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Derive Electric Potential Energy

  1. Feb 14, 2010 #1
    1. The problem statement, all variables and given/known
    For a a uniformly charged sphere of radius R with total charge Q, show that the potential energy of a point charge q' varies with r

    U(r) = Qq'/4[tex]\pi[/tex][tex]\epsilon[/tex]R * (3/2 - r2/2R2 if r < R

    2. Relevant equations
    [tex]\Delta[/tex]U = - [tex]\int[/tex]E(r)q'dr cos[tex]\theta[/tex]

    3. The attempt at a solution
    I used that the electric field of q' when r < R is Qr/4[tex]\pi[/tex][tex]\epsilon[/tex]R3. I tried to integrate from initial to final for r, but then I only get 1/2 as opposed to 3. I have no idea where the 3 comes from.

    Can someone help please?
  2. jcsd
  3. Feb 14, 2010 #2


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    What did you take as the initial and final r's? You have to integrate from infinity to r because potential energies are always stated with the potential at infinity as 0. (Of course, the electric field outside the sphere is not E=kQr/R^3, which adds another complication.)
  4. Feb 14, 2010 #3
    Ohh, I took the initial r to be R. Does that mean I have to add back the potential energy at R from infinity in order to get the expression for the potential energy at r when r < R?
  5. Feb 14, 2010 #4


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  6. Feb 14, 2010 #5
    Thank you!
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