Maxwell & Electromagnetic Waves

11.1 James Clerk Maxwell (1831–1879)

James Clerk Maxwell was born in Edinburgh to a wealthy Scottish family. A prodigious child, he published his first scientific paper at age 14 — on the geometry of oval curves. He studied at Edinburgh and Cambridge (where he became Second Wrangler), then held chairs at Aberdeen, King's College London, and finally Cambridge, where he founded the Cavendish Laboratory.

Maxwell made transformative contributions to at least three fields: electromagnetism, statistical mechanics (the Maxwell-Boltzmann distribution), and colour vision (the first colour photograph, 1861). Einstein called Maxwell's electromagnetic theory “the most profound and the most fruitful that physics has experienced since the time of Newton.”

He died of abdominal cancer at just 48 — the same age and the same disease that had killed his mother. Had he lived another decade, he might have witnessed Hertz's confirmation of electromagnetic waves.

11.2 Translating Faraday into Mathematics

Faraday had the physical intuition — lines of force, the electromagnetic field — but not the mathematical language to express it precisely. Most Continental mathematicians dismissed Faraday's ideas as vague and unrigorous. Maxwell alone saw their depth:

“As I proceeded with the study of Faraday, I perceived that his method of conceiving the phenomena was also a mathematical one, though not exhibited in the conventional form of mathematical symbols. I also found that these methods were capable of being expressed in the ordinary mathematical forms, and thus compared with those of the professed mathematicians.”— Maxwell, Preface to A Treatise on Electricity and Magnetism, 1873

Maxwell's programme was to translate Faraday's physical pictures into differential equations. He published his results in three great papers:

11.3 The Displacement Current: Maxwell's Stroke of Genius

The known laws of electricity and magnetism in the early 1860s were:

• • Gauss's law for electricity: charges create diverging electric fields
• • Gauss's law for magnetism: no magnetic monopoles
• • Faraday's law: changing magnetic fields create electric fields
• • Ampère's law: currents create circulating magnetic fields

Maxwell noticed a mathematical inconsistency in Ampère's law. Taking the divergence of both sides, the left side vanishes (the divergence of a curl is zero), but the right side \(\mu_0 \nabla \cdot \mathbf{J}\) does not vanish when charge is accumulating. Ampère's law, as stated, violated charge conservation!

Maxwell's fix was to add a new term — the displacement current \(\varepsilon_0 \frac{\partial \mathbf{E}}{\partial t}\) — to Ampère's law. This seemingly small correction had enormous consequences: it meant that a changing electric field produces a magnetic field, just as a changing magnetic field produces an electric field (Faraday's law).

11.4 Maxwell's Equations

The complete set of four equations, in the modern differential form due to Oliver Heaviside:

11.5 Light Is an Electromagnetic Wave

In free space (no charges or currents), Maxwell's equations reduce to wave equations for\(\mathbf{E}\) and \(\mathbf{B}\):

These are wave equations with speed:

Maxwell calculated this speed from the known values of \(\mu_0\) (measured in magnetic experiments) and \(\varepsilon_0\) (measured in electrostatic experiments). The result matched the measured speed of light exactly. Maxwell wrote:

“The agreement of the results seems to show that light and magnetism are affections of the same substance, and that light is an electromagnetic disturbance propagated through the field according to electromagnetic laws.”— Maxwell, A Dynamical Theory of the Electromagnetic Field, 1865

This was one of the greatest unifications in the history of science: electricity + magnetism + light = one phenomenon. Three seemingly unrelated areas of physics were revealed to be aspects of a single underlying reality.

11.6 The Electromagnetic Spectrum

Maxwell's equations predicted electromagnetic waves at all frequencies, not just visible light. The full spectrum, from lowest to highest frequency:

All are the same phenomenon — oscillating electric and magnetic fields propagating at speed \(c\) — differing only in wavelength and frequency.

11.7 Hertz: Experimental Confirmation (1887)

Heinrich Hertz (1857–1894) was a German physicist who, at age 30, provided the definitive experimental proof of Maxwell's theory. Using a spark-gap transmitter and a loop antenna receiver, Hertz generated and detected electromagnetic waves in his laboratory at the Karlsruhe Polytechnic.

Hertz demonstrated that his waves:

• • Propagated at the speed of light
• • Could be reflected by metal sheets
• • Could be refracted through prisms
• • Exhibited interference and diffraction
• • Were polarized — transverse waves, exactly as Maxwell predicted

When asked about the practical applications of his discovery, Hertz famously replied: “It's of no use whatsoever. This is just an experiment that proves Maestro Maxwell was right.” Within a decade, Marconi was transmitting messages across the Atlantic.

Hertz died of granulomatosis at age 36, having accomplished in his brief career one of the most important experimental verifications in the history of physics. The unit of frequency, the hertz (Hz), honours his name.

11.8 Heaviside, Lorentz & the Modern Framework

Maxwell originally wrote his equations as 20 equations in 20 unknowns using quaternion notation. The elegant four-equation form we use today is due to Oliver Heaviside (1850–1925), a self-taught English engineer who also invented vector calculus notation (\(\nabla\), div, curl, grad) and operational calculus.

Hendrik Lorentz (1853–1928) added the force law for a charged particle in electromagnetic fields:

Together, Maxwell's equations and the Lorentz force law form the complete theory of classical electrodynamics. They remain exactly correct in the relativistic regime — indeed, it was the attempt to make Newtonian mechanics compatible with Maxwell's equations that led Einstein to special relativity in 1905.

11.9 Maxwell's Legacy

Maxwell's electromagnetic theory:

“One scientific epoch ended and another began with James Clerk Maxwell.”— Albert Einstein