PHYS 515: Plasma Physics & MHD Grade Assigments for 10/3/00 Note: assigments are due on Wednesday, Oct. 11. You can put them in my mailbox, or hand them in in class. 1. a) Assume a proton-electron plasma. From the expression for the drift velocity in the case of grad(B) X B drift derive the drift current in terms of v_perpendicular, B, and n (eliminate gyro radius and gyro frequency). Take note of the 10/5 handout on gradient drift, and carefully separate the electron and proton components b) Assume that the kinetic energy in the gyro motion for protons and electrons is equal, w_perp. Derive: j_B = n c w_perp grad(1/B) X b, where B is the magnitude of the magnetic field, X again indicates the cross product, and b is the unit vector along the magnetic field. c) Now consider a current sheet with magnetic field in the z direction only: B_z = B_0 tanh(y/d), where y is the y coordinate, and d a parameter related to the thickness of the sheet. What is a typical length scale for the variation of the field in the y direction? Using this length scale, what is the criterion for the derivation in the handout of grad(B) X B drift to be valid? At what location is this criterion not fulfilled? 2. Consider the expression for the deflection angle in e-p, and e-e collisions as a function of the impact parameter b. a) For a fully Maxwellian velocity distribution, derive an expression for mu u_0^2 in terms of the temperature (mu is the reduced mass, u_0 the impact velocity). b) Given that expression derive r_0, the impact distance at which there is a 90 degree deflection. c) Compare the logarithmic term, ln(1+X_m), in the expression derived from integration over the impact parameter, b, (see handout) with the Coulomb logarithm derived from integration over the deflection angle. What is the exact relation between the two? d) Consider coronal plasma with density n = 10^9 cm^-3 and T = 10^6 K. Assume that b_max, the maximum impact distance equals d/2, where d is the mean interparticle distance. Calculate the value of the Coulomb logarithm in this case. What is the result for a cut-off at the Debye length?