An infinite line charge of uniform electric charge density `lambda` 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 `epsilon` 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 density j(t) at any point in the material ?

Options

A.
B.
C.
D.

Explanation

Suppose charger per unit length at any instant is `lamda`. Initial value of `lamda_0` is suppose Electric field at a distance at any instant is
`E=lamda/(2pi e r) , J=sigma E= sigma=lamda/(2pie r)`
`(d lamda l)/(dt)=- lamda/(2pi e r) times sigma 2 pi r l (q =lamda l)`
`int_lamda^lamda (dlamda)/lamda=-sigma/e int_a^t dt, lamda=lamda_0 e^(-a/e t)`
`J=sigma/(2pie r) lamda=(sigma lamda_0)/(2pi e) e^(- sigma/e t) =J_0 e^(- sigma/e t)`
Here, `J_0=(sigma lamda_0)/(2pi e r) therefore J(t)` decreases exponentially as shown in figure below.