What is Maxwell's equations?
Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern electrical and communications technologies. Maxwell's equations describe how electric and magnetic fields are generated and altered by each other and by charges and currents. They are named after the physicist and mathematician James Clerk Maxwell, who published an early form of those equations between 1861 and 1862.
The equations have two major variants. The "microscopic" set of Maxwell's equations uses total charge and total current, including the complicated charges and currents in materials at the atomic scale; it has universal applicability but may be infeasible to calculate. The "macroscopic" set of Maxwell's equations defines two new auxiliary fields that describe large-scale behaviour without having to consider these atomic scale details, but it requires the use of parameters characterizing the electromagnetic properties of the relevant materials.
The term "Maxwell's equations" is often used for other forms of Maxwell's equations. For example, space-time formulations are commonly used in high energy and gravitational physics. These formulations, defined on space-time rather than space and time separately, are manifestly compatible with special and general relativity. In quantum mechanics and analytical mechanics, versions of Maxwell's equations based on the electric and magnetic potentials are preferred.
Since the mid-20th century, it has been understood that Maxwell's equations are not exact but are a classical field theory approximation to the more accurate and fundamental theory of quantum electrodynamics. In many situations, though, deviations from Maxwell's equations are immeasurably small. Exceptions include nonclassical light, photon-photon scattering, quantum optics, and many other phenomena related to photons or virtual photons.