# Introduction

Radio emission from the ionosphere can produce a whistling sound in the audio frequency that can be heard[1]. The whistling sounds are described as groups of descending tones which are called the whistler mode. When lightning hits the southern hemisphere it produces a range of radio waves, some of which can travel along the earths magnetic field lines from the southern hemisphere to the northern hemisphere[1].These waves are called extraordinary waves .The extraordinary waves emit two solutions to the wave equation named L and R waves.The L and R refer to left and right hand circularly polarized. The waves that describe the whistling sound are R waves and they will be detected in the north and the different frequencies of these waves will travel at different speeds. For $$\omega<\frac{\omega_{ce}}{2}$$ the phase and group velocities increase with frequency, where $$\omega_{ce}=\frac{eB}{m}$$ is the electron cyclotron frequency[1]. Due to this, the lower frequencies will arrive at the northern hemisphere later than the higher frequencies will, causing the descending tone in the whistler mode. These R waves waves that travel along the magnetic field lines are called whistler waves and these waves can only propagate for $$\omega<\frac{\omega_ce}{2}$$. This lab seeks out to measure the dispersion relation of the whistler waves and to find the wave patterns theoretically and experimentally in the inductively coupled plasma device using Appletons equation.

# Theory

Appleton’s equation is given by

$\eta^2=1-\frac{\omega_{pe}^2}{\omega^2\left((1+\frac{i\nu}{\omega})-\frac{\omega_{ce}^2\sin^2\theta}{2\omega^2\left(1-\frac{\omega_{pe}^2}{\omega^2}\right)}\pm\sqrt{\frac{(\omega_{ce}^2\sin^2\theta)^2}{4(1-\frac{\omega_{pe}^2}{\omega^2})}+\frac{\omega_{ce}^2cos^2\theta}{\omega^2}}\right)}$

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, $$\omega_{ce}$$ 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 $n^2=1-\frac{\omega_{pe}^2}{\omega(\omega-\omega_{ce})}$.Noting that $$\omega<\omega_{ce}$$ then this results in an equation for index of refraction $n^2\approx\frac{\omega_{pe}^2}{\omega\omega_{ce}}$. 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 $$\frac{kc}{\omega}$$.