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Homework Help: Laplace's Equation in Cylindrical Coordinates

  1. Feb 16, 2008 #1


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    1. The problem statement, all variables and given/known data
    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.

    2. Relevant equations


    3. 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]\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.
  2. jcsd
  3. Feb 16, 2008 #2
    First of all you have to break the problem into two regions [itex]V_{in}[/itex] inside the pipe and [itex]V_{out}[/itex] outside the pipe. In order for [itex]V[/itex] to be finite to each region you can eleminate [itex]\alpha[/itex] or [itex]\beta[/itex] in eah region.
    The potential [itex]V[/itex] must be periodical for [itex]\phi[/itex] thus you know [itex]k[/itex].
    Lastly you have to use the superposition principle and Fourier analysis for the remaining constants.
  4. Feb 16, 2008 #3


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    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?
  5. Feb 16, 2008 #4
    Since the potential must be periodic in [itex]\phi[/itex] you must have [itex]k\in \mathbb{N}[/itex]. Thus the solution is

    [tex]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)[/tex]

    The boundrary conditions read

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

    The constants [itex]A_n,\,B_n[/itex] can be found by Fourier analysis. Can you do that?
  6. Feb 16, 2008 #5


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    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.
    Last edited: Feb 16, 2008
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