Léon Foucault

How to use this course

Léon Foucault had no degree in science until he was 33, held no professorship, and earned his living for years as a newspaper columnist. He was also the finest experimental physicist in France in his generation. His pendulum made the rotation of the Earth visible to anyone who stood under it. His rotating mirror showed that light is slower in water than in air and then measured its speed to better than one percent. His gyroscope, his discovery of the heating of conductors by eddy currents, and his method for making and testing silvered glass telescope mirrors are still part of everyday technology.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, derivations and exercises with folded solutions. Foucault was a close friend, partner and later rival of Hippolyte Fizeau; the Fizeau course tells the shared story from the other side.

Timeline

A publisher’s son, 1819–1839

Jean Bernard Léon Foucault was born in Paris on 18 September 1819, five days before Hippolyte Fizeau. His father, Jean Léon Fortuné Foucault, was a publisher. Poor health led the father to retire with his family to Nantes, where he died in 1829, when Léon was nine. Léon and his mother returned to Paris and lived at the corner of the rue de Vaugirard and the rue d’Assas, near the Luxembourg gardens. The house remained his home, and he died there in 1868.

He was a delicate child, educated partly at home. At the Collège Stanislas his teachers thought him lazy, and his mother engaged private tutors to bring him through his studies. His gift was in his hands: he was a patient and skilful builder of small mechanisms. At Stanislas he met Fizeau, who became a close friend.

Like Fizeau, he began in medicine. He entered the Paris medical school in 1839, but he could not bear the sight of blood and suffering, and he left after his first experience of hospital work.

Microscopes and daguerreotypes, 1839–1845

Foucault found his way into science through Alfred Donné, a physician who taught a course in microscopic anatomy. Donné engaged Foucault as his laboratory assistant for about three years. Foucault prepared the specimens and the projections for the lectures, and the two used the new daguerreotype process to photograph what the microscope showed. In 1845 they published an atlas of microscopy illustrated with engravings made from these daguerreotypes.

The daguerreotype had been made public in August 1839, and in September Foucault and Fizeau attended Louis Daguerre’s demonstrations in Paris. Projection microscopy and photography both need a powerful, steady light, and Foucault began to work on the carbon arc, in which an electric current crosses the gap between two carbon rods. The rods burn away and the gap widens, so the light flickers and dies unless the gap is regulated. Foucault built a regulator that adjusted the gap automatically, the first of many self-regulating mechanisms he designed.

In 1845 Donné handed over to Foucault his position as scientific writer for the Journal des débats, one of the most influential Paris newspapers. Foucault reported on the sessions of the Academy of Sciences for years. His reviews were clear and often sharp, and some academicians whose votes he would later need resented them.

Fizeau, Arago and the arc, 1845–1850

At the suggestion of François Arago, director of the Paris Observatory, Foucault and Fizeau photographed the image of the Sun formed by a telescope. Their daguerreotype of 1845 shows sunspots and the darkening of the solar disc towards its edge. They compared the brightness of sunlight with the light of the carbon arc and the limelight, and in 1847 they showed that the invisible radiant heat beyond the red end of the spectrum produces interference fringes like visible light. They also studied chromatic polarization and interference over long path differences.

In 1849 Foucault examined the spectrum of the carbon arc and found a bright double line at the position of the dark D lines in sunlight. When he passed sunlight through the arc, the D lines in the sunlight became darker. The arc therefore absorbed the same light it emitted. He published the observation, but it attracted little notice. Ten years later Gustav Kirchhoff, working independently, made the same observation (Lesson 04) the starting point of spectrum analysis and of the law relating emission and absorption.

By 1850 the partnership with Fizeau had become a rivalry. Both wanted to perform the experiment Arago had proposed in 1838: to compare the speed of light in air and in water. A sealed letter that Foucault deposited with the Academy on 27 May 1850 shows that the two had worked together that spring on whether moving air drags light, and found no effect. But on the air–water question they raced, and Foucault won (Lesson 05).

Arc light and the D lines, 1849

For the general reader. Sunlight spread into a spectrum is crossed by dark lines. In 1849 Foucault found that the light of an electric arc contains a bright double line at exactly the place of the dark D lines of the Sun, and that the arc, placed in the path of sunlight, made those dark lines darker. The same vapour that emits a colour also absorbs it. Ten years later Gustav Kirchhoff turned the same observation into the law that founded spectrum analysis and, with it, the chemistry of the stars.

