This chapter presents a comprehensive framework for describing external electromagnetic fields by reducing the vast amount of data required to map complex three-dimensional vector fields. It introduces spatial harmonic expansion as a natural, symmetry-based method for efficiently representing these fields. Using a multipolar motor as an example, the text shows that ideal fields decay rapidly with distance, whereas “fringe fields” caused by real-world non-idealities persist and dominate in the far field. These multipole expansions are essential tools for characterizing machine imperfections and predicting field behavior at a distance. The analysis moves from the basic electrostatics of point charges to advanced magnetostatics, detailing the use of Green’s functions to generate highly economical source descriptors. It also covers the physical foundations of magnetism, including the Lorentz force and the Hall effect, as well as Maxwell’s equations and the quasistatic limits for low-frequency applications. This theoretical synthesis provides the tools needed to characterize vacuum fields from any arbitrary source.