Electron flow over a conducting surface of variable resistivity

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fyzxfreak
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Hello fellow physics-people,

I was just thinking about the following setup:
We have a conducting surface (with smoothly varying resistivity) hooked up to some battery with the wires contacting the surface at two arbitrary points, A and B. How would we go about figuring out the current density?

Would the electrons flow in a simple curve from point A to B (to follow the path of least resistance) or would they spread out (as there might be a configuration in which parallel flow reduces mutual electric repulsion)?

Any ideas/suggestions? Thanks!
 
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I believe that the current would flow through a large area of the conductor. But I'm not sure.
 
Is there a systematic method of approaching this problem? i.e. application of Maxwell's equations, etc.

=EDIT= There'd definitely be some variational calculus involved (for the different paths from point A to point B). Yeah... this problem does not seem particularly simple anymore, haha.
 
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fyzxfreak said:
Is there a systematic method of approaching this problem? i.e. application of Maxwell's equations, etc.

I'm not knowledgeable enough to answer. I was basing my guess on knowing that the more conductor a current can flow through, the less the resistance overall. I would think this is balanced against the resistance caused by the longer path some of the current takes, resulting in an equilibrium somewhere. But all this is merely a guess.