The observation

Joseph von Fraunhofer had mapped the dark lines of the solar spectrum and noted that a lamp flame showed a bright double line at the position of his dark D line. Foucault, studying the arc for projection and photography, saw the bright D lines in its spectrum. He then sent sunlight through the arc and into the spectroscope. The solar D lines did not fill in; they became darker. When he sent light from the incandescent carbon tips, which give a continuous spectrum, through the arc, dark D lines appeared. He concluded that the arc is a medium that emits the D rays and absorbs them when they come from elsewhere.

Kirchhoff’s argument was thermodynamic: if some body absorbed less than it emitted at a given frequency, it could be used to drive heat from a cooler to a hotter body. The explicit form of \(B_\nu\) came only with Planck in 1900. The law explains why a strong absorber at a frequency is also a strong emitter there.

Bright or dark?

Proof. The transfer equation along the ray is \(dI_\nu/ds=-\alpha_\nu I_\nu+j_\nu\). With \(d\tau=\alpha_\nu ds\) and Kirchhoff’s law, \(dI_\nu/d\tau=-(I_\nu-B_\nu)\), whose solution is the formula. In Foucault’s experiment the background was sunlight or the white-hot carbon, brighter at the D lines than the arc vapour’s \(B_\nu\), so the lines appeared dark. Without a background, \(I_0=0\) and the lines appeared bright. The solar Fraunhofer lines arise in the same way, from cooler gas above hotter layers.

The arc regulator

An arc between carbon rods burns the rods away. As the gap lengthens, the current falls and the arc goes out. Foucault’s regulator used the arc current itself, through an electromagnet, to release a clockwork that moved the rods together whenever the current fell, a feedback mechanism.

Light in air and water, 1850

For the general reader. Is light a stream of particles or a wave? Both theories explained why a ray bends on entering water, but in opposite ways: Newton’s particles speed up in water, Huygens’s and Fresnel’s waves slow down. Arago proposed in 1838 to race light through air and water using a rapidly rotating mirror, an idea of Charles Wheatstone’s. Foucault made it work. In April 1850 he showed that light is slower in water.

Foucault’s arrangement

Light from an illuminated slit passes through a lens and falls on a plane mirror \(R\) that rotates rapidly about an axis in its own plane. The lens would form an image of the slit at a point on a concave mirror \(M\) whose centre of curvature lies on the axis of \(R\). Whatever the angle of \(R\), within the range for which the light strikes \(M\), the light returns along its own path to \(R\), and through the lens back to the slit. If \(R\) has turned during the round trip, the returning beam is deflected and the image of the slit is displaced. A glass plate at 45° reflects part of the returning light to a micrometer eyepiece.

A mirror turned through \(\alpha\) deflects a reflected beam through \(2\alpha\), and \(R\) turns through \(\omega\tau\) during the round trip. Foucault used two concave mirrors at equal distances from \(R\) and placed a 3 m tube of water in front of one of them. The rotating mirror, driven by a small steam turbine adapted from a siren of Charles Cagniard de la Tour, turned at 600 to 800 revolutions per second. Mirrors of tin amalgam broke up above about 200 revolutions per second, so he used silvered ones. The two return images, one from each path, appeared side by side. The image from the water path was displaced more. Each displacement was only a few tenths of a millimetre.

Foucault announced the result on 27 April 1850 and presented it to the Academy shortly after. Fizeau and Louis Breguet confirmed it on 17 June. Foucault concluded, more modestly than Arago had hoped, that the emission theory was incompatible with the facts. In 1853 he made the experiment the subject of his doctoral thesis at the Sorbonne. The experiment was qualitative: it compared two displacements and needed neither \(c\) nor the exact rotation rate. Pierre Duhem later used it as his example of why no single experiment can be strictly crucial between two theories, since it refutes only the emission theory together with its auxiliary assumptions.

The pendulum, 1851

For the general reader. A heavy pendulum, hung so that it can swing in any direction, keeps swinging in the same plane relative to the stars. The Earth turns beneath it. At the North Pole the floor would turn once a day under a fixed plane of swing. In Paris the effect is slower, a full turn in about 32 hours, but within minutes visitors under the dome of the Panthéon could see the pendulum’s line drifting clockwise. For the first time the rotation of the Earth was shown without looking at the sky.

History

In January 1851, in the cellar of his house, Foucault hung a 5 kg brass bob on a steel wire 2 m long and saw the plane of swing turn. Arago allowed him to repeat the experiment with an 11 m wire in the meridian room of the Paris Observatory, and on 3 February 1851 Arago presented Foucault’s note to the Academy. A few weeks later, with the support of the Prince-President Louis-Napoléon Bonaparte, a 28 kg bob was hung from the dome of the Panthéon on a wire 67 m long. From late March the public was admitted. A stylus beneath the bob cut a mark in a ring of sand at each swing.

