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Question:

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 in 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 best describes the subsequent variation of the magnitude of current density j(t) at any point in the material?

Solution:

Suppose linear charge density is λ₀ initially and λ at any time instant. Electric field at distance r is E = λ/2πεr. J = σE = σλ/2πεr. dq/dt = J(A) = -Jσ2πrl. dλ/dt = -λ/2πεrσ2πrl. λ = λ₀e⁻σt/ε. Thus, J = J₀e⁻σt/ε. This is exponentially decreasing curve with value 0 at t=∞. Answer is option D.