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Electric Field minimum in a spherical capacitor.

  1. Oct 20, 2009 #1
    1. The problem statement, all variables and given/known data

    The potential difference [tex]\Delta\phi[/tex] between the plates of a spherical capacitor is kept constant. Show that then the Electric Field at the surface of the inner sphere will be a minimum if [tex]a = \frac{1}{2}b[/tex]

    2. Relevant equations
    [tex]E = \frac{Q}{4\pi\epsilon_{0}r^{2}}[/tex]
    between the plates.


    3. The attempt at a solution
    Not really sure. I know that
    [tex]\Delta\phi = - \int E dl[/tex]
    and that it is constant. But not really sure how to proceed from here. I think I need to find some sort of new equation for E, then find the minima? Answer in the back of the book is E = 4 (delta)(phi) / b

    The answer latex didn't work, sorry.
     
    Last edited: Oct 21, 2009
  2. jcsd
  3. Oct 21, 2009 #2

    turin

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    Homework Helper

    Your line integral can be taken along a radial line, from a to b.

    BTW, I would use [itex]\Delta\phi[/itex] instead of [itex]\nabla\phi[/itex] for the total potential difference. It's just a notational convention, but it is a very common one, and I was a little bit confused at first.
     
  4. Oct 21, 2009 #3
    Err, yes I meant delta phi. Taking the line integral doesn't help with the solution. If I did take the line integral, I get the electric potential difference. But this isn't helpful because
    [tex]\Delta\phi = \frac{Q}{4\pi\epsilon_{0}}\left(\frac{1}{b}-\frac{1}{a}\right)[/tex]

    And then how would I find the minimum of the electric field from there?

    I already passed in the problem--probably done incorrectly. I used the mean value theorem to integrate over E from b to a, then divide by (b-a). Then I took the derivative with respect to a, and tried to set equal to 0 but there were no values for which that occurred. I did manage to get rid of one of the two terms when a = (1/2)b.
     
  5. Oct 21, 2009 #4

    turin

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    Homework Helper

    Consider Q as a function of a (or b, or their ratio).
     
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