James Clerk Maxwell

How to use this course

James Clerk Maxwell is remembered for four equations, but those equations were one achievement among many. In a career of under thirty years he settled what Saturn's rings are made of, laid the quantitative foundations of colour science, introduced probability into physics, unified electricity, magnetism and light, wrote the first mathematical analysis of a feedback controller, and set out the modern grammar of physical units. This course takes each of these in turn, alongside the story of his life.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, derivations and exercises with folded solutions. Formulas are written in modern SI notation and vector form. Maxwell himself used components, quaternions in places, and Gaussian-type units; Lesson 10 explains why the choice of units mattered so much to him.

Timeline

Edinburgh and Glenlair, 1831–1847

James Clerk Maxwell was born on 13 June 1831 at 14 India Street in Edinburgh's New Town. His father, John Clerk, had taken the additional surname Maxwell on inheriting the estate of Middlebie, and practised, without much enthusiasm, as an advocate. His mother was Frances Cay. Soon afterwards the family moved to Glenlair, the house his parents built on their land in Kirkcudbrightshire, in the south-west of Scotland.

The boy grew up in the country, endlessly curious. Family accounts describe a small child who wanted to know "the go of" every mechanism and bell-wire in the house. His mother taught him at first. She died of abdominal cancer in December 1839, when he was eight. A young tutor engaged afterwards proved a failure, and in November 1841 the boy was sent to live with his aunt in Edinburgh and attend the Edinburgh Academy.

The Academy

Arriving mid-term, in country clothes and with a Galloway accent, he was mocked by classmates who nicknamed him "Dafty". He made two lifelong friends: Lewis Campbell, later a classical scholar and his first biographer, and Peter Guthrie Tait, a year below him, later one of Britain's leading mathematical physicists. By his early teens he was winning school prizes in mathematics, English and verse.

In 1846, aged 14, he worked out a way to draw ovals with a pin-and-string construction generalizing the gardener's ellipse: curves on which a weighted sum of distances to two or more foci is constant. René Descartes had studied related curves, but Maxwell's construction was his own. Because he was considered too young to present it himself, James David Forbes read the paper to the Royal Society of Edinburgh on 6 April 1846.

The University of Edinburgh, 1847–1850

In November 1847, at 16, Maxwell entered the University of Edinburgh. He studied mathematics with Philip Kelland, natural philosophy with James David Forbes, and logic and metaphysics with the leading Scottish philosopher of the day, whose lectures left a lasting mark on his thinking about the foundations of knowledge. Forbes, an experimenter known for work on glaciers and the polarization of heat, let him use the class apparatus for his own experiments.

He spent long summers at Glenlair, with a home-made laboratory in a garret. There he examined strained glass and gelatine between crossed polarizers, watching coloured fringes reveal internal stresses. This photoelastic method fed into one of two papers he published in the Transactions of the Royal Society of Edinburgh before he was twenty: one on rolling curves, the other on the equilibrium of elastic solids. The second already shows his characteristic mix of careful experiment and mathematical theory.

Edinburgh gave him a strong grounding in experimental physics and in Scottish philosophy, but its mathematics was limited. After three years his father agreed that he should go to Cambridge, where the Mathematical Tripos was then the most demanding examination in mathematics in Britain.

Cambridge, 1850–1856

Maxwell went up to Peterhouse in October 1850 and moved to Trinity before the end of his first term, believing a fellowship would be easier to win there. His private coach was William Hopkins, nicknamed the "senior wrangler-maker" for the number of top students he trained. Hopkins is said to have remarked that Maxwell found it almost impossible to think incorrectly about physical problems. Tait later wrote that Maxwell arrived poorly drilled for examination tricks but with immense intellectual power.

He joined the Cambridge Apostles, the university's exclusive discussion society, and wrote essays and light verse. In April 1853 he went through an evangelical religious experience; he remained a devout Presbyterian all his life and later served as an elder of the Church of Scotland.

