deletions | additions       diff --git a/Theory.tex b/Theory.tex  index fe9dbbe..28a2259 100644  --- a/Theory.tex  +++ b/Theory.tex 
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\begin{tiny}{\[\eta^2=1-\frac{\omega_{pe}^2}{\omega^2\left((1+\frac{i\nu}{\omega})-\frac{\omega_{ce}^2\sin^2\theta}{2\omega^2\left(1-\frac{\omega_{pe}^2}{\omega^2}\right)}\pm\sqrt{\frac{(\omega_{ce}^2\sin^2\theta)^2}{4(1-\frac{\omega_{pe}^2}{\omega^2})}+\frac{\omega_{ce}^2cos^2\theta}{\omega^2}}\right)}\]}  \end{tiny}  for an infinite plasma[2]. This describes the index of refraction for a whistler wave where $\eta^2=\left(\frac{kc}{\omega}\right)^2$. $\omega_{pe}$ is the plasma frequency,$\omega_{ce}$ frequency, $\omega_{ce}$  is the electron cyclotron frequency, and $\nu$ is the rate of collisions in the plasma.The angle $\theta$ refers to the angle the waves make with respect to the background magnetic field $B_0$.The first assumption made is that the first experiment's waves are made with $\theta=0$ for waves parallel to $B_0$ and that damping is slight so that $\nu\approx0$ [2].Making these assumptions it is found that Appleton's equation reduces to \[n^2=1-\frac{\omega_{pe}^2}{(\omega+\omega_{ce})(\omega-\omega_{ce})}\].Further assumptions can be made to reduce this equation by noting that $\omega_{ci}>>\omega_{ce}$,$\omega>\omega_{ce}$, and that the wave frequency $\omega$ is less than $\omega_{ce}$.This results in an equation for index of refraction that is is simple provided the above assumptions \[n^2\approx\frac{\omega_{pe}^2}{\omega\omega_{ce}}\]. Appleton's equation can be derived from the plasma force equation,Maxwell's equations, and using fourier analysis to show pertubations of the magnetic field are in the form $\vec{B_1}(\vec{r},t)=Be^{i(\vec{k} \cdot \vec{r}-\omega t)}$.The plasma force equation is written assuming that the particles are cold so that there are no pressure gradients and that the quantity $\vec{v}\cdot\nabla\approx0$ in the convective derivative, then the force equation is $m\frac{\partial\vec v}{\partial t}=q(\vec{E}+\vec{v}\times\vec{B})-m\nu\vec{v}$ for electrons.The two Maxwell's equations used are $\nabla\times\vec{E}=-\frac{\partial\vec{B}}{\partial{t}}$ and $\nabla\times\vec{B}=\mu_0\vec{J}+\mu_0\epsilon_0\frac{\partial\vec{E}}{\partial t}$.Linearize equations involving $\vec{B}$ and solve the force equations for $\frac{kc}{\omega}$.