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kuruman

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You use spherical coordinates, but without boundary conditions, you will not be able to find the potential, so you need to specify them.

Barring time-dependent boundary conditions, "conducting" means that the entire sphere is an equipotential.
 
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So, my solution will satisfy:

[tex]\nabla^{2}\Psi = 0[/tex]

[tex]\Psi = \sum a \Psi[/tex]

Should I assume there will be no potential outside the spheroid (or whatever)? And do the boundary conditions determine what particular solution (there is a table of different ones) it will be?
 

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