Two magnetic mirrors of fixed throat field strength converge slowly
PHYSICS · Subjective · Fermi University Physics · Plasma Physics · The plasma state and single-particle motion · The plasma state and single-particle motion
Plasma Physics, Ch Slow changes and what survives them: adiabatic invariants, mirrors and trapped orbits
Problem
Two magnetic mirrors of fixed throat field strength converge slowly, halving the gap between them repeatedly while the field at the centre stays unchanged. The mirror ratio between the throats and the centre therefore stays fixed at Rmir=4 throughout. A proton bounces between the mirrors, starting (before any halving) with pitch angle α0=60∘ at the centre. Because B at the centre is unchanged, μ conservation keeps v⊥ (evaluated at the centre) fixed at each halving; because the bounce path shortens, conservation of the longitudinal invariant J=∮mv∥ds requires v∥ (at the centre) to double at each halving of the gap. Find after how many halvings of the gap the proton's pitch angle at the centre first falls inside the loss cone, so that it escapes.