Intrinsic symmetry of Ampère’s circuital law and other educational issues

Abstract

This paper explores Ampère's circuital law (ACL) from an educational perspective. The interchangeability of the ampèrian loop with the current loop, an intrinsic symmetry of ACL that is seldom addressed in the literature or textbooks, is illustrated here. It is verified that the symmetry axis of a circular current is an ampèrian loop. The attempt to apply ACL to a finite wire, a common source of student misunderstanding, is used to highlight the limitations of ACL. The generalisation of ACL is illustrated using an instructive example where the displacement current is unconfined and not spatially uniform. This work is primarily intended for teachers and more advanced undergraduate students, who may benefit from the ideas that are presented here.PACS Nos: 03.50. De, 01.40.Fk Résumé : Nous réétudions ici le théorème d'Ampère, dans une perspective pédagogique. Nous illustrons l'interchangeabilité de la boucle d'Ampère et la boucle de courant, une symétrie intrinsèque du théorème qui est rarement discutée dans la litté-rature ou les manuels. Nous vérifions que l'axe de symétrie d'un courant circulaire est une boucle d'Ampère. Afin d'illustrer les limitations du théorème, nous utilisons l'application du théorème à un fil de longueur finie, une source fréquente de malentendus chez les étudiants. Nous illustrons la généralisation du théorème d'Ampère à l'aide d'un exemple pédagogique où le courant de déplacement n'est pas confiné ni spatialement uniforme. Ce travail vise surtout les enseignants et les étudiants des 2 e et 3 e cycles qui peuvent profiter des idées présentées ici.[Traduit par la Rédaction]

Ampère–Maxwell law for a conducting wire: a topological perspective

“…Previously, [7,11] used a simple example of two opposite charges placed at the ends of a straight conducting segment to illustrate Ampère-Maxwell's law (referred to in [11] as the generalized Ampère circuital law). Abandoning the symmetry restrictions of [7,11], consider an arbitrarily-shaped conducting wire connecting two charges +Q and −Q (Q > 0), as illustrated in figure 1. At any point P on the conducting wire, a solid angle is subtended by an arbitrarily-shaped closed curve C. As explained in figure 1, such a choice of solid angle is not unique.…”

“…see [10] and references therein. In a recent study, Anacleto et al [11] explored the intrinsic symmetry of Ampère's circuital law from an educational perspective and illustrated the concept of displacement current. Although some concepts from topology were used within the context of the examples treated therein, Anacleto et al fell short of recasting Ampère-Maxwell's law from a topological perspective and, as far as is known to us, this remains a gap in the existing literature.…”

Comment on ‘Exact electromagnetic fields produced by a finite wire with constant current’

“…JCA also conclude that this non-magnetostatic problem is one of the few cases where an exact solution through retarded potentials can be found. Given the relevance of JCA's analysis of the finite wire problem both to earlier papers [2,3] and to subsequent ones [4][5][6][7][8], we revisited JCA's work, the aim of the present comment being to further reinforce their analysis by correcting a few points in their paper.…”