Inverse of method of image charges

[diegzumillo]

Hi all
What if instead of charges and a surface, we were given a set of charges and image charges and have to find the surface, how would you do that?

This is actually part of my homework but I'm pretty sure he doesn't want us to prove it mathematically (the case is obviously a sphere) so I think this forum is more appropriate than homework, as I'd like a more informal discussion on this rather than a direct solution.

I've seen similar problems to this in other fields, like the famous 'hearing the shape of a drum' problem. Inverse of boundary value problems are very interesting, I wonder if this is one of them.

edit: just to get the ball rolling, for simple cases solving \varphi=0 gives you the solution, but more complicated ones could give you several different surfaces (infinite maybe?) and it might be impossible to determine the shape of the surface. This is a guess, of course.

 

[mfb]

Surfaces are always areas of the same potential. If you have charges and image charges, those surfaces are easy to find. Every surface will work as solution, and every potential value will give one so the set is infinite.

 

[diegzumillo]

Surfaces are always areas of the same potential. If you have charges and image charges, those surfaces are easy to find. Every surface will work as solution, and every potential value will give one so the set is infinite.

Actually, I completely overlooked the fact that the solutions are unique. If you solve for phi=0 that surface is the only solution. Wow, boring.

 

[mfb]

You can solve for phi equal to some other value :).