Charles-Augustin de Coulomb

How to use this course

Every student of physics meets Coulomb’s law in the first week of electromagnetism. Engineers meet his name again in friction, in soil mechanics and in the theory of torsion. These are not separate careers. Coulomb was an army engineer whose method was always the same: build an instrument that turns a small force into a measurable angle, find the simplest law that fits, and check it against practice.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements in modern notation, derivations and exercises with folded solutions. The lessons follow his work in order: statics of structures (1773), friction (1781), torsion (1784), and the seven memoirs on electricity and magnetism (1785–1791). Where Coulomb’s own reasoning differs from the modern derivation, the lesson says so.

Timeline

Angoulême, Paris, Montpellier, 1736–1758

Charles-Augustin Coulomb was born in Angoulême on 14 June 1736 and baptised in the parish church of Saint-André. His father, Henry Coulomb, came from a Montpellier family of lawyers and administrators and served as an inspector of the royal domains. His mother, Catherine Bajet, came from a wealthy family.

The family moved to Paris while he was young, and he studied at the Collège Mazarin, also known as the Collège des Quatre-Nations. Its curriculum covered languages, literature and philosophy, and also mathematics, astronomy, chemistry and botany. He was drawn above all to mathematics.

His father then lost much of the family fortune in speculation and returned to Montpellier. Coulomb followed him in 1757. Montpellier had an active Society of Sciences, affiliated with the Paris Academy, and he joined it in March 1757, presenting his first papers there. These were apprentice pieces, but they gave him the habit of presenting results to a learned society that would mark the rest of his life.

The engineering school at Mézières, 1758–1761

A career in the Corps du Génie, the military engineers, offered a young man of modest gentry a profession built on mathematics. Entry to the royal engineering school at Mézières, in the Ardennes, required a demanding examination set by the academician Charles-Étienne-Louis Camus. Coulomb returned to Paris in October 1758 to prepare with tutors, passed, and entered the school in February 1760.

Mézières was one of the best technical schools in Europe. Students learned descriptive geometry, mechanics, hydraulics, surveying, fortification and the strength of materials, and spent much time on drawing and practical site work. Among the teachers was the mathematician Charles Bossut, who taught mechanics and hydraulics and later became Coulomb’s sponsor at the Academy. A few years later Gaspard Monge joined the school.

Coulomb graduated in November 1761 as a lieutenant in the Corps du Génie. The school’s training explains much of his later science. He thought like an engineer: forces were to be balanced, materials had limits, and a theory was good if it predicted where a wall would crack.

Fort Bourbon and the road to Paris, 1762–1781

After a first posting at Brest, Coulomb was sent to Martinique in February 1764. France had just lost much of its empire in the Seven Years’ War, and Martinique, returned by the 1763 peace treaty, needed defending. He was put in charge of building Fort Bourbon, a large masonry fortress on the heights above Fort-Royal (now Fort-de-France). The work lasted until June 1772. Steep ground, a large workforce and tropical fever made it an education in structures and in endurance. He fell seriously ill more than once, and his health never fully recovered.

The fortress gave him his questions. How much thrust does earth exert on a retaining wall? When does a masonry pier crush? How should an arch be proportioned? Back in France, posted to Bouchain in Flanders, he wrote these thoughts up. In 1773 he read to the Académie des Sciences his Essai sur une application des règles de maximis et minimis à quelques problèmes de statique relatifs à l’architecture, printed in 1776 in the Academy’s series for outside contributors (Lesson 04). On 6 July 1774 he was named a correspondent of the Academy, attached to Bossut.

Postings followed at Cherbourg, Rochefort and the Île d’Aix. At Cherbourg he wrote a memoir on the best way to make magnetic needles, entered for the Academy’s prize of 1777. He suspended the needle on a silk thread to eliminate pivot friction and noticed that the twist of the thread gave a restoring torque. The prize was shared with the Dutch physicist Jan Hendrik van Swinden. At Rochefort, from 1779, he used the naval dockyard to run a long series of friction experiments on timber, which became Théorie des machines simples (Lesson 05). It won the Academy’s double prize in 1781. On 12 December 1781 he was elected adjoint mécanicien of the Academy, and from then on he lived in Paris.

