ABSTRACT

This chapter studies in general how time-dependent electromagnetic fields are produced by arbitrary charges and currents, and recalls Maxwell’s equations in vacuum. The arbitrariness in the choice of potentials is called the gauge freedom of the theory, while the corresponding transformations are called gauge transformations. The chapter exploits this freedom in the process of solving the differential equations for the potentials. A pair of coupled second order differential equations is simplified by utilizing the gauge freedom in defining the potentials. Differential equations are solved for the potentials in the Lorentz gauge. Since the potentials are linearly related to their sources, they may be expressed in terms of a Green’s function. The retarded Green’s function is adopted to satisfy the correct time boundary condition. The chapter derives elementary generalizations of the potentials for electrostatics and magnetostatics, in which the generalizations reflect the finite propagation speed of light.