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Non-Uniform Surface Charge Spherical Shell

  1. Oct 14, 2014 #1
    1. The problem statement, all variables and given/known data
    A thin spherical shell of radius R carries a surface charge density of the form
    [itex]

    kcos3 \theta
    [/itex]
    Find the electric field inside and outside the sphere and demonstrate explicitly that its
    components satisfy the relevant boundary conditions at the surface.

    2. Relevant equations
    The solution to Laplace's Equation in spherical coordinates :


    \Sigma Al rl Pl(cos \theta) (r≤R)


    \Sigma Bl r-(l+1)Pl(cos \theta) (r≥R)




    3. The attempt at a solution
    I worked it through until


    Al= k/2ε0Rl-1∫cos3 \theta Pl cos \theta sin \theta d \theta


    Where do I go from here? Do I only need to consider the third Legendre polynomial?i.e.


    ##(5cos3\theta - 3 cos \theta)/2 ##
    Are all the other coefficients zero?

    EDIT: It seems TeX does not want to work for me, but I'm sure you get the idea
     
    Last edited: Oct 14, 2014
  2. jcsd
  3. Oct 20, 2014 #2
    Thanks for the post! Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
     
  4. Oct 21, 2014 #3
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