Magnetic mirror / electron motion and velocity

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(F. F. ~ Chen 2.20) The magnetic field along the axis of a magnetic mirror is \(B_(z) = B_0(1+\alpha^2z^2)\), where \(\alpha\) is a constant. Suppose that at \(z=0\) an electron has velocity \(v^2 = 3 v_{\parallel}^2 = \frac{3}{2}v_{\perp}^2\).

(a) Describe qualitatively the electron motion.

(b) Determine the values of \(z\) where the electron is reflected.

(c) Write the equation of motion of the guiding center for the direction parallel to \(\vec{B}\) and show that there is a sinusoidal oscillation. Calcule the frequency of the motion as a function of \(v\).