Charged particle in a magnetic field: from basic physics to Brownian motion

Abstract In university-level physics education, the motion of a charged particle in a magnetic field is typically restricted to idealized, non-viscous environments, focusing primarily on circular motion derived from the balance of magnetic and centripetal forces. This paper presents a more comprehensive approach based on Newton’s equations of motion, providing deeper physical insights and allowing for generalizations to more realistic scenarios. We investigate the particle’s dynamics in viscous media, accounting for Stokes friction alongside the Lorentz force, and explicitly calculate trajectories, displacements, and angular momentum. Furthermore, the framework is extended to Brownian motion using the classical Langevin equation with white thermal noise. Our findings reveal that the mean angular momentum (and consequently the magnetic moment) approaches a finite non-zero value in the long-time limit, leading to a discussion of how the results should be interpreted within the framework of classical statistical mechanics, in particular in relation to the assumptions underlying the Bohr–van Leeuwen theorem. This approach serves as an effective pedagogical tool, bridging concepts from classical mechanics, statistical physics, and stochastic processes while encouraging student debate on the foundations of classical magnetism.