Page 165 - Electromagnetics

Figure 3.16: A conducting edge.

Since the origin is included we cannot have negative powers of ρ and must put a ρ = 0.
The boundary condition (ρ, 0) = 0 requires B φ = 0. The condition (ρ, β) = 0 then
requires sin(k φ β) = 0, which holds only if k φ = nπ/β, n = 1, 2,.... The general solution
for the potential near the edge is therefore
N

nπ nπ/β
(ρ, φ) = A n sin φ ρ (3.108)
β
n=1
where the constants A n depend on the excitation source or system of conductors. (Note
that if the corner is held at potential V 0 = 0, we must merely add V 0 to the solution.)
The charge on the conducting surfaces can be computed from the boundary condition
on normal D. Using (3.30)we have
N N
1 ∂
nπ nπ/β
nπ nπ (nπ/β)−1
E φ =− A n sin φ ρ =− A n cos φ ρ ,
ρ ∂φ β β β
n=1 n=1
hence
N

nπ (nπ/β)−1
ρ s (x) =− A n x
β
n=1
on the surface at φ = 0. Near the edge, at small values of x, the variation of ρ s is dom-
inated by the lowest power of x. (Here we ignore those special excitation arrangements
that produce A 1 = 0.)Thus
ρ s (x) ∼ x (π/β)−1 .

The behavior of the charge clearly depends on the wedge angle β. For a sharp edge
(half plane)we put β = 2π and find that the field varies as x −1/2 . This square-root edge
singularity is very common on thin plates, fins, etc., and means that charge tends to
accumulate near the edge of a flat conducting surface. For a right-angle corner where
β = 3π/2, there is the somewhat weaker singularity x −1/3 . When β = π, the two
surfaces fold out into an infinite plane and the charge, not surprisingly, is invariant with
x to lowest order near the folding line. When β< π the corner becomes interior and we
find that the charge density varies with a positive power of distance from the edge. For
very sharp interior angles the power is large, meaning that little charge accumulates on
the inner surfaces near an interior corner.

© 2001 by CRC Press LLC