Maxima and minima in architecture

For the general reader. Will a retaining wall hold back the earth behind it? Coulomb imagined every possible wedge of soil that might slide, computed the push each one exerts, and said the wall must resist the worst of them. Finding the worst case is a maximum problem from calculus. His 1773 memoir did this for walls, cuttings, piers and arches, and its central idea is still taught in every course on soil mechanics.

Cohesion plus friction

Coulomb treated a material as failing by sliding along a plane. The resistance on that plane has two parts: a cohesion proportional to the area of the plane, and a friction proportional to the normal force pressing the two sides together. In modern notation, with \(\tau\) the shear stress and \(\sigma\) the normal stress on a plane:

Active pressure on a wall

Take a smooth vertical wall of height \(H\) holding back dry, cohesionless soil (\(c=0\)) of unit weight \(\gamma\), with a horizontal surface. Consider a wedge cut off by a plane through the foot of the wall at angle \(\beta\) to the horizontal. Per unit length of wall, three forces act on it:

• its weight \(W=\tfrac12\gamma H^2\cot\beta\), downward;
• the wall’s reaction \(P\), horizontal because the wall is smooth;
• the reaction \(R\) of the soil below the plane, inclined at \(\varphi\) to the plane’s normal, opposing downward slip.

Closing the force triangle gives \(P = W\tan(\beta-\varphi)\), so

A wedge with \(\beta\) close to \(\varphi\) does not want to slide, and a wedge with \(\beta\) close to \(90^\circ\) is too thin to matter. In between, \(P\) has a maximum, and this maximum is the thrust the wall must bear.

Derivation

Write \(f(\beta)=\cot\beta\,\tan(\beta-\varphi)\). Using \(\tan(\beta-\varphi)=\sin(\beta-\varphi)/\cos(\beta-\varphi)\),

by the product-to-sum formulas. This is increasing in \(s=\sin(2\beta-\varphi)\), so it is largest when \(s=1\), that is \(2\beta-\varphi=90^\circ\). Substituting \(s=1\) gives \(K_a=(1-\sin\varphi)/(1+\sin\varphi)\), and the identity \((1-\sin\varphi)/(1+\sin\varphi)=\tan^2(45^\circ-\varphi/2)\) gives the other form. Coulomb’s memoir also handled cohesion and wall friction. William Rankine reached the same \(K_a\) in 1857 by a different route, from the state of stress in the soil.

The theory of simple machines: friction

For the general reader. Friction decides whether a ship slides down its launching ways, whether a rope holds on a capstan, and how much effort a pulley wastes. Guillaume Amontons had proposed in 1699 that friction is proportional to load and independent of contact area. Coulomb tested this on a large scale at Rochefort, with heavily loaded sledges, and gave the rules the form engineers still use, together with their exceptions.

What Coulomb measured

His apparatus was a heavy oak bench along which a loaded sledge was pulled by a rope running over a pulley to a weight pan. He varied the woods and metals in contact, the lubrication with tallow, the area of contact, the load and the speed, and also the time the surfaces had rested in contact before being set in motion. His main findings, in modern language, were these.

He explained friction mostly by the interlocking of surface roughness, with a smaller contribution from adhesion. The modern explanation is different: real contact happens only at microscopic asperities whose total area is proportional to the load, and friction is the force needed to shear those junctions. That theory, due to Frank Bowden and David Tabor in the twentieth century, explains why Coulomb’s laws work so well.

The friction cone

The total contact force on a body at rest lies inside a cone around the normal with half-angle \(\varphi_s=\arctan\mu_s\). A block on an incline stays at rest exactly when the incline’s angle satisfies \(\theta\le\varphi_s\). This angle of repose is the same \(\varphi\) that appeared in Lesson 04: Coulomb used one idea for blocks and for soil.

Dry-friction damping

A mass \(m\) on a spring of stiffness \(k\), sliding on a surface with kinetic coefficient \(\mu\), obeys \(m\ddot x+kx=-\mu mg\,\operatorname{sgn}(\dot x)\). On each half-swing the friction is constant, so the motion is simple harmonic about a shifted centre \(\pm\Delta\), with \(\Delta=\mu mg/k\).

