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 have two major variants. The "microscopic" set of Maxwell's equations uses total charge and total current including the difficult-to-calculate atomic level charges and currents in materials. The "macroscopic" set of Maxwell's equations defines two new auxiliary fields that can sidestep having to know these 'atomic' sized charges and currents.
Maxwell's equations are named after the Scottish physicist and mathematician James Clerk Maxwell, since in an early form they are all found in a four-part paper, "On Physical Lines of Force," which he published between 1861 and 1862. The mathematical form of the Lorentz force law also appeared in this paper.
It is often useful to write Maxwell's equations in other forms; these representations are still formally termed "Maxwell's equations". A relativistic formulation in terms of covariant field tensors is used in special relativity, while, in quantum mechanics, a version based on the electric and magnetic potentials is preferred.
Conceptually, Maxwell's equations describe how electric charges and electric currents act as sources for the electric and magnetic fields. Further, it describes how a time varying electric field generates a time varying magnetic field and vice versa. (See below for a mathematical description of these laws.) Of the four equations, two of them, Gauss's law and Gauss's law for magnetism, describe how the fields emanate from charges. (For the magnetic field there is no magnetic charge and therefore magnetic fields lines neither begin nor end anywhere.) The other two equations describe how the fields 'circulate' around their respective sources; the magnetic field 'circulates' around electric currents and time varying electric field in Ampère's law with Maxwell's correction, while the electric field 'circulates' around time varying magnetic fields in Faraday's law.