Question

The cross-sectional area and length of cylinder conductor are $A$ and $l$ respectively. The specific conductivity varies as $\sigma \left(x\right)={\sigma }_{0}l\sqrt{x}$ where $x$ is the distance along the axis of the cylinder from one of its ends.
(i) Compute the resistance of the system along the cylindrical axis.
(ii) What is the electric field at each point in the cylinder in the above case?

Answer 1

$dR=f\left(x\right)\frac{dx}{A}$

$=\frac{1}{\sigma \left(x\right)}\frac{dx}{A}=\frac{x}{{\sigma }_0}\frac{dx}{lA}$

$R=\int dR=\frac{1}{{\sigma }_{0}lA}\underset{0}{\overset{l}{\int }}xdx=\frac{l}{2{\sigma }_{0}A}$

$\therefore$ Conductor is in steady state,

$i=\text{constant}=\frac{v_0}{R}=\frac{2v_0{\sigma }_{0}A}{l}$

Using ohm's law,

$j=\sigma \left(x\right)E\left(x\right)$

$j=\frac{i}{A}=\text{const}$

$E\left(x\right)=\frac{2v_{0}x}{l^2}$