Ionic resistance of a conical pore ($\mathcal{R}=\rho L/(\pi ab)$)

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.

For a point-like object with constant electrophoretic mobility, deterministic traversal is the integral of inverse local drift speed. In a conical frustum of length $L$ at pore voltage $V$, the one-dimensional field is $E(z)=Vab/[Lr(z{)}^2]$. Integrating gives $t=L^2(b/a+1+a/b)/(3\mu V)$ for positive $\mu V$. For two symmetric frusta of total length $2l$ and the same total pore voltage, replace $L$ by $2l$. The inequality $b/a+a/b\ge 2$ shows that this nonuniform field cannot produce a shorter deterministic point-particle transit than a uniform field at fixed total length and voltage. Reservoir forces, diffusion, adsorption and finite-object conformation are additional physics.