From compass needles to wireless telegraphy. The unified theory of electricity, magnetism, and light — and the first great triumph of mathematical physics.
Electricity and magnetism look like separate phenomena. They are not. Maxwell's 1865 paper showed they are aspects of one field — and that field's waves travel at exactly the speed of light. Light, then, is electromagnetism.
The unification of electricity and magnetism was the first great success of mathematical physics. It happened over the 19th century, peaking with James Clerk Maxwell's twenty equations (later compressed to four). The theory's predictions — including the existence of radio waves — were confirmed within a generation.
This deck traces the unification from Coulomb to Hertz, lays out Maxwell's equations and what they say, and follows electromagnetism into modern technology and physics.
Antiquity knew two strange phenomena. Lodestone (magnetite, Fe₃O₄) attracted iron — known to the Greeks (the name derives from Magnesia in Asia Minor) and to Chinese navigators (the south-pointing compass, perhaps 11th century). Amber rubbed with cloth attracted light objects — described by Thales of Miletus (c. 600 BC). The Greek word for amber, elektron, gave us "electricity."
For two millennia these were curiosities, not theory. William Gilbert's De Magnete (1600) was the first systematic study. Gilbert proposed that the Earth itself is a magnet — explaining why compasses point north. He distinguished electric attraction (rubbed amber) from magnetic attraction (lodestone) but considered them separate.
The 17th and 18th centuries built up empirical phenomenology — Otto von Guericke's friction generator, the Leyden jar (1745), Franklin's kite (1752, with the proposal that lightning is electrical) — without theoretical unification.
Charles-Augustin de Coulomb's torsion balance experiments (1785) established that the force between two electric charges falls as the inverse square of distance — exactly like gravity. The same year Cavendish independently arrived at similar results, but Cavendish was reclusive and his work was not published until Maxwell rediscovered the manuscripts a century later.
F = k·q₁q₂/r²
Coulomb's law made electricity quantitative. It also revealed the deep structural parallel with gravity — both are central forces falling as 1/r² — which would later turn out to be a clue. Both arise from massless mediating particles (photon, graviton) propagating in 3D space; the geometry forces 1/r².
The unit of charge, the coulomb, is named for him.
Galvani's 1780s frog-leg twitching experiments suggested "animal electricity." Alessandro Volta argued the electricity came from the contact between dissimilar metals, not the frog. To prove it, in 1800 he stacked discs of zinc and copper alternated with cloth soaked in brine — the voltaic pile, the first battery, the first source of steady electric current.
Before Volta, electricity meant brief sparks from accumulated static charge. After Volta, electricity meant continuous flow. This was the precondition for almost everything that followed — electrolysis (Davy isolated sodium and potassium), electromagnetism experiments (Ørsted, Ampère, Faraday), telegraph signals.
The volt, the unit of electric potential, is named for him. The voltaic pile was the most important experimental tool in physical chemistry for half a century.
April 1820. Hans Christian Ørsted, lecturing in Copenhagen, notices that a compass needle deflects when a nearby wire carries current. The current produces a magnetic field. Electricity and magnetism are connected.
Ørsted published the discovery in July 1820 (in Latin, four pages). Within months André-Marie Ampère in Paris had developed the mathematics: any current produces a circulating magnetic field; two parallel currents attract or repel each other depending on their direction.
∮ B·dl = μ₀I (Ampère's law)
The 1820s were the foundational decade. Ørsted's discovery, Ampère's law, the Biot-Savart law (current loops produce calculable magnetic fields), and Ohm's law (1827, V = IR). The conceptual framework of "current, voltage, resistance, magnetic field" was assembled.
If currents make magnetism, do magnets make currents? Michael Faraday, the bookbinder's apprentice turned Royal Institution prodigy, found the answer in 1831: a changing magnetic field induces an electric current in a nearby conductor.
EMF = -dΦ/dt (Faraday's law)
This is the principle behind every electric generator and every transformer — that is, almost the entire electrical economy. Move a magnet through a coil and a current flows. Spin a magnet inside a stationary coil (or vice versa) and you have a generator. Send AC through one coil and induce current in a neighbouring coil and you have a transformer.
Faraday also introduced the concept of the field — invisible lines of force pervading space. He couldn't do the math (he had no formal mathematics training); Maxwell would do it for him a generation later. Faraday's Experimental Researches in Electricity (three volumes, 1839-55) is one of the great laboratory records.
