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.

Faraday's law

$ \nabla \times \mathbf{E} = - \frac{1}{c} \frac{\partial \mathbf{B}}{\partial t} $

Ampére's law

$ \nabla \times \mathbf{B} = \frac{1}{4 \pi} \mathbf{J} + \frac{1}{c} \frac{\partial \mathbf{E}}{\partial t} $

Definition of current (summation is carried over all species)

$ \mathbf{J} = \sum_{\sigma} q_{\sigma} n_{\sigma} \mathbf{v}_{\sigma} $

Particle conservation

$ \frac{\partial n_{\sigma}}{\partial t} + \nabla \left( n_{\sigma} \mathbf{v}_{\sigma} \right) = 0 $

Momentum conservation

$ \frac{\mathbf{v_{\sigma}}}{\partial t} + \mathbf{v}_{\sigma} \cdot \nabla \mathbf{v}_{\sigma} - \frac{q_{\sigma}}{m_{\sigma}} \left( \mathbf{E} + \frac{v_{\sigma}}{c} \mathbf{B} \right) = 0 $

Cold, non - magnetized Plasma Edit

The only unperturbed variable that is different from zero is the density. We will denote it by $ n_{\sigma 0} $. We assume the perturbed variables vary as $ \exp \left[ i \left( \mathbf{k} \cdot \mathbf{r} - \omega t \right) \right] $, and denote the amplitude with subscript 1. To first order in the perturbation, the equations are

$ \mathbf{k} \times \mathbf{E}_1 = \frac{\omega}{c} \mathbf{B}_1 $

$ i \mathbf{k} \times \mathbf{B}_1 = \frac{4 \pi}{c} \mathbf{J} - i \frac{\omega}{c} \mathbf{E}_1 $

$ \mathbf{J} = \sum_{\sigma} q_{\sigma} n_{\sigma 0} \mathbf{v}_{\sigma 1} $

$ -i \omega \mathbf{v}_{\sigma 1} = \frac{q_{\sigma}}{m_{\sigma}} \mathbf{E}_1 $

We distinguish between two different cases: $ \mathbf{E}_1 \parallel \mathbf{k} $ (a.k.a longitudinal or electrostatic wave) and $ \mathbf{E}_1 \perp \mathbf{k} $ (a.k.a transverse or electromagnetic wave).

Electrostatic Wave Edit

In this case $ \mathbf{B}_1 = 0 $, so

$ \frac{4 \pi}{c} \sum_{\sigma} q_{\sigma} n_{\sigma 0} \frac{i q_{\sigma}}{m_{\sigma} \omega} \mathbf{E}_1 = i \frac{\omega}{c} \mathbf{E}_1 $

$ \omega^2 = 4 \pi \sum_{\sigma} \frac{q_{\sigma}^2 n_{\sigma 0}}{m_{\sigma}} = \omega_p^2 $

where $ \omega_p $ is called the plasma frequency.

Electromagnetic Wave Edit

In this case the magnetic field is orthogonal to both $ \mathbf{E}_1 $ and $ \mathbf{k} $. The dispersion equation in this case is

$ \omega^2 = \omega_p^2 + k^2 c^2 $

Cold, Magnetized Plasma Edit

Next, we repeat the previous calculation for a cold plasma in the presence of a uniform magnetic field $ \mathbf{B_0} = B_0 \hat{z} $. The only perturbed equation that changes is the conservation of momentum

$ -i \omega \mathbf{v}_{\sigma 1} = \frac{q_{\sigma}}{m_{\sigma}} \left( \mathbf{E_1} + \frac{1}{c} \mathbf{v}_{\sigma 1} \times \mathbf{B}_0 \right) $

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 $ \mathbf{B}_0 $, and then solve separately for $ \mathbf{v}_{\sigma 1} $, $ \mathbf{v}_{\sigma 1} \cdot \mathbf{B}_0 $ and $ \mathbf{v}_{\sigma 1} \times \mathbf{B}_0 $.

$ - i \omega \mathbf{v}_{\sigma 1} \cdot \mathbf{B}_0 = \frac{q_{\sigma}}{m_{\sigma}} \mathbf{E}_1 \cdot \mathbf{B}_0 $

$ - i \omega \mathbf{v}_{\sigma 1} \times \mathbf{B}_0 = \frac{q_{\sigma}}{m_{\sigma}} \left[ \mathbf{E}_1 \times \mathbf{B}_0 - \frac{1}{c} B_0^2 \mathbf{v}_{\sigma 1} + \frac{1}{c} \left( \mathbf{v}_{\sigma 1} \cdot \mathbf{B}_0 \right) \mathbf{B}_0 \right] $

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 $ \mathbf{k} \parallel \mathbf{E}_1 \parallel \mathbf{B}_0 $. From the Faraday equation we get $ \mathbf{B}_1 = 0 $, and hence $ \mathbf{v}_{\sigma 1} = \frac{q_{\sigma}}{m_{\sigma}} \mathbf{E}_1 $, 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. $ \mathbf{k} \parallel \mathbf{B}_0 \perp \mathbf{E}_1 $. In this case, it is easier to solve the conservation of momentum for the velocity

$ \mathbf{v}_{\sigma 1} = \frac{q_{\sigma}}{m_{\sigma}} \frac{i \omega \mathbf{E}_1 - \omega_{\sigma c} \left(\mathbf{E}_1 \times \mathbf{B}_0 \right)/B_0 }{\omega^2 - \omega_{\sigma c}^2} $

where $ \omega_{\sigma c} = \frac{q_{\sigma} B}{c m_{\sigma}} $ is the cyclotron frequency. One can verify that this result coincides with that of the non magnetized case in the limit $ B_0 \rightarrow 0 $

Substituting everything into Ampére's law yields

$ -i \frac{c}{\omega} k^2 \mathbf{E}_1 = \frac{4 \pi}{c} \sum\limits_{\sigma} q_{\sigma} n_{\sigma 0} \frac{q_{\sigma}}{m_{\sigma}} \frac{i \omega \mathbf{E}_1 - \omega_{\sigma c} \left( \mathbf{E}_1 \times \mathbf{B}_0/B_0 \right)}{\omega^2 - \omega_{\sigma c}^2} -i \frac{\omega}{c} \mathbf{E}_1 $

$ \left[ c^2 k^2 - \omega^2 + \omega^2 \sum\limits_{\sigma} \frac{\omega_{\sigma p}^2}{\omega^2 - \sigma_{\sigma c}^2} \right] \mathbf{E}_1 = i \sum\limits_{\sigma} \frac{\omega \omega_{\sigma p}^2 \omega_{\sigma c}}{\omega^2-\omega_{\sigma c}^2} \frac{\mathbf{E}_1 \times \mathbf{B}_0 }{B_0} $

Multiplying each side by its complex conjugate yields

$ c^2 k^2 - \omega^2 + \omega^2 \sum\limits_{\sigma} \frac{\omega_{\sigma p}^2}{\omega^2 - \omega_{\sigma c}^2} = \pm \omega \sum\limits_{\sigma} \frac{\omega_{\sigma p}^2 \omega_{\sigma c}}{\omega^2-\omega^2_{\sigma c}} $

$ c^2 k^2 = \omega^2 \left( 1 - \sum\limits_{\sigma} \frac{\omega_{\sigma p}^2 / \omega^2}{1 \pm \omega_{\sigma c}/\omega} \right) $