On the Electric Field Screening by Electron-Positron Pairs in the Pulsar   Magnetosphere II
  We study on the self-consistency of the pulsar polar cap model, i.e., the problem of whether the field-aligned electric field is screened by electron-positron pairs that are injected beyond the pair production front. We solve the one-dimensional Poisson equation along a magnetic field line, both analytically and numerically, for a given current density incorporating effects of returning positrons, and we obtain the conditions for the electric-field screening. The formula which we obtained gives the screening distance and the return flux for given primary current density, field geometry and pair creation rate at the pair production front.   If the geometrical screening is not possible, for instance, on field lines with a super-Goldreich-Julian current, then the electric field at the pair production front is constrained to be fairly small in comparison with values expected typically by the conventional polar cap models. This is because (1) positive space charge by pair polarization is limited to a small value, and (2) returning of positrons leave pair electrons behind. A previous belief that pair creation with a pair density higher than the Goldreich-Julian density immediately screens out the electric field is unjustified at least for for a super Goldreich-Julian current density. We suggest some possibilities to resolve this difficulty. 
S. Shibata, J. Miyazaki, F. Takahara