In January 1854 he graduated as Second Wrangler behind Edward Routh. In the harder Smith's Prize examination the two were bracketed equal. One question on that paper, set by George Gabriel Stokes, asked candidates to prove the theorem now called Stokes's theorem. Maxwell was elected a Fellow of Trinity on 10 October 1855.

Two early programmes

The two years after the Tripos set the agenda for much of his life. He began quantitative experiments on colour mixing with spinning discs (Lesson 05). And, encouraged by William Thomson, later Lord Kelvin, he read Michael Faraday's Experimental Researches in Electricity and set out to give Faraday's lines of force a mathematical form. On Faraday's lines of force was read to the Cambridge Philosophical Society in December 1855 and February 1856. It used the analogy of an incompressible fluid flowing through a resisting medium, and it was the first step toward the field equations of Lesson 07.

Early in 1856 his father fell ill. Maxwell applied for the chair of natural philosophy at Marischal College, Aberdeen, partly to be nearer home. His father died at Glenlair in April 1856, shortly before news of the appointment arrived. Maxwell was 24.

The rings of Saturn

For the general reader. In 1855 the University of Cambridge set Saturn's rings as the subject of its Adams Prize: could the rings be solid, liquid, or made of separate bodies, and would they be stable? Maxwell showed that a solid ring would break up and a fluid ring would break into drops. Only a swarm of separate particles, each orbiting Saturn on its own, could last. Spacecraft photographs confirmed this more than a century later.

The problem

Pierre-Simon Laplace had shown that a uniform rigid ring around a planet cannot be in stable equilibrium, and suggested that the rings were narrow solid rings with irregular mass distributions. Maxwell examined that loophole and closed it. He showed that a rigid ring could be stable only with a very lopsided mass distribution, most of its mass concentrated at one point of its circumference, which is inconsistent with what telescopes showed. He then considered fluid rings and found that waves in them grow, so a fluid ring would break into separate masses.

Keplerian shear

The simplest obstacle to a solid ring is differential rotation. A particle in a circular orbit of radius \(r\) about a mass \(M\) has

A rigid ring has one angular velocity. Its inner parts would move too slowly to orbit freely and its outer parts too fast, so the ring must sustain large internal stresses. Maxwell estimated those stresses and showed they exceed the strength of any known material for a ring of Saturn's size.

A ring of satellites

Maxwell's main model was a ring of \(n\) equal satellites of mass \(m\), equally spaced on a circle of radius \(a\) about a central mass \(M\), rotating with uniform angular velocity \(\Omega\). In a frame rotating with the ring, this configuration is an equilibrium. He perturbed each satellite radially and tangentially, wrote the perturbations as waves travelling around the ring,

and obtained for each wavenumber \(k\) a quartic equation for \(\omega\). The ring is stable if all roots \(\omega\) are real for every \(k\), so that every disturbance oscillates instead of growing. The hardest case is the shortest wave, where neighbouring satellites move in opposite directions.

The more satellites there are, the smaller the total ring mass must be relative to the planet. A ring that is massive compared with this bound breaks up through its own gravity. A real ring of countless particles is a limit in which the mutual attractions are negligible and each particle follows its own Keplerian orbit, disturbed by collisions and by the planet's other moons. Later analysis showed that Maxwell's approximation needs \(n\ge7\) to be self-consistent, and refined the constant, but his conclusion stands.

Confirmation

In 1895 James Keeler measured the Doppler shift of sunlight reflected from the rings and found that the inner parts move faster than the outer parts, as separate Keplerian orbits require. Spacecraft from Voyager onward showed the rings as a disk of ice particles ranging from dust to boulders. George Biddell Airy, the Astronomer Royal, called the essay one of the most remarkable applications of mathematics to physics he had seen.

Colour vision and the first colour photograph

For the general reader. Thomas Young had guessed in 1802 that the eye has three kinds of colour receptor, so that any colour can be matched by a mixture of three primaries. Maxwell turned the guess into measurement. With spinning tops and a box of prisms he wrote down colour equations, showed that colour perception is three-dimensional, measured how colour-blind people differ, and in 1861 projected the first photograph made by the three-colour method.

