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Answer the following questions:

1. If $\beta > 1$ then is $\frac{\omega_e}{\Omega_e} > 1$? ($\beta$ is the ratio of particle pressure, $p$, to magnetic pressure, $B^2/4\pi$: $\frac{p}{B^2/4\pi}$, $\omega_e$ is the electron plasma frequency, and $\Omega_e$ is the electron gyrofrequency.)

2. Outline a method that you could use to measure the distribution of wave energy with wave number.

3. Review how Young et al. [1973] (JGR, p. 1082) show that the wave energy for electron cyclotron harmonic (ECH) waves should be distributed over a wide energy of parallel wave numbers, $k_{\parallel}$, centered near $k_{\parallel}=0.5\Omega_e/v_{\mathrm{th}_{\parallel}}$ and a narrower range of perpendicular wave, $k_{\perp}$, centered near $k_{\perp}=2\Omega_e/v_{\mathrm{th}_{\perp}}$.

4. Using the resonance condition, $w-k_{\parallel}v_{\parallel}=-n\Omega_e$, where n $=0,\pm1,\pm2...$ explain why diffusion rates are generally largest for n=-1.

5. Use the above homework problems to show that diffusion into the loss cone from ECH waves with the dominant emissions at a frequency, $w$, near $1.5\Omega_e$ should be most rapid for electrons with $v \approx v_{\mathrm{th}_{\parallel}}$

6. Using Equation 2 in Lyons [1974]: $(v_{\parallel}-\omega/k_{\parallel})^2 + v_{\perp}^2=const$, show that the diffusion surfaces approach surfaces of constant energy as $v$ increases above 3 $v_{\mathrm{th}_{\parallel}}$ since for the 1.5$\Omega_e$ waves $\omega/k_{\parallel} \approx 3v_{\mathrm{th}_{\parallel}}$.