Lecture Notes on Maxwell's Equations

We introduce physical concepts of gradient, divergence, and curl, as a pre-requisite to understanding Maxwell’s equations. We then present the experimental laws of electricity and magnetism. These laws are described in terms of physical contents and their mathematical representations. Taken together, they suggest no new effects beyond the original experiments they represent. It is only when the displacement current is added that new physics emerges. This physics includes the prediction of the existence of electromagnetic waves which follow from Maxwell’s equations and transport energy and momentum through empty space by means of electromagnetic fields. We begin with electromagnetic wave equations in terms of electric scalar potential and magnetic vector potential, and then explain its physical meaning. Electromagnetic waves and their solutions are illustrated. Physical insight of magnetic vector potential is presented as electromagnetic momentum per unit charge of a test charge. We then describe spherical waves from a point source, electromagnetic waves in a dielectric medium, the complex refractive index, and finally the energy flow in the electromagnetic field.