Colour equations

Maxwell's colour top carried interlocking discs of coloured paper whose exposed sectors could be adjusted. Spun fast, the sectors fuse into one colour. An inner set of black and white discs was adjusted until the two regions matched. The result is an equation such as

where the \(x\) are sector fractions and \(\mathbf K\) is black. He found that such equations obey the rules of linear algebra: they can be added, scaled and combined, and different observers with normal vision obtain nearly the same equations. He reported this in Experiments on colour, as perceived by the eye, with remarks on colour-blindness, read to the Royal Society of Edinburgh in March 1855.

With his colour box of 1860, which mixed narrow bands of spectral light, Maxwell measured the colour-matching functions of the spectrum: how much of each primary is needed to match each wavelength. This is the experimental core of modern colorimetry. For this work the Royal Society awarded him the Rumford Medal in 1860.

The colour triangle

Divide by the total \(C_1+C_2+C_3\) to get chromaticity coordinates \(c_i=C_i/\sum C_j\), which lie in a plane. Maxwell drew the primaries at the corners of a triangle. A mixture then sits at the centre of mass of its components, weighted by their amounts. The spectral colours form a curve, mostly outside any triangle of real primaries, which is why three primaries cannot reproduce every colour with positive amounts.

Colour blindness

Maxwell measured colour-blind observers with the top and found their matches needed only two primaries: their colour space is two-dimensional. Each such observer confuses all colours along a family of lines in the triangle, which converge on the point representing the missing receptor. He suggested spectacles with coloured filters to help them tell red from green.

The tartan ribbon, 1861

In 1855 Maxwell proposed colour photography by three black-and-white photographs taken through red, green and blue filters, then projected through the same filters and superimposed. On 17 May 1861, in a lecture at the Royal Institution on the theory of three primary colours, he showed the method with a tartan ribbon photographed by Thomas Sutton. The result was recognizably coloured. It should not have worked as well as it did: the plates of the day were almost insensitive to red. In 1961 researchers at Kodak found the explanation. The red parts of the ribbon reflected ultraviolet, which passed the red filter and exposed the plate.

Kinetic theory and the Maxwell distribution

For the general reader. A gas is a vast number of molecules flying about and colliding. Rudolf Clausius had treated them as if all had the same speed. Maxwell asked how the speeds are distributed, and answered with the bell-shaped law now named after him and Ludwig Boltzmann. It was the first time a law of physics was a statement about probabilities. From it he predicted something that seemed absurd: the viscosity of a gas does not depend on its pressure. He then measured it and found he was right.

The 1860 argument

In Illustrations of the dynamical theory of gases, read to the British Association at Aberdeen in 1859 and published in 1860, Maxwell made two assumptions about the velocity distribution in a gas at rest in equilibrium: it is isotropic, and the three Cartesian components of velocity are statistically independent. Then the density \(f\) satisfies

Proof. Take logarithms: \(\psi(u)=\ln\varphi(\sqrt u)\) satisfies \(\psi(a)+\psi(b)+\psi(c)=\Psi(a+b+c)\) for \(a,b,c\ge0\). Setting \(c=0\) and comparing gives Cauchy's functional equation for \(\psi-\psi(0)\), whose continuous solutions are linear. So \(\ln\varphi(u)=\ln A-\beta u^2\), and integrability forces \(\beta>0\). The factor \(4\pi v^2\) is the area of the sphere of radius \(v\) in velocity space. Identifying \(\beta\) uses \(\tfrac12m\langle v^2\rangle=\tfrac32k_BT\), which links the distribution to the ideal gas law.

