Disperson relation and the dielectric tensor for magnetized plasmas with inhomogeneous magnetic field

We investigate the dispersion relation for a magnetized plasma with weak magnetic field gradients perpendicular to the ambient magnetic field. An explicit expression for the effective dielectric tensor is derived, incorporating the relevant contributions due to the inhomogeneity, which include corrections to all orders in the small parameter e, where e= [(1/Bo)(dBo/dz)]. It is shown that this effective dielectric tensor satisfies the required symmetry conditions and is the tensor which should be utilized in the dispersion relation, in order to describe correctly wave-particle interactions in media with inhomogeneous magnetic field. The case of high frequency oscillations propagating perpendicularly to the magnetic field in a Maxwellian plasma is considered as an example and the effect of inhomogeneities in the magnetic field upon the absorption coefficient and the optical depth of ordinary mode waves is discussed. A region of negative absorption coefficient is predicted near the electron cyclotron frequency for sufficiently high inhomogeneity. Moreover, it is shown that significant differences may exist between the absorption coefficient evaluated with the present formulation and results from other approaches found in the literature which do not exhibit correct symmetry properties