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Potential energy by concentric shells

  1. May 3, 2017 #1
    1. The problem statement, all variables and given/known data

    Concentric spherical shell of radius ##a## and ##b##, with ##b > a## carry charge ##Q## and ##-Q## respectively, each charge uniformly distributed. Find the energy stored in the E field of this system.

    2. Relevant equations


    3. The attempt at a solution

    Field between ##a## and ##b## is ##\displaystyle E = {Q \over R^2}## for ## a <R < b##.

    Field outside b should be zero as ##E_{t} = \dfrac{Q}{R^2} - \dfrac{Q}{R^2} = 0##.

    So I just need to calculate energy inside the b and outside a.

    $$\begin{align} U &= {1\over 8\pi} \int_\text{region} E^2 dv \\ &= {1\over 8\pi} \iiint_\text{region} E^2 R^2 \sin \theta dR d\theta d\phi \\ &= {Q^2 \over 8\pi} \int^{2\pi}_{0}\int^{\pi/2}_{-\pi/2}\int^{b}_{a} {1\over R^2} dRd\theta d\phi \\ &= {Q^2 \pi \over 4 }\left(\frac1a - \frac1b\right)\end{align}$$.

    Is this correct ?
     
  2. jcsd
  3. May 3, 2017 #2

    ehild

    User Avatar
    Homework Helper
    Gold Member

    Yes, it is correct.
     
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