MAXWELL AND THE SPEED OF LIGHT

In 1856, six years before Maxwell wrote down his unification of electricity and magnetism, two German physicists named Wilhelm Weber and Rudolf Kohlrausch performed a careful capacitor-discharge experiment in Göttingen. They were measuring the ratio between the electrostatic and electromagnetic units of charge — a number with no obvious connection to optics. The ratio they reported was approximately $3.107 \times 10^{8}$ metres per second. Weber, working in the deep tradition of action-at-a-distance Continental physics, did not know what to make of it. The number meant nothing to him. It would mean everything to James Clerk Maxwell.

By 1862 Maxwell had completed the equations that bear his name, and from them the wave equation for the electric field falls out in three lines:

The propagation speed of the wave is whatever sits in the coefficient of the Laplacian, square-rooted:

Plug in the modern SI values $\mu_{0} = 1.25663706212 \times 10^{-6}$ N/A² and $\varepsilon_{0} = 8.8541878128 \times 10^{-12}$ F/m and you get exactly the Weber-Kohlrausch ratio: $c = 299{,}792{,}458$ m/s. Maxwell's reaction in On Physical Lines of Force is now famous: "We can scarcely avoid the inference that light consists in the transverse undulations of the same medium which is the cause of electric and magnetic phenomena." Light, electricity, and magnetism: one phenomenon.

Maxwell's number was a derivation, not yet a measurement. The independent experimental anchor came that same year from Léon Foucault, working in a laboratory in Paris with an apparatus that fit on two long tables. A pulse of light leaves a fast-rotating mirror $R$, travels a known distance $L$ to a fixed mirror $M$, and returns. By the time it gets back, $R$ has turned through a small angle $\Delta\theta$, and the returning beam is deflected. The geometry gives

so a measurement of $\Delta\theta$ at known $\omega$ and $L$ determines $c$ directly. Foucault's 1862 figure — using $L \approx 20$ m and a mirror spinning at about 800 rev/s — was $c = 2.98 \times 10^{8}$ m/s, accurate to better than one percent. The same number Maxwell had pulled out of two electrostatic constants. Two completely different lines of evidence, the same answer.

The deflection at Foucault's screen was less than a millimetre, which is why the experiment had to wait for Foucault: he was the same man who had built ultra-stable rotating mirrors for his earlier Earth-rotation pendulum. The measurement was small but unambiguous. By 1862 the speed of light was known to a percent, and that number agreed with what Maxwell's equations said it had to be.

Maxwell died in 1879, sixteen years before the experiment that would have closed his loop. The man who closed it was Heinrich Hertz, working in Karlsruhe in 1887–1888 with apparatus that any modern radio amateur would recognise. A Ruhmkorff induction coil drove an oscillating current across the gap of a dipole transmitter; a few metres away, a small loop of wire with its own micro-gap acted as receiver. When the transmitter sparked, a sympathetic spark fired across the receiver gap. The wave was real, and it travelled at the same speed Maxwell predicted.

Hertz worked at frequencies near 50 MHz (wavelength about 6 m), well below visible light but obeying the same equation. He confirmed reflection, refraction, interference, and polarisation — every property optics had catalogued for two centuries. The phenomenon Maxwell had derived from $\mu_{0}$ and $\varepsilon_{0}$ now had bench experiments at radio wavelengths, optical bench experiments at visible wavelengths, and a single equation tying them together.

Here is the unsettling part, and the reason this topic belongs to the special-relativity branch and not just the electromagnetism one. The wave equation EQ.01 has a propagation speed $c$ baked into it. But a propagation speed is always measured with respect to some frame. Galilean invariance would say that if light travels at $c$ in one inertial frame, it travels at $c - v$ in a frame moving with velocity $v$ in the same direction — exactly the way water waves on a pond appear slower if you walk after them. So which frame is Maxwell's $c$?

The aether hypothesis was natural, but it had a testable consequence: the Earth orbits the Sun at about $30$ km/s, so a year-long experiment should detect Earth's motion through this stationary medium as a directional dependence in the measured speed of light. The arms of an interferometer pointing along Earth's velocity should give a different round-trip time than arms pointing perpendicular to it. The fringe-shift prediction was

with $L$ the arm length, $\lambda$ the wavelength of the source, and $v$ Earth's orbital speed. For a $11$-m arm at $\lambda = 590$ nm, the predicted shift is about $0.4$ fringes — entirely measurable.

That is the experiment Albert Michelson and Edward Morley designed and ran in 1887. The next topic is what they found.

Three threads are now in tension. First: $c$ is a real, measured number that falls out of two electromagnetic constants having nothing to do with optics. Second: Maxwell's equations are not Galilean-invariant — they pick out a special frame, the way the wave equation for sound picks out the rest frame of the air. Third: nobody, including Maxwell himself, asked which frame they were referring to. The aether hypothesis answered that question by fiat, and it predicted a consequence: the Earth's motion through the aether should be detectable.

The next four sections of §01 follow the wreckage. §01.3 watches Michelson and Morley get a null result they cannot explain. §01.4 watches Einstein, in 1905, take the null result as a postulate — light has the same speed in every inertial frame, regardless of source or observer — and let everything else fall out from there. §01.5 watches Newtonian time die on the page when two postulates collide.

For now, hold the picture clean: $c = 1/\sqrt{\mu_{0}\varepsilon_{0}}$ is a fact about vacuum electromagnetism. The frame in which it is true is the question that breaks classical physics open.