# Penetration depth of an electric field in a semi-infinite plasma

## TL;DR

This paper derives an explicit formula for the electric field penetration depth in a semi-infinite plasma, showing it follows an exponential decay modulated by oscillations, and depends on plasma dielectric properties and thermal velocity.

## Contribution

It provides a new explicit calculation of the electric field penetration depth in a semi-infinite plasma, including surface effects, which was not previously detailed.

## Key findings

- Penetration depth follows exponential attenuation with oscillations.
- Explicit formula for penetration depth involving dielectric function and thermal velocity.
- Surface contributions significantly affect the penetration behavior.

## Abstract

It is shown that the penetration of an oscillating electric field in a semi-infinite classical plasma obeys the standard exponential attenuation law $e^{-x/\lambda_{e}}$ (besides oscillations), where $x$ is the distance from the wall and $\lambda_{e}$ is the extinction length (penetration depth, attenuation length). The penetration depth is computed here explicitly; it is shown that it is of the order $\lambda_e \simeq [\mid\varepsilon \mid/(1-\varepsilon)]^{1/3}v_{th} / \omega$, where $\varepsilon$ is the dielectric function, $\omega$ is the frequency of the field and $v_{th}=\sqrt{T/m}$ is the thermal velocity ($T$ being the temperature and $m$ the particle (electron) mass). The result is obtained by including explicitly the contribution of the surface term.