Space charge limitation on the focus of electron beams

In cathode ray tubes or other focused beam devices the smallest spot size is obtained by focusing on a plane between the gun and screen. The equations for the electron paths of a homocentric uniform density electron beam under the influence of space charge are differentiated with respect to initial radial velocity. The resultant transcendental equation is solved numerically. A universal curve for the smallest spot size at the target is obtained. At high beam currents this differs significantly from available curves for the beam cross section at the point of zero radial velocity.