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Electromagnetic waves (dispersion)

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A plasma acts as a dielectric medium for electromagnetic waves. Waves with frequencies slow compared to the plasma frequency will be significantly damped. This is described by the dispersion relation :

$\omega^{2}-\omega_{p}^{2}=c^2 k^2$

This is a valid relation for homogenous unmagnetized cold plasmas.

We start off with some of Maxwell's equations :

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

And the vector relation :

$\nabla\times\left(\nabla\times\mathbf{A}\right)=\nabla\left(\nabla\cdot\mathbf{A}\right)-\nabla^{2}\mathbf{A}$

Taking the cross product of the first relation and using the vector identity:

$\nabla\times\left(\nabla\times\mathbf{E}\right)=\nabla\left(\nabla\cdot\mathbf{E}\right)-\nabla^{2}\mathbf{E}=-\frac{1}{c^2}\frac{\partial}{\partial t}\left(4\pi \mathbf{J}+\frac{\partial\mathbf{E}}{\partial t}\right)$

We can rewrite the current as:

$\mathbf{J}=n_{e}e\mathbf{v}$

And writing the Lorentz force equation

$m_{e}\mathbf{a}=-q\left(\mathbf{E}+\frac{\mathbf{v}}{c}\times\mathbf{B}\right)$

Since $v 1$ is a perturbed quantitiy and $v 0 = 0$ , we can assume it is small compared to c and neglect the magnetic contribution to the lorentz force:

$\frac{\partial\mathbf{J}}{\partial t}=n_{e}e\frac{e}{m_{e}}\mathbf{E}$

so that:

$\nabla\left(\nabla\cdot\mathbf{E}\right)-\nabla^{2}\mathbf{E}=-\frac{1}{c^2}n_{e}e\frac{e}{m_{e}}\mathbf{E}-\frac{1}{c^2}\frac{\partial^{2}\mathbf{E}}{\partial t^{2}}$

We can write the perturbed values in terms of plane waves moving in the $\hat{z}$ direction, for example:

$\mathbf{E}=\mathbf{E}_{1}\exp i\left(kz-\omega t\right)$

Here we have taken $E 0 = 0$ (no zero order electric field). Plugging in:

$\nabla\left(ik\hat{z}\cdot\mathbf{E}\right)+k^{2}\mathbf{E}_{1}=-\frac{1}{c^2}\frac{n_{e}e^{2}}{m_{e}}\mathbf{E}_{1}+\frac{1}{c^2}\omega^{2}\mathbf{E}_{1}$

The first term on the left must be zero, since the electric field must be perpendicular to the wave vector: