A discussion is presented of the limitation of current density by random irregularities of the cathode surface in an electron beam which is focused by a static electromagnetic field. The Langmuir‐Pierce law for the maximum attainable current density is profoundly altered by the effects of cathode roughness. It is found that the minimum beam diameter attainable in typical cathode‐ray tubes is unaffected by change of cathode temperature or proportional change in the potentials of all electrodes.