James Clerk Maxwell, in three papers (1855, 1861, 1865), translated Faraday's intuitive field concept into mathematics — and went beyond Faraday. The 1865 paper "A Dynamical Theory of the Electromagnetic Field" is the founding document.
Maxwell's key insight was the displacement current: a changing electric field acts like a current and produces a magnetic field. Adding this term to Ampère's law completed the symmetry between electric and magnetic phenomena and made the equations mathematically consistent.
The four Maxwell equations (in modern form, due to Heaviside):
∇·E = ρ/ε₀
∇·B = 0
∇×E = -∂B/∂t
∇×B = μ₀J + μ₀ε₀ ∂E/∂t
Translated: charges produce divergent electric fields; there are no magnetic monopoles; changing magnetic fields produce circulating electric fields (Faraday); currents and changing electric fields produce circulating magnetic fields (Ampère + displacement current).
Maxwell's equations have a stunning consequence. Combine them in a vacuum and you get a wave equation — for the electric and magnetic fields jointly. The wave's speed is determined entirely by the constants ε₀ and μ₀ in the equations:
c = 1/√(μ₀ε₀) ≈ 3 × 10⁸ m/s
That number is the speed of light. Light is an electromagnetic wave.
This was Maxwell's leap. The constants ε₀ and μ₀ were measured in static electricity and magnetism experiments — entirely terrestrial, lab-bench experiments. The fact that a wave built from them travels at the speed of light meant that visible light, infrared, ultraviolet — all of optics — is electromagnetic radiation.
Maxwell predicted that other electromagnetic waves would exist at frequencies above and below visible light. This prediction would be experimentally confirmed by Heinrich Hertz in 1887-88.
Maxwell died in 1879, age 48, before his prediction was confirmed. Heinrich Hertz, in Karlsruhe between 1886 and 1889, built spark-gap transmitters and resonant-loop receivers. He generated electromagnetic waves at radio frequencies, detected them at a distance, measured their wavelength, demonstrated that they reflect, refract, and polarise like light. He confirmed Maxwell completely.
Asked about applications, Hertz famously replied: "It's of no use whatsoever... we just have these mysterious electromagnetic waves that we cannot see with the naked eye. But they are there."
Hertz died in 1894 at age 36, before Marconi's commercial wireless telegraphy. The unit of frequency, the hertz (Hz, cycles per second), is his.
Within a decade of Hertz, Guglielmo Marconi had a working radio telegraph. By 1901 he transmitted across the Atlantic. The 20th century is built on Hertz's "useless" waves.
One phenomenon, distinguished by frequency. From low to high:
Radio (3 Hz – 300 GHz, wavelength m to cm). AM/FM broadcast, Wi-Fi, cellular, radar.
Microwave (300 MHz – 300 GHz, the upper radio band). Microwave ovens, mobile phones, satellite links.
Infrared (300 GHz – 430 THz). Heat radiation. Night vision. IR spectroscopy. Fibre optic communication.
Visible light (430 THz – 770 THz, wavelength 700-400 nm). Red through violet — a single octave on the spectrum.
Ultraviolet (770 THz – 30 PHz). Causes sunburn, sterilises water, ionises atmosphere.
X-rays (30 PHz – 30 EHz). Discovered by Röntgen 1895. Medical imaging, crystallography, astronomy.
Gamma rays (above 30 EHz). Nuclear decay, cosmic sources, oncology.
All the same phenomenon. The same equations describe radio waves and gamma rays.
Maxwell's equations have a peculiar feature. They predict a fixed speed of light, but say nothing about what frame of reference that speed is measured in. Newton's mechanics was Galilean — speeds add (a ball thrown forward from a moving train moves at train-speed plus throw-speed). Maxwell's equations weren't.
The Michelson-Morley experiment (1887) tried to detect the Earth's motion through the supposed luminiferous "ether" by measuring differences in light-speed in different directions. They found nothing. Light moved at c in every direction, regardless of motion.
Einstein's 1905 special relativity took this as a postulate: the speed of light is the same in all inertial frames. Everything else — time dilation, length contraction, E=mc² — followed.
The 1905 paper's title is "On the Electrodynamics of Moving Bodies." It's an electromagnetism paper. The relativity revolution was provoked by the inconsistency between Maxwell and Newton — and resolved by reformulating mechanics to match Maxwell, not the other way around.
Maxwell's equations are classical — they describe continuous fields. But light comes in discrete photons (Einstein 1905, photoelectric effect). The quantum theory of the electromagnetic field, quantum electrodynamics (QED), was developed in the 1940s by Tomonaga, Schwinger, Feynman, and Dyson. They shared the 1965 Nobel.
