Page 330 - Electromagnetics

For a plane-wave impressed field this reduces to
∞
˜ #
˜ 2π E 0 ν n
H z (ρ, φ, ω) = n j J ν n (k 0 ρ) cos ν n φ cos ν n φ 0 .
ψ η 0
n=0
Behavior of current near a sharp edge. In § 3.2.9 we studied the behavior of static
charge near a sharp conducting edge bymodeling the edge as a wedge. We can follow
the same procedure for frequency-domain fields. Assume that the perfectly conducting
wedge shown in Figure 4.26 is immersed in a finite, z-independent impressed field of a
sort that will not concern us. A current is induced on the surface of the wedge and we
wish to studyits behavior as we approach the edge.
Because the field is z-independent, we mayconsider the superposition of TM and TE
fields as was done above to solve for the field scattered bya wedge. For TM polarization,
if the source is not located near the edge we maywrite the total field (impressed plus
scattered) in terms of nonuniform cylindrical waves. The form of the field that obeys the
boundaryconditions at φ = 0 and φ = ψ is given by(4.368):
∞
#
˜
E z = A n sin ν n φJ ν n (k 0 ρ),
n=0
where ν n = nπ/ψ. Although the A n depend on the impressed source, the general behavior
of the current near the edge is determined bythe properties of the Bessel functions. The
current on the wedge face at φ = 0 is given by
˜
ˆ
˜
˜
ˆ ˜
J s (ρ, ω) = φ × [φH φ + ˆρH ρ ]| φ=0 =−ˆ zH ρ (ρ, 0,ω).
By(4.349) we have the surface current
∞
1 # ν n
˜
J s (ρ, ω) =−ˆ z A n J ν n (k 0 ρ).
Z TM k 0 ρ
n=0
For ρ → 0 the small-argument approximation (E.51) yields
∞
1 # 1 k 0 ν n ν n −1
˜
J s (ρ, ω) ≈−ˆ z A n ν n ρ .
Z TM k 0 (ν n + 1) 2
n=0
The sum is dominated bythe smallest power of ρ. Since the n = 0 term vanishes we
have
˜
J s (ρ, ω) ∼ ρ π ψ −1 , ρ → 0.
For ψ< π the current density, which runs parallel to the edge, is unbounded as ρ → 0.
A right-angle wedge (ψ = 3π/2) carries
˜
J s (ρ, ω) ∼ ρ −1/3 .
Another important case is that of a half-plane (ψ = 2π) where
1
˜ J s (ρ, ω) ∼ √ . (4.377)
ρ
This square-root edge singularitydominates the behavior of the current flowing parallel
to anyflat edge, either straight or with curvature large compared to a wavelength, and
is useful for modeling currents on complicated structures.

© 2001 by CRC Press LLC