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Multipole expansion on a sphere

  1. Nov 7, 2009 #1
    1. The problem statement, all variables and given/known data
    A sphere of radius R, centered at the origin, carries charge density
    ρ(r,θ) = (kR/r2)(R - 2r)sinθ,
    where k is a constant, and r, θ are the usual spherical coordinates. Find the approximate potential for points on the z axis, far from the sphere.

    2. Relevant equations
    The multipole expansion of V and some manipulations on the charge density?

    3. The attempt at a solution
    So I thought I had solved the problem, but both the monopole and the dipole vanished and I believe the answer given was approximated by the dipole term. And when I decided to try the quadrupole, it got really messy. I ended up with (3πkR5/64εoz3. Is this right at all? I tried to change the charge density in terms of the vector, r', from the origin to the points on/in the sphere but I don't seem to be getting anywhere.
    Here's what I did to find the dipole:

    V(r,θ) = 1/4πεo * 1/r2 ∫∫∫ r'cosθ'kR(R-2r')sin2θ' dr'dθ'dφ'

    Everything is fine until I integrate with respect to θ:

    0πsin2θ'cosθ' dθ' = 1/4sin4θ' |0π = 0

    The monopole did the same thing, but I expected that.
    Last edited: Nov 7, 2009
  2. jcsd
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