QED treats charged particles and photons as excitations of underlying fields. Interactions are described by Feynman diagrams — pictures of particles exchanging photons that translate into specific mathematical expressions for probability amplitudes.
QED is the most accurately tested theory in physics. The electron's anomalous magnetic moment, predicted by QED to twelve decimal places, agrees with measurement at that precision. Nothing in any other field of science approaches this level of agreement.
QED is the prototype for the entire Standard Model. The same mathematical structure (gauge field theory) describes the strong and weak nuclear forces. The Standard Model is, in a sense, three QEDs stitched together.
Faraday's law turned into industrial civilisation. A coil rotated inside a magnetic field generates AC current proportional to the rotation rate. Mechanical energy → electrical energy.
The 19th century saw the development of dynamos (DC), alternators (AC), motors (the inverse — AC into rotation). The "war of currents" (1880s-90s) between Edison's DC and Westinghouse-Tesla's AC was decided by economics: AC can be transformed to high voltage for long-distance transmission, dramatically reducing transmission losses. Niagara Falls (1895) was the first major hydroelectric AC station.
The 20th century built out the global grid. Today's grid is a vast Faraday-law machine — coal, gas, nuclear, hydro, wind, and solar generation, transformed and redistributed across thousands of kilometres, all running on Maxwell's equations.
Solar PV is the exception — it generates DC directly from photon-electron interactions (the photoelectric effect, again). Inverters convert it to AC for the grid.
An antenna is a length of conductor whose electron motion radiates electromagnetic waves at frequencies tuned to its dimensions (roughly a half-wavelength). Maxwell's equations connect oscillating current to radiated EM waves.
Marconi's 1901 transatlantic transmission used long-wavelength radio (kilometres), reflected off the ionosphere. Modern Wi-Fi at 2.4 and 5 GHz uses centimetre-wavelength waves. 5G millimetre-wave at 28-39 GHz uses millimetre-wavelength waves. Each requires antenna designs scaled to the wavelength.
The 21st-century antenna design problem is dense, multi-band integration. A modern smartphone has roughly 10 antennas — cellular (multiple bands), Wi-Fi, Bluetooth, GPS, NFC, UWB. Beamforming and MIMO arrays use multiple antennas working together to direct radio energy.
The same equations Maxwell wrote down in 1865.
If light is electromagnetic, optics is electromagnetic engineering. Lenses, mirrors, prisms, polarisers, fibre optics, lasers — all explicable from Maxwell.
Refraction (Snell's law). Light slows in dense media; the wave fronts bend. The refractive index n = c/v is what determines the bending angle.
Diffraction. Waves passing through narrow apertures spread out. The diffraction pattern depends on aperture size relative to wavelength. Diffraction limits microscope and telescope resolution (Rayleigh criterion).
Polarisation. EM waves are transverse — the field oscillates perpendicular to propagation. Polarising filters select one orientation. Polarised sunglasses, LCD displays.
Interference. Waves add coherently. Interferometers (Michelson, Mach-Zehnder, LIGO) measure tiny path differences. LIGO detects gravitational waves by measuring sub-proton-diameter mirror motions via interference.
Einstein's 1917 paper on stimulated emission predicted that an excited atom can be triggered by an incoming photon to emit an identical photon — same frequency, same phase, same direction. The first laser (Light Amplification by Stimulated Emission of Radiation) was Theodore Maiman's ruby laser, 1960.
Lasers produce coherent light — fixed phase relationship across the beam, allowing tight focusing, low divergence, high spectral purity. Applications: fibre optic communication (telecom-band lasers at 1550 nm), DVD/Blu-ray, barcode scanners, laser surgery, laser cutting and welding, range-finding, rifle sights, atomic clocks, optical lattices for quantum simulation, gravitational wave detection.
2018 Nobel went to Ashkin (optical tweezers), Mourou and Strickland (chirped pulse amplification — the basis of high-power femtosecond lasers).
Maxwell's equations built the laser; the laser built the optical communications era; the optical communications era built the internet.
The classical theory of magnetism is incomplete because it doesn't explain why some materials are magnetic. The answer is quantum mechanical.
Spin — the intrinsic angular momentum of electrons — is the source of microscopic magnetic moments. In most materials, electron spins pair up (Pauli exclusion) and cancel; the material is non-magnetic. In ferromagnetic materials (iron, nickel, cobalt, certain rare earths), exchange interactions align unpaired spins across domains, producing macroscopic magnetism.
