Maxwell's equations

Maxwell's equations are the four fundamental equations of classical electromagnetism, formulated by James Clerk Maxwell in 1865, that describe how electric and magnetic fields are generated and altered by each other and by charges and currents. They are the most elegant and complete field theory in classical physics, and they remain the foundation of all electrical engineering, optics, and plasma physics.

The four equations, in their modern differential form, are:

Gauss's law for electricity: ∇ · E = ρ/ε₀ — electric charges produce electric fields that diverge from them.

Gauss's law for magnetism: ∇ · B = 0 — there are no magnetic monopoles; magnetic field lines always form closed loops.

Faraday's law of induction: ∇ × E = −∂B/∂t — a changing magnetic field induces an electric field.

Ampère-Maxwell law: ∇ × B = μ₀J + μ₀ε₀ ∂E/∂t — electric currents and changing electric fields produce magnetic fields.

The last term, the displacement current μ₀ε₀ ∂E/∂t, was Maxwell's original contribution. It predicted that a changing electric field — even in a vacuum, with no charges present — produces a magnetic field, just as a changing magnetic field produces an electric field. This symmetry between electric and magnetic fields implied the existence of self-propagating electromagnetic waves, and Maxwell showed that these waves travel at a speed determined by the ratio of the electric and magnetic constants: c = 1/√(μ₀ε₀). This speed matched the measured speed of light, and Maxwell concluded that light itself is an electromagnetic wave.

The equations are not merely a description of electric and magnetic phenomena. They are a field theory: they describe fields that exist throughout space, not just at the locations of charges and currents. The electric and magnetic fields are the primary entities; the charges and currents are their sources. This inversion of the Newtonian picture — where forces act at a distance between particles — was the first field theory in physics, and it became the template for all subsequent field theories, including general relativity and quantum field theory.

In the relativistic formulation, Maxwell's equations are unified into two tensor equations involving the electromagnetic field tensor Fμν and the four-current Jμ. This formulation reveals that the separation of the field into electric and magnetic components is observer-dependent: what one observer calls an electric field, another observer in relative motion calls a mixture of electric and magnetic fields. This is the origin of the magnetic force, which is not a separate fundamental force but a relativistic effect of the electric force observed from a moving frame.

Maxwell's equations are the point where physics became a theory of fields rather than a theory of particles. They are the first demonstration that the fundamental entities of nature are not material bodies but dynamical fields that permeate space. The equations are not merely true; they are the prototype of what a fundamental physical theory looks like. Every subsequent theory of force — general relativity, quantum electrodynamics, the Standard Model — is a generalization of the pattern that Maxwell discovered: write down the field equations, identify the symmetries, and let the solutions reveal the phenomena. Maxwell's equations are not just about electricity and magnetism. They are about what it means to do physics.