Time evolution for different geometrical configurations of charged particles in a time-varying magnetic field

Through rescaling and obtaining invariant equations, we study the expansion of charged particles immersed in a time-varying magnetic field B(t) = B₀(1+Ωt)^-β. We develop the theory for a configuration of point particles confined in a plane with an initial cylindrical symmetric configuration. It is found that the system expands according to (1+Ωt)^2β/3 for β ≤ 1 and it is conjectured that in a properly rescaled space and time, it reaches asymptotically an inhomogeneous (in space) Brillouin flow with rigid rotation at half the cyclotron frequency (ω = ½ω_c). Particular attention is given to the ‘pivot’ case β = 1. It is shown then that the radius of the rescaled configuration is now dependent on the ratio Ω/ω_c, and grows indefinitely with time for Ω/> 3.2^-3/2. Numerical experiments are presented which confirm these theoretical predictions and conjectures. Finally, the properties of this system are compared with those of single-particle and ‘rod’ electrostatic models.