 Ionic resistance of a conical pore

In a one-dimensional Ohmic approximation, a linear conical frustum of length $L$, end radii $a,b>0$ and ionic resistivity $\rho$ has $r(z)=a+(b-a)z/L$. Conservation of electric current gives $d\mathcal{R}=\rho dz/[\pi r(z{)}^2]$. Integration yields the displayed resistance. Two identical opposed frusta, each of length $l$, have resistance $2\rho l/(\pi ab)$, whether their central junction is the wider or narrower end. Their local field profiles nevertheless differ. The approximation neglects surface conduction and resistance outside the defined pore, and is not an exact three-dimensional field solution for arbitrary wall slope.