A solid cylinder of length l and cross- sectional area A is made of a material whose resistivity depends on the distance r from the axis of the cylinder as  $\rho =\frac{k}{r^2}$  where k is constant. Then the resistance of the cylinder is  $\frac{x\pi kl}{A^2}$. Find x?

Consider a cylindrical shell of radius $r$  and thickness  $dr.$
Its resistance is  $dR=\frac{\rho l}{dA}=\frac{k}{r^2}\frac{1}{2\pi rdr}=\frac{kl}{2\pi r^{3}dr}$
Since, all such shells are in parallel, we have
 $\begin{array}{l}\frac{1}{R}=\int \frac{1}{dR}=\frac{2\pi }{kl}\underset{0}{\overset{a}{\int }}{r}^{3}dr=\frac{\pi a^4}{2kl}\\ Now,A=\pi a^2{a}^4=A^2/{\pi }^2\\ \therefore R=\frac{2kl}{\pi a^4}=\frac{2kl}{\pi (A^2/{\pi }^2)}=\frac{2\pi kl}{A^2}\end{array}$
(b)  $E=\frac{V}{l}=\frac{ir}{l}=\frac{I}{l}\times \frac{2\pi kl}{A^2}=\frac{2\pi kl}{A^2}$