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Potential inside NON-Conducting hollow spherical shell

  1. Sep 15, 2011 #1
    Hi Guys,
    Suppose we have a spherical shell with charge density on the surface [itex]\sigma[/itex] and radius R. The potential inside the shell is given by:

    V_(x,y,z) = [itex]\frac{V0}{R^{2}}(6z^2+ax^2+by^2)[/itex]
    It is assumed, that the potential is rotational symmetric around the z-axis inside and outside the shell, and goes to 0 far away from the shell. There's no charge inside and outside the shell and no outer field.

    How do i determine the constants a and b?

    Mabye change to spherical coordinates and solve the equation:
    [itex]\frac{\partial{V}}{\partial{\theta}}=0[/itex]
    for a or b. But i can't figure out any other conditions if this is right.
     
  2. jcsd
  3. Sep 15, 2011 #2

    Born2bwire

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    The Laplacian of the voltage is related to the charge. So your boundary conditions would be:

    [tex] \nabla^2 V = -\frac{\sigma}{\epsilon} [/tex]

    On the surface of the shell. So for ease of analysis you would want to do this in spherical coordinates.
     
  4. Sep 15, 2011 #3
    I can't really see that this is a boundary condition? And would solving possions equation on the surface help me determine a and b, i can't really follow?
     
  5. Sep 15, 2011 #4
    Aah sorry im really tired now. I take the laplacian, and this would relate it all, beautiful, Thanks. Just so used to try finding the potential :)
    And sorry for the language. I'm from Denmark.
     
  6. Sep 15, 2011 #5

    clem

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    If the pot inside is axially symjmetric, and b must be equal.
    Since DEl^2 V=0 inside, the angular distribution must be a Legendre polynomial.
    Since the power of x,y,and z is 2, it must be
    [tex]P_2(\cos\theta)=(3\cos^2\theta-1)/2[/tex].
    This means a=b=-3.
     
    Last edited: Sep 15, 2011
  7. Sep 15, 2011 #6
    That's gold thank you.
     
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