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(A) The electrostatic energy $U$ of a system of charges is given by $U = \int \frac{1}{2} \epsilon_0 E^2 dV$.For a conducting sphere of radius $a$ wit

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$\frac{5}{12} \frac{kQ^2}{a}$

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(A) The electrostatic energy $U$ of a system of charges is given by $U = \int \frac{1}{2} \epsilon_0 E^2 dV$.For a conducting sphere of radius $a$ with charge $Q$,the electric field $E$ is:$1$. $E = 0$ for $r $2$. $E = \frac{kQ}{r^2}$ for $a $3$. $E = 0$ for $2a $4$. $E = \frac{kQ}{r^2}$ for $r > 3a$ (since the shell is neutral,the total charge enclosed is $Q$).The energy stored in the region $a The energy stored in the region $r > 3a$ is $U_2 = \int_{3a}^{\infty} \frac{1}{2} \epsilon_0 (\frac{kQ}{r^2})^2 (4 \pi r^2 dr) = \frac{kQ^2}{2} \int_{3a}^{\infty} \frac{1}{r^2} dr = \frac{kQ^2}{2} [-\frac{1}{r}]_{3a}^{\infty} = \frac{kQ^2}{6a}$.Total energy $U = U_1 + U_2 = \frac{kQ^2}{4a} + \frac{kQ^2}{6a} = \frac{3kQ^2 + 2kQ^2}{12a} = \frac{5kQ^2}{12a}$.

$A$ conducting sphere of radius $a$ has charge $Q$ on it. It is enclosed by a neutral conducting concentric spherical shell having inner radius $2a$ and outer radius $3a$. Find the electrostatic energy of the system.

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