Skip to content

Commit

Permalink
Corrected the bug and improved a lot the RF fundamental part.
Browse files Browse the repository at this point in the history
  • Loading branch information
Julien Hillairet committed Jul 14, 2017
1 parent d452681 commit 254841a
Show file tree
Hide file tree
Showing 4 changed files with 1,113 additions and 819 deletions.
6 changes: 3 additions & 3 deletions Dispersion_Relation.tex
Original file line number Diff line number Diff line change
@@ -1,5 +1,5 @@
\section{Dispersion Relation}
\subsection{Vacuum dispersion relation}
\chapter{Dispersion Relation}
\section{Vacuum dispersion relation}
In a non-dispersive isotropic homogeneous non-lossy dielectric medium, such as vacuum (or air), the wavevector $\mathbf{k}=k\mathbf{\hat{k}}$ direction is given by:
\begin{equation}
\mathbf{\hat{k}}\cdot\mathbf{E}=0
Expand All @@ -11,7 +11,7 @@ \subsection{Vacuum dispersion relation}
\mathbf{k} \times \mathbf{k} \times \mathbf{E} + k^{2} \mathbf{E} = \mathbf{0}
\end{equation}

The condition for that system of 3 equations and 3 unknowns $(E_{x},E_{y},E_{z})$ to have a non-trivial solution, consists in solving the \emph{dispersion relation} between the wavevector $\mathbf{k}$ and the angular frequency $\omega$, i.e. $\mathbf{k}(\omega)$. In this medium, this relation is simple (TODO demo): $k=\sqrt{\mu\varepsilon}\omega=\frac{c}{n}\omega$\parencite[(7.4)]{Jackson1998}\footnote{While in a lossy medium $k^{2}=\mu\varepsilon\omega^{2}-j\omega\mu\sigma$ \parencite[§8.2]{Bladel2007}.}.
The condition for that system of 3 equations and 3 unknowns $(E_{x},E_{y},E_{z})$ to have a non-trivial solution, consists in solving the \emph{dispersion relation} between the wavevector $\mathbf{k}$ and the angular frequency $\omega$, i.e. $\mathbf{k}(\omega)$. In this medium, this relation is simple (TODO demo): $k=\sqrt{\mu\varepsilon}\omega=\frac{c}{n}\omega$\parencite[(7.4)]{Jackson1998}\footnote{While in a lossy medium $k^{2}=\mu\varepsilon\omega^{2}-j\omega\mu\sigma$ \parencite[sec. 8.2]{Bladel2007}.}.


% Dans un plasma froid magnétisé, la situation est radicalement différente,
Expand Down
Loading

0 comments on commit 254841a

Please sign in to comment.