Cold-Plasma Dielectric Permittivity

(5.12) | ||

(5.13) |

where is the equilibrium electron number density. Substitution of plane-wave solutions of the type (5.1) into the previous equations yields

Let

be the

(5.20) |

where .

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

where , et cetera. In solving for , it is helpful to define the vectors

(5.22) | ||

(5.23) |

Here, are a set of mutually orthogonal, right-handed unit vectors. It is easily demonstrated that

(5.24) | ||

(5.25) |

It follows that

where , et cetera.

The conductivity tensor is diagonal in the ``circular'' basis . In fact, its elements are the coefficients of and in Equations (5.26) and (5.21), respectively. Thus, the dielectric permittivity tensor, defined in Equation (5.8), takes the form

(5.27) |

where

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.

The previous expressions are only approximate because the small mass-ratio ordering has already been incorporated 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 (5.28)-(5.30) and (5.31)-(5.33) are equivalent in the limit . Furthermore, Equations (5.31)-(5.33) generalize in a fairly obvious manner to plasmas consisting of more than two particle species.

In order to obtain the standard expression for dielectric permittivity tensor, it is necessary to transform 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

(5.34) |

where the unitary transformation matrix is written

(5.35) |

The dielectric permittivity in the Cartesian basis is then

(5.36) |

We obtain

where

(5.38) |

and

(5.39) |

represent the sum and difference of the right- and left-handed dielectric permittivities, respectively.