Q. 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
A) $\frac{Q(R^{2} + r^{2})}{4\pi\varepsilon_{0}(R + r)}$
B) $\frac{QR}{R + r}$
C) Zero
D) $\frac{Q(R + r)}{4\pi\varepsilon_{0}(R^{2} + r^{2})}$
Explanation
(d) $q_{1} + q_{2} = Q$ and $\frac{q_{1}}{4\pi r^{2}} = \frac{q_{2}}{4\pi R^{2}}$ [Given]
$q_{1} = \frac{Qr^{2}}{R^{2} + r^{2}}$ and $q_{2} = \frac{QR^{2}}{R^{2} + r^{2}}$
Potential at common centre
$\frac{1}{4\pi\varepsilon_{0}}\left( \frac{Qr^{2}}{\left( R^{2} + r^{2} \right)r} + \frac{QR^{2}}{\left( R^{2} + r^{2} \right)R} \right) = \frac{Q(R + r)}{4\pi\varepsilon_{0}(R^{2} + r^{2})}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.