Foucault stated that the plane turns at a rate proportional to the sine of the latitude. The mathematician Jacques Binet gave a derivation to the Academy on 17 February, and Giovanni Plana followed in Turin. Two centuries earlier, Vincenzo Viviani, a pupil of Galileo, had noticed that a pendulum’s swing drifts, but he did not connect it with the Earth’s rotation.

Equations of motion

Use local coordinates with \(x\) east, \(y\) north and \(z\) up, at latitude \(\phi\). The Earth rotates with angular velocity \(\boldsymbol\Omega\) of magnitude \(\Omega=7.292\times10^{-5}\ \text{rad/s}\), whose vertical component is \(\Omega_z=\Omega\sin\phi\). For small oscillations of a pendulum of length \(\ell\), with \(\omega_0^2=g/\ell\), the Coriolis acceleration \(-2\boldsymbol\Omega\times\mathbf v\) gives, neglecting vertical motion and terms of order \(\Omega^2\) in the centrifugal force,

Proof. The two equations combine into \(\ddot z+2i\Omega_z\dot z+\omega_0^2z=0\). Substitute \(z=e^{-i\Omega_zt}w\): then \(\dot z=e^{-i\Omega_zt}(\dot w-i\Omega_zw)\) and \(\ddot z=e^{-i\Omega_zt}(\ddot w-2i\Omega_z\dot w-\Omega_z^2w)\). Adding, the terms in \(\dot w\) cancel and \(\ddot w+(\omega_0^2+\Omega_z^2)w=0\). If the bob is released from rest in the rotating frame, \(w\) moves on a very thin ellipse, essentially a line, and that line turns as \(e^{-i\Omega_zt}\).

At the poles the plane turns once per sidereal day; at the equator it does not turn at all. In Paris, at latitude 48°51′, it turns about 11.3° per hour. In practice a pendulum develops a slight elliptical motion from imperfect release or asymmetric suspension, and an elliptical orbit precesses on its own at a rate proportional to its area. Long wires, heavy bobs and careful release keep this spurious precession small; Heike Kamerlingh Onnes analysed it in his doctoral thesis of 1879. Modern museum pendulums often use a Charron ring or an electromagnetic drive to suppress it.

The gyroscope, 1852

For the general reader. The pendulum shows only the component of the Earth’s rotation about the vertical, and slowly. Foucault looked for a more direct demonstration. A rapidly spinning, well-balanced wheel, mounted in gimbals so that no force can twist it, keeps its axis pointing in a fixed direction in space. Relative to the turning Earth the axis slowly appears to move. Foucault called the device a gyroscope, from Greek words for “rotation” and “to see”.

Foucault presented the gyroscope in September 1852. A heavy, carefully balanced rotor was spun up to high speed by a gear train, then placed in a gimbal mounting pivoted at its centre of mass. Through a microscope he watched the axis move relative to a scale fixed to the laboratory. The motion matched the Earth’s rotation and lasted only a few minutes before friction slowed the wheel, but it required no assumption about pendulum suspensions.

Part (b) follows from \(d\mathbf L/dt=\boldsymbol\tau\): a torque perpendicular to \(\mathbf L\) changes its direction without changing its size, at rate \(\tau/L\).

Pointing north

Foucault also described in 1852 what happens if the axis is constrained to stay horizontal. The Earth’s rotation then forces the axis towards the meridian, about which it oscillates. This is the principle of the gyrocompass, which Hermann Anschütz-Kaempfe made into a practical ship’s instrument in the first decade of the twentieth century.

Sketch. In the inertial frame the constraint must keep turning the horizontal plane, which rotates about the north horizontal axis at rate \(\Omega\cos\phi\). The component of that rotation that changes the direction of \(\mathbf L\) is \(\Omega\cos\phi\sin\alpha\), and the constraint supplies a torque of size \(L\Omega\cos\phi\sin\alpha\) about the horizontal, whose reaction acts about the vertical and drives \(\alpha\) towards 0.

Eddy currents, 1855

For the general reader. Move a lump of metal through a magnetic field and electric currents swirl inside it. They oppose the motion and heat the metal. Foucault showed this in 1855 with a copper disc spun between the poles of an electromagnet: when the magnet was switched on, the disc became hard to turn and grew hot. In France these eddy currents are still called courants de Foucault. They brake trains and roller coasters, heat induction cookers, and waste energy in transformers.

Arago had found in 1824 that a spinning copper disc drags a magnetic needle suspended above it, and Michael Faraday explained the effect by induction in 1831. Foucault reversed the arrangement in September 1855. He spun a copper disc by hand through a gear train between the poles of an electromagnet. With the magnet on, much more effort was needed to keep it turning, and the work done reappeared as heat: the disc grew noticeably warm within minutes.

