Space charge effect in an accelerated beam

It is usually assumed that the space charge effects in relativistic beams scale with the energy of the beam as γ^-2, where γ is the relativistic factor. We show that for a beam accelerated in the longitudinal direction there is an additional space charge effect in free space that scales as E/γ, where E is the accelerating field. This field has the same origin as the “electromagnetic mass of the electron” discussed in textbooks on electrodynamics. It keeps the balance between the kinetic energy of the beam and the energy of the electromagnetic field of the beam. We then consider the effect of this field on a beam generated in an rf gun and calculate the energy spread produced by this field in the beam.