Proof. Release from rest at \(x_0>\Delta\). While \(\dot x\lt0\), friction is \(+\mu mg\), so \(x\) oscillates about \(+\Delta\) and turns at \(x_1=2\Delta-x_0\), giving \(|x_1|=x_0-2\Delta\). The next half-swing oscillates about \(-\Delta\) and loses another \(2\Delta\). Viscous damping, by contrast, removes a fixed fraction per cycle.

The torsion of wires

For the general reader. Twist a thin wire and it twists back. Coulomb found in 1784 that the restoring torque is exactly proportional to the angle of twist, over a useful range, and that very fine wires resist with tiny torques. A fine wire with a pointer therefore becomes a balance able to weigh forces far smaller than any spring scale could detect. This instrument made his electrical work possible.

His 1777 memoir had guessed a cube law in the diameter. The 1784 experiments corrected it to the fourth power. He measured \(\kappa\) dynamically: a cylinder or rod hung from the wire oscillates with a period independent of amplitude, so the torque is linear in the angle.

Derivation from elasticity

In a twisted cylinder, a thin tube at radius \(r\) is sheared by the angle \(\gamma=r\theta/\ell\). With shear modulus \(G\), the shear stress is \(\tau=G\gamma\). The torque is

So Coulomb’s constant is \(\mu=\pi G/32\). The fourth power has a simple meaning: doubling the radius doubles both the strain at the rim and the lever arm, and quadruples the area.

The torsion pendulum

A body of moment of inertia \(I\) hanging from the wire obeys \(I\ddot\theta=-\kappa\theta\), so

Measuring a period and computing \(I\) from the shape of the bob calibrates the balance absolutely. Henry Cavendish used exactly this method in 1797–98 to weigh the Earth.

The inverse-square law of electricity

For the general reader. By the 1760s several people suspected that electric forces, like gravity, weaken as the square of the distance. Coulomb was the first to measure the force directly and publish the result. With his torsion balance he showed in 1785 that halving the distance between two charged balls quadruples their repulsion. The law that bears his name is the foundation of electrostatics.

Predecessors

Franz Aepinus considered an inverse-square law in the late 1750s. Joseph Priestley inferred it in 1767 from the absence of electric force inside a charged hollow vessel, by analogy with Newton’s theorem for gravity. John Robison measured an exponent near 2.06 in 1769 but did not publish it until much later. Henry Cavendish established the law with great precision in the early 1770s and left it unpublished. Coulomb knew none of the unpublished work. His contribution was a direct measurement of force against distance.

The repulsion experiment (first memoir, 1785)

Inside a glass cylinder, a horizontal straw coated with sealing-wax hung from a fine metal wire, with a small pith ball at one end. A second pith ball, fixed on an insulating stem, could be placed against the first. A torsion micrometer at the top of the wire set the twist, and a scale around the cylinder read the angle between the balls. Charging the two touching balls together gave them equal charges of the same sign, and they flew apart.

He reported three readings. With no added twist, the balls separated to \(36^\circ\). Twisting the micrometer by \(126^\circ\) brought them to \(18^\circ\), a total torsion of \(126+18=144^\circ\). Twisting it by \(567^\circ\) brought them to \(8\tfrac12^\circ\), a total of \(575\tfrac12^\circ\).

Coulomb never measured charge independently, since there was no unit of charge. He varied it by touching a charged ball with an identical uncharged one, which halves the charge. Halving one ball’s charge halved the force, which supports the product law \(F\propto q_1q_2\).

Attraction by oscillation (second memoir)

Attraction is unstable in a torsion balance: the balls rush together. Coulomb therefore hung a small horizontal needle of shellac, with a charged gilt-paper disc at one end, near a large charged globe, and timed its oscillations. If the force on the disc is \(F\propto1/r^2\), the restoring torque is proportional to \(F\), so the frequency satisfies \(\omega^2\propto F\) and the period \(T\propto r\). His timings followed this proportionality, after correction for leakage.

The best tests do not use balances at all. If the law were \(1/r^{2+\epsilon}\), a charged closed conductor would induce a small field inside it. Cavendish used this null method around 1773 (see the Cavendish module). In 1971 Edwin Williams, James Faller and Henry Hill bounded \(|\epsilon|\) below about \(3\times10^{-16}\).

