Page 331 - Electromagnetics

In the case of TE polarization the magnetic field near the edge is, by(4.375),

∞
#
˜
H z (ρ, φ, ω) = B n cos ν n φJ ν n (k 0 ρ), ρ < ρ 0 .
n=0
The current at φ = 0 is
˜
˜
ˆ
˜
J s (ρ, ω) = φ × ˆ zH z | φ=0 = ˆρH z (ρ, 0,ω)
or
∞
˜
#
J s (ρ, ω) = ˆρ B n J ν n (k 0 ρ).
n=0
For ρ → 0 we use (E.51) to write
∞ 1 k 0 ν n
˜
#
J s (ρ, ω) = ˆρ B n ρ .
ν n
(ν n + 1) 2
n=0
The n = 0 term gives a constant contribution, so we keep the first two terms to see how
the current behaves near ρ = 0:
π
˜
J s ∼ b 0 + b 1 ρ .
ψ
Here b 0 and b 1 depend on the form of the impressed field. For a thin plate where ψ = 2π
this becomes
√
˜ J s ∼ b 0 + b 1 ρ.
This is the companion square-root behavior to (4.377). When perpendicular to a sharp
edge, the current grows awayfrom the edge as ρ 1/2 . In most cases b 0 = 0 since there is
no mechanism to store charge along a sharp edge.

4.11.8 Propagation of spherical waves in a conducting medium

We cannot obtain uniform spherical wave solutions to Maxwell’s equations. Anyfield
dependent onlyon r produces the null field external to the source region, as shown in
§ 4.11.9. Nonuniform spherical waves are in general complicated and most easilyhandled
using potentials. We consider here onlythe simple problem of fields dependent on r and
θ. These waves displaythe fundamental properties of all spherical waves: theydiverge
from a localized source and expand with finite velocity.
Consider a homogeneous, source-free region characterized by ˜ (ω), ˜µ(ω), and ˜σ(ω).
˜
˜
We seek wave solutions that are TEM r in spherical coordinates (H r = E r = 0) and
φ-independent. Thus we write
˜
ˆ ˜
ˆ ˜
E(r,ω) = θE θ (r,θ,ω) + φE φ (r,θ,ω),
ˆ ˜
˜
ˆ ˜
H(r,ω) = θH θ (r,θ,ω) + φH φ (r,θ,ω).
To determine the behavior of these fields we first examine Faraday’s law
1 ∂ 1 ∂ 1 ∂
˜
ˆ
˜
˜
˜
∇× E(r,θ,ω) = ˆ r [sin θ E φ (r,θ,ω)] − θ ˆ [r E φ (r,θ,ω)] + φ [r E θ (r,θ,ω)]
r sin θ ∂θ r ∂r r ∂r
˜
=− jω ˜µH(r,θ,ω). (4.378)
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