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  $V_0=[{\left(V_{wholesphere}\right)}_O-{\left(V_{cavity}\right)}_O]$
$$\begin{gathered} \Rightarrow V_0=\frac{3}{8\pi {\epsilon }_0}\frac{Q}{R}-\frac{1}{4\pi {\epsilon }_0}\frac{\left(\frac{Q}{R}\right)}{\frac{R}{2}}\Rightarrow V_0=\frac{5}{16\pi {\epsilon }_0}\frac{Q}{R} \\ \Rightarrow V_0=\frac{5}{16\pi {\epsilon }_0}\times \left(\frac{4}{3}\pi R^3\rho \right)\Rightarrow V_0=\frac{5}{12}\frac{R^2\rho }{{\epsilon }_0} \end{gathered}$$