Kirchhoff`s Forgotten Contributions to Electromagnetism: Continuity Equation versus Displacement Currents

TL;DR

This paper revisits Kirchhoff's 1857 work, highlighting his early use of the continuity equation in electromagnetism, predating Maxwell's displacement current concept, and argues for the fundamental role of the continuity equation in field dynamics.

Contribution

The paper demonstrates that Kirchhoff's 1857 work introduced the continuity equation for electromagnetic propagation, emphasizing its fundamental importance over the displacement current concept.

Findings

Kirchhoff's 1857 work predates Maxwell's displacement current.

The continuity equation is more fundamental than displacement current.

Field dynamics rely on the continuity equation of matter.

Abstract

In 1857, Kirchhoff published two seminal papers on the motion of electricity in wires. In that work, he was the first to derive what we now call the telegrapher`s equations, which describes the propagation of electromagnetic signals along a cable at the speed of light, in some conditions. How was Kirchhoff able to describe electromagnetic propagation as early as 1857, when the notion of displacement current which is believed to be the essential ingredient for the propagation of electric and magnetic fields was not introduced by Maxwell until 1861 and fully explained later in 1865? In this paper, we show that Kirchhoff was the first, in his 1857 paper, to introduce the continuity equation when discussing electromagnetic propagation. We argue that the continuity equation used by Kirchhoff is a more fundamental concept now as it was in 1865 than displacement current. In our view, the dynamics of charged particles can be formulated without invoking fields, but the dynamics of fields cannot be formulated without a continuity equation describing the properties of matter.