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Maxwell's equations on a plaque on his statue in Edinburgh. Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, electric and magnetic circuits.
The covariant formulation of classical electromagnetism refers to ways of writing the laws of classical electromagnetism (in particular, Maxwell's equations and the Lorentz force) in a form that is manifestly invariant under Lorentz transformations, in the formalism of special relativity using rectilinear inertial coordinate systems.
The four equations we use today appeared separately in Maxwell's 1861 paper, On Physical Lines of Force: Equation (56) in Maxwell's 1861 paper is Gauss's law for magnetism, ∇ • B = 0. Equation (112) is Ampère's circuital law, with Maxwell's addition of displacement current.
The first equation listed above corresponds to both Gauss's Law (for β = 0) and the Ampère-Maxwell Law (for β = 1, 2, 3). The second equation corresponds to the two remaining equations, Gauss's law for magnetism (for β = 0) and Faraday's Law (for β = 1, 2, 3).
Electromagnetic or magnetic induction is the production of an electromotive force (emf) across an electrical conductor in a changing magnetic field. Michael Faraday is generally credited with the discovery of induction in 1831, and James Clerk Maxwell mathematically described it as Faraday's law of induction.
The two Maxwell equations, Faraday's Law and the Ampère–Maxwell Law, illustrate a very practical feature of the electromagnetic field. Faraday's Law may be stated roughly as "a changing magnetic field inside a loop creates an electric voltage around the loop". This is the principle behind the electric generator.
This is simply the Lorentz force law on a per-unit-charge basis — although Maxwell's equation first appeared at equation in "On Physical Lines of Force" in 1861, [6] 34 years before Lorentz derived his force law, which is now usually presented as a supplement to the four "Maxwell's equations".
The Maxwell stress tensor (named after James Clerk Maxwell) is a symmetric second-order tensor in three dimensions that is used in classical electromagnetism to represent the interaction between electromagnetic forces and mechanical momentum.