On the interaction of point charges in an arbitrary domain

The electrostatic field energy due to two fixed pointlike charges shows some peculiar features concerning the distribution in space of the field energy density of the system. Here we discuss the evaluation of the field energy and the mathematical details that lead to those peculiar and non-intuitive physical features.

A novel energy minimization formulation of electrostatics that allows computation of the electrostatic energy and forces to any desired accuracy in a system with arbitrary dielectric properties is presented. An integral equation for the scalar charge density is derived from an energy functional of the polarization vector field. This energy functional represents the true energy of the system even in non-equilibrium states. Arbitrary accuracy is achieved by solving the integral...

Studies on nanoscale materials merit careful development of an electrostatics model concerning discrete point charges within dielectrics. The discrete charge dielectric model treats three unique interaction types derived from an external source: Coulomb repulsion among point charges, direct polarization between point charges and their associated surface charge elements, and indirect polarization between point charges and surface charge elements formed by other point charges. ...

We compute the force acting on a free, static electric charge outside a uniform dielectric sphere. We view this force as a self-interaction force, and compute it by applying the Lorentz force directly to the charge's electric field at its location. We regularize the divergent bare force using two different methods: A direct application of the Quinn-Wald Comparison Axiom, and Mode-Sum Regularization.

In this paper, we study the static Born-Infeld equation $$ -\mathrm{div}\left(\frac{\nabla u}{\sqrt{1-|\nabla u|^2}}\right)=\sum_{k=1}^n a_k\delta_{x_k}\quad\mbox{in }\mathbb R^N,\qquad \lim_{|x|\to\infty}u(x)=0, $$ where $N\ge3$, $a_k\in\mathbb R$ for all $k=1,\dots,n$, $x_k\in\mathbb R^N$ are the positions of the point charges, possibly non symmetrically distributed, and $\delta_{x_k}$ is the Dirac delta distribution centered at $x_k$. For this problem, we give explicit qua...

A simple mathematical procedure is introduced which allows redefining in an exact way divergent integrals and limits that appear in the basic equations of classical electrodynamics with point charges. In this way all divergences are at once removed without affecting the locality and the relativistic covariance of the theory, and with no need for mass renormalization. The procedure is first used to obtain a finite expression for the electromagnetic energy-momentum of the syste...

In this work, we provide the mathematical elements we think essential for a proper understanding of the calculus of the electrostatic energy of point-multipoles of arbitrary order under periodic boundary conditions. The emphasis is put on the expressions of the so-called self parts of the \Ewald\, summation where different expressions can be found in literature. Indeed, such expressions are of prime importance in the context of new generation polarizable force field where the...

In this work, we present an explanation of the electric charge quantization based on a semi-classical model of electrostatic fields. We claim that in electrostatics, an electric charge must be equal to a rational multiple of the elementary charge of an electron. However, the charge is quantized if the system has certain boundary conditions that force the wavefunction representing an electric field to vanish at specific surfaces. Next, we develop the corresponding model for th...

We calculate the self-force of a point charge in rectilinear motion, using a local method and compare our results with those from the literature.

In this pedagogical work we point out a subtle mistake that can be done by undergraduate or graduate students in the computation of the electrostatic energy of a system containing charges and perfect conductors if they naively use the image method. Specifically, we show that the naive expressions for the electrostatic energy for these systems obtained directly from the image method are wrong by a factor 1/2. We start our discussion with well known examples, namely, point char...