Abstract
In the non-self-consistent theories of a free-electron maser thederivation of the dispersion equation of the travelling wave in the device hastwo approaches: one is based on the wave equation of the transverse componentsof the electrical fields, where the effect of the gradient of the chargedensity is not taken into account; the other is based on the wave equation ofthe longitudinal component of the magnetic fields. In this paper we rigorouslyshow within the Vlasov frame that these two approaches are equivalent and thecontribution of the gradient of the perturbed charge density to the dispersionequation is trivial for both the transverse electric mode and thetransverse magnetic mode.
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