Page 335 - Electromagnetics

4.11.9 Nonradiating sources
We showed in § 2.10.9 that not all time-varying sources produce electromagnetic waves.
In fact, a subset of localized sources known as nonradiating sources produce no field
external to the source region. Devaneyand Wolf [54] have shown that all nonradiating
time-harmonic sources in an unbounded homogeneous medium can be represented in the
form

ˇ nr
ˇ
2 ˇ

J (r) =−∇ × ∇× f(r) + k f(r) (4.386)
ˇ
where f is anyvector field that is continuous, has partial derivatives up to third order,
ˇ
ˇ
and vanishes outside some localized region V s . In fact, E(r) = j ˇωµf(r) is preciselythe
ˇ nr
phasor electric field produced by J (r). The reasoning is straightforward. Consider the
Helmholtz equation (4.203):
ˇ
2 ˇ
ˇ
∇× (∇× E) − k E =− j ˇωµJ.
By(4.386) we have
2 ˇ ˇ
∇× ∇× −k E − j ˇωµf = 0.
ˇ
ˇ
Since f is zero outside the source region it must vanish at infinity. E also vanishes at
ˇ
ˇ
infinitybythe radiation condition, and thus the quantity E − j ˇωµf obeys the radiation
condition and is a unique solution to the Helmholtz equation throughout all space. Since
the Helmholtz equation is homogeneous we have
ˇ
ˇ
E − j ˇωµf = 0
ˇ
ˇ
ˇ
everywhere; since f is zero outside the source region, so is E (and so is H).
An interesting special case of nonradiating sources is
ˇ
∇!
ˇ f =
k 2
so that
ˇ
∇!
ˇ

J ˇ nr =− ∇× ∇× −k 2 =∇!.
k 2
ˇ
ˇ
Using !(r) = !(r), we see that this source describes the current produced byan oscillat-
ing spherical balloon of charge (cf., § 2.10.9). Radially-directed, spherically-symmetric
sources cannot produce uniform spherical waves, since these sources are of the nonradi-
ating type.
4.12 Interpretation of the spatial transform

Now that we understand the meaning of a Fourier transform on the time variable, let
us consider a single transform involving one of the spatial variables. For a transform over
z we shall use the notation
z
ψ (x, y, k z , t) ↔ ψ(x, y, z, t).

© 2001 by CRC Press LLC