Maxwell knew the independence assumption was the weak point. In On the dynamical theory of gases (1867) he gave a second derivation based on collisions: in equilibrium, the rate of collisions taking pairs of velocities \((\mathbf v,\mathbf w)\) to \((\mathbf v',\mathbf w')\) must equal the rate of the reverse process, which requires \(f(\mathbf v)f(\mathbf w)=f(\mathbf v')f(\mathbf w')\) whenever energy and momentum are conserved. Boltzmann generalized this argument in 1868 and 1872.

Characteristic speeds

From \(g(v)\) one finds

in the ratio \(1:1.128:1.225\).

Mean free path and viscosity

Clausius had introduced the mean free path \(\lambda\) in 1858. For hard spheres of diameter \(d\), number density \(n\) and Maxwellian velocities,

Viscosity is the transport of momentum across layers of a sheared gas. A molecule crossing a plane carries the mean flow velocity of the layer about one free path away, so the stress is roughly \(\tfrac13 n m\langle v\rangle\lambda\) times the velocity gradient:

A denser gas has more carriers of momentum, but each travels a proportionally shorter distance. Maxwell found the prediction so surprising that he tested it himself. In experiments at his London home, with oscillating discs in a sealed chamber and with the help of his wife Katherine, he found the viscosity of air independent of pressure over a wide range, as predicted (Bakerian Lecture, 1866). He also found it roughly proportional to \(T\) instead of \(\sqrt T\). That led him in 1867 to molecules repelling with a force proportional to the inverse fifth power of distance, "Maxwell molecules", for which the collision integrals can be computed exactly.

The electromagnetic field and light

For the general reader. Electric charges and magnets push and pull at a distance. Faraday pictured the space around them filled with lines of force; Maxwell took the picture seriously and wrote equations for it. He added one new term, the displacement current, and found that the equations then predict waves travelling at a speed fixed by electrical measurements on the laboratory bench. That speed was the speed of light. Light, he concluded, is an electromagnetic wave.

Three papers and a treatise

On Faraday's lines of force (1855–56) was a geometrical analogy with fluid flow. On physical lines of force (1861–62) built a mechanical model of the ether: rotating molecular vortices for magnetism, with small "idle wheel" particles between them carrying electric current. To make the model elastic, he let the idle wheels be displaced slightly in insulators. That displacement, changing in time, acts as a current. Computing the speed of transverse waves in this elastic medium from the Weber–Kohlrausch measurement of the ratio of electrostatic to electromagnetic units, he found it almost exactly equal to Hippolyte Fizeau's measured speed of light. He wrote that one could scarcely avoid the inference that light is a transverse undulation of the same medium.

In A dynamical theory of the electromagnetic field (read 8 December 1864, published 1865) he discarded the vortices and derived the same results from general dynamical principles in Lagrange's form. The full theory appeared in A Treatise on Electricity and Magnetism (1873). Oliver Heaviside and Heinrich Hertz later recast it in the four vector equations used today.

Why the displacement current is needed

Ampère's law in its original form, \(\nabla\times\mathbf B=\mu_0\mathbf J\), implies \(\nabla\cdot\mathbf J=0\), since the divergence of a curl vanishes. But charge conservation says \(\nabla\cdot\mathbf J=-\partial\rho/\partial t\), which is not zero when a capacitor charges. Using Gauss's law, \(\partial\rho/\partial t=\varepsilon_0\,\nabla\cdot\partial\mathbf E/\partial t\), so \(\mathbf J+\varepsilon_0\partial\mathbf E/\partial t\) is divergence-free. Maxwell's added term makes Ampère's law consistent with conservation of charge. He arrived at it through his mechanical model, not through this argument, but the logic is the same.

Waves

In vacuum (\(\rho=0,\ \mathbf J=0\)), take the curl of Faraday's law and use \(\nabla\times(\nabla\times\mathbf E)=\nabla(\nabla\cdot\mathbf E)-\nabla^2\mathbf E\):

The plane-wave solutions \(\mathbf E=\mathbf E_0\cos(\mathbf k\cdot\mathbf x-\omega t)\) with \(\omega=c|\mathbf k|\) are transverse (\(\mathbf k\cdot\mathbf E_0=0\)), with \(\mathbf B=\hat{\mathbf k}\times\mathbf E/c\). Both constants \(\varepsilon_0\) and \(\mu_0\) are measured with charges, currents and coils, without any reference to light.

