Charge densities that change linearly with time are effectively electrostatic

Abstract

We consider classical electromagnetic systems in which the current density $\mathbf{J}$ 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.

After posting this article, I discovered that its result was already reported in Griffiths and Heald, Am. J. Phys. 59, 111 (1991)