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Homework Help: Express electrostatic energy in terms of both charges

  1. Sep 2, 2011 #1
    1. The problem statement, all variables and given/known data
    For two concentric conducting spheres (radius a and b, b>a) that form a capacitor with charge q on the inner sphere and -q on the outer sphere, express the electrostatic energy in terms of q and -q and the potential difference between them.

    2. Relevant equations
    Gauss's Law, the equation for electrostatic potential, the equation for the energy stored in a static electric field.

    3. The attempt at a solution
    I have the field
    \vec{E}=\frac{q}{4\pi\epsilon_0 r^2}\hat{r}
    between the conductors, but when I calculate the energy, should I only integrate between the spheres?
    W=\frac{\epsilon_0}{2}\int E^2 d\tau=\frac{q^2}{8\pi\epsilon_0}\int_{a}^{b}\frac{1}{r^2}dr=\frac{q^2}{8\pi\epsilon_0}\left( \frac{1}{a}-\frac{1}{b} \right)

    Then to express it in terms of the original charges and the potential difference,
    but how would I write this in terms of the charges? Does it want me to split it up like
    and put this in the above equation?

    Thanks in advance,
  2. jcsd
  3. Sep 3, 2011 #2

    I like Serena

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    Homework Helper

    Isn't the electric field zero at every other point?

    You have calculated a 1-dimensional integral here.
    Shouldn't you integrate over 3-dimensional space?

    The phrasing of your problem implies that q and -q are equal and opposite.
    It's a bit weird that the problem asks for you to use q and -q, since "just" q should suffice.
  4. Sep 3, 2011 #3
    Yeah that seems reasonable, from Gauss' law.

    I integrated the phi and theta parts in the background, sorry I wasn't more explicit, the constants out front should already reflect those integrals being done.

    yeah, I don't know how to interpret the question about the q and -q, that's why I brought it here lol.
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