Energy and momentum

The field carries energy density \(u=\tfrac12\varepsilon_0E^2+\tfrac1{2\mu_0}B^2\). John Henry Poynting, one of Maxwell's students, showed in 1884 that its flux is \(\mathbf S=\mathbf E\times\mathbf B/\mu_0\). In the Treatise Maxwell predicted that light exerts pressure: an absorbing surface receives momentum flux \(I/c\) for intensity \(I\). Pyotr Lebedev, and Ernest Nichols and Gordon Hull, measured it around 1900–1901. Hertz generated and detected electromagnetic waves in 1887–88, eight years after Maxwell's death.

The demon and statistical thermodynamics

For the general reader. The second law of thermodynamics says heat flows from hot to cold, never the reverse on its own. Maxwell imagined a tiny being guarding a trapdoor between two boxes of gas, letting fast molecules through one way and slow ones the other. Without doing work, it would make one side hot and the other cold. His point was that the second law is a statistical truth about vast numbers of molecules, not an absolute law. Explaining why the demon fails took another century and led to the physics of information.

The demon

Maxwell described the idea to Tait in a letter of 11 December 1867, and published it in Theory of Heat (1871). He called the agent a being whose faculties are sharp enough to follow every molecule. William Thomson named it a "demon" in 1874. In a letter to John William Strutt, later Lord Rayleigh, Maxwell compared the second law to the impossibility of recovering a tumbler of water once it has been thrown into the sea.

Leo Szilard analysed a one-molecule engine in 1929: knowing which half of a box holds the molecule lets one extract \(k_BT\ln2\) of work by isothermal expansion. Rolf Landauer and Charles Bennett placed the unavoidable cost in erasing memory, not in measuring. Experiments since 2012 with colloidal particles and single electrons have measured the Landauer bound and built working information engines, none of which violates the second law.

Maxwell's relations

In Theory of Heat Maxwell derived four identities among thermodynamic derivatives, using geometrical arguments. In modern notation they follow from the first and second laws, \(dU=T\,dS-p\,dV\), and the equality of mixed partial derivatives of the potentials \(U,\ H=U+pV,\ F=U-TS,\ G=H-TS\):

For example \(dF=-S\,dT-p\,dV\) gives \(S=-\partial F/\partial T\) and \(p=-\partial F/\partial V\); equating \(\partial^2F/\partial V\partial T\) both ways gives the third relation. Their value is that they turn unmeasurable entropy derivatives into measurable ones involving \(p,V,T\).

The thermodynamic surface and the equal-area rule

After reading Josiah Willard Gibbs's papers of 1873, Maxwell modelled Gibbs's energy–entropy–volume surface for water in clay and plaster in 1874 and sent a cast to Gibbs at Yale. In 1875 he resolved a defect of Johannes van der Waals's equation of state, whose isotherms wiggle below the critical temperature. The liquid and vapour coexist at the pressure \(p_*\) for which the horizontal line cuts off equal areas above and below the isotherm, because the Gibbs free energies of the two phases must be equal:

On governors: the birth of control theory

For the general reader. A steam engine's governor uses spinning weights to open or close the steam valve and keep the speed steady. Engineers knew that governors sometimes made engines hunt, surging back and forth. In 1868 Maxwell showed how to predict this from the equations of motion: write the system's linear differential equation, and check whether the roots of its characteristic polynomial all have negative real parts. That question, and the feedback viewpoint behind it, became control theory.

