Question

An infinite line charge of uniform electric charge density $\lambda$ lies along the axis electrically conducting infinite cylindrical shell of radius $R.$ At time $t=0,$ the space in the cylinder is filled with a material of permittivity $\epsilon$ and electrical conductivity $\sigma .$ The electrical conduction in the material follows Ohm's law. Which one of the following best describes the subsequent variation of the magnitude of current density $j\left(t\right)$ at any point in the material?

• A.
• B.
• C.
• D.

Answer 1

The correct option is D

Suppose linear charge density is ${\lambda }_0$ initially and $\lambda$ at any time instant.

Electric field at distance r is

$E=\frac{\lambda }{2\pi {ϵ}_{0}r}$
current density of the function $J=\sigma \frac{\lambda }{2\pi ϵr}$
rate of flow of charge is given by,
$\frac{dq}{dt}=J\left(A\right)=-J2\pi rl$ (Here, - sign shows reduction rate of charge)
$\frac{dq}{dt}=2\pi rl\left[\frac{-\sigma \lambda }{2\pi {ϵ}_{0}r}\right]\frac{dq}{dt}=\frac{-\sigma l\lambda }{{ϵ}_0}$

$\frac{ld\lambda }{dt}=\frac{-\sigma l\lambda }{{ϵ}_0}$
$\Rightarrow \frac{d\lambda }{\lambda }=\frac{-\sigma }{{ϵ}_0}dt$
On integrting the above function,
${\int }_{{\lambda }_0}^{\lambda }\frac{d\lambda }{\lambda }={\int }_0^t\frac{-\sigma }{{ϵ}_0}dt$
$\Rightarrow \lambda ={\lambda }^{}{e}^{-\sigma t/ϵ}$
this function shows the charge density of a body.

Here, charge density is continuously decreasing so current density will also decress simultaneously.

hence, option D is correct.