(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

A spherical shell with potential:

[tex]V(r=R)=V_0cos\theta[/tex]

(a) Please solve the potential inside and outside the shell.

(b) if there is a point charge at the center and the potential of the shell keeps the same, solve the potential inside and outside.

2. Relevant equations

Laplace equation:

[tex]\nabla^2V=0[/tex]

3. The attempt at a solution

Part a) I can do. Using the separation of variables in spherical coordinates I found that:

[tex]\frac{V_0 r}{R} cos\theta ; r\leq R[/tex]

and

[tex]\frac{V_0 R^2}{r^2} cos\theta ; r\seq R[/tex]

Part b) I'm having some trouble.

I think we should apply the uniqueness theorem since the potential is specified on the boundary (the sphere) but not sure how. Does that mean that the potential is the same as before inside the sphere? I was told that this is not the case by upperclassman who can't tell me why. They only know the answer is:

[tex]V(r,\theta)=\frac{V_0}{R} cos \theta +\frac{q}{4 \pi \epsilon_0}(\frac{1}{r}-\frac{1}{R})[/tex]

I think that this is indeed the right answer since it satisfy the boundary condition, but I want to know why. It looks like that the first term accounts for the charge distribution on the sphere, but if that is the case shouldn't the second term be the point charge in ther center namely:

[tex]\frac{q}{4 \pi \epsilon_0}\frac{1}{r}[/tex]

instead?

THe outside is even more tricky. Can I even apply the uniqueness theorem? The boundary where the potential is defined doesn't enclose the area outside the sphere. What can I do?

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# Electrostatics ofa shell question

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