PHYSICS · Multi-part · Fermi University Physics · Plasma Physics · The plasma state and single-particle motion · Fluid and magnetohydrodynamic description
Plasma Physics, Ch From particles to fluids: the distribution function, its moments and the two-fluid model
Problem
A cylindrical plasma column of initial radius r0, initial length L0, initial density n0 and initial temperature T0=100eV (so p0=n0T0) is compressed slowly compared with a gyration period and with a particle's transit time along the column, so that each particle's magnetic moment μ and longitudinal invariant J are conserved as appropriate, and fast compared with a collision time, so that collisions cannot share the compressional work among all three velocity directions. Two separate compressions are considered, starting from the same initial state.
((a)) (i) Radial compression: an increasing external axial field compresses the column radius from r0 to r1=r0/2 at fixed length L0 and fixed particle number, conserving each particle's μ. (ii) Axial compression: end pistons reduce the column length from L0 to L1=L0/4 at fixed radius r0 and fixed particle number, conserving each particle's J. State the density ratio n1/n0 produced by each compression, and, for each, state which value of N and hence which ratio of specific heats γs=(N+2)/N applies, justifying the choice from the invariant conserved and the number of velocity components that absorb the compressional work.
((b)) Using p∝nγs (equivalently T∝nγs−1) with the γs identified in part (a), find T1/T0 and p1/p0 for the radial compression (i) and for the axial compression (ii).
((c)) Check the radial result independently using flux conservation, B×(cross-sectional area)=const, together with conservation of μ. Then state, as a multiple, by how much the pressure rise in the axial compression would be under-predicted if the radial (perpendicular) closure were mistakenly applied to it instead of the correct one.