Page 105 - Electromagnetics

Substituting ∂H/∂t from (2.311) and rearranging, we get
2
∂ ∂ ∂ ∂ ∂J
¯ ¯ −1 ¯ ¯ ¯ ¯ −1
∇− ξ · ¯µ · ∇+ ζ + ¯ · E =− ∇− ξ · ¯µ · J m − .
∂t ∂t ∂t 2 ∂t ∂t
(2.312)
This is the general wave equation for E. Using an analogous set of steps, and assuming
¯
¯ and ζ are independent of time, we can find
2
∂ ∂ ∂ ∂ ∂J m
¯ ¯ −1 ¯ ¯ ¯ ¯ −1
∇+ ζ · ¯ · ∇− ξ + ¯ µ · H = ∇+ ζ · ¯ · J − .
∂t ∂t ∂t 2 ∂t ∂t
(2.313)
This is the wave equation for H. The case in which the constitutive parameters are
time-dependent will be handled using frequency domain techniques in later chapters.
Wave equations for anisotropic, isotropic, and homogeneous media are easily obtained
from (2.312) and (2.313) as special cases. For example, the wave equations for a homo-
¯
¯
¯
¯
geneous, isotropic medium can be found by setting ζ = ξ = 0, ¯µ = µI, and ¯ =
I:
2
1 ∂ E 1 ∂J
¯
¯
¯
∇· (∇· E) +
=− ∇· J m − ,
µ ∂t 2 µ ∂t
2
1 ∂ H 1 ∂J m
¯
¯
¯
∇· (∇· H) + µ = ∇· J − .

∂t 2
∂t
Returning to standard curl notation we find that these become
2
∂ E ∂J
∇× (∇× E) + µ
=−∇ × J m − µ , (2.314)
∂t 2 ∂t
2
∂ H ∂J m
∇× (∇× H) + µ
=∇ × J −
. (2.315)
∂t 2 ∂t
In each of the wave equations it appears that operations on the electromagnetic fields
have been separated from operations on the source terms. However, we have not yet
invoked any coupling between the fields and sources associated with secondary interac-
tions. That is, we need to separate the impressed sources, which are independent of
the fields they source, with secondary sources resulting from interactions between the
sourced fields and the medium in which the fields exist. The simple case of an isotropic
conducting medium will be discussed below.
Wave equation using equivalent sources. An alternative approach for studying
wave behavior in general media is to use the Maxwell–Boffi form of the field equations
∂B
∇× E =− , (2.316)
∂t
B ∂
0 E
∇× = (J + J M + J P ) + , (2.317)
µ 0 ∂t
∇· (
0 E) = (ρ + ρ P ), (2.318)
∇· B = 0. (2.319)
Taking the curl of (2.316) we have
∂
∇× (∇× E) =− ∇× B.
∂t

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