Cold-Plasma Dielectric Permittivity

(461) | |||

(462) |

Substitution of plane wave solutions of the type (450) into the above equations yields

Let

be the

(469) |

The parallel component of the above equation is readily solved to give

(471) | |||

(472) |

Here, are a set of mutually orthogonal, right-handed unit vectors. Note that

(473) |

where ,

The conductivity tensor is *diagonal* in the basis
. Its elements are given by the coefficients of and
in Eqs. (474) and (470), respectively. Thus,
the dielectric permittivity (457) takes the form

(475) |

Here, and represent the permittivities for right- and left-handed circularly polarized waves, respectively. The permittivity parallel to the magnetic field, , is identical to that of an unmagnetized plasma.

In fact, the above expressions are only approximate, because the small
mass-ratio ordering has already been folded into the
cold-plasma equations. The exact expressions, which are most easily
obtained by solving the individual charged particle equations
of motion, and then summing to obtain the
fluid response, are:

Equations (476)-(478) and (479)-(481) are equivalent in the limit . Note that Eqs. (479)-(481) generalize in a fairly obvious manner in plasmas consisting of more than two particle species.

In order to obtain the actual dielectric permittivity, it is necessary
to transform back to the Cartesian basis
.
Let
, for ease of notation. It follows that
the components of an arbitrary vector in the Cartesian basis are
related to the components in the ``circular'' basis via

(482) |

(483) |

(484) |

where

(486) |

(487) |