The vector potential of a steady azimuthal current density. Once again.

Authors

DOI:

https://doi.org/10.31349/RevMexFis.23.020210

Keywords:

Magnetostatics, Vector potential, Azimuthal current density

Abstract

We give an integral expression for the vector potential of a time-independent, steady azimuthal current density. Our derivation is substantially simpler and somewhat more general than others given in the literature. As an illustration, we recover the results for the vector potential of a circular current loop as an orthogonal expansion in spherical and cylindrical coordinates. Additionally, we obtain closed analytical expressions for the vector potential and the magnetic induction of a circular current loop in terms of Legendre functions of the second kind, that are simpler than the results in terms of complete elliptic integrals given in textbooks.

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References

J. D. Jackson, Classical Electrodynamics, 3rd Edition, (John Wiley & Sons, New York, N.Y. 1999)

A. Zangwill, Modern Electrodynamics, Cambridge Univ. Press, Cambridge, UK (2012)

A. Vasilyev, Vector Potential and Magnetic Field of Axially Symmetric Currents, arXiv:1208.4983 [physics.class-ph] (2012), https://doi.org/10.48550/arXiv.1208.4983

Wolfram Research Inc., Mathematica, Version 14.2, Champaign, IL (2025)

A. Erdelyi et al., Higher Trascendental Functions, Vol. II, McGraw-Hill, New York, NY, (1953)

Wikipedia contributors, Elliptic integral, in Wikipedia, The Free Encyclopedia, (2025), https://en.wikipedia.org/wiki/Elliptic_integral

A. Erdelyi et al., Tables of Integral Transforms, Vol. I, McGraw-Hill, New York, NY, (1954)

A. Erdelyi et al., Higher Trascendental Functions, Vol. I, McGraw-Hill, New York, NY, (1953)

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Published

2026-07-01

How to Cite

[1]
A. Bouzas, “ Once again”., Rev. Mex. Fis. E, vol. 23, no. 2, pp. 1–6, Jul. 2026.

Issue

Vol. 23 No. 2 (2026): Revista Mexicana de Física E

Section

Education in Physics

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Copyright (c) 2026 Antonio Bouzas

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