A proton gyrates in a field with speed across the field and speed along it

PHYSICS · Multi-part · Fermi University Physics · Mechanics & Dynamical Systems · Plasma Physics · The plasma state and single-particle motion

Action and Symmetry, Ch Adiabatic invariants, canonical perturbation theory and integrable systems

Problem

A proton gyrates in a field

B_0 = 0.010\,\mathrm{T}

with speed across the field

v_{\perp0} = 2.0\times10^5\,\mathrm{m\,s^{-1}}

and speed along it

v_{\parallel0} = 3.0\times10^5\,\mathrm{m\,s^{-1}}

. It moves along the field line towards a region where the field rises slowly to

B_1 = 0.040\,\mathrm{T}

. ((a)) Find the speed across the field that the proton would need at

B_1

to keep

\mathcal M = mv_\perp^2/2B

fixed, and compare it with the proton's total speed

v

. ((b)) Hence decide whether the proton reaches the region where the field is

B_1

. If it does not, find the field

B_{\rm turn}

at which it turns back.