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Derivation of the Field Due to a Magnetic Dipole Without Use of the Vector Potential

Proctor, John E.; Gould, Harvey Thomas

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Authors

Harvey Thomas Gould



Abstract

The mathematical form of the magnetic field due to a current loop, and the fact that it is identical to the electric field due to an electric dipole in the far field, are fundamental to our understanding of electromagnetism. While undergraduate level electromagnetism textbooks usually derive the electric field from an electric dipole, few derive the magnetic field from a current loop. Most simply state it without proof, or perform the derivation for simpler cases such as the on-axis field. Those that perform the derivation use the magnetic vector potential, a relatively advanced concept that most undergraduate students would not encounter until their final year of study, if at all. Here, a simple derivation to obtain the magnetic field due to a current loop in the far-field approximation is presented. The derivation begins from the Biot–Savart law and does not require the vector potential. The problem is set up so that only a single integration is necessary (from angle α = 0 to α = 2π around the current loop), and the result is compared with that for the electric field surrounding an electric dipole.

Citation

Proctor, J. E., & Gould, H. T. (2023). Derivation of the Field Due to a Magnetic Dipole Without Use of the Vector Potential. Physics Teacher, 61(1), 40-42. https://doi.org/10.1119/5.0077127

Journal Article Type Article
Acceptance Date Apr 1, 2022
Online Publication Date Dec 28, 2022
Publication Date Jan 1, 2023
Deposit Date Jan 5, 2023
Publicly Available Date Jan 5, 2023
Journal The Physics Teacher
Print ISSN 0031-921X
Electronic ISSN 1943-4928
Publisher AIP Publishing
Volume 61
Issue 1
Pages 40-42
DOI https://doi.org/10.1119/5.0077127
Keywords General Physics and Astronomy, Education
Publisher URL https://doi.org/10.1119/5.0077127
Additional Information Additional Information : This article may be downloaded for personal use only. Any other use requires prior permission of the author and AIP Publishing. This article appeared in John E. Proctor and Harvey Thomas Gould , "Derivation of the Field Due to a Magnetic Dipole Without Use of the Vector Potential", The Physics Teacher 61, 40-42 (2023) https://doi.org/10.1119/5.0077127 and may be found at https://aapt.scitation.org/doi/10.1119/5.0077127

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