# The force on a point charge source of the classical electromagnetic field

## TL;DR

This paper derives a well-defined electromagnetic force on a point charge within classical field theories that satisfy certain regularity conditions, avoiding infinities and third-order derivatives present in traditional formulations.

## Contribution

It establishes a rigorous method to define the electromagnetic force on a point charge without infinities or ad hoc assumptions, applicable to certain nonlinear field equations.

## Key findings

- The force expression is derived from momentum conservation under regularity conditions.
- The approach avoids the third-order derivative problem of the Abraham-Lorentz-Dirac equation.
- It applies to nonlinear theories like Maxwell-Bopp-Lande-Thomas-Podolsky and Maxwell-Born-Infeld.

## Abstract

It is shown that a well-defined expression for the total electromagnetic force $f^{em}$ on a point charge source of the classical electromagnetic field can be extracted from the postulate of total momentum conservation whenever the classical electromagnetic field theory satisfies a handful of regularity conditions. Amongst these is the generic local integrability of the field momentum density over a neighborhood of the point charge. This disqualifies the textbook Maxwell-Lorentz field equations, while the Maxwell-Bopp-Lande-Thomas-Podolsky field equations qualify, and presumably so do the Maxwell-Born-Infeld field equations. Most importantly, when the usual relativistic relation between the velocity and the momentum of a point charge with bare rest mass $m_b \neq 0$ is postulated, Newton's law $\dot{p} = f$ with $f = f^{em}$ becomes an integral equation for the point particle's acceleration; the infamous third-order time derivative of the position which plagues the Abraham-Lorentz-Dirac equation of motion does not show up. No infinite bare mass renormalization is invoked, and no ad hoc averaging of fields over a neighborhood of the point charge. The approach lays the rigorous microscopic foundations of classical electrodynamics.