In electron beam devices the thermal velocities of the electrons leaving the cathode limit the current density which can be attained with given cathode current density, cathode temperature, potential, and angular spread of arriving electrons. By using Liouville's theorem, limiting expressions are obtained which are valid for any point focus or line focus concentrating system using steady electric and magnetic fields. Current densities in perfectly focused beams are also evaluated. Curves are presented which relate the nearness of approach to the maximum attainable current density with the fraction of cathode current necessarily lost in the concentrating system. Use of the expressions obtained is illustrated by applying them to cathode ray tubes and electron multipliers.