In this chapter, we study the motion of charges and electromagnetic waves. After studying static charges, uniformly moving charges, and the standard electrostatic method of the mirror image charges, we consider the multipole expansion of the electric and magnetic fields. The electric field is generated by monopole (electric charge) and higher multipole, and magnetic field by dipole and higher multipoles. Electromagnetic waves are then studied. For arbitrary moving charges, we calculate the retarded potentials, and in particular the Lienard–Wiechert forms. We then show that we need at least dipoles to generate electromagnetic waves. We end by describing Maxwell duality.

We study dualities in various dimensions. We start with the example of Maxwell duality in 3+1 dimensions from the point of view of a transformation with a master action. We continue with the particle–vortex duality in 2+1 dimensions, also through a master action, and then the particle–string duality in 3+1 dimensions. Finally, we describe the general Poincaré duality in d dimensions, and we end with applications in 3+1 dimensions.

We define the Dirac monopole as a simple consequence of extending Maxwell duality to the Maxwell equations with sources, and we show that the resulting gauge fields are only defined on patches. We write formulas in terms of p-form language, and define the magnetic charge in terms of the gauge fields on patches. Then, from the quantization of the first Chern number, a topological number, we obtain Dirac quantization for the product of electric and magnetic charges. One obtains an unphysical Dirac string singularity, and its unphysical nature leads again to Dirac quantization. Finally, semiclassical nonrelativistic considerations also lead to the same Dirac quantization.