Q 12-02-167JEE MainJEE Main 2019 (10 Jan, Shift 1)Medium
A charge $Q$ is distributed over three concentric spherical shells of radii $a, b, c\ (a < b < c)$ such that their surface charge densities are equal to one another.
The total potential at a point at distance $r$ from their common centre, where $r < a$, would be:
Answer: (D) $\dfrac{Q(a+b+c)}{4\pi\epsilon_0(a^2+b^2+c^2)}$
Equal surface densities mean each shell's charge is proportional to its area: $q_a = \dfrac{Qa^2}{a^2+b^2+c^2}$, and similarly for $b$ and $c$.
Inside all shells each contributes its surface potential $\dfrac{q}{4\pi\epsilon_0 R}$:
$$V = \frac{1}{4\pi\epsilon_0}\cdot\frac{Q(a + b + c)}{a^2+b^2+c^2}$$
Solution by Sreeraj P, M.Sc Physics