A point charge $q$ is located at a distance $r$ from the centre $O$ of an uncharged conducting spher

Question

A point charge $q$ is located at a distance $r$ from the centre $O$ of an uncharged conducting spherical layer whose inside and outside radii are equal to $R_1$ and $R_2$ respectively. Find the electric potential at the point $O$ if $r < R_1$.

Professor PC Thomas Classes · Answer

The potential at the center $O$ is the scalar sum of the potentials due to the point charge $q$, the induced charge on the inner surface of the layer, and the induced charge on the outer surface. The point charge $q$ creates a potential $V_q = \frac{q}{4\pi\epsilon_0 r}$ at $O$. Due to electrostatic induction, a total charge $-q$ is distributed on the inner surface (radius $R_1$) and a total charge $+q$ on the outer surface (radius $R_2$). Since all points on the inner surface are at a distance $R_1$ from $O$, the potential due to the inner induced charge at $O$ is $V_1 = -\frac{q}{4\pi\epsilon_0 R_1}$, regardless of whether the charge distribution is uniform (it is non-uniform due to the off-center point charge). Similarly, the potential due to the outer induced charge is $V_2 = +\frac{q}{4\pi\epsilon_0 R_2}$ (this distribution is uniform because the electric field is zero within the bulk of the conductor, isolating the outer surface from the asymmetrical field of the point charge). The total potential is $V = V_q + V_1 + V_2 = \frac{q}{4\pi\epsilon_0} \left( \frac{1}{r} - \frac{1}{R_1} + \frac{1}{R_2} \right)$.