Abstract
We demonstrate how to derive Maxwell’s equations, including Faraday’s law and Maxwell’s correction to Ampère’s law, by generalizing the description of static electromagnetism to dynamical situations. Thereby, Faraday’s law is introduced as a consequence of the relativity principle rather than an experimental fact, in contrast to the historical course and common textbook presentations. As a by-product, this procedure yields explicit expressions for the infinitesimal Lorentz and, upon integration, the finite Lorentz transformation. The proposed approach helps to elucidate the relation between Galilei and Lorentz transformations and provides an alternative derivation of the Lorentz transformation without explicitly referring to the speed of light.
Highlights
The theory of electromagnetism, usually taught after a course on mechanics, introduces as a new concept the electromagnetic field
An electromagnetic field can be considered as the coexistence of two independent physical entities, namely an electric field generated by a static charge density ρ(r) and a magnetic field generated by a stationary and divergence-free charge current density j(r) with ∇ · j(r) = 0
If we switch to the rest frame S′ of the point charge with the help of the Galilei transformation with u = v, we find that neither the electric field nor the magnetic field contributes to the Lorentz force, F′ = 0, in contradiction to Eq (11)
Summary
The starting point for deriving Maxwell’s equation is the description of static electromagnetism, summarized by the static field equations (in SI units). An electromagnetic field can be considered as the coexistence of two independent physical entities, namely an electric field generated by a static charge density ρ(r) and a magnetic field generated by a stationary and divergence-free charge current density j(r) with ∇ · j(r) = 0. We make the charge density ρ, the current density j, the electric field strength E, and the magnetic field strength B time dependent. The Lorentz force is still given by Eq (2), but with time-dependent electric and magnetic field strengths. A much stronger argument to include them comes, from checking the consistency with fundamental principles
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