Page 389 - Electromagnetics

we have not derived this condition: we have merely postulated it. As with all postulates
it is subject to experimental verification.
The radiation condition implies that for points far from the source the potentials
behave as spherical waves:
e − jkr
˜
ψ(r,ω) ∼ , r →∞.
r
Substituting this into (5.96) and (5.97) we find that the radiation condition is satisfied.
With S ∞ →∞ we have

˜ ˜
ψ(r,ω) = S(r ,ω)G(r|r ; ω) dV −
V
˜
∂G(r|r ; ω) ∂ψ(r ,ω)

˜

− ψ(r ,ω) − G(r|r ; ω) dS ,
∂n ∂n
S 0
which is the expression for the potential within an infinite medium having source-
excluding regions. As S 0 → 0 we obtain the expression for the potential in an unbounded
medium:

˜ ˜
ψ(r,ω) = S(r ,ω)G(r|r ; ω) dV ,
V
as expected.
The time-domain equation (5.71) may also be solved (at least for the lossless case) in
a bounded region of space. The interested reader should see Pauli [143] for details.
5.3 Transverse–longitudinal decomposition
We have seen that when only electric sources are present, the electromagnetic fields
in a homogeneous, isotropic region can be represented by a single vector potential Π e .
Similarly, when only magnetic sources are present, the fields can be represented by a
single vector potential Π h . Hence two vector potentials may be used to represent the
field if both electric and magnetic sources are present.
We may also represent the electromagnetic field in a homogeneous, isotropic region us-
ing two scalar functions and the sources. This follows naturally from another important
field decomposition: a splitting of each field vector into (1) a component along a certain
pre-chosen constant direction, and (2) a component transverse to this direction. Depend-
ing on the geometry of the sources, it is possible that only one of these components will
be present. A special case of this decomposition, the TE–TM field decomposition, holds
for a source-free region and will be discussed in the next section.

5.3.1 Transverse–longitudinal decomposition in terms of fields
Consider a direction defined by a constant unit vector ˆ u. We define the longitudinal
component of A as ˆ uA u where

A u = ˆ u · A,

and the transverse component of A as
A t = A − ˆ uA u .

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