Conductors, surface charge, leakage

For the general reader. Where does electricity sit on a charged metal object? Coulomb showed experimentally that it sits entirely on the surface, that it crowds toward sharp edges, and that two touching bodies share it in proportions set by their shapes. He also measured how charge slowly leaks into the air. These were the data that Siméon-Denis Poisson turned into the first mathematical theory of electrostatics.

The proof plane

To sample the charge at a point of a conductor, Coulomb touched it with a small gilt-paper disc on an insulating handle, then measured the disc’s charge in the torsion balance. The disc, lying flat against the surface, becomes part of it and carries away a charge proportional to the local surface density \(\sigma\). With this plan d’épreuve he found no charge inside hollow conductors and mapped how \(\sigma\) varies over spheres, cylinders and plates.

Statement (i) follows from Gauss’s law applied to any closed surface inside the metal. Statement (iii), often called Coulomb’s theorem, follows from Gauss’s law applied to a thin pillbox straddling the surface. Coulomb himself argued from the inverse-square law and from mutual repulsion of the electric fluid: each element pushes the others outward until none remains inside.

Curvature and sharing

In his fifth and sixth memoirs Coulomb measured how charge divides between spheres of different sizes placed in contact, and how density varies along a long cylinder, rising toward the ends. A clean model shows the trend. Two spheres of radii \(R_1,R_2\), far apart and joined by a thin wire, are at the same potential:

The smaller sphere has the higher density. This is why points discharge into the air and why lightning rods are pointed. For spheres in contact the ratio is less simple. Poisson solved that problem in 1812 with series expansions and found good agreement with Coulomb’s figures.

Leakage

In his third memoir Coulomb measured how a charged ball in the balance loses charge to the surrounding air and along its supports. Over short intervals he found that the rate of loss was proportional to the charge present, with a coefficient that depended on the humidity of the day. That is the law \(dq/dt=-q/\tau\), with exponential solution \(q(t)=q_0e^{-t/\tau}\). He used it to correct his inverse-square measurements and found that shellac insulated better than silk.

Magnetism and magnetic molecules

For the general reader. Coulomb began his research career with compass needles and ended his great series of memoirs with magnetism. He showed that magnetic poles also attract and repel as the inverse square of distance. He also argued that magnetism belongs to every tiny particle of a magnet, each a small magnet in its own right, which is why breaking a magnet in two gives two magnets.

Needles and oscillations

In the 1777 prize memoir he suspended compass needles on silk threads, showing that the thread’s torsion was small and calculable, and that a needle so hung tracked the daily variation of the Earth’s magnetic declination better than a pivoted one. A needle of magnetic moment \(m\) and moment of inertia \(I\) in a horizontal field \(B\) oscillates for small angles with

So \(B\propto1/T^2\) for a given needle. Coulomb used this to compare fields, and to map the strength of magnetization along a bar by bringing a small needle near different parts of it.

He tested this with long, thin magnetized needles, whose poles are concentrated near the ends, so that the distant pole could be neglected or corrected for. Modern physics treats isolated magnetic poles as fictions: the field of a magnet comes from currents and spins. The pole model still gives correct fields outside a magnet, because the far field of any small current loop is that of a pair of opposite poles.

Dipoles from poles

Place poles \(\pm p\) at \(z=\pm\ell/2\) and take a point on the axis at distance \(r\gg\ell\). The axial field is

On the perpendicular bisector the field is \(\frac{\mu_0}{4\pi}\frac{m}{r^3}\), half as large and antiparallel to \(m\). The \(1/r^3\) law of a dipole is how an inverse-square law of poles shows up in practice.

Magnetic molecules

Coulomb favoured two magnetic fluids, austral and boreal, but he observed that if they could flow freely through a bar, breaking it would give one north piece and one south piece. Since every fragment is a complete magnet, he concluded in his seventh memoir that the fluids never leave the individual particles. Each particle, a molécule aimantaire, becomes polarized, and the bar’s magnetism is the sum of these polarizations. He also found that a bundle of thin magnetized wires could be stronger than a single solid bar of the same weight, which he explained by the weakening effect each part has on its neighbours. These ideas anticipate magnetization as dipole density and the demagnetizing field.

