Electrostatic Potential Outside a Conducting Shell in a Charge Density with Angular Dependence

1. Oct 19, 2014

hepcats pajamas

1. The problem statement, all variables and given/known data
A spherical conducting shell of radius R is held at a potential V0. Outside the shell,
the charge density is ρ(r) = ρ0sinθcosφ for R < r < 2R. Find the electrostatic potential
everywhere outside the shell.

2. Relevant equations
Green's function in spherical coordinates between radii a < r < b:
G(r,r') = $\sum\limits_{l,m} {\frac{4π/(2l+1)}{l-(\frac{a}{b})^{2l+1}} Y^*_{lm}(θ',φ')Y_{lm}(θ,φ)(r^l_< - {\frac{a^{2l+1}}{r^{l+1}_<}})({\frac{1}{r^{l+1}_>}}-{\frac{r^l_>}{b^{2l+1}}})}$

The solution for the potential using Green's function with Dirichlet Boundary Conditions (V is the Volume, S is the boundary surface, n is the normal to the surface):
$Φ(\textbf{r'} ∈ V) = {\int\limits_{V} d\textbf{r} ρ(\textbf{r})G(\textbf{r',r})} - ε_0 {\int\limits_{S} R^2 \: dΩ \: V (\textbf{r}){\frac{∂G(\textbf{r',r})}{∂n}}}$

3. The attempt at a solution
I know the potential at the boundary surface:
$Φ_{S}(\textbf{r}) = V_0$ where r = R

The normal direction on the surface is r, a = R, b = 2R, so:
at r = R, ${\frac{∂G(\textbf{r',r})}{∂n}} = \sum\limits_{l,m} {\frac{4π/(2l+1)}{l-(\frac{1}{2})^{2l+1}} Y^*_{lm}(θ',φ')Y_{lm}(θ,φ)((2l+1)R^{l-1})({\frac{1}{r'^{l+1}}}-{\frac{r'^l}{R^{2l+1}}})}$

The charge density can be expressed using spherical harmonics:
$ρ = ρ_0 {\frac{1}{2}} {\sqrt{\frac{8π}{3}}} (Y_{1,-1}(θ,φ) - Y_{1,1}(θ,φ))$

So:
$Φ(\textbf{r'}) = {\int\limits_{Ω}}{\int\limits_{R}^{2R}} r^2 dr \: dΩ \: [ ρ_0 {\frac{1}{2}} {\sqrt{\frac{8π}{3}}} (Y_{1,-1}(θ,φ) - Y_{1,1}(θ,φ)) ]$ ...
... $[ \sum\limits_{l,m} {\frac{4π/(2l+1)}{l-(\frac{1}{2})^{2l+1}} Y^*_{lm}(θ',φ')Y_{lm}(θ,φ)(r^l_< - {\frac{R^{2l+1}}{r^{l+1}_<}})({\frac{1}{r^{l+1}_>}}-{\frac{r^l_>}{(2R)^{2l+1}}})} ]$ ...
...$- ε_0 {\int\limits_{S} R^2 \: dΩ \: V_0 \: \sum\limits_{l,m} {\frac{4π/(2l+1)}{l-(\frac{1}{2})^{2l+1}} Y^*_{lm}(θ',φ')Y_{lm}(θ,φ)((2l+1)R^{l-1})({\frac{1}{r'^{l+1}}}-{\frac{r'^l}{R^{2l+1}}})} }$

Am I barking up the wrong tree? I am looking for some key simplification. Maybe my approach is over-complicating it, and I should use a Gaussian surface. The issue with that is that I don't see where the electric field would be constant, leading to a simplification.

Last edited: Oct 19, 2014
2. Oct 24, 2014