Delayed radiation reaction effects on the stabilization of driven cyclotron motion and the magnetic response of an extended charge system

Abstract

This study investigates the cyclotron motion of a uniformly charged spherical shell confined in a harmonic potential and driven by an oscillating electric field in the presence of a steady magnetic field. The analysis focuses on the orbital angular momentum and the resulting magnetic response in the steady state, where radiative energy dissipation is exactly balanced by external energy input. By formulating the dissipation term using velocity differences at distinct times instead of the conventional third-order time derivative, the unphysical runaway solutions associated with the Abraham–Lorentz force are eliminated. For sufficiently small delay times, the extended-charge model smoothly reduces to the point-charge limit. In contrast, as the delay time increases, the system exhibits a distinct resonance behavior characterized by multiple reversals of the rotational direction as a function of the driving frequency. Furthermore, the proposed approach enables a detailed analysis of the transient dynamics leading to a stable limit cycle by solving a delay differential equation, offering new insights into the stability of dynamical systems with delayed interactions.