Physicist Solves Electric Field and Charge Density Together as One Nonlinear Problem

A theoretical paper published in the journal Discover Fluid Mechanics on 11 May 2026 proposes a way of solving electrostatic problems that departs from the textbook tradition. Instead of taking either a charge distribution or an electric field as given input and computing the other, Peter Vadasz formulates the stationary equilibrium of a continuously distributed charge so that both the electric field and the charge density emerge together as parts of the solution. The price of this generality is mathematical: a problem that is normally linear becomes nonlinear, and the analysis shows that two kinds of equilibrium can exist in principle, only one of which the author argues can be physically real.

The starting point is a line of work the author published in Physics of Fluids in 2025, in which an electric charge distributed continuously in space is treated as an inviscid compressible fluid. Under that postulate, the charge has mass density as well as charge density, and its motion obeys an inviscid, compressible form of the Navier–Stokes momentum equation. Maxwell’s equations and the charge and mass continuity equations supply the kinematic constraints, while the momentum equation governs the dynamics. The formulation applies to continuously distributed charges moving freely in space or carried by a fluid, connecting to magneto-fluid-dynamics research.

The mathematical core of the paper is a proof about signs. Within the volume occupied by a distributed charge that creates a field, the product of charge density and electric field is always positive: where the charge density is positive the field points outward positively, and where it is negative the field is negative. This follows directly from Coulomb’s law in field form, which relates the divergence of the electric field to the charge density, combined with the fact that the sign of the charge does not change within the distributed region. The same reasoning extends to a test charge outside the region, whose product with the field is positive if it carries the same sign as the source charge and negative otherwise.

At a stationary equilibrium the charge velocity is zero, and neglecting gravity, which the author notes is vastly weaker than electromagnetic effects, the momentum equation reduces to a balance between the pressure gradient and the electrostatic force term. Introducing a linear relation between pressure and mass density, with a constant propagation speed for pressure waves, and assuming a constant ratio of mass density to charge density, the problem collapses to a single nonlinear ordinary differential equation for the electric field under spherical symmetry. The author notes this equation can be converted into the familiar Liouville equation and may admit multiple solutions.

One particular solution can be written down analytically. For a freely distributed charge of either sign, the radial electric field at equilibrium falls off as one over the distance from the center, scaled by the mass-to-charge density ratio and the square of the pressure-wave speed. The corresponding charge density falls off as one over the distance squared, carrying the sign of the source charge. The electrostatic force within the distributed charge acts outward, a repulsion pushing inner layers away from the center, while by spherical symmetry the outer layers exert no net force on the inner ones.

The sign convention carries real weight in this framework. The Lorentz force terms in the momentum equation are written here with a negative sign, opposite to the gravitational term, contrary to what the author identifies as the mainstream convention in plasma dynamics and magneto-fluid-dynamics. The paper argues that adopting the conventional positive sign would produce a charge density solution with the wrong sign, an impossible inconsistency, and demonstrates that the negative sign is necessary. The author notes that in prior works using the other convention the derivations and conclusions are not affected by the sign change.

The paper then applies the framework to a configuration resembling the electron-proton system of a hydrogen atom: a continuously distributed negative charge surrounding a distributed positive charge confined to a nucleus, with the electron cloud behaving as an inviscid compressible fluid. Solving the equilibrium equations with spherical symmetry yields an analytic electron charge density that falls off as one over the distance squared and is everywhere negative, together with an electron electric field that combines an inverse-square term from the proton with an inverse-distance term.

One consequence of the solution is that the total electron charge integrated over an infinite domain would diverge, so the distribution must terminate at a finite outer radius. Requiring the integrated charge to equal one electron charge relates that outer radius to the pressure-wave speed. If the outer radius is taken as twice the Bohr radius, about 1.06 times ten to the minus ten meters, the implied wave speed is about 1.093 million meters per second. The analysis also shows that the combined force per unit volume on the electron, due to both its own field and the proton field, points inward toward the nucleus, an attractive result the author describes as expected.

Beyond the static solutions, the paper derives the wave equations for the electromagnetic field consistent with these equilibria. The magnetic field satisfies the classical wave equation, while the electric field obeys a wave equation coupled to the charge density through a nonlinear term. Oscillations around the non-trivial equilibrium are therefore possible, and the author flags the stability of these stationary equilibria and the resulting oscillatory solutions as a task for future study. Oscillations around the trivial equilibrium, by contrast, are rejected on physical grounds.

The rejection of the trivial equilibrium is the paper’s most striking physical argument. If the equilibrium values of the field and charge density were both zero, any oscillation would drive the charge density and electric field through zero, forcing them to change sign. An electron-like charge would periodically become positive and back again, a charge conjugation built into the oscillation. The author argues such behavior has never been observed and is highly unlikely: an electron can annihilate with a positron, producing gamma particles at low energy, but does not oscillate into a positron and back. The trivial equilibrium is therefore discarded as non-realistic, leaving the non-trivial equilibrium as the only viable stationary state, should it prove unique. The work remains theoretical, with no experimental verification reported, and its assumptions, including the fluid treatment of charge and the constant mass-to-charge density ratio, define the scope within which these analytic equilibria hold.

Subject of Research: Theoretical equilibrium solutions for the electric field and charge density of a continuously distributed electric charge treated as an inviscid compressible fluid.

Article Title: Equilibrium solutions for the electric field and charge density of a continuously distributed charge

Article References: Vadasz, P. (2026). Equilibrium solutions for the electric field and charge density of a continuously distributed charge. Discover Fluid Mechanics, 2(1), Article 4. https://doi.org/10.1007/s44369-026-00008-4

Image Credits: AI Generated

DOI: 10.1007/s44369-026-00008-4

Keywords: electrostatics, charge density, electric field, Maxwell equations, Navier-Stokes, nonlinear equilibrium, electromagnetic fluid dynamics, hydrogen atom, Lorentz force, stationary equilibrium, spherical symmetry, theoretical physics