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Laplace's Equation in Cylindrical Coordinates

  1. Feb 16, 2008 #1

    G01

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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

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


    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]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.
     
  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

    G01

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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

    G01

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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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