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## Homework Statement

A long copper pipe, with it's axis on the z axis, is cut in half and the two halves are insulated. One half is held at 0V, the other at 9V. Find the potential everywhere in space.

## Homework Equations

[tex]\nabla^2V=0[/tex]

## The Attempt at a Solution

Alright. This is a laplace's equation problem in two dimensions, since the potential should be independent of z because the pipe is infinite. Using separation of variables( [tex]V=R\Phi[/tex]) the solution to the two dimensional Laplace's EQ in cylindrical coordinates is:

[tex]R(r)=ar^k+br^-k[/tex]

[tex]\Phi(\phi)=A\sin(k\phi) + B\cos(k\phi)[/tex]

w/ [tex]V=R(r)\Phi(\phi)[/tex]I am getting confused when I try to apply boundary conditions to this problem. Here is what I think the boundary conditions should be:

1) V should go to zero as r goes to infinity. (I don't think this is right, since the pipe is infinite, but I am not sure.)

2)V=0 for phi between 0 and pi, r=radius of pipe

3)V=9 for phi between 0 and 2pi, r=radius of pipe.

I can't figure out how to actually use these to eliminate any of the constants. Any help and hints would be greatly appreciated.