# Physics Cup – TalTech 2021 – Pr. 4, Hint 2.

First, please notice that the magnetic field is here only to make the problem essentially one-dimensional: the thermal energy $kT$ is much smaller than the gap between the energy levels $\hbar \omega_B$ ($\omega_B=Be/m$), and hence, following the Boltzmann distribution, almost all the electrons stay at the lowest Landau level.

Second, in this sparse electron gas, we are asked to neglect the scattering of electrons on electrons and so we can consider the motion of a single electron along its one-dimensional motion, along the magnetic field. Since the potential barrier of the shock wave is high, initially the electrons are trapped between the shock and the wall. However, the adiabatic invariant is conserved (as the shock wave moves slowly) and so the energy of the electrons starts growing until they can cross the barrier. In order to solve the problem, you need to study the speed distribution of the electrons which have crossed the barrier, and what happens to their speed afterwards. Don’t forget the Hint 1.

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