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Potential of Spherical Shell with Nonunifor Surface Charge

  1. Oct 18, 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 kcos 3 θ .
    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

    3. The attempt at a solution
    I solved Laplace's equation using separation of variables, and got to where I would integrate to find the coefficients of the Fourier series, but that's where I'm stuck. I get that my coeffcient Al = k/2εRl-1∫cos3θ Pl(cos θ) sin θ dθ where Pl(cos θ) are the Legendre polynomials.

    I don't know how to integrate this, but intuitively shouldn't all terms that aren't P3 give me zero?
     
  2. jcsd
  3. Oct 18, 2014 #2
    You need to write [tex]
    \cos^3\theta
    [/tex]
    In terms of legendre polynomials to do this integral easily. See wikipedia for the P polynomials http://en.wikipedia.org/wiki/Associated_Legendre_polynomials.

    I'll try and get you started.
    We start with out cos^3theta term, and now we need to find the corresponding legendre polynomial.
    [tex]
    P^{0}_{3}(\cos\theta)=\frac{1}{2}(5\cos^3\theta-3\cos\theta)
    \\
    \frac{1}{5}(2P^{0}_3(\cos\theta)+3\cos\theta)=\cos^3\theta
    \\
    now\, use\, the\, fact\, that\, P^{0}_{1}(\cos\theta)=\cos\theta\, and\, sub\, in
    \\
    \frac{1}{5}(2P^{0}_3(\cos\theta)+3P^{0}_{1}(\cos\theta))=\cos^3\theta
    [/tex]

    now you can sub in and integrate!
     
  4. Oct 19, 2014 #3
    Yes this is exactly what I was looking for thank you! When I integrate, I should get that all terms that are not P1 and P3 drop out due to the orthogonality of the Legendre Polynomials, yes?
     
  5. Oct 19, 2014 #4
    Yes, the orthogonal P terms will cancel out!
     
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