Heat above the Curie temperature destroys the alignment. Iron's Curie point is 770°C. Strong external fields can magnetise initially-disordered domains.
The strongest commercial permanent magnets are neodymium-iron-boron (Nd₂Fe₁₄B), invented in 1982. They power hard drive read heads, electric vehicle motors, wind turbine generators. Rare earth supply (China dominates) is now a strategic concern.
Heat a gas enough and atoms ionise — electrons separate from nuclei. The result is a plasma, a soup of charged particles. Plasmas conduct electricity, respond to magnetic fields, and exhibit collective behaviour described by magnetohydrodynamics (MHD) — the marriage of fluid dynamics and Maxwell's equations.
99% of visible matter in the universe is plasma. Every star, the solar wind, the interstellar medium, planetary magnetospheres. The aurora borealis is solar-wind plasma channelled by Earth's magnetic field.
On Earth: lightning, neon signs, fluorescent tubes, the inside of fusion reactors. ITER (under construction in France) and the new generation of compact fusion reactors (Commonwealth Fusion's SPARC, TAE, Helion) all confine plasma magnetically.
Fusion's hardest problem is plasma instability. Plasmas spontaneously develop turbulent and unstable behaviours that classical MHD doesn't fully predict. Modern fusion engineering is plasma engineering.
Below a critical temperature, certain materials lose all electrical resistance. Superconductivity was discovered by Heike Kamerlingh Onnes (1911) in mercury at 4 K. The BCS theory (Bardeen, Cooper, Schrieffer, 1957) explained it: electrons pair up via lattice vibrations and condense into a coherent macroscopic quantum state.
High-temperature superconductors (Bednorz and Müller, 1986, copper-oxide ceramics above 90 K — above liquid nitrogen temperature, easier to cool) revolutionised the field. The mechanism is still not fully understood. Recent progress on hydride superconductors (H₃S at 15°C and ~150 GPa, 2015) approaches room temperature, though only at extreme pressures.
Superconductors enable: MRI scanners (superconducting magnets, ~1.5-3 T routinely), particle accelerators (LHC magnets at 8 T), maglev trains, the qubit lattice in IBM and Google quantum computers (Josephson junctions).
Room-temperature, ambient-pressure superconductors remain a major open goal.
The SI second is defined by the cesium-133 hyperfine transition: 9,192,631,770 cycles per second of microwave radiation. Atomic clocks based on this transition are accurate to roughly 10⁻¹⁵ — losing or gaining about a second per 30 million years.
Optical lattice clocks based on strontium or ytterbium transitions reach 10⁻¹⁹ — a second per 30 billion years (more than the age of the universe). They're now sensitive enough to detect gravitational time dilation from elevation differences of a centimetre.
The 2026 redefinition of the second (under discussion at the BIPM) will likely move from cesium microwave to optical-lattice transitions. This is what relativistic geodesy and the Global Navigation Satellite Systems run on.
GPS satellites carry atomic clocks. Their orbital relativity (special and general relativistic time dilation) requires ~38 microseconds per day correction; without it, GPS positions would drift by kilometres per day.
Charged particles in magnetic fields move on circles; in electric fields, accelerate. Combine the two and you have a particle accelerator. The cyclotron (Lawrence, 1929), the synchrotron, the Stanford linear accelerator (SLAC, 3 km), the Large Hadron Collider (LHC, 27 km circumference at CERN).
The LHC accelerates protons to 6.8 TeV — 99.999999% of the speed of light. Superconducting dipole magnets at 8.3 tesla bend the beams; superconducting RF cavities at 400 MHz accelerate them. Maxwell's equations at the most extreme empirical regime humans have engineered.
Discoveries: the W and Z bosons (CERN 1983), the top quark (Fermilab 1995), the tau neutrino (Fermilab 2000), the Higgs boson (ATLAS and CMS, LHC, 2012).
The proposed Future Circular Collider (FCC) at CERN would be 91 km. Its rationale and budget remain hotly contested in the physics community.
The Earth has a magnetic field of about 25-65 microtesla at the surface, generated by the geodynamo — the convecting liquid iron outer core, rotating with Earth, sustaining a self-amplifying current loop.
The field has reversed polarity many times in geological history. The last full reversal was the Brunhes-Matuyama, ~780,000 years ago. The field weakens during reversals; it's currently weakening at 5% per century. We are not in a reversal — yet.
