Paper 2, Section I, D

Electromagnetism
Part IB, 2013

Use Maxwell's equations to obtain the equation of continuity

ρt+J=0\frac{\partial \rho}{\partial t}+\nabla \cdot \mathbf{J}=0

Show that, for a body made from material of uniform conductivity σ\sigma, the charge density at any fixed internal point decays exponentially in time. If the body is finite and isolated, explain how this result can be consistent with overall charge conservation.