The method

On governors appeared in the Proceedings of the Royal Society in 1868. Maxwell's interest came from the regulators used in the British Association's measurement of electrical resistance (Lesson 10), which needed a coil spinning at constant speed, and from designs by William Thomson and Fleeming Jenkin. He linearized the equations about the steady motion. Each deviation then behaves as a sum of terms \(e^{st}\), where \(s\) runs over the roots of a polynomial \(p(s)\). The motion is stable if and only if every root has negative real part.

Proof sketch for the cubic. Positive coefficients are necessary, since \(p(s)=a_0\prod(s-s_i)\) has positive coefficients when all roots lie in the left half-plane. Assume positivity; the roots move continuously with the coefficients, so stability can be lost only when a root crosses the imaginary axis. A real root cannot cross at 0 because \(a_3>0\). A pair crosses at \(s=\pm i\omega\) when \(-a_0\omega^3+a_2\omega=0\) and \(-a_1\omega^2+a_3=0\), that is \(\omega^2=a_2/a_0=a_3/a_1\), or \(a_1a_2=a_0a_3\). At \(a_3\to0^+\) the roots are near \(0\) and the roots of \(a_0s^2+a_1s+a_2\), all stable. So the region \(a_1a_2>a_0a_3\) is stable and the other side unstable.

Maxwell could not solve the general case. He posed it to the mathematical community, and Edward Routh answered it in his Adams Prize essay of 1877. Adolf Hurwitz found an equivalent determinant criterion independently in 1895. The result is the Routh–Hurwitz criterion.

Moderators and governors

Maxwell distinguished two kinds of device. A moderator applies a correcting force proportional to the speed error; it leaves a steady error under a constant load. A true governor also responds to the accumulated error, the integral of the deviation, and so drives the steady error to zero. In modern language these are proportional and integral control. Adding integral action raises the order of the system, and that is exactly where hunting appears.

Units and dimensional analysis

For the general reader. Every measured quantity is a number times a unit. Maxwell insisted that all units be built from a few basic ones, length, mass and time, and that every physical formula be balanced in these units. From this bookkeeping alone one can often guess the form of a law. The same bookkeeping revealed that the ratio of two systems of electrical units is a speed, a clue that led him to light.

Maxwell and units

From 1862 Maxwell served on the British Association committee on standards of electrical resistance. With Fleeming Jenkin and Balfour Stewart he measured the absolute value of the resistance unit, the forerunner of the ohm, by spinning a coil in the Earth's magnetic field. With Jenkin he wrote the 1863 report on the elementary relations between electrical measurements, which set out derived units systematically. In 1871 and in the preliminary chapter of the Treatise (1873) he wrote the dimensions of any quantity as a product of powers of the three fundamental units, using the symbols \([L]\), \([M]\), \([T]\), the notation still in use. Joseph Fourier had used dimensional exponents in 1822, and John William Strutt, Lord Rayleigh, made the method a standard tool.

Proof idea. A monomial \(\prod q_i^{\alpha_i}\) is dimensionless exactly when the exponent vector \(\alpha\) lies in the kernel of the \(k\times n\) dimension matrix, which has dimension \(n-r\). Since the law is unchanged when base units are rescaled, one can use \(r\) of the quantities as local units and set them to 1; the remaining freedom is expressed through the kernel's basis monomials.

Electrical units

In the electrostatic system, charge is defined by Coulomb's law \(F=q^2/r^2\) in vacuum; in the electromagnetic system, current is defined by the force per length \(F/\ell=2I^2/d\) between parallel wires. Maxwell emphasized that the resulting units for the same quantity differ in dimension by a velocity, and that the measured ratio of the units (Weber and Kohlrausch, 1856) is numerically a speed close to that of light. His own electrical theory explained why it is exactly \(c\).

Aberdeen, London, Glenlair, the Cavendish, 1856–1879

Aberdeen. Maxwell took up the chair at Marischal College in November 1856. There he wrote the Saturn essay and began kinetic theory. He married Katherine Mary Dewar, daughter of the college's principal, in 1858; they had no children, and she assisted in several of his experiments. In 1860 Marischal merged with King's College to form the University of Aberdeen, and as the junior of two professors of natural philosophy he lost his post. He applied that year for Forbes's chair at Edinburgh; Tait was appointed.

