A total amount of positive charge Q is spread into a non-conducting flat circular annulus of inner radius a and outer radius 3a. The charge is distributed, so that the charge density is given  by  $\sigma =\frac{k}{r^3}$, where r is the  distance from the centre of the annulus to any point on it. The  potential at the centre of annulus is $\frac{Q}{n\pi {\epsilon }_{0}a}.$  find n.

$$\begin{gathered} Q=\underset{a}{\overset{3a}{\int }}\frac{k}{r^3}2\pi rdr \Rightarrow Q=2\pi k{\left[\frac{-1}{r}\right]}_a^{3a} \\ \Rightarrow Q=\frac{4\pi k}{3a} \Rightarrow k=\frac{3aQ}{4\pi } \end{gathered}$$

 Potential at the center,

$$\begin{gathered} dV=\left(\frac{1}{4\pi {\epsilon }_0}\right)\frac{dq}{r} \Rightarrow dV=\frac{1}{4\pi {\epsilon }_0}\frac{\sigma 2\pi rdr}{r} \\ \Rightarrow \int dV=\frac{1}{4\pi {\epsilon }_0}\underset{a}{\overset{3a}{\int }}\frac{\left(\frac{k}{r^3}\right)2\pi rdr}{r}\Rightarrow \int dV=\frac{k}{2{\epsilon }_0}{\left[\frac{1}{(-2)r^2}\right]}_a^{3a} \\ \Rightarrow V=\frac{2k}{9{\epsilon }_0{a}^2}=\frac{2\left(\frac{3aQ}{4\pi }\right)}{9{\epsilon }_0{a}^2}\Rightarrow V=\frac{Q}{6\pi {\epsilon }_{0}a} \end{gathered}$$