this is for holding javascript data
Chris Spencer edited Theory.tex
about 10 years ago
Commit id: 38c29e11791c1681c9b5fe46e42d08ecdbc11bb2
deletions | additions
diff --git a/Theory.tex b/Theory.tex
index 78f1f0f..18d8af0 100644
--- a/Theory.tex
+++ b/Theory.tex
...
\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.
Then the dielectric tensor has terms \[\kappa_{xx}=\kappa_{yy}=1-\frac{\omega_{pe}^2}{\omega^2-\omega_{ce}^2}\]