Beams and piers

For the general reader. The 1773 memoir did more than retaining walls. Coulomb asked how a loaded beam breaks and how a stone pillar crushes. For the beam he put the stretched and compressed fibres in the right places, something earlier writers had got wrong. For the pillar he showed that crushing is really sliding along an inclined plane, governed by the same cohesion-plus-friction rule as soil.

The cantilever

Galileo had assumed that a cantilever breaks by rotating about its bottom edge, with the whole section in tension. Edme Mariotte and Jakob Bernoulli improved this, but uncertainty about the position of the neutral line persisted. Coulomb wrote the equilibrium of the section properly: the stresses over the section must have zero resultant force and a resultant moment equal to the moment of the load. With tension above and compression below varying linearly, the neutral axis falls at the centroid.

Derivation. At height \(y\) above the neutral axis the stress is \(\sigma(y)=2\sigma y/h\). Zero net force holds by symmetry. The moment is \(\int_{-h/2}^{h/2}\sigma(y)\,y\,b\,dy=\frac{2\sigma b}{h}\cdot\frac{h^3}{12}=\frac{\sigma bh^2}{6}\).

The pier

A prism of masonry with cross-section \(A\) carries a vertical load \(P\). Coulomb supposed it fails by the upper part sliding along a plane inclined at \(\theta\) to the horizontal. The plane has area \(A/\cos\theta\). The load resolves into a component \(P\sin\theta\) along the plane and \(P\cos\theta\) across it. By the criterion of Lesson 04, slip requires

Derivation. Rearranging, slip occurs when \(P/A\ge c/[\cos\theta(\sin\theta-\cos\theta\tan\varphi)]\). Multiply numerator and denominator by \(\cos\varphi\): the denominator becomes \(\cos\theta\sin(\theta-\varphi)/\cos\varphi=[\sin(2\theta-\varphi)-\sin\varphi]/(2\cos\varphi)\). It is largest at \(2\theta-\varphi=90^\circ\), giving \(\sigma_c=2c\cos\varphi/(1-\sin\varphi)\), which equals \(2c\tan(45^\circ+\varphi/2)\).

Academician through the Revolution, 1781–1806

In Paris Coulomb combined Academy research with engineering commissions. One of these went badly. Sent to assess a project for navigable canals in Brittany in 1783–84, he reported against it and spent a week in prison in November 1783 after a dispute with the provincial authorities. He served as intendant of the royal waters and fountains, and in 1787 travelled to England with the surgeon Jacques Tenon to study hospital design. Their report on the Royal Naval Hospital at Stonehouse, near Plymouth, recommended its plan of separate pavilions to the French government.

Between 1781 and 1806 he read some twenty-five memoirs to the Academy and its successor. He sat on many Academy commissions, examining inventions, machines and proposals. The Revolution ended this world gradually. He retired from the Corps du Génie in April 1791 with the rank of lieutenant-colonel. He worked on the new system of weights and measures until the commission was purged in December 1793. He then withdrew, with the engineer and naval officer Jean-Charles de Borda, to La Justinière, his small property near Blois, and continued experiments there.

When the Institut de France replaced the abolished academies, Coulomb was elected in December 1795 to its section of experimental physics and returned to Paris. Around 1800 he presented experiments on the resistance of fluids to very slow motion, using a disc oscillating on a torsion wire in water and oil. He found that the resistance contained a term proportional to the velocity and one proportional to its square, and that it hardly changed when he coated the disc with tallow and then sprinkled it with sand, evidence that the fluid layer in contact with the disc moves with it. This is an early measurement of what is now called viscosity.

Under the Consulate he was appointed in 1802 one of the inspectors general of public instruction, and he worked hard to set up the new lycées across France. He was also made a member of the Legion of Honour. In 1802 he married Louise Françoise LeProust Desormeaux, the mother of his two sons, born in 1790 and 1797. He died in Paris on 23 August 1806, aged 70, after a long period of weak health, and his funeral was held at Saint-Germain-des-Prés.

Legacy: a unit, a law, a method

Coulomb’s measurements gave the French school of mathematical physics its starting point. Pierre-Simon Laplace and Siméon-Denis Poisson assumed his inverse-square laws, and Poisson’s memoirs of 1812 and 1824 on electricity and magnetism compared their predictions with Coulomb’s numbers. The two-fluid theory he favoured did not survive the discovery of the electron, but the force law did. Carl Friedrich Gauss and Wilhelm Weber later built absolute systems of electrical and magnetic units on exactly the kind of torsion and oscillation measurements Coulomb pioneered.

