Potential energy by concentric shells

[Buffu]

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 ?

 

[ehild]

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.