First slide

Electrostatic potential

Question

If on the concentric hollow spheres of radii $r$ and $R (> r)$ the charge $Q$ is distributed such that their surface densities are same then the potential at their common centre is

Moderate

A

$\frac{Q (R^{2} + r^{2})}{4 π \in_{0} (R + r)}$
B

$\frac{Q R}{R + r}$
C

$Z e r o$
D

$\frac{Q (R + r)}{4 π \in_{0} (R^{2} + r^{2})}$

Solution

$L e t s a y c h a r g e q_{1} i s o n i n n e r s p h e r e a n d c h a r g e q_{2} i s o n o u t e r s p h e r e q_{1} + q_{2} = Q a n d \frac{q_{1}}{4 π r^{2}} = \frac{q_{2}}{4 π R^{2}} (g i v e n) q_{1} = \frac{Q r^{2}}{R^{2} + r^{2}} a n d q_{2} = \frac{Q R^{2}}{R^{2} + r^{2}} P o t e n t i a l a t c o m m o n c e n t r e = V = V_{1} + V_{2} = \frac{1}{4 π \in_{0}} [\frac{Q r^{2}}{(R^{2} + r^{2}) r} + \frac{Q R^{2}}{(R^{2} + r^{2}) R}] = \frac{Q (R + r)}{4 π \in_{0} (R^{2} + r^{2})} (∵ P o t e n t i a l i n s i d e s h e l l = p o t e n t i a l o n t h e s h e l l)$

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