Question

Question 1. Consider a cylindrically symmetric, time-varying magnetic field that varies parabolically with axial distance z as B &B(t)(1 + Z2/12). Assume that B() increases slowly from the value Bo at time t0 to Bi at t . A charged particle of mass m located at z 0 has perpendicular energy Wio and parallel energy Wzo at t 0. Assume that the guiding center equations of motion are valid and that mag const. a) Give the final perpendicular energy Wii at z 0 (after a time t). b) Write the equation for the motion along z, assuming that the motion is fast compared to the time variation of B(t). Show that the motion is a sinusoidal oscillation along z, and calculate the oscillation frequency Db. This shows that the particle is confined axially in the magnetic field. c) (c) Assume now that Wzo 0 and Rt) I, where R(t) is the radial distance of the guiding center of the particle from the z axis. By using Faraday's law to find the induced electric field EM) and calculating the resulting E x B drift, show that B(r)R。 = const during the slow change from Bo to Bi

plasma question

Answer 1

"Arial distance: - z as B - z B (t) (H z^2/L^2) t = 0 to B_1 at t = t_1 z = 0 perpendicular energy w_io and partial energy w_oo at t = 0 to prove: - (a) Given the final perpendicular energy w_i at z = 0 (after a time e_1) proof "": - Given B = B (e)/1 + 2^2/i^2 B (t) salty varying field we assume B (t) = at + b (slow varying field) at t = 0, B (0) = B_0 t = 0 B (0) = a_0 + b b_0 = b b = B ? 0 t = t_1"