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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 $$v$$.