Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2012 / S16 / 1 / 3 / Solution

For the shifted beam Maxwell-Boltzmann distribution,On the rising flank put $u=(V-v)/{\sigma }_b>0$. The positive slope is proportional to $u{e}^{-u^2/2}$, whose derivative is $(1-u^2)e^{-u^2/2}$. Its unique positive maximum is at $u=1$, givingThis is the region where beam drive can most readily overcome the negative plasma slope when that background changes little over the narrow beam width. It is an estimate for the first resonant unstable wave, not a proof that every beam-plasma dispersion relation has its exact marginal phase velocity at this value. Both the total slope and the real dispersion relation determine the exact marginal mode.