A charge $Q$ is distributed over two concentric conducting thin spherical shells radii $r$ and $R(R>r)$. If the surface charge densities on the two shells are equal, the electric potential at the common centre is :
Correct Option: , 4
(4) Let $\sigma$ be the surface charge density of the shells.
Charge on the inner shell, $Q_{1}=\sigma 4 \pi r^{2}$
Charge on the outer shell, $Q_{2}=\sigma 4 \pi R^{2}$
$\therefore$ Total charge, $Q=\sigma 4 \pi\left(r^{2}+R^{2}\right)$
$\Rightarrow \sigma=\frac{Q}{4 \pi\left(r^{2}+R^{2}\right)}$
Potential at the common centre,
$V_{C}=\frac{K Q_{1}}{r}+\frac{K Q_{2}}{R}$ $\left(\right.$ where $\left.K=\frac{1}{4 \pi \varepsilon_{0}}\right)$
$=\frac{K \sigma 4 \pi r^{2}}{r}+\frac{K \sigma 4 \pi R^{2}}{R}=K \sigma 4 \pi(r+R)$
$=\frac{K Q 4 \pi(r+R)}{4 \pi\left(r^{2}+R^{2}\right)}=\frac{1}{4 \pi \varepsilon_{0}} \frac{(r+R) Q}{\left(r^{2}+R^{2}\right)}$
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