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Very Hard Integral (from int Coulomb) with -3/2 power + others

  1. Feb 16, 2013 #1
    1. The problem statement, all variables and given/known data

    I am trying to directly calculate the electric field (using Coulomb) at some arbitrary point P(0,0,z). The charge is evenly distributed over the surface of a sphere (radius R, charge density σ). Here I use θ for the polar angle and p for the azimuthal angle.

    I will leave out the messy details, but I know by symmetry only projection onto z-axis is relevant. I also determined the angle ψ (that between separation vector π and the z-axis) in terms of z,R,θ, and π.

    2. Relevant equations

    E(alongz) = (4∏ε0) ∫02∏0 [σR^2 sinθ (z - Rcosθ)] / (R^2 + z^2 - 2Rzcosθ)^(3/2) dθ dp

    3. The attempt at a solution

    ∫dp → 2∏
    removing 2∏R^2σ constants out from integrand

    0 [(z - Rcosθ)sinθ] / (R^2 + z^2 - 2Rzcosθ)^(3/2) dθ

    using u-substitution:
    u = cosθ du= -sinθ dθ
    θ = 0 → u = 1
    θ = ∏ → u =-1

    and reversing the limits of integration gives (ignoring constants out front):

    -11 (z - Ru) / (R^2 + z^2 - 2Rzu)^(3/2) du​
    (#1)

    →according to solutions manual→ this works out to:

    z^-2 [(z-R) / |z-R| - (-z-R) / |z+R|]​
    (#2)

    The manual says:
    Does anyone have any idea how you would use partial fractions to go from (1) to (2)??
     
    Last edited: Feb 16, 2013
  2. jcsd
  3. Feb 16, 2013 #2

    haruspex

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    I don't see how to use partial fractions here either, but how about a substitution t2 = R2+z2-2Rzu?
     
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