The electrons inside a system of two coaxial magnetic mirrors can be described by the so-called loss-cone distribution function

where v_∥ and v_⊥ denote the magnitudes of the electron velocities in the directions parallel and perpendicular to the magnetic bottle axis, respectively, and where α_∥² = 2kT_∥/m and α_⊥² = 2kT_⊥/m.

(a) Verify that the number density of the electrons in the magnetic bottle is given by n₀.

(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 f(v) as a function of v_∥ and V_⊥.

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f(v) = 2 TV/2no, () exp [-(-_) - (2)