Hint for part (a): The fact that the F=-pe seems a bit self-evident, but we're trying to prove it. So, you need to make use of a theorem that is related to the divergence theorem.

EXERCISE 4.19 When the thermal motions of the electrons in a plasma are important, the dispersion rela- tion for the plasma waves becomes more complex and allows for both propagation and damping of the waves. In the simplest case of a nonrelativistic plasma in which the electron-electron collision frequency is very high and the electron-ion collision frequency can be ignored, the thermal motions contribute just an isotropic pressure term to the equa- tions of motion. In the following we ignore heat conduction, viscosity, and other dissipa- tive processes. (a) Consider a group of Ne - constant = ne V electrons occupying a (macroscopic cally very small) volume V, where n, is the electron density. Show that the net pressure force on this volume of electrons is -VV pe, where pe = neksTe is the electron pressure, kg Boltzmann's constant, and T. the electron temperature. Show that the equation of motion for the mean velocity v of the electrons is dv nem dt = -VPe - neq Vo (4.285) where m is the electron mass and d the scalar potential, and the total time deriva- tive is d / dt = a/at + v . V. (b) Consider the internal energy - NeksTe of this same group of electrons and the work dW = ped V they do on the surrounding electrons as they expand the vol- ume they occupy. Use the first law of thermodynamics (conservation of energy) to show that ZKBTe) = kBTe dt dne ne di (4.286) or, equivalently, that dpe dne dt KBTe dt (4.287)