∇ × J + σ∇× E.
∂t 2
Since ∇× E =−∂B/∂t and ∇· H =∇ · B/µ = 0,wehave
2
∂H ∂ H i
2
∇ H − µσ − µ
=−∇ × J . (2.326)
∂t ∂t 2
This is the wave equation for H.

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