A general realistic treatment of the disk paradox

TL;DR

This paper generalizes the disk paradox in electromagnetism by demonstrating total angular momentum conservation in realistic finite systems, providing explicit formulas and a formalism applicable to educational settings.

Contribution

It introduces a comprehensive formalism expressing mechanical and field angular momentum in terms of charges and magnetic fluxes, extending previous idealized analyses to realistic finite systems.

Findings

Explicit conservation of total angular momentum in finite systems

Formulas relating angular momentum to charges and magnetic fluxes

Potential for improved educational demonstrations

Abstract

Mechanical angular momentum is not conserved in systems involving electromagnetic fields with non-zero electromagnetic field angular momentum. Conservation is restored only if the total (mechanical and field) angular momentum is considered. Previous studies have investigated this effect, known as "Feynman's Electromagnetic Paradox" or simply "Disk Paradox" in the context of idealized systems (infinite or infinitesimal solenoids and charged cylinders etc). In the present analysis we generalize previous studies by considering more realistic systems with finite components and demonstrating explicitly the conservation of the total angular momentum. This is achieved by expressing both the mechanical and the field angular momentum in terms of charges and magnetic field fluxes through various system components. Using this general expression and the closure of magnetic field lines, we demonstrate explicitly the conservation of total angular momentum in both idealized and realistic systems (finite solenoid concentric with two charged cylinders, much longer than the solenoid) taking all the relevant terms into account including the flux outside of the solenoid. This formalism has the potential to facilitate a simpler and more accurate demonstration of total angular momentum conservation in undergraduate physics electromagnetism laboratories.