Kirchhoff's Current Law with Displacement Current

TL;DR

This paper revisits Kirchhoff's Current Law, emphasizing the role of displacement current in high-speed circuits and biological ionic channels, and analyzes nanosecond switching using quantum mechanics.

Contribution

It extends Kirchhoff's law to include displacement current explicitly and applies quantum mechanical analysis to fast electronic switching.

Findings

Displacement current is integral to Kirchhoff's law in high-speed circuits.

Kirchhoff's current remains constant in biological ionic channels.

Quantum analysis reveals insights into nanosecond switching behavior.

Abstract

Kirchhoff's Current Law is an essential tool in the design of circuits that operate very quickly, faster than nanoseconds. But Kirchhoff's current is often identified as the flow of particles. The continuity equation or the Maxwell-Ampere law shows that the sum of displacement current $\textbf{plus}$ particle current is conserved by Maxwell's equations and Kirchhoff's law. Kirchoff included the displacement current in the current of his law, from early on. This Kirchhoff current (including the displacement current) does not vary with spatial location in the ionic channels of biology. Electronic circuits switching in nanoseconds are analyzed using the Bohm representation of quantum mechanics including particle and displacement current.