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Chris Spencer edited Theory.tex
about 10 years ago
Commit id: a66896282203584a287bf1e776ae70cf537a1bad
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\section{Theory}
\newcommand{\tensor}[1]{\stackrel{\leftrightarrow}{#1}}
Gaus's Law is in free space is \[\nabla\cdot\vec{E}=\frac{\rho}{\epsilon_0}\]. In a plasma $\epsilon_0$ becomes $\tensor{\epsilon}$ and the relation becomes \[\nabla\cdot(\epsilon\vec{E})=\rho\] and noting that $\vec{E}=-\nabla\phi$ and tensor $\kappa=\tensor{\epsilon}\epsilon_0$, Gaus's Law can be rewritten as \[\vec{D}=\epsilon_0\tensor{\kappa}\cdot\vec{E}\]
where $\tensor{\kappa}$ is the dielectric tensor.
For a cold plasma in an electric field, there will be no contribution to the to the dielectric tensor due to the ions having low thermal velocity in comparison to the electrons.