Inverse of method of image charges

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Discussion Overview

The discussion revolves around the concept of determining a surface from a given set of charges and image charges, particularly in the context of inverse boundary value problems. Participants explore the implications of this problem in relation to potential theory and its applications, with a focus on informal reasoning rather than strict mathematical proofs.

Discussion Character

  • Exploratory
  • Debate/contested
  • Homework-related

Main Points Raised

  • One participant suggests that finding the surface from charges and image charges could lead to multiple possible surfaces, especially in more complex cases, indicating a potential for infinite solutions.
  • Another participant asserts that surfaces correspond to areas of the same potential and claims that every surface could serve as a solution, suggesting an infinite set of surfaces based on different potential values.
  • A later reply challenges the uniqueness of the solution by stating that solving for a specific potential value (phi=0) yields a unique surface, which one participant finds uninteresting.
  • Another participant counters that one can solve for phi equal to other values, implying that there are indeed multiple surfaces that can be derived.

Areas of Agreement / Disagreement

Participants express differing views on the uniqueness of the surface solutions, with some arguing for the existence of multiple surfaces while others suggest that a specific potential leads to a unique solution. The discussion remains unresolved regarding the implications of these differing perspectives.

Contextual Notes

Participants note that the problem may involve complexities that could lead to multiple valid surfaces, but the exact conditions and assumptions under which these solutions hold are not fully explored.

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

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