A hydrogen plasma is a completely ion- ' ized gas, i.e., the electrons e~ of charge e and mass m, have been torn away from the protons p of charge +e and mass m,,. At rest, the numbers of protons and electrons per unit volume, n, and r,, respectively, have the same value ny, meaning that the plasma is locally neutral. We study the propagation within this plasma of a plane electromagnetic wave in the direction X, described by E=E | (x,t)X=Eycos(wtkx)X and B=B;(x,1). (1) The propagation of this wave at a given location x and time sets the charges in mo- tion, and we denote ve(x, ) = vie(x, f)X and vp(x, ) = v1p(x, ) X the local velocity field of the electrons and of the protons, respectively. That is, electrons at x and have velocity Vie(x, t) along X, for example. Because of these motions, the numbers of charges per unit volume change as well, and we define ne = no+ nie(x, t) and np, = ng+ n1p(x, t) as these local values for electrons and pro- tons, respectively. We also assume that we are in the linear limit, i.e., small disturbances such that ng > ny, and ng > np, All quantities that are indexed by 1 (e.g., e, V1p...) have zero temporal mean and are to be treated as being of the same order of magnitude in your calculations. Whenever we ask for a result, limit your calculations to these first-order terms, in order to remain consistent with the linear approximation. 5.1. b2, 5.3. 5.4. 3.9. Show that B = 0. Note: this means that we are not studying the same EM waves as we did in the lecture, which existed in free space! Give the expression for the current density J as a function of e, n;p, ne, v1, and vye. Deduce that OE, noe =g(ulpvle)- (2) Show that e B, ) and derive the equivalent equation for the protons. Use them to describe the relative motions of protons and electrons, and discuss their relative influence on the current density. Note: Eqn. (3) is actually only valid in the linear limit, but you may approach this question in a \"naive\" way and not worry about this subtlety. From the expressions you have derived so far, derive the dispersion relation for these waves, i.e., find an expression for w as a function of the other parameters of the system. Hint: introducing the quantity p = mpmel/ (mp+ me) should simplify the final expres- sion. Compute the time-averaged EM energy density