Where his name appears

• Units: the coulomb, now the SI unit of electric charge, was adopted with the volt, ampere and ohm by the International Electrical Congress in Paris in 1881.
• Physics: Coulomb’s law; the Coulomb potential and Coulomb gauge; Coulomb scattering and the Coulomb barrier in nuclear physics.
• Engineering: Coulomb friction and Coulomb damping; the Mohr–Coulomb criterion; Coulomb’s wedge theory of earth pressure. The British journal Géotechnique carries an illustration from his 1776 memoir on its cover.
• Instruments: the torsion balance, which Cavendish used for gravity, and which in refined forms tested the equivalence principle (Loránd Eötvös) and continues to probe gravity at short range.
• Honours: his name is one of the 72 inscribed on the Eiffel Tower.

Did the experiment work?

Coulomb reported only a few readings, and historians have asked how hard his balance was to use. In the early 1990s Peter Heering rebuilt the balance from Coulomb’s description. He found consistent results hard to obtain, with disturbances from the experimenter’s own charge, air currents and leakage, and argued that the published figures are unlikely to be a plain record of single runs. Coulomb was confident of the law from several methods, and he reported the readings that best displayed it, as was common practice at the time. The episode is a useful lesson about what a published experiment does and does not show.

Connections

People linked to Charles-Augustin de Coulomb across this collection. A name without a link has no page here yet. See Charles-Augustin de Coulomb in the network of all 337 lives →

Teachers and advisors

• Charles Bossut 1760-1761

Influenced by

• Benjamin Franklin
• Joseph Priestley
• Franz Aepinus
• Guillaume Amontons

Influenced

• Gaspard-Gustave Coriolis
• Henry Cavendish
• Jean-Victor Poncelet
• Siméon Denis Poisson

Colleagues and collaborators

• Henry Cavendish
• Pierre-Simon Laplace
• Jacques Tenon 1787
• Jean-Charles de Borda 1793-1795

Rivals

• Lazare Carnot 1777-1781
• Jan Hendrik van Swinden 1777

Also mentioned in this life

• Galileo Galilei
• Carl Friedrich Gauss
• Jacob Bernoulli
• Gaspard Monge
• Isaac Newton
• Wilhelm Weber

Fields

• electrostatics 4 lives
• magnetism 5 lives
• applied mechanics 3 lives
• elasticity 11 lives

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Charles Augustin de Coulomb, MacTutor History of Mathematics.
2. C. S. Gillmor, Coulomb, Charles Augustin, Dictionary of Scientific Biography.
3. C. S. Gillmor, Coulomb and the Evolution of Physics and Engineering in Eighteenth-Century France, Princeton University Press, 1971.
4. Charles-Augustin de Coulomb, Wikipedia.
5. EDF, Charles-Augustin Coulomb (booklet, in French).

Physics

1. C.-A. Coulomb, Essai sur une application des règles de maximis et minimis à quelques problèmes de statique relatifs à l’architecture, Mémoires de mathématique et de physique présentés à l’Académie royale des sciences par divers savans 7 (1776).
2. C.-A. Coulomb, Théorie des machines simples, Mémoires … par divers savans 10 (1785).
3. C.-A. Coulomb, Recherches théoriques et expérimentales sur la force de torsion et sur l’élasticité des fils de métal, Mémoires de l’Académie royale des sciences for 1784.
4. C.-A. Coulomb, Premier à septième mémoires sur l’électricité et le magnétisme, Mémoires de l’Académie royale des sciences for 1785–1789.
5. Coulomb’s law, Wikipedia (history and modern tests).
6. Europa-Universität Flensburg, Coulomb’s torsion balance (replica and description).
7. P. Heering, On Coulomb’s inverse square law, American Journal of Physics 60 (1992) 988.
8. E. R. Williams, J. E. Faller and H. A. Hill, New experimental test of Coulomb’s law: a laboratory upper limit on the photon rest mass, Physical Review Letters 26 (1971) 721.
9. J. Heyman, Coulomb’s Memoir on Statics: An Essay in the History of Civil Engineering, Cambridge University Press, 1972 (with translation).