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Capacitance of Concentric Shells (different charges)

  1. Apr 10, 2009 #1
    1. The problem statement, all variables and given/known data

    A thin conducting shell of radius a has charge q. Concentric to this is another shell of radius b (b > a) with a different charge Q. How is the charge distributed and what is the capacitance of the two shells. No hint to the relative polarity of the charges is given.

    2. Relevant equations

    [tex]C = \frac{Q}{V}[/tex]

    3. The attempt at a solution

    The difficulty lies in the difference in the charges (I'm sure I'm missing something). Any reference to capacitance I see requires equal and opposite charges, leading me to think the the distribution must be something like:

    Inside surface of small shell: no charge
    Outside surface of small shell: +q

    Inside surface of larger shell: -q (?)
    Outside surface of larger shell: q + Q (?)

    If I assume that it the charges are -q on inner and +q on outer then I think you could find it by:

    [tex]E = \frac{1}{4\pi\epsilon_0}\frac{q}{r^2}\hat{r}[/tex]

    [tex]V = -\frac{q}{4\pi\epsilon_0} \int_a^b \frac{1}{r^2} = \frac{q}{4\pi\epsilon_0} \left(\frac{1}{a} - \frac{1}{b}\right)[/tex]

    The just find the capacitance simply by [tex]C = \frac{q}{V}[/tex]

    But there is no mention of Q which seems wrong. Any help or suggestions very much appreciated.
  2. jcsd
  3. Apr 10, 2009 #2


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    Homework Helper

    When you connect the charged concentric shells, both must have the same potential.
    The common potential is given by ( total charge/total capacity)
    The capacitance of the spherical shell is proportional to its radius.
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