Calculation of the electric field in space plasma near a conducting sphere injecting an electron beam

The problem is considered of determining the electric field induced in the vicinity of a conducting spherical body that is at rest in a collisionless plasma and at the surface of which there is a prescribed sink of negative charge. The problem is solved for the general relativistic case under the assumption that the electron velocity in the neutralizing current is comparable with the speed of light. An integrodifferential equation is derived that describes the radial behavior of the electric field potential in the vicinity of the injector. A simplified method for determining the potential in the perturbed region is developed. The method implies that the problems of the potentials in a space charge region of radius R* (with a prescribed boundary potential φ*) adjacent to the body and in the outer region r > R* are solved separately and then the solutions obtained are matched at the boundary between these regions