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?

Q: 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?

c Suppose linear charge density is $\lambda_0$ initially and $\lambda$ at any time instant. Electric field at distance $r$ is $E =\frac{\lambda}{2 \pi \in r }$ $J =\sigma \frac{\lambda}{2 \pi \in r }$ $\frac{ dq }{ dt }= J ( A )=- J \sigma 2 \pi rl$ $\frac{ d \lambda}{ dt }=-\frac{\lambda}{2 \pi \in r } \sigma 2 \pi rl$ $\lambda=\lambda_0 e ^{-\sigma t / \in}$ Thus, $J=J_0 e^{-\sigma t / \in}$ This is exponentially decreasing curve with value o at $t=\infty$

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IIT 2016, Advanced

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