5.3 The electrons inside a system of two coaxial magnetic mirrors can be described by the so-called loss-cone distribution function
\a±J     \c*\\/ -
where i;,, and v± denote the magnitudes of the electron velocities in the directions parallel and perpendicular to the magnetic bottle axis, respectively, and where af{ = 2fcT||/m and a\ = 2kT±/m.
(a) Verify that the number density of the electrons in the magnetic bottle is given by no-
(b) Justify the applicability of the loss-cone distribution function to a magnetic mirror bottle by analyzing its dependence on v]{ and v±. Sketch, in a three-dimensionsal perspective view, the surface for /(v) as a function of i>n and v±.
6.3 For the loss-cone distribution function of problem 5.3 (in Chapter 5), show that
= s/2°2— ( —) exP
6.7 Consider (5.6.4), which is the solution of the Boltzmann equation with the relaxation model for the collision term, in the absence of external forces and spatial gradients, and when /Q0 and the relaxation time r are time-independent. Show that, according to this simplified equation, we have
Ga(t) = Ga0 + [Ga(0) - GaQ] exp {-t/r)
where
Ga(t) = / fa X d3v = na < x >q
J V
Ga0 =      fa0X d3V = Ua < X >a0 J v
Thus, according to the relaxation model for the collision term, every average value < x >a approaches equilibrium with the same relaxation time.
/a(v, t) = fa0 + [/a(v, 0) - /q0] e"'
(6.4)