Question

A cylindrical solid rod of length $L$ and radius $a$ is connected across a source of emf $V$ and negligible internal resistance as shown in the figure. The resistivity of the rod at point $P$ at a distance $x$ from the left end is given by $\rho =bx$ (where $b$ is a positive constant). The electric field at point $P$ is $\left(\frac{nV}{L^2}\right)x$ . The value of $n$ is

Answer 1

Answer: 2

Resistance, $R=\int \frac{\rho dx}{A}=\frac{\underset{0}{\overset{L}{\int }}bxdx}{A}=\frac{b{L}^2}{2A}$
Current, $I=\frac{V}{R}=\frac{2VA}{b{L}^2}$
Current density, $j=\frac{I}{A}=\frac{2V}{b{L}^2}$
$\Rightarrow j=\sigma E$
Electric field, $E=\rho j=\frac{\left(bx\right)2V}{b{l}^2}=\frac{2V}{L^2}x$