Charge sharing between two conductors

[NFuller]

This is a question born out of a homework thread that lead to a discussion between @haruspex, @rude man, and myself. Link here https://www.physicsforums.com/threads/what-is-the-charge-of-each-conductor-afterwards.923909/ . I feel this question deserves its own thread and hopefully we can get some more members to share their ideas on it.

Consider two arbitrarily shaped but identical perfect conductors. One conductor possesses charge $Q$ and the other conductor is uncharged. The two conductors are brought into contact such that they may exchange charge and are separated thereafter. Do both conductors now share equal charge $Q/2$ or does the amount of charge on each conductor depend on where contact was made?

I am looking for a mathematical proof of equal or non-equal sharing of charge or a specific case of unequal charge sharing.

 

[vanhees71]

It's difficult to answer in this general form. You have to solve the electrostatic Neumann problem for the shape of the combined conductors. Then you can calculate, how much charge is on the parts of the surface made up by each of the two bodies.

 

[NFuller]

You have to solve the electrostatic Neumann problem for the shape of the combined conductors.

Right, but I'm wondering if given that the potential is constant on both surfaces whether this is enough information to solve the general problem.

 

[Nidum]

 

[NFuller]

So in the diagram above, I think you can use symmetry to argue that the total charge is equal for the second and third configuration. The first configuration is not symmetric about the point of separation, so a symmetry argument can't be made but maybe there are other arguments that could be used to find the charge on each conductor without actually calculating the surface charge density.