Self‐consistent solutions for the problem of a collisionless plasma streaming across a magnetic field whose dimensions are small compared to(rcerci)1/2show that the electrons can accompany the ions across the field (over distances large compared torce) provided&ohgr;ce < &agr; &ohgr;pe, where&agr; ≤ 1is a function of the electron distribution function (&agr; = 1when the electron thermal velocities are neglected). This appears to explain the threshold,&ohgr;ce ≃ 0.9 &ohgr;pe, observed in recent experiments on plasma penetration across a radial magnetic field.