Consider a mirror machine in the following Figure, which has the following magnetic field in a cylindrical coordinate, B(r, 2) = B. (1+)2 - B, ) Plasma B pax Answer the following questions. [30 points] (a) Consider a particle moving along the z-axis (r-0). Evaluate z(t) for the motion of the guiding center if the position and velocity at time t-0 are z(t = 0) = 0 v(t = 0) = vo1f + Voz2 where vo1 and voz are the constants. [10 points] (b) Assume that there are two particles A and B, as described in (a). At z-0, the particle A has vo1 = Vov1/3 and voz = Vov2/3, while the particle B has vo1 = Vov2/3 and voz = Vov1/3. Before approaching to the location of z = 1, are the particle A and B trapped (is there reflecting point)? Assume that the particle is not trapped if its reflecting points are z > L.Find the solution of (b). Show that the motion is sinusoidal, and calculate its frequency. Use the initial conditions as the conditions (z-0 at t=0) in (a). [10 points] (c) What is the magnetic field gradient drift (VB drift) at z = 0 and r = 1/2 ? [10 points]