Dispersion equation of electron cyclotron masers

In non-self-consistent theories of the electron cyclotron maser, the derivation of the dispersion equation of the traveling wave in the device has two approaches: one is based on the wave equation of the transverse components of the electric fields, without the effect of the gradient of the charge density being taken into account; the other is based on the wave equation of the longitudinal component of the magnetic fields. In this paper, we rigorously show within the Vlasov frame that these two approaches are equivalent and the contribution of the gradient of the perturbed charge density to the dispersion equation is trivial.