The exact relation between the displacement current and the conduction current: Comment on a paper by Griffiths and Heald

TL;DR

This paper derives an exact relation between displacement current and conduction current, revealing local and non-local components, and clarifies how displacement current can serve as a surrogate for conduction currents at different locations.

Contribution

It provides the first exact mathematical expression linking displacement and conduction currents, including local and retarded non-local terms, clarifying their physical relationship.

Findings

Displacement current includes local and non-local terms.

Non-local term involves retarded values of conduction current.

Displacement current can act as a surrogate for conduction currents at other points.

Abstract

I introduce the exact relation between the displacement current $\epsilon_0 \partial \v{E}/\partial t$ and the ordinary current $\v{J}$. I show that $\epsilon_0 \partial \v{E}/\partial t$ contains a local term determined by the present values of $\v{J}$ plus a non-local term determined by the retarded values of $\v{J}$. The non-local term implements quantitatively the suggestion made by Griffiths and Heald that the displacement current at a point is a surrogate for ordinary currents at other locations.