
#1
Mar2711, 05:50 AM

P: 33

I'm looking to find an expression for the Faraday rotation of a wave in a magnetized plasma, propagating parallel to a magnetic field.
It's more math help than physics help I need here though. I know that I'll have to start with the dispersion relations for a right and left circularly polarized waves: Note that capital omegas [tex]\Omega[/tex] represent cyclotron frequencies, small omegas [tex]\omega[/tex] represent plasma frequencies, and the i/e subscripts refer to electrons or ions. The [tex]\omega[/tex] with no subscript is the frequency of the propagating wave. [tex] \frac{k_R^2 c^2}{\omega^2}=1\frac{\omega_{pe}^2}{\omega (\omega  \Omega_e)}\frac{\omega_{pi}^2}{\omega (\omega + \Omega_i)} [/tex] [tex] \frac{k_L^2 c^2}{\omega^2}=1\frac{\omega_{pe}^2}{\omega (\omega + \Omega_e)}\frac{\omega_{pi}^2}{\omega (\omega  \Omega_i)} [/tex] Note that I'm dealing with an Alfven wave, so when manipulating the above formulae, the following assumption can be made: [tex] \omega << \Omega_i << \Omega_e [/tex] The angle that the wave's plane of polarization rotates through is [tex] \Theta = \frac{(k_Rk_L)\Delta z}{2} [/tex] The problem lies in finding the [tex] k_Rk_L [/tex] term from the first two equations listed. Using the relationship [tex] \frac{\omega_{pi}^2}{\Omega_i} = \frac{\omega_{pe}^2}{\Omega_e} [/tex] I can simplify the term for the Rwave to get [tex] \frac{k_R^2 c^2}{\omega^2} \approx 1\frac{\omega}{\Omega_i}+\frac{\Omega_i^2}{\omega_{pi}^2} [/tex] I know this is correct, but repeating the same process for the Lwave doesn't yield a term that can easily be combined with this one to get the [tex] k_Rk_L [/tex] expression I'm looking for. If anyone could guide me through this, it would be much appreciated! 


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