Maxwell’s equations in vacuum and with sources
Maxwell’s four equations describe how charge and current shape electric and magnetic fields, and how changing fields propagate as electromagnetic waves.
The four equations in SI units
Gauss’s law gives ∇·E = ρ/ε₀. Gauss’s law for magnetism gives ∇·B = 0. Faraday’s law gives ∇×E = −∂B/∂t. The Ampère–Maxwell law gives ∇×B = μ₀J + μ₀ε₀∂E/∂t. Here ρ is charge density and J is current density.
What each equation measures
Divergence describes net outward field flux per unit volume. Curl describes local circulation. A point charge produces electric flux through a closed surface, while magnetic field lines have zero net flux through a closed surface.
A charging capacitor
Between ideal capacitor plates there is no conduction current through the gap. The changing electric field still contributes to the Ampère–Maxwell law. In empty space, combining the two curl equations with ρ = J = 0 produces a wave equation with speed 1/√(μ₀ε₀).