This page contains the derivation of dispersion relations for electromagnetic waves in collision - less plasmas. It essentially involves perturbation theory to the equations of magnetohydrodynamics.
Definition of current (summation is carried over all species)
Cold, non - magnetized Plasma Edit
The only unperturbed variable that is different from zero is the density. We will denote it by . We assume the perturbed variables vary as , and denote the amplitude with subscript 1. To first order in the perturbation, the equations are
We distinguish between two different cases: (a.k.a longitudinal or electrostatic wave) and (a.k.a transverse or electromagnetic wave).
Electrostatic Wave Edit
In this case , so
where is called the plasma frequency.
Electromagnetic Wave Edit
In this case the magnetic field is orthogonal to both and . The dispersion equation in this case is
Cold, Magnetized Plasma Edit
Next, we repeat the previous calculation for a cold plasma in the presence of a uniform magnetic field . The only perturbed equation that changes is the conservation of momentum
The magnetic field breaks the symmetry of the problem and complicates it somewhat. One way of solving this equation is by applying scalar and vector product by , and then solve separately for , and .
To continue, we examine each orientation (of the rays and the electric field) separately.
Propagation Parallel to the Ambient Magnetic Field Edit
In this section we will consider waves the propagate along the magnetic field.
Electrostatic Waves Edit
In this section we consider waves where both the propagation direction and electric field are both parallel to the ambient magnetic field . From the Faraday equation we get , and hence , so we get the same oscillations as in the non - magnetized case.
Electromagnetic Waves Edit
In this section we consider waves that propagate parallel to the ambient magnetic field, but polarised in the perpendicular direction, i.e. . In this case, it is easier to solve the conservation of momentum for the velocity
where is the cyclotron frequency. One can verify that this result coincides with that of the non magnetized case in the limit
Substituting everything into Ampére's law yields
Multiplying each side by its complex conjugate yields