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PhysicsSingle correct+3 / -1Question 5 of 54

JEE Advanced 2016 · Paper 1, Physics question 5

An infinite line charge of uniform electric charge density λ\lambda lies along the axis of an electrically conducting infinite cylindrical shell of radius RR. At time t=0t=0, the space inside the cylinder is filled with a material of permittivity ε\varepsilon and electrical conductivity σ\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)j(t) at any point in the material?

Choose one option
Answer and worked solution

Official answer: C

Charge relaxation. In an ohmic medium, ∇⋅J⃗=−∂ρ/∂t\nabla\cdot\vec J=-\partial\rho/\partial t with J⃗=σE⃗\vec J=\sigma\vec E and ∇⋅E⃗=ρ/ε\nabla\cdot\vec E=\rho/\varepsilon. These give
∂ρ∂t=−σερ.\frac{\partial\rho}{\partial t}=-\frac\sigma\varepsilon\rho.
The charge on the line leaks outward to the shell as λ(t)=λe−σt/ε\lambda(t)=\lambda e^{-\sigma t/\varepsilon}.

Current density. The field inside follows the enclosed charge, E∝λ(t)E\propto\lambda(t), so
j=σE∝e−σt/ε.j=\sigma E\propto e^{-\sigma t/\varepsilon}.
It is largest at t=0t=0 and decays exponentially to zero.

(C)

Topics:Ohm's law and resistivityElectric flux and Gauss's lawRC circuits
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