Consider two metal shells. The smaller of the two (radius $a$) in encapsulated by the larger of the two (radius $b$). Between them is a conducting material with conductivity $\sigma$. The inner sphere is held at a potential $V$ and the outer one is grounded.

Starting from Laplace’s equation, find the current, $I$, that flows between the spheres. What does the divergence of the current density, $\mathbf{J}$, say about conservation of current in this situtation?