An approximate solution of the paraxial ray equation for cylindrical dense electron beams was obtained by means of the Taylor expansion. The method was applied to the problems such as space charge divergence of a cylindrical beam in field-free space

design of transition region for Brillouin flow as well as determination of focal length of a thick magnetic lens. Approximate expressions were obtained in each case. Comparison with experimental results showed that this method is accurate enough to meet the general engineering design requirements.