The Effective Longitudinal Dielectric Constant for Plasmas in Inhomogeneous Magnetic Fields

We have recently discussed the correct form of the dispersion relation for electrostatic waves in inhomogeneous plasmas, considering for simplicity the particular case in which the magnetic field is homogeneous and other plasma parameters can be inhomogeneous, and deriving a general expression for the dielectric constant, valid for arbitrary direction of propagation [1-3] The situation in which the magnetic field is homogeneous has been chosen for these studies because it features a relatively simple geometry, which has been useful for the discussion of basic features, like the symmetry properties of the effective dielectric tensor. More general situations which also feature inhomogeneity in the magnetic field introduce considerable difficulty to the derivation of the effective dielectric tensor and to the effective dielectric constant, since the inhomogeneity in the magnetic field affects the resonance condition [4,5]. From the derivation presented in Refs. [3] and [2], it is important to remember here that the dispersion relation for electrostatic waves in inhomogeneous plasmas can be written in a general form which is well known from the literature [6], kεl − ik · ( ∇· ε ) = 0 , (1)