# Conducting Spherical Shell Capacitor

1. Feb 5, 2014

### BeBattey

1. The problem statement, all variables and given/known data
A conducting spherical shell is divided into upper and lower halves with a narrow insulating ring between them. The top half is at 10V and the bottom half is at -10V. Write down the appropriate expansion for Φ and use symmetry and the expected behavior at the origin to identify which coefficients are zero. Then solve for the nonzero coefficients which make Φ satisfy the values given at r = a. You will undoubtedly have to express the coefficients in integral form.

2. Relevant equations
No charge inside, so Laplace's equation applies:
$\nabla^{2}\phi=0$
Given the general solution for solving Laplaces equation in spherical coordinates:
$\phi (r,\theta,\varphi)= \sum^{\infty}_{n=0}(A_{n}r^{n}+\frac{B^{n}}{r^{n+1}})P_{n}(cos\theta)$

3. The attempt at a solution
I've only concluded so far that the B coefficients must all be 0 due to requiring finite potential at r=0. Past that I'm at a loss on how to tackle the function. I know:
$\phi (r,\theta,\varphi)= \sum^{\infty}_{n=0}A_{n}r^{n}P_{n}(cos\theta)$
But I don't know how I can tackle the boundary condition of plus and minus 10 at radius a, depending on angle theta.