An infinite line charge of uniform electric charge density λ 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 ε and electrical conductivity σ. 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?
(A)(see structure)
(B)(see structure)
(C)(see structure)✓ Correct
(D)(see structure)
Answer: (C)
Solution
The electric field at distance r from the line charge is E=2πεrλ.
The current density is j=σE=2πεrλσ.
As current flows radially outward, charge leaks from the line, so λ decreases with time. The rate of charge loss is:
dtdλ=−ελσ
Solving this differential equation:
λ(t)=λ0e−(σ/ε)t
Therefore:
j(t)=2πεrλ0σe−(σ/ε)t
This is an exponential decay starting from an initial value j0=2πεrλ0σ, which corresponds to graph (C).
JEE Advanced · 2016 · 22 May 2016 · Paper 1 · Single correct option
