Electromagnetism

Michael Faraday, a bookbinder's apprentice turned experimenter with almost no mathematics, asked the reverse question: can magnetism make electricity? In 1831 he found that it can, but only when something changes. Switching a current on in one coil, or moving a magnet through another, drives a brief current. Joseph Henry in Albany found the same effect independently.

To think about it, Faraday imagined space around magnets and charges filled with lines of force, visible in the patterns iron filings make. Most physicists treated the lines as a picture. Faraday insisted they were real, and that the space between objects is where the physics happens. The field was the most consequential idea of the century.

James Clerk Maxwell turned Faraday's pictures into mathematics. In modern notation, his equations say:

$$\nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0}, \quad \nabla \cdot \mathbf{B} = 0, \quad \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}, \quad \nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t}.$$

The last term was Maxwell's own addition. With it, a changing electric field makes a magnetic field and a changing magnetic field makes an electric one, so the fields can sustain each other as a wave travelling through empty space. Its speed is $1/\sqrt{\mu_0\varepsilon_0}$, calculated from laboratory measurements of coils and capacitors, and it came out equal to the measured speed of light. "We can scarcely avoid the inference," Maxwell wrote, "that light consists in the transverse undulations of the same medium which is the cause of electric and magnetic phenomena."

Maxwell's equations contain two constants that can be measured on a laboratory bench, with no light involved. The permeability $\mu_0$ sets how strongly a current produces a magnetic field, and so the force between two wires carrying current. The permittivity $\varepsilon_0$ sets how strongly charges push on each other, and so how much charge a capacitor holds. In modern units:

$$\mu_0 = 4\pi \times 10^{-7} \text{ N/A}^2 , \qquad \varepsilon_0 = 8.854 \times 10^{-12} \text{ F/m} .$$

The equations predict waves of electric and magnetic field travelling at

$$v = \frac{1}{\sqrt{\mu_0 \varepsilon_0}} = \frac{1}{\sqrt{4\pi \times 10^{-7} \times 8.854 \times 10^{-12}}} \approx 2.998 \times 10^8 \text{ m/s} .$$

That is the speed of light. In 1856 Wilhelm Weber and Rudolf Kohlrausch had measured the corresponding ratio of electrical units, by discharging a capacitor through a galvanometer, and found about $3.1 \times 10^8$ m/s. Hippolyte Fizeau had measured the speed of light in 1849 with a spinning toothed wheel and a mirror about 8.6 km away, and got about $3.1 \times 10^8$ m/s as well. Maxwell saw that the agreement could not be a coincidence.

Nothing in the experiments on currents and charges involved light, optics or astronomy. Yet the speed of light fell out of them. Light is an electromagnetic wave, and so, Maxwell predicted, there should be others at every wavelength. Hertz found radio waves twenty years later.

The calculation also carries the puzzle that led to relativity. The formula gives one speed, with no mention of who is measuring it or how fast they are moving. Since 1983 the metre has been defined by fixing the speed of light at exactly 299,792,458 m/s.

In 1887–88 Heinrich Hertz made Maxwell's waves with a spark gap and detected them across his lab. Radio was born, and the theory seemed complete.

But the equations contained a puzzle. They give one speed for light, $c$. Relative to what? The natural answer was a medium, the luminiferous ether, at rest in absolute space, with light moving at $c$ relative to it. The Earth moves through the ether at about 30 km/s around the Sun, so light should travel slightly faster in some directions than others. Albert Michelson and Edward Morley built an interferometer sensitive enough to detect the difference. It found almost nothing.

FitzGerald and Hendrik Lorentz proposed that objects moving through the ether shrink along their direction of motion by exactly enough to hide it. The fix worked but explained nothing. The deeper answer, that there is no ether and that $c$ is the same for everyone, needed a new idea of time. That idea is special relativity, born at the seam between this field and classical mechanics.