Question

A charge $Q$ is uniformly distributed over the surface of two conducting concentric spheres of radii $R$ and $r \, \left(R > r\right).$ Then potential at the common centre of these spheres

• A. $\frac{k Q \, \left(\right. R + r \left.\right)}{R r}$
• B. $\frac{k Q \, \left(\right. R + r \left.\right)}{\left(\right. R^{2} + r^{2} \left.\right)}$
• C. $\frac{k Q}{\sqrt{R^{2} + r^{2}}}$
• D. $kQ\left(\frac{1}{R} - \frac{1}{r}\right)$

Answer 1

Answer: (B)

Let the surface charge density is $\sigma .$
And, $q_{A}$ and $q_{B}$ are charges on shell $A$ and $B$

$\therefore \, \, \, Q=q_{A}+q_{B}$
$or \, Q=4\pi r^{2}\sigma +4\pi R^{2}\sigma $
$\therefore \, \, \, \sigma =\frac{Q}{4 \pi \, \left(r^{2} + R^{2}\right)} \, $
$\therefore \, \, \, V_{C}=V_{A}+V_{B}$
$\therefore \, \, \, V_{C}=\frac{q_{A}}{4 \pi ϵ_{0} r}+\frac{q_{B}}{4 \pi ϵ_{0} R}$
$=\frac{1}{4 \pi ϵ_{0}} \, \left[\right. \frac{q_{A}}{r} + \frac{q_{B}}{R} \left]\right.=k \, \left(\right. \frac{4 \pi r^{2} \sigma }{r} + \frac{4 \pi R^{2} \sigma }{R} \left.\right)$
$=k \, \left(4 \pi r \sigma + 4 \pi R \sigma \right)$
$=4\pi k \, \left[\frac{Q r}{4 \pi \, \left(R^{2} + r^{2}\right)} + \frac{Q R}{4 \pi \, \left(r^{2} + R^{2}\right)}\right]$
$=\frac{k Q \, \left(\right. R + r \left.\right)}{\left(\right. R^{2} + r^{2} \left.\right)} \, \, \, \left(H e r e , \, \, \, k = \frac{1}{4 \pi ϵ_{0}}\right)$