Laplace's Equation in Cylindrical Coordinates

[G01]

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

$$\nabla^2V=0$$

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( $$V=R\Phi$$) the solution to the two dimensional Laplace's EQ in cylindrical coordinates is:

$$R(r)=ar^k+br^-k$$

$$\Phi(\phi)=A\sin(k\phi) + B\cos(k\phi)$$

w/ $$V=R(r)\Phi(\phi)$$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.

 

[Rainbow Child]

First of all you have to break the problem into two regions $V_{in}$ inside the pipe and $V_{out}$ outside the pipe. In order for $V$ to be finite to each region you can eleminate $\alpha$ or $\beta$ in eah region.
The potential $V$ must be periodical for $\phi$ thus you know $k$.
Lastly you have to use the superposition principle and Fourier analysis for the remaining constants.

 

[G01]

OK. It seems the part screwing me up is finding k. The potential is periodic in Phi, but it seems more like a step function. I guess I'm saying that I don't understand how can represent that potential as a sine or cosine term, no matter what k happens to be.

Could you elaborate on that part of the problem?

 

[Rainbow Child]

Since the potential must be periodic in $\phi$ you must have $k\in \mathbb{N}$. Thus the solution is

$$V_{in}(r,\phi)=\frac{A_0}{2}+\sum_{n=1}^\infty r^n\left(A_n\,\cos(n\,\phi)+B_n\,\sin(n\,\phi)\right)$$

The boundrary conditions read

$$u(R,\phi)=\left\{\begin{matrix} 0 & 0<\phi<\pi\cr 9 & \pi<\phi<2\,\pi \end{matrix}$$

The constants $A_n,\,B_n$ can be found by Fourier analysis. Can you do that?

 

[G01]

Yes, I am fine with Fourier analysis. The boundary condition were just confusing the heck out of me. I don't quite no why either. I guess I just did not recognize that I had to use Fourier anaylsis. Thank you! I think I got it now.