Physical basis of general dispersion equation of free-electron masers

In the non-self-consistent theories of a free-electron maser the derivation of the dispersion equation of the travelling wave in the device has two ap proaches: one is based on the wave equation of the transverse components of the electrical fields, where the effect of the gradient of the charge dens ity is not taken into account; the other is based on the wave equation of t he longitudinal component of the magnetic fields. In this paper we rigorous ly show within the Vlasov frame that these two approaches are equivalent an d the contribution of the gradient of the perturbed charge density to the d ispersion equation is trivial for both the transverse electric mode and the transverse magnetic mode.