Maxwell's equations
The four coupled partial differential equations (∇·E = ρ/ε₀, ∇·B = 0, ∇×E = −∂B/∂t, ∇×B = μ₀J + μ₀ε₀ ∂E/∂t) that fully describe classical electromagnetism. Every electromagnetic phenomenon below the quantum scale follows from them.
Definition
Maxwell's equations are the four coupled partial differential equations that encode all of classical electromagnetism. In the modern vector-calculus form (due to Heaviside, 1884–1885): Gauss's law ∇·E = ρ/ε₀ (electric flux sources are charges); no-monopole law ∇·B = 0 (magnetic flux has no point sources); Faraday's law ∇×E = −∂B/∂t (changing magnetic fields produce curling electric fields); and Ampère–Maxwell's law ∇×B = μ₀J + μ₀ε₀ ∂E/∂t (currents and changing electric fields produce curling magnetic fields). Combined with the Lorentz force F = q(E + v×B) governing how fields act on matter, these four equations describe every classical electromagnetic phenomenon from static charges to radio waves to the optics of crystals.
The integral forms — ∮E·dA = Q/ε₀, ∮B·dA = 0, ∮E·dℓ = −dΦ_B/dt, ∮B·dℓ = μ₀(I + ε₀ dΦ_E/dt) — are the forms most often applied to specific geometries and are what Maxwell originally wrote in 1865. The differential forms are what Heaviside distilled from Maxwell's original twenty quaternion equations and what every physics textbook uses today. In source-free regions (ρ = 0, J = 0), the equations decouple into the wave equation for both E and B, with propagation speed c = 1/√(ε₀μ₀) — which matched the measured speed of light, identifying light itself as an electromagnetic wave.
Maxwell's equations are Lorentz-covariant: they take the same form in every inertial frame under the Lorentz transformation, a fact that Einstein took as the starting point for special relativity in 1905. They unify the electric and magnetic fields into a single antisymmetric tensor F^μν in four-vector form, which makes their symmetry structure transparent. Extended to general relativity, they couple to spacetime curvature via ∇_μ F^μν = (1/c) J^ν and describe electromagnetism on arbitrary curved backgrounds. At the quantum level, they become the classical limit of quantum electrodynamics (QED), where photons mediate the electromagnetic interaction and Maxwell's four equations re-emerge as the expectation values of the QED field operators. Every practical piece of electrical and optical engineering traces back to these four lines.