The magnetic field shields Earth from the solar wind. Without it, atmospheric ionisation and erosion would be much greater (this is one hypothesis for what happened to Mars's atmosphere — Mars has no global magnetic field today).
The field also enables animal navigation. Migratory birds, sea turtles, sharks, and possibly some mammals use magnetic sensing for orientation. The mechanism is not fully understood but appears to involve cryptochrome proteins in the retina coupling to the geomagnetic field.
Classical electromagnetism is closed. Maxwell's equations have not needed correction in 160 years. QED is closed at the empirically accessible energy scales. So what remains?
Magnetic monopoles. Maxwell's equations are nearly symmetric between electric and magnetic, except there are electric monopoles (charges) but no magnetic ones. Dirac (1931) showed that even a single magnetic monopole would explain why electric charge is quantised. Despite extensive searches, no monopole has been found. They are predicted by many GUT theories.
Strong-field QED. At fields above the Schwinger limit (~10¹⁸ V/m), the vacuum becomes nonlinear and pair production from vacuum fluctuations becomes significant. Modern petawatt lasers (the Mourou/Strickland regime) approach but haven't reached this. Forthcoming facilities like ELI may.
Room-temperature superconductivity. The 1986 cuprate revolution and the 2015 hydride results suggest the door isn't closed. The South Korean LK-99 claim (2023) was retracted, but the search continues.
↑ 3Blue1Brown · divergence and curl, the language of Maxwell's equations
Watch · Electromagnetism explained in simple words
Watch · Faraday's law of electromagnetic induction
Three paths.
For non-physicists. The Feynman Lectures Vol. II are accessible to anyone willing to follow the math. Falk's Light: A Radiant History for the historical narrative. The Mechanical Universe DVD lectures are still the best filmed introduction.
For physics undergraduates. Griffiths's Introduction to Electrodynamics is the universal undergraduate textbook. Then Jackson's Classical Electrodynamics for the graduate-level treatment — notorious for its difficulty but irreplaceable.
For QED and beyond. Feynman's QED for the conceptual intro. Then Peskin and Schroeder's An Introduction to Quantum Field Theory, Srednicki's Quantum Field Theory, or Schwartz's Quantum Field Theory and the Standard Model — pick one based on style preference.
For history. Crowther's British Scientists of the Nineteenth Century. Pais's Inward Bound. Forbes and Mahon's Faraday, Maxwell, and the Electromagnetic Field.
Three claims.
It is the cleanest example of theoretical unification. Two phenomena — electricity and magnetism — that looked separate were shown to be one. The third phenomenon — light — was shown to be the same theory's wave solutions. Predictions (radio waves, the speed of light) followed from the math. This is the model for what physics is supposed to do.
It is the foundation of modern technology. The electrical grid, electric motors, generators, transformers, radio, TV, Wi-Fi, mobile phones, computers (electron flow in transistors), MRI, lasers, fibre optics, GPS, electric vehicles. The 19th-century unification became the 20th-century material economy and the 21st-century information economy.
It is the prototype for the Standard Model. The mathematical structure of EM (gauge invariance, fields with associated quantum particles, renormalisation) is the structure of all the fundamental forces. EM was the model. Everything since has been modeled on it.
Where electromagnetism research is moving.
Photonic computing. Light-based logic gates and routing avoid the heat dissipation limits of silicon. Several startups (Lightmatter, Luminous, PsiQuantum) are commercialising. The substrate problem (silicon photonics integration) is hard but progressing.
Metamaterials. Engineered structures with effective electromagnetic properties not found in nature — negative refractive index, perfect absorbers, electromagnetic cloaking demonstrations. Most applications are in narrow frequency bands.
6G research. Sub-terahertz wireless (100 GHz – 1 THz), massive MIMO at scale, integrated sensing and communication. Much of the engineering is at the edge of what Maxwell's equations need to be discretised on.
Compact fusion. Commonwealth Fusion's SPARC and the new wave of high-field tokamaks rely on REBCO-tape high-temperature superconducting magnets. The whole programme depends on Faraday-law confinement of plasma at unprecedented field strengths.
Maxwell's equations are 160 years old. Their applications are not running out.
Electromagnetism — Volume III, Deck 13 of The Deck Catalog. Set in EB Garamond italic with monospace metadata. Cream paper #fbf7ee; copper and steel-blue accents.
Twenty-eight leaves on the unified theory of electricity, magnetism, and light. The first great triumph of mathematical physics — and still the model for the rest.
↑ Vol. III · Sci. · Deck 13