Classical Electrodynamics: A Tutorial on its Foundations

TL;DR

This tutorial systematically derives Maxwell's equations from fundamental axioms of charge conservation, Lorentz force, and magnetic flux conservation, emphasizing the foundational structure of classical electrodynamics.

Contribution

It provides a clear derivation of Maxwell's equations from basic physical principles and discusses constitutive assumptions needed for a complete system.

Findings

Maxwell's equations derived from fundamental axioms

Discussion of constitutive relations and metric dependence

Clarification of the foundational structure of electrodynamics

Abstract

We will display the fundamental structure of classical electrodynamics. Starting from the axioms of (1) electric charge conservation, (2) the existence of a Lorentz force density, and (3) magnetic flux conservation, we will derive Maxwell's equations. They are expressed in terms of the field strengths $(E,{\cal B})$, the excitations $({\cal D},H)$, and the sources $(\rho,j)$. This fundamental set of four microphysical equations has to be supplemented by somewhat less general constitutive assumptions in order to make it a fully determined system with a well-posed initial value problem. It is only at this stage that a distance concept (metric) is required for space-time. We will discuss one set of possible constitutive assumptions, namely ${\cal D}\sim E$ and $H\sim {\cal B}$. {\em file erik8a.tex, 1999-07-27}