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 [itex]\varphi=0[/itex] 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 :).

 

[sardar]

Interesting question! The inverse method of image charges is a technique used to solve electrostatic problems by creating an image charge that mimics the behavior of a real charge. In this case, the image charge is usually placed on a surface to simplify the problem.

Now, if we are given a set of charges and image charges and have to find the surface, there are a few different approaches we could take. One way would be to use the method of images in reverse, by starting with the known charges and image charges and working backwards to determine the surface. This could involve solving for the potential or electric field at various points and using that information to determine the surface.

Another approach could be to use techniques from inverse boundary value problems, as you mentioned. These types of problems involve determining the shape or properties of a boundary based on known information about the interior of a system. In this case, the known information would be the charges and image charges, and we would have to determine the surface that would produce these charges.

As you mentioned, for simple cases, solving for the potential or electric field could give us the solution. However, for more complex cases, there could be multiple surfaces that could produce the same set of charges and image charges. This could make it difficult or even impossible to determine the exact shape of the surface.

Overall, the inverse method of image charges is a fascinating technique that can be applied to various problems in physics and other fields. It highlights the importance of understanding the relationship between charges and surfaces in electrostatics and the challenges involved in solving inverse problems.