Maxwell's equations are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. The equations provide a

In the electric and magnetic field formulation there are four equations that determine the fields for given charge and current distribution. A

In a region with no charges (ρ = 0) and no currents (J = 0), such as in a vacuum, Maxwell's equations reduce to:
Taking the curl (∇×) of the curl equations, and using the

The Maxwell equations can also be formulated on a spacetime-like Minkowski space where space and time are treated on equal footing. The direct spacetime formulations make manifest that the Maxwell equations are relativistically invariant.

Gauss's law
Gauss's law describes the relationship between a static electric field and electric charges: a static electric field points away from positive charges and towards negative charges, and the net outflow of the electric field through a

The invariance of charge can be derived as a corollary of Maxwell's equations. The left-hand side of the modified Ampere's Law has zero divergence by the div–curl identity.

The above equations are the microscopic version of Maxwell's equations, expressing the electric and the magnetic fields in terms of the (possibly atomic-level) charges and currents