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An infinitely long line charge of uniform linear charge density lambda runs along the axis of a long conducting cylindrical shell of radius R. At t = 0 the region inside the cylinder is suddenly filled with a homogeneous material of permittivity epsilon and electrical conductivity sigma, and the conduction obeys Ohm's law. How does the magnitude of the conduction current density j(t) at a fixed point inside the material vary with time t thereafter?

1. It decays exponentially with time as j(t) = j0 * e^(-(sigma/epsilon) t)
2. It stays constant in time
3. It grows exponentially with time as j(t) = j0 * e^(+(sigma/epsilon) t)
4. It decreases linearly with time and reaches zero at a finite instant

Correct answer: It decays exponentially with time as j(t) = j0 * e^(-(sigma/epsilon) t)

Solution

Inside a homogeneous ohmic medium the free charge density obeys a relaxation equation. Continuity gives d(rho)/dt + div(j) = 0, with j = sigma*E and div(E) = rho/epsilon, so d(rho)/dt = -(sigma/epsilon)*rho, giving rho(t) = rho0 * e^(-(sigma/epsilon)t). Since E is proportional to the enclosed line charge and j = sigma*E, the current density at any fixed point falls off with the same exponential factor. The decay constant is sigma/epsilon, the reciprocal of the charge-relaxation time tau = epsilon/sigma.

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