dielectric constant of a collisionless plasma

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Cold electrons

Consider a quasi-neutral plasma in which the electrons are not bound to the much heavier ions, which are taken to be stationary. Suppose an electric field $\mathbf{E} = \mathbf{E}_0 e^{i \omega t}\,$ exists. The force on an electron is given by

$m_e \frac{d \mathbf{v}}{dt} = -e \mathbf{E}\,$ ,

and so there is a polarization current

$\mathbf{J}_{p} = - n_e e \mathbf{v} = \frac{n_e e^2}{m_e} \frac{1}{i\omega} \mathbf{E}\,$ ,

equal to

$\mathbf{J}_p = \frac{\partial \mathbf{P}}{\partial t}\,$ .

This leads to

$\mathbf{P} = -\frac{n_e e^2}{m_e \omega^2} \mathbf{E}\,$ .

Recalling that the electric polarization in a linear medium is defined via

$\varepsilon \mathbf{E} = \varepsilon_0 \mathbf{E} + \mathbf{P}\,$ ,

so we have a permittivity

$\varepsilon = \epsilon_0 \left(1 - \frac{\omega_{ep}^2}{\omega^2}\right)\,$ ,

where

$\omega_{ep}^2 = \frac{n_e e^2}{\varepsilon_0 m_e}\,$

is the electron plasma frequency.

dielectric constant of a collisionless plasma

From Physics wiki

Cold electrons

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