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 $t=\infty$