Description
Problems
β
1. A corner:
Consider the potential in the region bounded by two
half-planes, which meet at the origin at an angle β as shown
in the figure. Near the origin there are no charges, although
we presume there are other charges (not shown) away from
the origin. For simplicity, we shall also assume that nothing
depends on the variable z. Then the potential near the
origin can be expressed in cylindrical coordinates, and it
satisfies the Laplace equation, ∆φ = 0, with a solution
of the form
φ(ρ, ϕ) = a0 + b0ϕ + (c0 + d0ϕ) ln ρ +
X∞
n=1
h
ρ
νn
(an cos νnϕ + bn sin νnϕ)
+ρ
−νn
(cn cos νnϕ + dn sin νnϕ)
i
.
In this case, however, the parameters νn are not integers, since ϕ doesn’t run periodically from 0 to 2π in the charge-free region. We have also included terms linear in ϕ,
which may appear if the solution is not assumed periodic in ϕ.
(a) (20 pts) Assume that the half-planes are grounded, so that φ(ρ, 0) = φ(ρ, β) =
0. Use this to obtain the possible values of νn and to determine which of the
coefficients are nonzero.
(b) (20 pts) Since we are interested in the field near the corner, we include the point
ρ = 0 in the region and assume that the potential is finite as ρ → 0 (no charge
singularities). Write the general solution for the potential near the corner. What
is the leading behavior of the potential as ρ → 0? (We can assume that the
coefficient of the leading term is nonzero due to the presence of the other charges
away from the origin.)
(c) (20 pts) Keeping only the leading term of the solution near ρ = 0, determine the
E~ field and surface charge density. Discuss the behavior of the field and surface
charge near the corner as ρ → 0 for the values of β ≈ 0, β = π and β ≈ 2π.
2. A cylinder (Jackson, 3.9, 40 pts):
A hollow right circular cylinder of radius b has its axis coincident with the z axis and
its ends at z = 0 and z = L. The potential on the end faces is zero, while the potential
on the cylindrical surface is given as V (ϕ, z). Using the appropriate separation of
variables in cylindrical coordinates, find a series solution for the potential anywhere
inside the cylinder.
