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Spherical Charge Distribution

  1. Feb 25, 2008 #1
    1. The problem statement, all variables and given/known data
    There is a charge density rho that exists in a spherical region of space defined by 0 < r < a.
    [tex]\rho (r) = Ke^{-br}[/tex]
    How do you find the electric field if a charge density varies as such?

    3. The attempt at a solution

    I found Q total = [tex]\int \int \int \rho dV[/tex]
    Now I need to find E.

    My real question is can I just put Q (as a function of r) into E = kQ/r^2? Or do I need to reevaluate the integral using dq = [tex]\rho r^2 sin(\theta) dr d\theta d\phi[/tex]

    I get two different answers, (and I would have thought they should be the same) so which method is correct? I would have thought either would work.
    Last edited: Feb 25, 2008
  2. jcsd
  3. Feb 25, 2008 #2
    What do you mean? Q is the integral of the charge density over the volume. Also, I think you mean [itex]dr = rho r^2 sin(\theta) dr d\theta d\phi[/itex]. What did you do for your integral?
  4. Feb 26, 2008 #3
    Why [tex]sin(\theta)[/tex]? rho depends only on r so [tex]dQ = 4\pi Kr^{2}e^{-br}dr[/tex]
  5. Feb 27, 2008 #4
    Oh whoops, I shouldn't have had rho in there, and I missed it when you had it. You were right about the dq I was questioning. dq= rho *spherical jacobian (i.e. spherical integration differentials), which is what you had.

    Yes, [tex]dQ = 4\pi Kr^{2}e^{-br}dr[/tex]

    This is the way you want to go. I don't really understand what other way you would have gone about it.
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