Maxwell's Electromagnetic Theory
Fresnel's spot proved light behaves as a wave. It took James Clerk Maxwell, working entirely within electricity and magnetism, to show light had to be a wave — by combining a handful of laboratory-measured constants and finding, almost by accident, that they predicted the exact speed of light.
Unifying electricity and magnetism
By the middle of the nineteenth century, physicists had a good working understanding of electricity and magnetism as separate phenomena, plus a growing body of evidence — from Hans Christian Ørsted's discovery that a current-carrying wire deflects a nearby compass needle, to Michael Faraday's discovery of electromagnetic induction — that the two were deeply linked. What was missing was a single, unified mathematical framework tying every electric and magnetic observation together.
In the 1860s, James Clerk Maxwell provided exactly that. He assembled the existing laws of electricity and magnetism into a compact set of coupled equations relating electric and magnetic fields to their sources (charges and currents) and to each other. In working through the mathematical consequences of his own equations, Maxwell found something nobody had been looking for: the equations permitted a solution in which oscillating electric and magnetic fields could exist and propagate through empty space, entirely on their own, with no charges or currents needed to sustain them once started.
The structure of an electromagnetic wave
The wave Maxwell's equations predicted has a very specific structure. It consists of an oscillating electric field and an oscillating magnetic field, arranged so that the electric field, the magnetic field, and the wave's direction of travel are all mutually perpendicular to one another. As the wave propagates, a changing electric field generates a changing magnetic field (as Maxwell's extension of Ampère's law describes), and that changing magnetic field in turn generates a changing electric field (as Faraday's law describes) — each field continuously regenerating the other, a self-sustaining process that needs no material medium to travel through at all. This was itself a radical claim: unlike sound or water waves, an electromagnetic wave can propagate through a total vacuum.
Earlier wave theories of light, including Huygens' and (initially) Young's and Fresnel's, generally assumed light needed some material medium to travel through — a hypothetical substance usually called the "luminiferous ether," filling all of space. Maxwell's mathematics showed that electromagnetic waves don't need a medium at all: the oscillating fields sustain each other directly. The ether hypothesis nonetheless persisted for decades after Maxwell, and its eventual, dramatic failure is covered later in this module.
The speed of an electromagnetic wave
The most striking feature of Maxwell's result is that his equations don't just predict that electromagnetic waves exist — they predict exactly how fast such a wave must travel through a vacuum, in terms of two constants that, at the time, had nothing obviously to do with light or optics at all:
Here $\varepsilon_0$ is the permittivity of free space, a constant that sets how strong the electric field is around an isolated charge in a vacuum (it appears, for instance, in Coulomb's law for the force between two point charges). And $\mu_0$ is the permeability of free space, a constant that sets how strong the magnetic flux density is around a current-carrying wire in a vacuum (it appears in the formula for the magnetic field near a long straight wire). Both constants can be measured in entirely ordinary laboratory experiments involving charged objects, currents, and forces — nothing to do with light, lenses, or optics.
Maxwell's calculation was a purely theoretical argument: it showed that light's known speed matched the speed predicted for electromagnetic waves in general, which is strong circumstantial evidence that light is an electromagnetic wave — but it did not yet directly demonstrate that electromagnetic waves other than light could be produced and detected at all. That direct experimental confirmation, and an independent terrestrial measurement of $c$ to compare against, are both covered in the next lesson.
Maxwell's theory also explained, in one stroke, why light, radio waves, X-rays and every other form of electromagnetic radiation share the same fundamental character: they are all oscillating electric and magnetic fields, differing from one another only in frequency and wavelength, all travelling at the same speed $c$ through a vacuum. This unification — showing that light, electricity and magnetism are really three faces of a single underlying phenomenon — ranks among the most significant theoretical achievements in the whole of classical physics.
Key takeaways
- Maxwell unified the existing laws of electricity and magnetism into a single set of equations in the 1860s.
- These equations predicted self-sustaining electromagnetic waves: oscillating electric and magnetic fields, mutually perpendicular and perpendicular to the direction of travel, continuously regenerating each other with no material medium required.
- The predicted wave speed in a vacuum is $c = 1/\sqrt{\mu_0\varepsilon_0}$, where $\varepsilon_0$ is the permittivity of free space and $\mu_0$ is the permeability of free space.
- Both constants are measurable in ordinary electrical/magnetic laboratory experiments, entirely independent of optics.
- The calculated speed matched the already-measured speed of light almost exactly — far too precise a coincidence to ignore.
- This strongly implied that light itself is an electromagnetic wave, unifying optics with electricity and magnetism into a single theory.
- Maxwell's result was theoretical; direct experimental confirmation (generating and detecting electromagnetic waves other than light) came later, with Hertz.