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Suppose you have as many electrically charged particles as needed (even countably many) and consider the open unit ball centered at some point in space. For every continuous real valued function on the unit ball, is there a configuration of the particles (outside of the ball) that would generate an electric potential corresponding to said function on the ball?

Does an answer somehow trivially follow from Gauss' law?

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Gauss' law would impose that the given function must always be a harmonic function inside the empty ball, cf. en.wikipedia.org/wiki/Harmonic_function –  Qmechanic Jan 13 '12 at 17:29
    
Interesting question! I suggest a clarification: Are you allowed a countable set of charged particles, and each one can have a different arbitrary (positive or negative) charge? –  Andrew Moylan Jan 14 '12 at 3:46
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1 Answer

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First of all, according to Earnshaw's theorem one can not create a potential well in the ball.

Second, the function should fit Laplace's equation: $$ \Delta \varphi = 0. $$

In fact the second point is enough to prove the first one.

So the answer is:

  1. No, this can not be made for every continuous function.
  2. Yes, this follows from Gauss's law.
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However, remember there are en.wikipedia.org/wiki/Earnshaw%27s_theorem#Loopholes –  endolith Jan 16 '12 at 17:23
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