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

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)) ##

##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:
Thanks for the post! Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?

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