Page 118 - Electromagnetics

For large values of ρ/v,
τ

E z (ρ, t m ) ≈ 2E 0 ln 1 + .
2ρ/v
Using ln(1 + x) ≈ x when x 1, we find that

2τv
E z (ρ, t m ) ≈ E 0 .
ρ
Thus, as ρ →∞ we have E×H ∼ 1/ρ and the flux of energy passing through a cylindrical
surface of area ρ dφ dz is independent of ρ. This result is similar to that seen for spherical
2
waves where E × H ∼ 1/r .

2.10.8 Propagation of spherical waves in a lossless medium

In the previous section we found solutions that describe uniform cylindrical waves
dependent only on the radial variable ρ. It turns out that similar solutions are not
possible in spherical coordinates; fields that only depend on r cannot satisfy Maxwell’s
equations since, as shown in § 2.10.9, a source having the appropriate symmetry for the
production of uniform spherical waves in fact produces no field at all external to the region
it occupies. As we shall see in Chapter 5, the fields produced by localized sources are in
general quite complex. However, certain solutions that are only slightly nonuniform may
be found, and these allow us to investigate the most important properties of spherical
waves. We shall find that spherical waves diverge from a localized point source and
expand outward with finite velocity, carrying energy away from the source.
Consider a homogeneous, lossless, source-free region of space characterized by permit-
tivity
and permeability µ. We seek solutions to the wave equation that are TEM r in
spherical coordinates (H r = E r = 0), and independent of the azimuthal angle φ.Thus
we may write
ˆ
ˆ
E(r, t) = θE θ (r,θ, t) + φE φ (r,θ, t),
ˆ
ˆ
H(r, t) = θH θ (r,θ, t) + φH φ (r,θ, t).
Maxwell’s equations show that not all of these vector components are required. Faraday’s
law states that
1 ∂ 1 ∂ 1 ∂
ˆ
∇× E(r,θ, t) = ˆ r [sin θ E φ (r,θ, t)] − θ ˆ [rE φ (r,θ, t)] + φ [rE θ (r,θ, t)]
r sin θ ∂θ r ∂r r ∂r
∂H(r,θ, t)
=−µ . (2.350)
∂t
Since we require H r = 0 we must have
∂
[sin θ E φ (r,θ, t)] = 0.
∂θ
This implies that either E φ ∼ 1/ sin θ or E φ = 0. We shall choose E φ = 0 and investigate
whether the resulting fields satisfy the remaining Maxwell equations.
In a source-free region of space we have ∇· D =
∇· E = 0. Since we now have only a
θ-component of the electric field, this requires
1 ∂ cot θ
E θ (r,θ, t) + E θ (r,θ, t) = 0.
r ∂θ r

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