The electron beam in a cathode-ray tube

The beam in a cathode-ray tube is described by a "paraxial" Liouville equation. This equation takes into account the thermal spread of the electrons as well as the influence of the space charge of the beam. The solution of this equation is given. An ordinary differential equation is derived for the beam radius. The equation has been solved in the field-free region outside the gun. The result is presented in graphs which can be applied to the determination of the spot size on the screen of a cathode-ray tube.