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The Great Synthesis: Four Equations, One Field

Step back and see electromagnetism the way Maxwell did — not a pile of separate rules but one field with sources, whose ripples turn out to be light itself.

One theory, not five

In Volume I you met electromagnetism as a collection of independent rules: Coulomb's law for the force between charges, Gauss's law for flux, Faraday's law for induction, Ampère's law and the Biot–Savart law for magnetism. They worked, but they felt like separate crafts. Maxwell's achievement was to see that all of them are facets of one object: the electromagnetic field, described by Maxwell's equations.

The cast: two fields, two sources

At every point sit two vector fields: the electric field \mathbf{E} and the magnetic field \mathbf{B}. Two quantities stir them: the charge density \rho and the current density \mathbf{J}. A test charge dropped into the field feels the Lorentz force — the one law that tells fields how to push matter.

\mathbf{F} = q\,\big(\mathbf{E} + \mathbf{v}\times\mathbf{B}\big)

The Lorentz force: the electric part pushes along E; the magnetic part pushes sideways to the velocity, doing no work.

Note the split of labour. Maxwell's four equations govern how the sources make the fields; the Lorentz force governs how the fields act back on charges. Together they close the loop between matter and field. The magnetic term q\,\mathbf{v}\times\mathbf{B} is perpendicular to the motion, so a static magnetic field never changes a particle's kinetic energy — a fact worth remembering.

The four laws at a glance

Each of Maxwell's four equations paired with the physical statement it makes about charge, magnetism, and induction.

In words: (1) electric charge is the source from which \mathbf{E} diverges; (2) there are no magnetic charges, so \mathbf{B} never diverges — its lines always close; (3) a changing magnetic field wraps an electric field around itself; (4) currents and a changing electric field wrap a magnetic field around themselves. Read the fourth again — that second clause is Maxwell's own addition, and it is why light exists.

\begin{aligned} \nabla\cdot\mathbf{E} &= \dfrac{\rho}{\varepsilon_0}, & \nabla\times\mathbf{E} &= -\dfrac{\partial \mathbf{B}}{\partial t}, \\[4pt] \nabla\cdot\mathbf{B} &= 0, & \nabla\times\mathbf{B} &= \mu_0\mathbf{J} + \mu_0\varepsilon_0\,\dfrac{\partial \mathbf{E}}{\partial t} \end{aligned}

Maxwell's equations in vacuum (microscopic form), SI units. Left column: divergences (sources). Right column: curls (dynamics).

Why this rewrote physics

Combine the two curl equations in empty space and each field obeys a wave equation. The wave speed is built entirely from two lab-measured constants — the permittivity \varepsilon_0 and permeability \mu_0 — and comes out equal to the measured speed of light. The conclusion was inescapable: light is an electromagnetic wave, and all of optics is a chapter of electromagnetism.

c = \dfrac{1}{\sqrt{\mu_0\,\varepsilon_0}} \approx 2.998\times 10^{8}\ \mathrm{m/s}

The speed of light emerges from purely electric and magnetic constants — Maxwell's stunning bridge from electromagnetism to optics.

There is a deeper crack here too. Because c is fixed by the equations and does not reference any particular observer, Maxwell's theory is quietly incompatible with Galilean relativity — the same c in every frame is exactly the seed Einstein grew into special relativity and the invariance of the speed of light. We will make that structure explicit later with the covariant form of Maxwell's equations; for now, notice that these four lines already contain relativity in embryo.