# Laplace Equation, Non-Concentric Spheres

1. Dec 18, 2012

### gparker267

1. The problem statement, all variables and given/known data

General solution for eccentric spheres, smaller sphere (radius, b) completely embedded within larger sphere of radius c. The centers of both spheres lie on z-axis, distance a, apart (note: c>b+a). Problem is symmetric, so consider θ=[0,∏], r=[0,c]. The inner sphere is grounded while the outer sphere is held at a potential f(c,θ).

2. Relevant equations
General solution (see Morse & Feshbach, 1953) is given in coordinates of larger sphere as:

V(r,θ)=Ʃ1((r/a)^s×Ʃ2(A_n×(a/b)^n×[(-1)^(n-s)n!/s!(n-s)!])-(a/r)^(s+1)×Ʃ3(A_n×(b/a)^(n+1)×[s!/n!(s-n)!]))P_s(cosθ)

V(c,θ)=f(c,θ), θ=[0,∏], r=[0,c]

Limits of summations
Ʃ1, s=0 to s=s_max (can use s_max=50)
Ʃ2, n=s to n=n_max (can use n_max=50)
Ʃ3, n=0 to s=s

3. The attempt at a solution
I have expanded the boundary function, f(c,θ) as a series of Legendre polynomials and computed the unknown coefficients A_n. For the special case when the offset (a) is zero,the solution behaves well (and converges nicely to prescribed boundary conditions, BCs). However, in cases where the offset is greater than zero, the solution diverges in the region where radii r<a. Any ideas on what might be going wrong?

Last edited: Dec 18, 2012