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.This article is part of the theme issue ‘Celebrating 125 years of Oliver Heaviside's ‘Electromagnetic Theory’’.
Highlights
This research paper is written in the celebration of 125 years of Oliver Heaviside’s work Electromagnetic
Scientists are well versed in using static laws to derive time-dependent partial differential equations
Given we already have Maxwell’s equations (3.5) and (3.24), we can take them in primed form together with the primed version of (3.31) and the mathematical identity (3.24) to derive the wave-equation for E propagating through vacuum (i.e. ρ = 0 and J = 0) which is of the form ∇ 2E − μ0ε0∂2E /∂t2 = 0
Summary
This research paper is written in the celebration of 125 years of Oliver Heaviside’s work Electromagnetic. Students of electromagnetism are introduced to Maxwell’s equations and taught that they are generally true, not least because of the overwhelming body of experimental data that validates them Do they describe the E-fields and B-fields from charges and currents in vacuum but by considering the charges and currents produced in materials, they describe the fields produced by all the important technologically useful materials and an enormous range of physical phenomena in the world around us. Undergraduate textbooks derive the electrostatic and magnetostatic differential equations mathematically from Coulomb’s Law and Ampère’s Law. to arrive at Maxwell’s time-dependent equations, students follow the heuristic approach. We speculate about possible sources of experimental evidence for the breakdown of Maxwell’s equations
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