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Electric Potential Energy Spherical Shells

  1. Sep 18, 2011 #1
    1. The problem statement, all variables and given/known data

    A solid sphere of radius R has a uniform charge density ρ and total charge Q. Derive an expression for its total electric potential energy. Suggestion: Imagine that the sphere is constructed by adding successive layers of concentric shells of charge dq = (4[itex]\pi[/itex] r[itex]^{2}[/itex] dr) ρ and let dU = Vdq. (Use any variable or symbol stated above along with the following as necessary: ke.)

    2. Relevant equations

    U = [itex]\int[/itex]4[itex]\pi[/itex]r[itex]^{2}[/itex]k[itex]_{e}[/itex][itex]\frac{q}{r}[/itex]dr

    [itex]\rho[/itex]=[itex]\frac{Q}{\frac{4}{3}\pi r^{3}}[/itex]



    3. The attempt at a solution

    The sum of all dq is Q.

    U = qV - q is test charge
    U = q k[itex]_{e}[/itex][itex]\frac{Q}{r}[/itex] - equation of voltage substituted

    dQ = dq k[itex]_{e}[/itex][itex]\frac{Q}{r}[/itex] -small potential energy with respect to small charge

    dQ = 4[itex]k_{e}\pi\rho\frac{Q}{r} r^2 dr[/itex] - dq plugged in

    Then I integrated both sides.
     
  2. jcsd
  3. Sep 19, 2011 #2

    Delphi51

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    Homework Helper

    I'm having a little trouble following that.
    It seems to me the dQ for the spherical shell is 4πR²ρ*dR.
    The work done to bring dQ in from infinity to R is dU = kQ/R*dQ.
    And Q up to radius R is 4/3*πR³ρ.
    Combined, dU = 16/3π²k ρ²R⁴dR
    Check carefully; I make mistakes.
     
  4. Sep 20, 2011 #3
    Thank you
     
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