A point charge $q$ is placed at a distance $a$ from the center of an uncharged conducting spherical shell of inner radius $R_1$ and outer radius $R_2$, where $a>R_2$. The potential at the center of the shell is

Solution

1. The potential at the center of the shell is given by $V=V_q+V_{\text{shell}}$, where $V_q$ is the potential due to the point charge and $V_{\text{shell}}$ is the potential due to the induced charges.
2. The potential due to the point charge $q$ at distance $a$ from the center is $V_q=\frac{q}{4\pi {ϵ}_{0}a}$.
3. For an uncharged conducting shell, the total induced charge is zero, and the potential at any point inside the shell due to these induced charges is constant and equal to the potential at the surface $R_2$.
4. The potential at the surface $R_2$ due to the point charge is $\frac{q}{4\pi {ϵ}_{0}a}$, and the induced charges redistribute to ensure the shell is an equipotential, resulting in a total potential at the center of $V=\frac{q}{4\pi {ϵ}_0{R}_2}$.
5. Hence the answer is (C).