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Insulating Spherical Shell Potential Problem question

  1. Apr 19, 2010 #1
    1. The problem statement, all variables and given/known data

    I have a question about an upper-division E&M HW
    problem I'm stuck on. Any help would be GREATLY APPRECIATED!!!

    Problem Statement:

    An insulating spherical shell of radius R was set up to have a potential
    on its surface of V=A*cos^2(theta), where A is a constant. The potential
    at a distance very far away from the shell is defined to be zero.

    (a) Find the potential V(r,theta) both inside and outside the shell.



    2. Relevant equations

    E=-(grad)V

    3. The attempt at a solution

    Since E=-(grad)V , I took the gradient in spherical coordinates of the V expression above
    and got

    E={[(2*A)/R]cos(theta)sin(theta)}theta-hat


    for the E field on the surface of the shell. I then let R become the
    variable r and tried to do the line integral from "infinity" to R
    to get the potential outside the shell. But this involves a dot product with dr
    which is orthogonal to theta-hat so I got ZERO!! which certainly
    doesn't make sense...I would REALLY appreciate any advice someone might have on
    this problem!!!
     
  2. jcsd
  3. Apr 19, 2010 #2
    Think carefully about what you're doing here.

    Here's how I have read the question: You have a charged insulating spherical shell and no other sources of charge. You know the potential on the shell; deduce the potential at all other points.

    Now the equation E=-grad(V) tells you how to get the electric field from the potential. I couldn't follow exactly what you were doing but it seemed like you calculated the electric field along the surface of a shell (you don't know what it is perpendicular to the shell yet) trying to add together infinitely many spherical shells to find the electric field of a solid sphere of infinite radius?

    I think you need to consider general techniques for obtaining the potential due to a charged spherical shell, knowing the potential on the surface of the shell.
     
  4. Apr 19, 2010 #3
    THANK YOU very much for your response.

    Yes, I agree. I think I need to solve Laplace's Equation in spherical subject to the
    boundary conditions:

    (i) Vin = Vout
    (ii) (dVin/dr - dVout/dr)|r=R = -(surface charge density)/epsilon-knot
    (iii) As r-> infinity , V-> 0

    Does this sound right to you?

    Thank you so much again in advance!
     
  5. Apr 19, 2010 #4
    I solved the problem using Laplace's Equation - the only problem is the Dielectic aspect of the shell. I suppose my boundary condition (ii) above:

    (ii) (dVin/dr - dVout/dr)|r=R = -(surface charge density)/epsilon-knot

    is wrong? Should I just change the epsilon-knot to the epsilon of the dielectric surface?
    Although that doesn't seem to make too much sense to me. If there's no dielectric
    VOLUME then does it not "matter" and I should just use epsilon-knot for the vacuum around???

    Anyway, thank you so much again for your help! I REALLY appreciate it!!!!!
     
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