A Steady Loop Current Does Not Radiate

TL;DR

This work addresses whether a steady loop current radiates by extending the Edwards–Kenyon–Lemon decomposition to arbitrarily shaped loops with a fluid-model charge distribution. It shows that the retarded electric and magnetic fields for a steady loop can be written as a sum of a static-field loop integral and an exact differential, and that the integral of the exact differential vanishes, yielding no radiation; the remaining terms reproduce Coulomb's law and Biot–Savart law in loop form. The key contribution is a rigorous, exact retarded-field treatment for general loop geometries, clarifying the role of retardation and static components in preventing radiation and providing a pedagogical framework for teaching retarded fields. This advances the classical understanding of steady currents by offering a precise decomposition and matching known static-law forms, with potential utility for teaching and deeper theoretical insight into electromagnetic retardation.

Abstract

According to classical electrodynamics, a steady loop current does not radiate. Edward, Kenyon, and Lemon show that the present-time approximation of the retarded electric field (the approximation of the magnetic one is trivial) of a steady loop current can be partitioned into a loop integral of a static field expression and a loop integral of an exact (also called total, full, or perfect) differential. Because the latter is zero, no radiation is emitted. Inspired by their work, we do the same for the retarded electric and magnetic fields without approximation, and show that for a steady loop current, the loop integrals of the static field expressions with and without approximation are exactly equal, recovering Coulomb's law and the Biot-Savart law for a steady loop current.

Figures (3)

• Figure 1: This paper in the context of current literature.
• Figure 2: (a) A circular steady loop current with labeled point charges that are evenly spaced. (b) The retarded positions of the labeled point charges seen by the observer at position $obs$. Plots are generated with the computer code listed in Section \ref{['sec:code']}. Note that if the moving charges are negative, $I$ takes negative value.
• Figure 3: A general loop carrying steady current $I$. Note that if the moving charges are negative, $I$ takes negative value. The loop is not necessarily in a plane. It can be any 3-dimensional closed loop. $O$ is the origin of the coordinate system and $obs$ is the position of the observer. (a) A present-time segment $\delta {\mathbf l}_k$ of the loop is labeled. (b) The corresponding retarded segment $\delta{\mathbf l}_k^\prime$ of the same loop, containing the same charge, is labeled. The offset between $\delta{\mathbf l}_k^\prime$ and $\delta {\mathbf l}_k$ are greatly exaggerated for normal situations. They are labeled on the same loop, not on two separate loops.