A solid sphere having uniform charge density $\rho$ and radius R is shown in figure. A spherical cavity of radius $\frac{R}{2}$ is made in it. The potential at point O is $\frac{5R^2\rho }{n{\in }_0}$, then n is ___

Charge of the whole sphere is  $Q=\left(\frac{4}{3}\pi R^3\right)\rho$
Charge in the cavity region is  $q=\frac{4}{3}\pi {\left(\frac{R}{2}\right)}^2\rho =\frac{Q}{8}$
Potential at point O is given as $V_0=[{\left(V_{wholesphere}\right)}_O-{\left(V_{cavity}\right)}_O]$
$$\begin{gathered} \Rightarrow V_O=\frac{3}{8\pi {\epsilon }_0}\frac{Q}{R}-\frac{1}{4\pi {\epsilon }_0}\frac{\left(\frac{Q}{8}\right)}{\frac{R}{2}} \\ \Rightarrow V_O=\frac{5}{16\pi {\epsilon }_0}\frac{Q}{R} \\ \Rightarrow V_O=\frac{5}{16\pi {\epsilon }_{0}R}\times \left(\frac{4}{3}\pi R^3\rho \right) \\ \Rightarrow V_O=\frac{5}{12}\frac{R^2\rho }{{\epsilon }_0} \end{gathered}$$