this is for holding javascript data
Chris Spencer edited Theory.tex
about 10 years ago
Commit id: 690f101e0e61016eb9d8b4c5310c066827042aac
deletions | additions
diff --git a/Theory.tex b/Theory.tex
index b4b3917..c2d457f 100644
--- a/Theory.tex
+++ b/Theory.tex
...
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 components \[\kappa_{xx}=\kappa_{yy}=1-\frac{\omega_{pe}^2}{\omega^2-\omega_{ce}^2}\]
\[\kappa_{xy}=\kappa_{yx}=\frac{i\omega_{ce}\omega_{ce}^2}{\omega^2-\omega_{ce}^2}\]
\[\kappa_{\parallel}=1-\frac{\omega_{pe}^2}{\omega^2}\]
It is needed to define a potential for an oscillating point charge in this system. Define $\rho$ from Gauss's law as
\[\rho=qe^{-i\omega \[\rho_{ext}=qe^{-i\omega t}\sigma(\vec{r})\]
where $\sigma(\vec{r})$ is the delta function at $\vec{r}$ at zero. Use fourier analysis on Gaus's law and note that $E=-\nabla\phi$ to solve for the potential.It is obtained that $\phi(r,z)=\frac{q}{4\pi\epsilon_{o}\sqrt{\rho^2+z^2}}$. The resonance cone phenomena is described by electric fields so take the negative gradient of $\phi$ in cylindrical coordinates and the electric field is in the radial direction is \[E_r=-\frac{qe^{i\omega t}}{4\pi\epsilon_{0}\kappa_{\perp}\sqrt{\kappa_{\parallel}}}\left(\frac{\rho}{(\frac{z^2}{\kappa_{\parallel}}+\frac{\rho^2}{\kappa_{\perp}})^{3/2}}\right)\] where $\kappa_{\perp}$ is perpendicular to the background magnetic field.