  We consider charge transport in nanopores where the dielectric constant
inside the nanopore is much greater than in the surrounding material, so that
the flux of the electric fields due to the charges is almost entirely confined
to the nanopore. That means that we may model the electric fields due to charge
densities in the nanopore in terms of average properties across the nanopore as
solutions of one dimensional Poisson equations. We develop basic equations for
an M component system using equations of continuity to relate concentrations to
currents, and flux equations relating currents to concentration gradients and
conductivities. We then derive simplified scaled versions of the equations. We
develop exact solutions for the one component case in a variety of 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 utilised 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.
