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Homework Help: Grounded conducting cylinder using Laplace

  1. Feb 9, 2005 #1
    Hi, I'm having trouble applying Laplace's equation solution in cylindrical coordinates to the problem of the grounded conducting cylinder of radius a in a uniform external field. The cylinder axis is the z axis, and the external electric field is E0 in the x direction. I need to find the potential V(r) and the induced surface charge. Thanks.
     
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  3. Feb 9, 2005 #2

    dextercioby

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    What did u do exactly,did u separate variables and got ODE-s for all three variables...?

    Daniel.
     
  4. Feb 9, 2005 #3
    My professor in class said to use separation of variables.

    [tex] \Phi = \Phi(\rho, \phi, z) [/tex]
    [tex] \nabla^2\Phi = \frac{1}{\rho} \frac{\partial}{\partial\rho} (\rho\frac{d\Phi}{d\rho}) + \frac{1}{\rho^2}\frac{\partial^2\Phi}{\partial\phi^2} + \frac{\partial^2\Phi}{\partial z^2}[/tex]
    which is the Laplace's Equation in cylindrical coordinates. And I think he said that we can ignore the z term because this case is z independent.

    Then I'm not sure how to obtain the solutions to this equation. And after I get the solutions, how do I apply it to this problem?
     
  5. Feb 9, 2005 #4

    dextercioby

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    Well,u have to come up with so-called limit conditions.The general solution will not be good for anything,if u can't use the limit conditions...

    Daniel.
     
  6. Feb 9, 2005 #5
    I think we call them boundary conditions. Well in this case, since the cylindrical conductor is grounded, the limit condition must be that V(a) = 0.
     
  7. Feb 9, 2005 #6

    dextercioby

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    Okay,then,separate varaibles and integrate each equation.Though i think you may need another condition.You must fix 2 integraton constants,after all...

    Daniel.
     
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