Abstract:
The longitudinal and transverse dispersion relations using four-dimensional
covariant Maxwell's equations and three dimensional Maxwell's field
equations are derived. Their equivalence is found be in agreement in both
the formulations.
Employing ambient weak magnetic fields, collective modes of electron
plasma are derived. Two different approaches are used.
(i) In first case, a covariant form of Maxwell's equations is used along
with Vlasov model and 4-dimentional polarization tensor is derived. Using
temporal axial gauge A0=0 and employing ultra-relativistic Maxwellian
distribution function, dispersion relations for both parallel and perpendicular
propagations are derived. Some limiting cases are also discussed.
(ii) In second case, Maxwell's field equations are used along with Vlasov
model and 3-dimentional polarization tensor is derived. It is observed that
the response of polarization tensor obtained by using temporal gauge A0=0
in the covariant form of magnetized dielectric tensor and the polarization
tensor obtained from the Maxwell's fields equations is same. The problem is
extended for isotropic relativistic Maxwellian velocity distribution function
and generalized relativistic dispersion relations for weakly magnetized
electron plasma are obtained. The integrals (called Meijer G functions)
which arise due to relativistic effects are studied in various limits and
dispersion relations are derived for the non-relativistic, weakly-relativistic,
strongly-relativistic and ultra-relativistic Maxwellian velocity distributions.
It is generally observed that the propagation domains of the modes are
enlarged as one proceeds from the non-relativistic to the highly relativistic
regime. Resultantly, due to the relativistic effects, the Whistler mode is
suppressed in the R-wave, the non-propagation band of X-mode is reduced
and the X-mode itself approaches the O-mode. Further, the results derived in
the ultra-relativistic and non-relativistic limits found to be in agreement with
the earlier calculations.
