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(A) Let the total excess charge on the conducting shell be $q$. Due to the point charge $+Q$ at the center,an induced charge of $-Q$ appears on the in

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$-5Q$

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(A) Let the total excess charge on the conducting shell be $q$. Due to the point charge $+Q$ at the center,an induced charge of $-Q$ appears on the inner surface of the shell. The charge density on the inner surface is $\sigma_{in} = \frac{-Q}{4 \pi a^2}$. Since the total charge on the shell is $q$,the charge on the outer surface is $q_{out} = q - (-Q) = q + Q$. The charge density on the outer surface is $\sigma_{out} = \frac{q + Q}{4 \pi (2a)^2} = \frac{q + Q}{16 \pi a^2}$. For the charge densities to be equal,we set $\sigma_{in} = \sigma_{out}$: $\frac{-Q}{4 \pi a^2} = \frac{q + Q}{16 \pi a^2}$ $-Q = \frac{q + Q}{4}$ $-4Q = q + Q$ $q = -5Q$.

$A$ solid spherical conducting shell has an inner radius $a$ and an outer radius $2a$. At the center of the shell is located a point charge $+Q$. What must the excess charge of the shell be in order for the charge density on the inner and outer surfaces of the shell to be exactly equal?

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