Waves in two-fluid magnetized plasmas

1.  Show that waves become longitudinal, \( \vec{E}\parallel \vec{k} \) when \( n^2\rightarrow \infty \) (is this condition necessary or sufficient or both ?) and find frequencies of these oscillations (resonance frequencies). 

2.  Derive dispersion relations for an electron-positron plasma.

3.  Let a plasma consist of electrons and two ion species. Derive the dispersion relation for waves propagating in the direction parallel to the magnetic field. What is new ?