Response of Average Electron Velocity Vector under AC Electric and DC Magnetic Fields in a Constant-Collision-Frequency Model

In order to study fundamental features of electron transport in magnetized plasmas, the average electron velocity $\bf V$ in gas under uniform AC electric and DC magnetic fields, $\bf E$ and $\bf B$, crossed at a right angle is theoretically derived assuming a constant collision frequency $\nu$. When ${\bf E}=(0,-E\sin\omega_Et,0)$ and ${\bf B}=(0,0,B)$, the analytical solution of ${\bf V}=(V_x,V_y,V_z)$ in periodical steady state is $V_x=[2a\nu\omega_E\omega_B/\Omega]\cos\omega_Et+[a\omega_B(\omega_E^2-\omega_B^2-\nu^2)/\Omega]\sin\omega_Et$, $V_y=[a\nu(\omega_E^2+\omega_B^2+\nu^2)/\Omega]\sin\omega_Et-[a\omega_E(\omega_E^2-\omega_B^2+\nu^2)/\Omega]\cos\omega_Et$ and $V_z=0$. Here, $a=eE/m$, $\omega_B=eB/m$, $\Omega=[(\omega_E+\omega_B)^2+\nu^2][(\omega_E-\omega_B)^2+\nu^2]$, and $e$ and $m$ are the electronic charge and mass. Although this model ignores the dependence of the collisions on electron energy, it is a merit that basic $\bf V$ responses at various $E$ and $B$ are predictable from the solution. $\bf V$ draws an ellipse in the $V_xV_y$-plane synchronously to $\bf E$ and the tilt of its major axis represents the time-averaged Hall deflection angle of $\bf V$. This depiction is informative to understand the electron swarm response under AC $\bf E$ and DC $\bf B$ fields.