The response of the edge plasma to an applied voltage is analysed. When the anomalous transport is ambipolar, the radial current is determined by neoclassical fluxes. An equation is given for the neoclassical radial current which is valid across the transition between the plateau and the collisional viscous regimes. The current at first increases with the applied field, E, and then decreases due either to the resonant velocity band being pushed to the tail of the ion velocity distribution or to a decrease in viscous damping. Thus the same current can be driven by two different electric fields, E1 or E2. This allows an equilibrium field profile, where E changes abruptly from E1 to E2 at some intermediate radius, while maintaining current continuity. It is shown unambiguously that a continuous electric field profile becomes unstable when the applied voltage exceeds a critical value, the instability being driven by negative incremental resistivity. The profile then bifurcates to one with discontinuous electric field. Qualitatively, the predicted behaviour agrees well with that observed in tokamaks where an H-mode has been induced using a biased electrode. Quantitative comparisons are made with TEXTOR results, and the predictions found to be in qualitative agreement with measurement. The implications for spontaneous L-H transitions are discussed.