Is magnetic moment variant or invariant in a plasma?

For many years one of the basic tenets of plasma physics has been the invariance, or constancy, of the magnetic moment of a charged particle in a magnetic field that varies slowly in time or space. However it is proposed here that this is invalid and that the magnetic moment is not constant: it is a function of the magnetic flux density. It is shown that there are contradictions within the conventional theory, and that this is due to a missing term in the derivation. A new equation for the variation of the magnetic moment in a collision free plasma is derived. The implication of this new equation for the loss cone in magnetic mirrors is considered. Introduction A charged particle moving in a magnetic field executes Larmor rotations, and so has a magnetic moment. The invariance of the magnetic moment of a particle in a collision free plasma, in a varying magnetic field, was first proposed by Alfven [1] in 1950, and appears in many texts and papers on Plasma Physics: see for example Boyd and Sanderson (2003) [2] or Yi et al (2005) [3]. A key consequence of this conventional theory is that the magnetic flux Φ, through the particle’s gyro-orbit must also be constant, as shown for example by Chen (1984) [4] or as indicated here in equation (6). The way that a charged particle moves in a varying magnetic field is generally considered in two ways. The first case is when the field is time varying only and the second when it is spatially varying only. In both cases the conventional theory concludes that the magnetic moment of the particle is constant, provided that the field changes slowly or adiabatically. A similar approach is adopted here – and in both cases it is suggested that a missing term in the conventional theory has produced an incorrect conclusion. Definitions and standard formulas The radius, r, of the Larmor orbit, where B is the magnetic field, m the mass of the particle and e is its charge is given by the standard formula: