Poisson's equation in cylindrical symmetry is solved numerically assuming that at each electrode there is an unlimited supply of charges of zero energy; positive charges at the positive electrode and negative charges at the negative electrode. A solution is obtained corresponding to zero electric field at each electrode surface; this defines the maximum possible current flow of each charge polarity. The numerical technique is based on a perturbation of the solution appropriate to parallel-plane geometry and is carried out for a ratio of electrode radii ranging from 1 to 400. In the case of the external emitter configuration, the increase in charged-particle current afforded by bipolar flow in comparison to unipolar flow can be significant. Application to the electron-beam pumping of vacuum ultraviolet excimer lasers is briefly considered.