Maxwell's equations

Maxwell's equations are a set of four coupled equations for electric and magnetic fields. In their macroscopic form, they relate the fields to electric charge density and current density in materials; the exact constitutive relations depend on the medium. The equations are commonly written in either differential form, which describes fields locally, or integral form, which relates fluxes and circulations over finite surfaces and loops.

Gauss's law for electricity relates electric flux through a closed surface to the charge enclosed. Gauss's law for magnetism states that the net magnetic flux through any closed surface is zero, expressing the absence of observed isolated magnetic monopoles in classical electromagnetism. Faraday's law states that a changing magnetic flux induces an electric field. The Ampère–Maxwell law relates magnetic-field circulation to electric current and to a changing electric field; Maxwell's displacement-current term makes the relation consistent for situations such as a charging capacitor.

In a source-free, homogeneous medium, Maxwell's equations imply wave equations for the electric and magnetic fields. Their wave speed is determined by the medium's permeability and permittivity; in vacuum it equals the speed of light. This connection led Maxwell to identify light as an electromagnetic wave. Heinrich Hertz later generated and detected electromagnetic waves in laboratory experiments, supporting the prediction.

James Clerk Maxwell developed a unified field theory from earlier experimental laws of electricity and magnetism. His 1865 paper, “A Dynamical Theory of the Electromagnetic Field,” presented the theory in mathematical form. The compact four-equation vector-calculus formulation used today reflects later mathematical work, notably by Oliver Heaviside. The equations are used throughout classical electromagnetism, including analysis of circuits, antennas, optics, and wave propagation.