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Spherically symmetric charge density given electric potential

  1. Oct 9, 2012 #1
    1. A spherically symmetric charge distribution results in an electric potential of the form
    pt1.jpg

    What is the charge distribution?




    2.
    Hint: consider the difference in electric field between two values of r
    pt2.jpg

    Show that the answer is of the form
    pt3.jpg





    3. I have attempted several solutions but haven't gotten anywhere.
     
  2. jcsd
  3. Oct 10, 2012 #2
    Did you at least calculate the electric field?

    E = - ∇ V where the gradient is in spherical coordinates.
     
  4. Oct 10, 2012 #3
    I calculated E using the first equation. E=dV/dr
     
  5. Oct 13, 2012 #4
    Correction E=-dv/dr <-- I am unsure to just have dr=dr or dr = (the derivative of everything in the 1st equations brackets[] ) - I decided to just have it equal dr

    I then plugged the E value into the difference equation with r = r, and r' = (r+dr). I took the difference and set it equal to the right side of the same equation, and then solved for p(r). I hoped this would give me something in the form of charge density noted. However, I ended up with a large number of variables which would not simplify to this form.

    Any Insights?
     
    Last edited: Oct 13, 2012
  6. Oct 13, 2012 #5

    vela

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    You have the right approach. You can neglect the higher-order terms. Keep only the terms proportional to dr.
     
  7. Oct 13, 2012 #6
    You can have r'=0 and just integrate the right side of the equation from 0 to r. You now have Gauss' Law, and you can just solve algebraically for [itex]\rho[/itex].
     
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