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 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. The equations have solutions that describe waves propagating in vacuum at a fixed speed. Maxwell already noted that this speed matched the speed of light, and correctly guessed that light, like radio waves and X-rays, is a form of electromagnetic radiation in a specific wavelength range.

Maxwell's equations describe how electric and magnetic fields are generated and altered by each other and by charges and currents. The 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.

It is often useful to write Maxwell's equations in other forms which are often still termed "Maxwell's equations". There are several natural formulations defined on four dimensional spacetime, rather than space and time separately, which are manifestly compatible with special and general relativity. Such four dimensional formulations are commonly used in high energy and gravitational physics. In quantum mechanics, versions based on the electric and magnetic potentials are preferred.

Since the Maxwell equations imply a fixed speed of light, they were long believed to be only valid for an observer at rest with respect to a postulated "ether". Einstein, in the special theory of relativity, postulated instead that the Maxwell equations are valid for arbitrary observers, and showed that this implies that separate notions of space and time have no observer-independent physical reality. Since the mid-20th century, it has been understood, however, that Maxwell's equations are not exact laws of the universe but an approximation to the more accurate and fundamental theory of quantum electrodynamics.