Page 122 - Electromagnetics

So H obeys the homogeneous wave equation everywhere, and H = 0 follows from causality.
The laws of Ampere and Faraday may also be combined with (2.359) to show that

2
1 ∂ 1
(∇× ∇×) + E(r, t) + ∇ψ(r, t) = 0
2
c ∂t 2
0
for all r. By causality
1
E(r, t) =− ∇ψ(r, t) (2.361)

0
everywhere. But since ψ(r, t) = 0 external to V , we must also have E = 0 there.
Note that E =−∇ψ/
0 is consistent with Ampere’s law (2.360) provided that H = 0
everywhere.
We see that sources having spherical symmetry such that
2
∂ψ(r, t) ∂ ψ(r, t)
J(r, t) = ˆ rJ r (r, t) =∇ = ˆ r
∂t ∂r∂t
obey (2.359) and are therefore nonradiating. Hence the fields associated with any outward
traveling spherical wave must possess some angular variation. This holds, for example,
for the fields far removed from a time-varying source of finite extent.
As pointed out by Lindell [113], nonradiating sources are not merely hypothetical.
The outflowing currents produced by a highly symmetric nuclear explosion in outer
space or in a homogeneous atmosphere would produce no electromagnetic field outside
the source region. The large electromagnetic-pulse effects discussed in § 2.10.6 are due
to inhomogeneities in the earth’s atmosphere. We also note that the fields produced
r
by a radiating source J (r, t) do not change external to the source if we superpose a
nr
r
nonradiating component J (r, t) to create a new source J = J nr + J . We say that the
r
two sources J and J are equivalent for the region V external to the sources. This presents
difficulties in remote sensing where investigators are often interested in reconstructing an
unknown source by probing the fields external to (and usually far away from) the source
region. Unique reconstruction is possible only if the fields within the source region are
also measured.
For the time harmonic case, Devaney and Wolf [54] provide the most general possible
form for a nonradiating source. See § 4.11.9 for details.

2.11 Problems
2.1 Consider the constitutive equations (2.16)–(2.17) relating E, D, B, and H in a
bianisotropic medium. Using the definition for P and M, show that the constitutive
equations relating E, B, P, and M are

1

¯
¯
¯
P = P −
0 I · E + L · B,
c

¯ ¯ 1 ¯
M =−M · E − cQ − I · B.
µ 0
Also find the constitutive equations relating E, H, P, and M.
© 2001 by CRC Press LLC