King's College London, 1860–1865. In 1860 he nearly died of smallpox. That year he took the chair of natural philosophy at King's College London. These five years were the most productive of his life: the colour photograph, Physical lines of force, the Dynamical theory, the resistance standard, the viscosity experiments, and a paper on reciprocal figures and frameworks that founded graphical statics. In London he came to know Faraday, then nearly seventy, at the Royal Institution. He was elected FRS in 1861.

Glenlair, 1865–1871. In 1865 he resigned and returned to Glenlair to manage the estate and write. From there came On the dynamical theory of gases, On governors, the demon, Theory of Heat, and most of the Treatise. He also served as an examiner and moderator for the Cambridge Tripos, which he helped to modernize by adding heat, electricity and magnetism.

The Cavendish. In 1871 the University of Cambridge created a professorship of experimental physics, funded with a new laboratory by its chancellor, William Cavendish, 7th Duke of Devonshire. William Thomson and Hermann von Helmholtz declined; Maxwell, after hesitation, was elected in March 1871. He designed the laboratory, which opened in June 1874, and bought much of its apparatus with his own money. Among those who studied or worked with him in Cambridge were John Henry Poynting, George Chrystal, Horace Lamb, Richard Glazebrook and Arthur Schuster. He spent five years editing the unpublished electrical papers of Henry Cavendish, which showed that Cavendish had anticipated Coulomb's law and Ohm's law. The edition appeared in 1879. He also wrote Matter and Motion (1876), a short book for students.

Death and legacy

In the spring of 1879 Maxwell began to have difficulty swallowing. He finished the Easter term's lectures and went to Glenlair for the summer, where his health declined. On 8 October he returned to Cambridge, barely able to walk. He died there of abdominal cancer, the disease that had killed his mother at the same age, on 5 November 1879. He was 48. His physician, George Paget, wrote of the calm with which he met death. He was buried in the churchyard of Parton Kirk, near Castle Douglas in Galloway. John William Strutt, Lord Rayleigh, succeeded him as Cavendish Professor.

Lewis Campbell and William Garnett published The Life of James Clerk Maxwell in 1882, and his Scientific Papers appeared in two volumes in 1890, edited by William Davidson Niven.

Legacy

• Electromagnetism: his theory predicted radio waves, confirmed by Hertz in 1887–88, and became the foundation of electrical engineering, optics and telecommunications. Its invariance group led Hendrik Lorentz and Albert Einstein to special relativity. Einstein kept portraits of Faraday and Maxwell, alongside Newton, on his study wall.
• Statistical physics: the Maxwell distribution, transport theory and the statistical reading of the second law were taken up by Boltzmann and Gibbs; the demon became a founding problem of the physics of information.
• Field theory: the idea that the field is a physical system with its own energy and momentum is the template for every modern gauge theory.
• Engineering and measurement: control theory, colorimetry, graphical statics and the dimensional framework of units all trace to his papers.
• Honours: the Smith's Prize (1854), the Adams Prize (1857), the Rumford Medal (1860), the Keith Medal of the Royal Society of Edinburgh and the Hopkins Prize of the Cambridge Philosophical Society. The James Clerk Maxwell Telescope on Mauna Kea, the Maxwell Montes on Venus and the Maxwell Gap in Saturn's C ring are named after him, as is the CGS unit of magnetic flux.

