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A conducting sphere $A$ of radius $a$, with charge $Q$, is placed concentrically inside a conducting shell $B$ of radius $b$. $B$ is earthed. $C$ is the common centre of the $A$ and $B$.
The field is a distance $r$ from $C$, where $a \leq r \leq b,$ is $\frac{1}{{4\pi \,{\varepsilon _0}}}\,\frac{Q}{{{r^2}}}$
The potential at a distance $r$ from $C$, where $a \leq r \leq b,$ $\frac{1}{{4\pi \,{\varepsilon _0}}}\,Q\,\left( {\frac{1}{r}\, - \,\frac{1}{b}} \right)$
The potential difference between $A$ and $B$ is $\frac{1}{{4\pi \,{\varepsilon _0}}}\,Q\,\left( {\frac{1}{a}\, - \,\frac{1}{b}} \right)$
all of the above
Solution
$a \leq r \leq b, E=k \frac{Q}{r^{2}}$
$V_{A}-V_{B}=\frac{1}{4 \pi \in_{0}}\left[\left\{\frac{Q}{a}+\frac{-Q}{b}\right\}-\left\{\frac{Q}{b}+\frac{-Q}{b}\right\}\right]$
$=\frac{1}{4 \pi \epsilon_{0}} Q\left[\frac{1}{a}-\frac{1}{b}\right]=k Q\left(\frac{1}{a}-\frac{1}{b}\right)$
$V=\frac{1}{4 \pi \in 0}\left[\frac{Q}{r}+\frac{-Q}{b}=k Q\left(\frac{1}{r}-\frac{1}{b}\right)\right.$
