An infinite line charge of uniform electric charge density i. lies along the axis of an electrically conducting infinite cylindrical shell of radius $R$. At time $t=0$, the space inside the cylinder is filled with a material of permittivity $\varepsilon$ and electrical conductivity $\sigma$. The electrical conduction in the material follows Ohm's law. Which one of the following graphs best describes the subsequent variation of the magnitude of current censity $j(t)$ at any point in the material?
IIT 2016, Advanced
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Suppose linear charge density is $\lambda_0$ initially and $\lambda$ at any time instant.
Electric field at distance $r$ is
$E =\frac{\lambda}{2 \pi \epsilon r }$
$J =\sigma \frac{\lambda}{2 \pi \epsilon r }$
$\frac{ dq }{ dt }= J ( A )=- J \sigma 2 \pi rl$
$\frac{ d \lambda}{ dt }=-\frac{\lambda}{2 \pi \epsilon r } \sigma 2 \pi rl$
$\lambda=\lambda_0 e ^{-\sigma t / \epsilon}$
Thus, $J=J_0 e^{-\sigma t / \epsilon}$
This is exponentially decreasing curve with value o at $t=\infty$
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