Dynamic magneto conductivity tensor for free electrons a metal with a concentration n of free electrons of charge – e is in a static magnetic field B. The electric current density in the xy plane is related to the electric field by j_x = σ_xxE_x + σ_xyE_y;                  j_y = σ_yzE_T + σ_yyE_y.

Assume that the frequency w >> w₀, and where w >> 1/τ, where w_c = eB/mc and τ is the collision time. 

(a) Solve the drift velocity equation (51) to find the components of the magneto conductivity tensor:

σ_xx = σ_yy + iw²_p/4πw;               σ_xy = – σ_yx = wcw²_p/4πw², where w²_p = 4πne²/m. 

(b) Note from a Maxwell equation that the (dielectric function tensor of the medium is related to the conductivity tensor as ε = 1 + i(4π/w)σ. Consider an electromagnetic wave with wave vector k = kz. Show that the dispersions relation for this wave in the medium is c²k² = w² – w²_p ± w_c w²_p/w 

At a given frequency there are two modes of propagation with different wave vectors and different velocities. The two modes correspond to circularly polarized waves. Because a linearly polarized wave can he decomposed into two circularly polarized waves, it follows that the plane of polarization of a linearly polarized wave will he rotated by the magnetic field.