Maxwell's Equations

The four equations that unify electricity and magnetism and predict electromagnetic waves traveling at the speed of light.

Category: Electromagnetism · Created: 2026-08-17 · Updated: 2026-08-17

Maxwell's equations are four partial differential equations describing how electric and magnetic fields are produced by charges and currents and how the two fields generate each other. In SI units, with E the electric field, B the magnetic field, ρ the charge density, and J the current density:

∇·E = ρ/ε₀ — Gauss's law: electric charges are the sources of electric fields.
∇·B = 0 — no magnetic monopoles exist; magnetic field lines are closed.
∇×E = −∂B/∂t — Faraday's law: a changing magnetic field produces an electric field.
∇×B = μ₀J + μ₀ε₀∂E/∂t — Ampère–Maxwell law: currents and changing electric fields produce magnetic fields.

The equations unify electricity and magnetism into a single theory. Maxwell's crucial addition to Ampère's law was the displacement-current term μ₀ε₀∂E/∂t, required for consistency with charge conservation. Combining the equations yields a wave equation for the fields with speed c = 1/√(μ₀ε₀), which numerically equals the measured speed of light — the first hint that light is an electromagnetic wave, confirmed experimentally by Hertz in 1887.

The equations are linear, so fields superpose, and in the quasi-static limit they reduce to the rules of circuit theory: Kirchhoff's laws follow from them, and the design of three-phase electric power systems, motors, generators, and transformers all rest on them.

The theory also shaped physics itself. Its prediction that light travels at the same speed in every inertial frame was the empirical starting point of special relativity, and in covariant form Maxwell's equations are the prototype of modern gauge theories. Their practical reach is total: radio, optics, power engineering, and every electronic device operate under these four equations.

Maxwell's Equations

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