We consider a collisionless plasma with E (the electric field) parallel to the y-axis of a cartesian frame of reference and B (the magnetic induction) parallel to the z- axis. When there is a gradient, orthogonal to B, of the quantity &egr;E2, where &egr;=1+4&pgr; &rgr;m c2/B2, where &rgr;m is the plasma mass density and c the speed of light, there is driven a (dielectric) current ( j) given in the following two cases, by j=[∇(&egr;E2)>c/B. (i) When ∇(&egr;E2) is parallel to E, j is in the direction of E¿B, and (ii) when ∇(&egr;E2) is in the direction E¿B, j is in the direction of E. In the former cases it follows that, in a steady state, p⊥+B2/8&pgr;-&egr;E2/8&pgr;= constant along a line in the y-direction, and in the latter case p⊥ +B2/8&pgr;+&egr;E2/8&pgr;=constant on a line in the x-direction where p⊥ is the pressure perpendicular to B. The currents related to &egr;E2 are geophysically significant in the region of interaction of the solar wind and geomagnetic field in the low latitude boundary layer and geomagnetic tail. The last written equation enables a realistic estimate of the croos-tail electric field to be made.