Force on a point charge source of the classical electromagnetic field

It is shown that a well-defined expression for the total electromagnetic force ${\mathit{f}}^{\mathrm{em}}$ 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. Among 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-Landé-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 $m_{\mathrm{b}}\ne 0$ is postulated, Newton’s law $\frac{\mathrm{d}}{\mathrm{dt}}\mathit{p}=\mathit{f}$ with $\mathit{f}={\mathit{f}}^{\mathrm{em}}$ 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 with point charges.

Figure 1

The velocity of a point charge, starting from rest in a strong, constant applied electrostatic field ${\mathcal{E}}^{\mathrm{hom}}$, as per test particle theory (dashed curve), and as per BLTP electrodynamics with $O({\varkappa }^2)$ friction only (continuous curve).

Figure 2

The velocity of a point charge, starting from rest in a very weak, constant applied electrostatic field ${\mathcal{E}}^{\mathrm{hom}}$, as per test particle theory (dashed curve) and as per BLTP electrodynamics with $O({\varkappa }^2)$ friction only (continuous curve).

Figure 3

The velocity of a point charge, starting from rest in a constant applied electrostatic field ${\mathcal{E}}^{\mathrm{hom}}$ of slightly larger-than-critical field strength, as per test particle theory (dashed curve) and as per BLTP electrodynamics with $O({\varkappa }^2)$ friction only (continuous curve).