Charge transport in one dimension: Dissipative and non-dissipative space-charge-limited currents

We consider charge transport in a pore where the dielectric constant inside the pore is much greater than that in the surrounding material, so that the flux of the electric fields due to the charges is almost entirely confined to the pore. We develop exact solutions for the one component case for the Dirichlet and Neumann boundary conditions using a Hopf-Cole transformation, Fourier series, and periodic solutions of the Burgers equation. These are compared with a simpler model in which the scaled diffusivity is zero so that all charge motion is driven by the electric field. In this non-dissipative case, recourse to an admissibility condition is used to obtain the physically relevant weak solution of a Riemann problem concerning the electric field. It is shown that the admissibility condition is Poynting's theorem.