A derivation of Maxwell’s equations using the Heaviside notation

Hampshire, D.P.

Abstract

Maxwell's four differential equations describing electromagnetism are among the most famous equations in science. Feynman said that they provide four of the seven fundamental laws of classical physics. In this paper, we derive Maxwell's equations using a well-established approach for deriving time-dependent differential equations from static laws. The derivation uses the standard Heaviside notation. It assumes conservation of charge and that Coulomb's law of electrostatics and Ampere's law of magnetostatics are both correct as a function of time when they are limited to describing a local system. It is analogous to deriving the differential equation of motion for sound, assuming conservation of mass, Newton's second law of motion and that Hooke's static law of elasticity holds for a system in local equilibrium. This work demonstrates that it is the conservation of charge that couples time-varying E-fields and B-fields and that Faraday's Law can be derived without any relativistic assumptions about Lorentz invariance. It also widens the choice of axioms, or starting points, for understanding electromagnetism.

Citation

Hampshire, D. (2018). A derivation of Maxwell’s equations using the Heaviside notation. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 367(2134), Article 20170447. https://doi.org/10.1098/rsta.2017.0447

• Journal Article Type: Article
• Acceptance Date: Jun 4, 2018
• Online Publication Date: Oct 29, 2018
• Publication Date: Dec 13, 2018
• Deposit Date: Jul 16, 2015
• Publicly Available Date: Sep 17, 2018
• Journal: Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences.
• Print ISSN: 1364-503X
• Electronic ISSN: 1471-2962
• Publisher: The Royal Society
• Peer Reviewed: Peer Reviewed
• Volume: 367
• Issue: 2134
• Article Number: 20170447
• DOI: https://doi.org/10.1098/rsta.2017.0447
• Public URL: https://durham-repository.worktribe.com/output/1402312