Connections

People linked to James Clerk Maxwell across this collection. A name without a link has no page here yet. See James Clerk Maxwell in the network of all 337 lives →

Teachers and advisors

• James David Forbes 1847-1850
• William Hopkins 1851-1854

Students

• George Chrystal
• Horace Lamb
• John Henry Poynting

Influenced by

• André-Marie Ampère
• Augustin-Jean Fresnel
• George Biddell Airy
• George Gabriel Stokes
• George Green
• Henry Cavendish
• James Watt
• John Herschel
• Michael Faraday
• Rudolf Clausius
• Thomas Young
• Wilhelm Weber
• William Rowan Hamilton
• William Thomson, Lord Kelvin

Influenced

• Albert Einstein
• Guglielmo Marconi
• Heinrich Hertz
• Hendrik Lorentz
• Hermann von Helmholtz
• J. J. Thomson
• Johannes Diderik van der Waals 1874-1875
• John William Strutt, Lord Rayleigh
• Josef Stefan
• Ludwig Boltzmann
• Oliver Heaviside

Colleagues and collaborators

• George Gabriel Stokes
• Hermann von Helmholtz
• John William Strutt, Lord Rayleigh
• Josiah Willard Gibbs
• Osborne Reynolds 1879
• William Crookes
• William Thomson, Lord Kelvin
• Fleeming Jenkin 1862-1864

Rivals

• Edward Routh 1854

Friends

• Peter Guthrie Tait 1841-1879

Also mentioned in this life

• Augustin-Louis Cauchy
• Charles-Augustin de Coulomb
• René Descartes
• Christian Doppler
• Hippolyte Fizeau
• Joseph Fourier
• Carl Friedrich Gauss
• Hermann Grassmann
• Adolf Hurwitz
• Johannes Kepler
• Joseph-Louis Lagrange
• Pierre-Simon Laplace
• Isaac Newton
• Georg Simon Ohm
• Count Rumford

Fields

• electromagnetism 12 lives
• kinetic theory 4 lives
• statistical mechanics 8 lives
• celestial mechanics 16 lives

On CoursesHub

• Maxwell & Vector Calculus · History of Mathematics & Physics
• Classical Electrodynamics
• Statistical Mechanics

Sources and further reading

Biography

1. J. J. O'Connor and E. F. Robertson, James Clerk Maxwell, MacTutor History of Mathematics.
2. James Clerk Maxwell, Wikipedia (with extensive references).
3. L. Campbell and W. Garnett, The Life of James Clerk Maxwell, Macmillan, 1882.
4. B. Mahon, The Man Who Changed Everything: The Life of James Clerk Maxwell, Wiley, 2003.
5. C. W. F. Everitt, James Clerk Maxwell: Physicist and Natural Philosopher, Scribner, 1975.
6. P. M. Harman (ed.), The Scientific Letters and Papers of James Clerk Maxwell, 3 vols, Cambridge University Press, 1990–2002.

Physics

1. J. C. Maxwell, On the Stability of the Motion of Saturn's Rings, Macmillan, 1859; summary in MNRAS 19 (1859).
2. MacTutor, James Clerk Maxwell on the nature of Saturn's rings.
3. Linear stability of the n-gon relative equilibria of the (1+n)-body problem, arXiv:1505.05246 (on Maxwell's ring model).
4. J. C. Maxwell, Experiments on colour, as perceived by the eye, Trans. R. Soc. Edinburgh 21 (1855); On the theory of compound colours, Phil. Trans. 150 (1860).
5. J. C. Maxwell, Illustrations of the dynamical theory of gases, Phil. Mag. (1860); On the dynamical theory of gases, Phil. Trans. 157 (1867).
6. J. C. Maxwell, On physical lines of force, Phil. Mag. (1861–62); A dynamical theory of the electromagnetic field, Phil. Trans. 155 (1865).
7. J. C. Maxwell, A Treatise on Electricity and Magnetism, Clarendon Press, 1873.
8. J. C. Maxwell, Theory of Heat, Longmans, 1871.
9. J. C. Maxwell, On governors, Proc. R. Soc. 16 (1868).
10. Maxwell's demon, Wikipedia; C. H. Bennett, The thermodynamics of computation, Int. J. Theor. Phys. 21 (1982).
11. Dimensional analysis, Wikipedia (history section).
12. O. Darrigol, Electrodynamics from Ampère to Einstein, Oxford University Press, 2000.