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Potential in Concentric Spherical Shells

  1. Sep 29, 2015 #1
    1. The problem statement, all variables and given/known data
    Two grounded spherical conducting shells of radii a and b (a < b) are arranged concentrically. The space between the shells carries a charge density ρ(r) = kr^2. What are the equations for the potential in each region of space?


    2. Relevant equations
    Poisson's and LaPlace's in Spherical Coordinates

    3. The attempt at a solution
    I solved Poisson's Equation for the space between the shells, in spherical coordinates, and arrived at:
    V(r) = (1/ε)kr^2/6 - (C1)/r + (C2)
    where C1 and C2 are the constants of integration.
    What would be the general solution for the potential in the other regions where ρ=0? Would I simply apply Laplace's equation in those regions, than apply the suitable boundary conditions?
     
  2. jcsd
  3. Sep 30, 2015 #2

    TSny

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    Your first term on the right does not have the correct dimensions for electric potential.
    Yes, that will work.
     
  4. Oct 1, 2015 #3
    I made a mistake, the equation in-between a and b should read: V(r) = kr^4/20 - c1/r +c2
    The boundary conditions should be Vinside(a) = Vbetween(a) and Vbetween(b) = Vout(B) for continuity; am I missing any other conditions? I know there's the discontinuous derivative of potential = some surface charge, but I am not given such a surface charge.
     
  5. Oct 1, 2015 #4

    TSny

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    I believe the sign of the first term is incorrect.
    I don't understand your boundary conditions. Perhaps it's the notation you are using. The usual interpretation of "grounding a conductor" is to set the potential of the conductor to 0.

    The potential is continuous everywhere. As you say, the derivative of V will be discontinuous at a surface containing surface charge.

    You will be able to determine the surface charges after you find V.
     
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