Weak Maxwell does not authorize quadratic self-energy

Given: A nonempty open time interval $I$, a bounded Lipschitz polyhedron $\mathrm{\Omega }\subseteq {\mathbb{R}}^3$ containing $x_0$, a point charge $q\mathrm{\ne }0$ in coulombs, a position $x_0$ in metres, and ${\epsilon }_0>0$ in $\mathrm{F}{\mathrm{m}}^{-1}$. The ideal field $E_q$ has units $\mathrm{V}{\mathrm{m}}^{-1}$.

$E_q(x)=\frac{q(x-x_0)}{4\pi {\epsilon }_0\mathrm{\mid }x-x_0{\mathrm{\mid }}^3}(x\mathrm{\ne }{x}_0)\mathrm{.}$

For a stationary point source, $E_q$ is locally integrable, smooth off $x_0$, and satisfies $\mathrm{div}{E}_q=(q\mathrm{/}{\epsilon }_0){\delta }_{x_0}$ and $\mathrm{curl}{E}_q=0$ distributionally; the value assigned at $x_0$ is immaterial. The localized Newtonian potential and the ideal point-charge Coulomb field

For $0<\delta <R$, the truncated ordinary electric-field energy is

$\frac{{\epsilon }_0}{2}{\int }_{\{\delta \le \mathrm{\mid }x-x_0\mathrm{\mid }\le R\}}\mathrm{\mid }{E}_q(x){\mathrm{\mid }}^{2}d{\lambda }_3(x)=\frac{q^2}{8\pi {\epsilon }_0}(\frac{1}{\delta }-\frac{1}{R}),$

and this tends to $+\mathrm{\infty }$ as $\delta \downarrow 0$ when $q\mathrm{\ne }0$. Divergence of the point-charge field energy and the limits of mass inference

The finite-energy boundary model requires $E(t),H(t)\in L^2(\mathrm{\Omega };{\mathbb{R}}^3)$ and defines its energy by the corresponding finite quadratic integral. Finite-energy PEC and impedance boundary formulation

The adopted distributional vacuum Maxwell equations apply to the point-source model with $\mathrm{div}E=\rho \mathrm{/}{\epsilon }_0$, $\mathrm{div}B=0$, $\mathrm{curl}E=-{\mathrm{\partial }}_{t}B$, and $\mathrm{curl}B={\mu }_{0}J+c^{-2}{\mathrm{\partial }}_{t}E$; ${\mu }_0$ has units $\mathrm{H}{\mathrm{m}}^{-1}$ and $c$ has units $\mathrm{m}{\mathrm{s}}^{-1}$. Distributional vacuum Maxwell equations for the declared point and stationary-sheet sources Framework-local SI units, the vacuum constants and the exact/measured status of their values

A stationary point charge has $\rho =q{\delta }_{x_0}$ and $J=0$; $q$ is measured in coulombs under the framework's point-source and SI conventions. Distributional Maxwell equations with point and stationary-sheet sources Framework-local SI units, the vacuum constants and the exact/measured status of their values

$E$ and $B$ are the electric and magnetic spatial vector fields of the framework, measured in $\mathrm{V}{\mathrm{m}}^{-1}$ and tesla. The electric and magnetic fields as fixed-time spatial vector fields Framework-local SI units, the vacuum constants and the exact/measured status of their values