## FANDOM

167 Pages

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

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

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