Potential energy by concentric shells

  • #1
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Homework Statement



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.

Homework Equations




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 ?
 

Answers and Replies

  • #2
ehild
Homework Helper
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Homework Statement



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.

Homework Equations




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 ?
Yes, it is correct.
 
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