Charge densities that change linearly with time are effectively electrostatic

TL;DR

This paper demonstrates that in certain classical electromagnetic systems with time-varying charge densities, the electromagnetic fields can be exactly computed using Coulomb and Biot-Savart laws evaluated at the current time, simplifying analysis.

Contribution

It reveals that for systems with steady current but changing charge density, the fields depend on present-time sources, providing an exact solution framework.

Findings

Fields are given by Coulomb and Biot-Savart laws at current time.

Charge density can change linearly with time while fields depend on present sources.

Provides exact electromagnetic field expressions for specific charge dynamics.

Abstract

We consider classical electromagnetic systems in which the current density ${\bf J}$ does not change with time, but has nonzero divergence, so that electric charge does build up steadily over time. We show that for such a system, the \emph{exact} electromagnetic fields are given by the Coulomb and Biot-Savart laws with the sources evaluated at the \emph{present} (rather than retarded) time, even though the charge density is changing over time.