The structure of the electromagnetic field as derived from first principles

The physical origin of the coupling between the electromagnetic field and electrically charged particles is rooted in quantum physics, and is thus not directly accessible to classical electrodynamics. We provide a physical discussion of the electromagnetic potentials that bridges this gap. Starting from an examination of the Aharonov-Bohm effect, we recognize the fundamental role of the electromagnetic potentials in the coupling process, and recover the mathematical origin of gauge invariance in electrodynamics. Then, we identify the dynamical components of the electromagnetic field, give an account of the concept of gauge fixing, and finally discuss solutions of Maxwell's equations in terms of gauge potentials.