$(a)\;\large\frac{\rho R^2}{6 \in_{0}}\qquad(b)\;\large\frac{\rho R^2}{4\in_{0}} \qquad(c)\;\large\frac{\rho R^2}{3 \in_{0}}\qquad(d)\;\large\frac{\rho R^2}{2 \in_{0}}$

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Answer : (a) $\large\frac{\rho R^2}{6 \in_{0}}$

Explanation :

Potential at centre = $\; V_{C}|_{r=0}=\large\frac{kQ}{2R^3}\;(3R^2-r^2)|_{r=0}$

$V_{C}=\large\frac{3kQ}{2R}$

$V_{surface}=\large\frac{kQ}{R}$

$\bigtriangleup V=\large\frac{kQ}{2R}=\large\frac{k}{2R}\;\rho\large\frac{4}{3} \pi R^3$

$\bigtriangleup V= \large\frac{\rho R^2}{6 \in_{0}}\;.$

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