1. Waves in magnetized plasma
Introduction – some definitions
Self-consistency, linear theory
Transverse waves in unmagnetized plasma
Longitudinal waves, Landau damping
Waves in magnetized plasma
Beam-plasma instabilities, plasma masers

2. Dispersion relation Dielectric permittivity tensor Linearization
Consider cold plasma
Equation of motion
Linearization
Linearization

3. Cyclotron (Larmor) frequency
Omit for simplicity (for a moment) index 

5. Current density

6. Dielectric permittivity tensor

7. Dispersion relation isotropic medium Now, consider waves
Particular cases:
isotropic medium
With regard to the field perpendicular to the external magnetic field, fully magnetized plasma behaves as vacuum
Now, consider waves
Dispersion relation
General case is rather complicated. Let us consider waves propagating parallel to and perpendicular to the external magnetic field

8. Waves propagating along the magnetic field
2 kinds of solution correspond to different orientations of the wave electric field with respect to the external magnetic field
Plasma oscillations along the external magnetic field are the same as in the unmagnetized plasma

9. What is the physical sense of different signs in the dispersion relations?
Waves are circularly polarized.
Signs correspond to different directions of the electric field vector rotation
Ordinary wave
Dispersion depends on the direction of rotation
Extraordinary wave

10. Faraday effect
Consider linearly polarized wave propagating in plasma along the external magnetic field
The wave remains linearly polarized but the polarization plane rotates
The rotation angle linearly depends on the distance

11. Rotation of the polarization plane is known in optics
Rotation of the polarization plane is known in optics. Media, in which the rotation occurs, are called optically active media. An example is a solution of chiral molecules (e.g., sugar). There is, however, a fundamental difference that concerns the direction of rotation.
The direction of rotation with respect to the wave propagation in the magnetized plasma depends on the direction of the wave propagation with respect to the magnetic field
Direction of external magnetic field is a separated direction
When the wave passes through a slab of optically active medium, reflects from a mirror, and comes back, the polarization plane rotation on the opposite path compensates that on the direct path. In the magnetized plasma, the rotation angles on the direct and opposite paths add together.

12. Faraday rotation is an important tool in astrophysics
Faraday rotation is an important tool in astrophysics. In particular, Faraday rotation measurements of polarized radio signals from extragalactic radio sources shadowed by the solar corona allows for estimating both the electron density distribution and the direction and strength of the magnetic field in the coronal plasma.
Similarly, Faraday rotation measurements of the polarized light beam are used for the diagnostics of the dense plasma in experiments on plasma implosion (Z-machine, etc.)

13. Return to the dispersion relation
Analytic solutions are possible only at some assumptions
Slightly modified dispersion of the transverse wave in isotropic plasma
Cutoff frequency
This solution is valid if

14. Very low-frequency waves
If one assumes RHS is zero, the difference between the ordinary and extraordinary waves is lost
Quasineutrality
Alfven velocity
Alfven wave

15. Alfven wave is a type of magnetohydrodynamic wave in which ions oscillate in response to a restoring force provided by an effective tension on the magnetic field lines. It is like a wave travelling along a stretched string – the magnetic field line tension is analogous to string tension.
Alfven velocity is not necessarily the velocity of an Alfven wave
Account correctly for the RHS in the dispersion relation
Lower sign – ordinary wave – Alfven wave
Upper sign – extraordinary wave – fast magnetosonic wave

16. Intermediate frequency
It is possible to omit the terms relating to the ion component
Suppose now that
Solution exists only for the upper sign
Extraordinary wave
non-contradictory assumption

17. This wave is called whistler or helicon
This wave is called whistler or helicon. Terrestrial whistlers are electromagnetic waves occurring at audio frequencies. They are produced by lightning strokes and travel along the Earth’s magnetic field lines.
Dispersion of group velocity
Perceived as a descending tone  due to the slower velocity of the lower frequencies 

18. Example of spectrogram

20. Summary: 5 branches of oscillations propagating along the magnetic field
Electromagnetic waves
Langmuir oscillations
There are asymptotic solutions:
Whistler wave
Magnetohydrodynamic waves

21. Waves propagating across the magnetic field
Again, 2 kinds of solution correspond to different orientations of the wave electric field with respect to the external magnetic field

22. Transverse waves. Magnetic field does not influence
Waves are neither transverse, nor longitudinal purely
Dispersion relation

23. Again, solutions are possible under some assumptions
Magnetohydrodynamic wave

24. Summary: 4 branches of oscillations propagating across the magnetic field
Asymptotic solutions:
Upper hybrid
Lower hybrid