11.6
Physics Foundations
§11 · Electromagnetism

11.6 Maxwell’s equations and electromagnetic waves

One term is still missing. The four field equations as assembled so far are inconsistent with charge conservation until Maxwell adds a second source for the magnetic field: a changing electric field. That single addition completes the equations, makes them symmetric, and forces a startling consequence — the fields can propagate through empty space as waves, travelling at the speed of light.

The displacement current

Ampère’s law ∇×B=μ0J\nabla\times\mathbf{B} = \mu_0\mathbf{J} cannot be the whole story. Taking its divergence gives ∇⋅J=0\nabla\cdot\mathbf{J} = 0, which contradicts charge conservation whenever charge accumulates — as it does on a charging capacitor, where current flows in but no conduction current crosses the gap.

▶ Maxwell's fix Derivation

Charge conservation requires ∇⋅J=−∂ρ/∂t\nabla\cdot\mathbf{J} = -\partial\rho/\partial t. Using Gauss’s law ρ=ε0∇⋅E\rho = \varepsilon_0\nabla\cdot\mathbf{E},

∇⋅J=−∂∂t(ε0∇⋅E)=−∇⋅ ⁣(ε0∂E∂t).\nabla\cdot\mathbf{J} = -\frac{\partial}{\partial t}(\varepsilon_0\nabla\cdot\mathbf{E}) = -\nabla\cdot\!\left(\varepsilon_0\frac{\partial\mathbf{E}}{\partial t}\right).

So J+ε0 ∂E/∂t\mathbf{J} + \varepsilon_0\,\partial\mathbf{E}/\partial t is divergence-free, and this combination is what should source B\mathbf{B}. Adding the extra term to Ampère’s law,

∇×B=μ0J+μ0ε0∂E∂t.\nabla\times\mathbf{B} = \mu_0\mathbf{J} + \mu_0\varepsilon_0\frac{\partial\mathbf{E}}{\partial t}.

The new term ε0 ∂E/∂t\varepsilon_0\,\partial\mathbf{E}/\partial t is the displacement current: a changing electric field produces a magnetic field, just as Faraday’s changing magnetic field produces an electric one. The two induction laws are now reciprocal.

The complete equations

∇⋅E=ρε0,∇⋅B=0,\nabla\cdot\mathbf{E} = \frac{\rho}{\varepsilon_0}, \qquad \nabla\cdot\mathbf{B} = 0, ∇×E=−∂B∂t,∇×B=μ0J+μ0ε0∂E∂t.\nabla\times\mathbf{E} = -\frac{\partial\mathbf{B}}{\partial t}, \qquad \nabla\times\mathbf{B} = \mu_0\mathbf{J} + \mu_0\varepsilon_0\frac{\partial\mathbf{E}}{\partial t}.

These four equations, with the Lorentz force, are the whole of classical electromagnetism. Everything electric, magnetic, and optical follows from them.

Electromagnetic waves

In empty space (ρ=0\rho = 0, J=0\mathbf{J} = 0) the two curl equations feed each other, and the fields obey a wave equation.

▶ The wave equation from Maxwell Derivation

Take the curl of Faraday’s law and substitute Ampère’s law with no current:

∇×(∇×E)=−∂∂t(∇×B)=−μ0ε0∂2E∂t2.\nabla\times(\nabla\times\mathbf{E}) = -\frac{\partial}{\partial t}(\nabla\times\mathbf{B}) = -\mu_0\varepsilon_0\frac{\partial^2\mathbf{E}}{\partial t^2}.

Using ∇×(∇×E)=∇(∇⋅E)−∇2E\nabla\times(\nabla\times\mathbf{E}) = \nabla(\nabla\cdot\mathbf{E}) - \nabla^2\mathbf{E} with ∇⋅E=0\nabla\cdot\mathbf{E} = 0,

∇2E=μ0ε0∂2E∂t2.\nabla^2\mathbf{E} = \mu_0\varepsilon_0\frac{\partial^2\mathbf{E}}{\partial t^2}.

This is the wave equation, with propagation speed

c  =  1μ0ε0  ≈  3.0×108 m/s.c \;=\; \frac{1}{\sqrt{\mu_0\varepsilon_0}} \;\approx\; 3.0\times10^8\,\text{m/s}.
where
cc
speed of light in vacuum m/s
μ0\mu_0
permeability of free space H/m
ε0\varepsilon_0
permittivity of free space F/m

The speed is built entirely from two electrostatic and magnetostatic constants — quantities measured with capacitors and coils, nothing about light. That their combination equals the measured speed of light revealed light itself to be an electromagnetic wave. The waves are transverse (E\mathbf{E} and B\mathbf{B} perpendicular to each other and to the propagation direction), they carry energy at density 12ε0E2+12μ0B2\tfrac12\varepsilon_0 E^2 + \tfrac{1}{2\mu_0}B^2, and their properties — dispersion, impedance, reflection — are exactly those of the waves chapter, now realised in the electromagnetic field.

⏳ The history — Maxwell, Faraday's lines of force, and the electromagnetic theory of light

Michael Faraday, largely self-taught and no mathematician, pictured electric and magnetic effects as lines of force filling space — a physical field rather than action at a distance. Most contemporaries treated this as a heuristic; James Clerk Maxwell took it literally. Between 1861 and 1865 he translated Faraday’s field picture into mathematics, and in doing so found that Ampère’s law was incomplete. Adding the displacement current to restore charge conservation, he obtained a set of equations whose empty-space solutions were waves travelling at 1/μ0ε01/\sqrt{\mu_0\varepsilon_0}.

That number, assembled from purely electrical and magnetic measurements, matched the measured speed of light. Maxwell concluded that light is an electromagnetic wave — unifying optics with electromagnetism in a single stroke. He did not live to see it confirmed: Heinrich Hertz generated and detected electromagnetic waves in the laboratory in 1887, eight years after Maxwell’s death, measuring their speed and showing they reflected, refracted, and interfered exactly as light does.