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.

Answer

A

Explanation

Let `lamda(t)` represent the linear density of charge as a function of timee on the inner wire. `lamda_(0)` be the charge density at `t=0` on the inner wire. Let `beta(t)` represent the linear density on the outer cylinder.
`lamda(t)+beta(t)=lamda_(0)` by conservation of charge. the electric field at P as a function of time `t` the `E(t)` then `E(r,t)=(lamda(t))/(2piepsi_(0).epsir)`
`thereforej(t)=sigma(r,t)=((sigma)/(2piepsi_(0)epsi))(lamda(t))/(r)`
`thereforej(t)proplamda(t)`
clearly `lamda(t)to0` as `ttoinfty` as all the charge will eventually migrate to the outer surface.
Let the cross-sectional radius of the inner wire be
`r_(0)`
Potential difference between the inner and outer cylinder
`(lamda(t))/(2piepsi_(0)epsi)int_(r_(0))^(R)(dr)/(r)=(lamda(t))/(2piepsi_(0)epsi)log((R)/(r_(0)))`
Now consider the cylindrical shell of thickness `dr` let the length of the cylinder be unit elementary resistnace
`=(lamda(t))/(2piepsi_(0)epsi)int_(r_(0))^(R)(dr)/(r)=(lamda(t))/(2piepsi_(0)epsi)log((R)/(r_(0)))`
total resistance `=(log((R)/(r_(0)))/(2pisigma)` [per unit length]
`-(lamda(t))/(2piepsi_(0)epsi)log((R)/(r_(0)))=(log(R)/(r_(0)))/(2pisigma)lamda(t)`
`implieslamda(t)=-clamda(t)`
Which is an exponentially