equationsMaxwells

Maxwell's equations provide the fundamental description of electric and magnetic fields in classical physics. They describe how charges and currents generate fields and how time-varying fields propagate and interact, forming a unified framework for electricity, magnetism, and light.

In differential form and SI units, the equations are: ∇·E = ρ/ε0, ∇·B = 0, ∇×E = −∂B/∂t, and ∇×B = μ0 J + μ0 ε0 ∂E/∂t. Here E is the electric field, B the magnetic field, ρ the charge density, J the current density, ε0 the vacuum permittivity, and μ0 the vacuum permeability.

In integral form, they are expressed as: Gauss's law ∮ E·dA = Qenc/ε0; Gauss's law for magnetism ∮ B·dA = 0; Faraday's law ∮ E·dl = − d/dt ∫ B·dA; and the Ampère–Maxwell law ∮ B·dl = μ0 Ienc + μ0 ε0 d/dt ∫ E·dA.

In vacuum with ρ = 0 and J = 0, Maxwell's equations yield wave equations for E and B, predicting electromagnetic waves that propagate at speed c = 1/√(μ0 ε0). This demonstrates that light is an electromagnetic phenomenon.

Generalizations to materials involve constitutive relations such as D = ε E and B = μ H, incorporating polarization P and magnetization M and leading to boundary conditions that describe field behavior at interfaces. Maxwell's equations underpin modern physics and technologies, from radio and telecommunications to optics and beyond.