# Magnetic mirrors / Fermi acceleration

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(F. F. Chen ~ 2.12, Fermi acceleration of cosmic rays).

A cosmic ray proton is trapped between two moving magnetic mirrors with mirror ratio $$R_m=5$$. Initially its energy is $$W=1$$ keV and $$v_\perp = v_\parallel$$ at the midplane. Each mirror moves toward the midplane with a velocity $$v_m=10$$ km/s and the initial distance between the mirrors is $$L=10^{10}$$ km.

(a) Using the invariance of $$\mu$$, find the energy to which the proton is accelerated before it escapes.

(b) How long does it take to reach that energy? Suggestions: i) suppose that the $$B$$ field is approxiamtely uniform in the space between the mirrors and changes abruptly near the mirrors, i.e., treat each mirror as a flat piston and show that the velocity gained at each bounce is $$2v_m$$; ii) compute the number of bounces necessary; iii) assume that the distance between the mirrors does not change appreciably during the acceleration process.