deletions | additions       diff --git a/Theory.tex b/Theory.tex  index e61033c..67c9723 100644  --- a/Theory.tex  +++ b/Theory.tex 
...
\[\kappa_{xy}=\kappa_{yx}=\frac{i\omega_{ce}\omega_{ce}^2}{\omega^2-\omega_{ce}^2}\]  \[\kappa_{\parallel}=1-\frac{\omega_{pe}^2}{\omega^2}\]  It is needed to define a potential for an oscillating point charge in this system. Define $\rho$ from Gauss's law as \[\rho_{ext}=qe^{-i\omega t}\sigma(\vec{r})\]  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 $\phi(r,z)=\frac{q}{4\pi\epsilon_{o}\sqrt{\rho^2+z^2}}$ where now $\rho$ is referring to radius. 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. The resonance cones that are observed are described by the electric field in the radial direction. The resonance cone angle can be is  found from geometry as $\tan^2\theta=\frac{\rho}{z}$. when looking for at what point does the electric field becomes infinite.This occurs when the condition $(\frac{z^2}{\kappa_{\parallel}}+\frac{\rho^2}{\kappa_{\perp}}=0$