PHY731
HW#2
due date March 26, 2025 (in class)
1. Using the method of images, discuss the problem of a point charge q inside a hollow,
grounded, conducting sphere of inner radius a. Find
(a) The potential inside the sphere.
(b) The induced surface-charge density.
(c) The magnitude and direction of the force acting on q.
2. Three perpendicular, infinite, conducting planes at potential 𝑉0 form a cubic corner.
Let them be the planes x = 0, y = 0 and z = 0. Let
ν be the region x > 0, y > 0, z > 0.
(a) Show that Φ(𝑥, 𝑦, 𝑧) = (𝑎𝑥 + 𝑏)(𝑐𝑦 + 𝑑)(𝑒𝑧 + 𝑓) obeys Laplace’s equation in
(b) Show that the potential in
ν
ν is given by Φ(𝑥, 𝑦, 𝑧) = 𝑉0 + 𝐶𝑥𝑦𝑧, where C is a
constant.
(c) Find the surface charge density at (x, y, 0).
3. The planes in the last problem are now grounded, and a point charge +q is placed at
(a, a, a).
(a) Describe a set of image charges that may be used to find the potential Φ(x,y,z).
(b) What is the total induced surface charge on the x = 0 plane?
(c) If Φ is expanded as a power series in r for 𝑟 > √3𝑎, which term, i.e., which
power of r will dominate at large r? Why? (Do not try to calculate the term)
4.
(a) Show that the Green function 𝐺(𝑥, 𝑦; 𝑥 ′ , 𝑦′) appropriate for Dirichlet boundary
conditions for a square two-dimensional region, 0 ≤ 𝑥 ≤ 1, 0 ≤ 𝑦 ≤ 1 , has an
expansion
∞
′
𝐺(𝑥, 𝑦; 𝑥 , 𝑦′) = 2 ∑ g 𝑛 (𝑦, 𝑦 ′ ) sin(𝑛𝜋𝑥) sin(𝑛𝜋𝑥 ′ )
𝑛=1
Where 𝑔𝑛 (𝑦, 𝑦′) satisfies
𝜕2
(𝜕𝑦 ′2 − 𝑛2 𝜋 2 ) g 𝑛 (𝑦, 𝑦′) = −4𝜋𝛿(𝑦 − 𝑦 ′ ) and g 𝑛 (𝑦, 0) = g 𝑛 (𝑦, 1) = 0
(b) Taking for g 𝑛 (𝑦, 𝑦′) appropriate linear combinations of sinh(𝑛𝜋𝑦′) and
cosh(𝑛𝜋𝑦′) in the two regions, 𝑦 ′ < 𝑦 and 𝑦 ′ > 𝑦, under the boundary conditions and
the discontinuity in slope required by the source delta function, show that the explicit
form of 𝐺 is
∞
′
𝐺(𝑥, 𝑦; 𝑥 , 𝑦′) = 8 ∑
𝑛=1
1
sin(𝑛𝜋𝑥) sin(𝑛𝜋𝑥′) sinh(𝑛𝜋𝑦< ) sinh[𝑛𝜋(1 − 𝑦> )]
𝑛 sinh(𝑛𝜋)
Where 𝑦< (𝑦> ) is the smaller (larger) of 𝑦 and 𝑦′.
5.
Consider a potential problem in the half-space defined by z ≥ 0, with Dirichlet
boundary conditions on the plane z = 0 (and at infinity).
(a) Write the appropriate Green function G(x,x′) in cylindrical coordinates.
(b) If the potential on the plane z = 0 is specified to be Φ = V inside a circle of radius
a centered at the origin, and Φ = 0 outside that circle, find an integral expression
for the potential at the point P specified in terms of cylindrical coordinates
(ρ,φ,z).
(c) Show that, along the axis of the circle (ρ = 0), the potential is given by