Q 12-02-155JEE MainJEE Main 2020 (2 Sep, Shift 2)Medium
A charge $Q$ is distributed over two concentric conducting thin spherical shells of radii $r$ and $R$ $(R > r)$. If the surface charge densities on the two shells are equal, the electric potential at the common centre is:
Answer: (D) $\dfrac{1}{4\pi\varepsilon_0}\dfrac{(R+r)}{(R^{2}+r^{2})}Q$
With equal surface density $\sigma$: $q_1 = 4\pi r^{2}\sigma$, $q_2 = 4\pi R^{2}\sigma$, so $\sigma = \dfrac{Q}{4\pi(r^{2}+R^{2})}$.
Potential at the centre:
$$V = \frac{1}{4\pi\varepsilon_0}\left(\frac{q_1}{r} + \frac{q_2}{R}\right) = \frac{1}{4\pi\varepsilon_0}\,4\pi\sigma(r+R) = \frac{1}{4\pi\varepsilon_0}\frac{(R+r)Q}{R^{2}+r^{2}}$$
Solution by Sreeraj P, M.Sc Physics