The effects of space charge in long, magnetically focused electron beams directed parallel to positively charged sheath electrodes are determined from a simple analysis. The main effects of space charge are: (a) to introduce departure from the potential distribution of the electrostatic case; (b) to set an upper limit for the beam current; and (c) to introduce instabilities and hysteresis phenomena in the behavior of the tube. It is shown that in a long beam the longitudinal potential gradient is negligible throughout the major portion of the beam length so that the upper limit for beam current is independent of beam length and of potentials of end electrodes, but depends only upon the relative transverse dimensions of the beam and of the surrounding sheath electrode and upon the sheath-electrode potential. The analysis gives expressions for maximum beam current, for space-potential distribution and for the minimum value of the focusing magnetic field, in terms of the beam and sheath-electrode dimensions and the sheath-electrode potential. The use of multicellular sheath electrodes to minimize the effects of space charge is discussed. Thin "sheet" beams and "thin-walled" tubular beams are treated separately by a simplified "capacitance" method and expressions for maximum current, space potential, and for minimum-focusing magnetic field are derived. The effect of space-charge interaction between multiple beams inside a common sheath electrode is considered and a method of control of current in one beam by the current in the adjacent beam is analyzed and also illustrated by experimental data.