Conservation of Magnetic Moment for Charged Particle Motion in a Time-Dependent Uniform Magnetic Field

The adiabatic magnetic moment invariant for the motion of a charged particle in a spatially uniform, time-dependent magnetic field B(t) is studied numerically. The robustness of the magnetic moment invariant $\mu $ = mv$_{\bot }^{2}$/2B is explored for slowly varying and rapidly varying magnetic field B(t), for a charged particle moving in a long solenoid with time-varying current. The effects of various functional forms of B(t) on the conservation of the adiabatic magnetic moment invariant are examined numerically. In the case of a slowly varying magnetic field, where the time-scale of the change in the magnetic field is much larger than the particle gyroperiod, it is shown numerically that the adiabatic magnetic moment $\mu $ is asymptotic to a recently discovered exact magnetic moment invariant M, which is conserved even for rapidly varying magnetic fields [1, 2]. [1] R.C. Davidson and H. Qin, \textit{Physics of Intense Charged Particle Beams in High Energy Accelerators} (World Scientific, 2001). [2] H. Qin and R.C. Davidson, An Exact Magnetic Moment Invariant of Charged Gyromotion, submitted for publication (2005).