In the frame of the sheet the field moves, and Faraday’s law gives an electric field \(\mathbf E'=\mathbf v\times\mathbf B\) in the field region. Ohm’s law gives \(\mathbf J=\sigma\mathbf E'\), and the power is \(\int J^2/\sigma\,dV\). The currents close through the field-free part of the sheet, and charges at the edges of the field region reduce the current, typically by a factor of order 2, which the simple model ignores. The force \(\mathbf J\times\mathbf B\) opposes the motion, as Lenz’s law requires. At high speed the eddy currents’ own field partly expels the applied field, and the drag falls off.

Laminations

In a conductor exposed to an alternating field \(B_0\sin\omega t\), eddy currents waste power. For a lamination of thickness \(d\) with the field parallel to its faces, the loss per unit volume at low frequency is

averaged over a cycle. The loss falls as \(d^2\), which is why transformer and motor cores are built from thin insulated sheets.

Silvered glass and the knife edge, 1857–1862

For the general reader. Before Foucault, large telescope mirrors were cast in speculum metal, a brittle alloy of copper and tin that reflected only about two thirds of the light and tarnished quickly. Foucault made mirrors of glass, which can be ground and polished accurately, and coated them with a thin chemical layer of silver that can be renewed. Then he invented a simple test, using only a light, a razor edge and the naked eye, that reveals errors in a mirror’s shape of a fraction of a wavelength. Every large telescope until the 1930s, and amateur telescope-making to this day, rests on these two ideas.

Sphere and paraboloid

The paraboloid focuses parallel light by the focus–directrix property. The sphere expansion is \(R-\sqrt{R^2-r^2}=\frac{r^2}{2R}+\frac{r^4}{8R^3}+\cdots\). For a 40 cm mirror with \(R=5\) m, the edge correction is \(0.2^4/(8\times125)=1.6\ \mu\text{m}\), about three wavelengths of light. The figure must be correct to a fraction of this.

The knife-edge test

Place a point source near the centre of curvature of the mirror, and bring a straight edge into the returning cone of light just at its focus. To an eye behind the edge, a perfect sphere with the source at its centre darkens uniformly all at once. If the edge is slightly inside or outside focus, a shadow sweeps across the mirror from one side or the other. A surface error that tilts the local normal sends that zone’s light past the edge or onto it, so the mirror appears as a relief map lit from the side, with hills and hollows of a few nanometres made visible.

Proof. At radius \(r\) the normal to \(z=r^2/(2R)\) has direction \((-r/R,1)\), so it meets the axis at \(z=r^2/(2R)+R=R+r^2/(2R)\). A source at that point lies on the normal and is returned to itself, which gives the first case. With the source fixed at \(z=R\), the source sits a distance \(r^2/(2R)\) in front of the normal’s crossing point. By the law of reflection, for small angles the reflected ray crosses the axis at the mirror-image point on the other side, a further \(r^2/(2R)\) beyond, so at \(R+r^2/R\) to leading order. The optician measures the knife position at which each zone darkens and compares with these values.

Foucault presented a 10 cm silvered-glass reflector on 16 February 1857, unaware that Carl August von Steinheil had made a similar proposal in Munich the year before. The silver was deposited chemically from solution. He published the knife-edge test in 1858 and a full memoir on silvered-glass telescopes in 1859. A 40 cm reflector was mounted at the Observatory in 1859, and in 1862 he completed a mirror of 80 cm aperture (78 cm clear) and 4.5 m focal length, then the largest of its kind. It was installed at the Marseille Observatory, where Édouard Stephan later used it to test Fizeau’s method of stellar interferometry and to discover galaxies. Foucault also invented the polarizing prism that bears his name (1857) and, in 1866, a solar eyepiece with a semi-transparent silver film on the objective.

The speed of light and the size of the solar system, 1862

For the general reader. In 1862 Foucault turned his rotating mirror into an absolute measuring instrument. With the light path folded into the space of a room, he measured the speed of light as 298 000 km/s, nearly 4% less than the value then accepted. The result mattered to astronomers. Combined with the aberration of starlight, it gave the speed of the Earth in its orbit and hence the distance to the Sun, and it agreed with Le Verrier’s calculations from the motions of the planets.

The measurement

The 1850 apparatus compared two paths. To measure \(c\), Foucault needed an exact rotation rate and a known path. He folded the path with five concave mirrors to a total one-way length of about 20 m, so that the return image was displaced by less than a millimetre. The rotation rate was fixed by comparing the mirror with a toothed disc driven by a clock. He presented the result to the Academy on 22 September 1862 and described the apparatus on 24 November.

From c to the Sun

James Bradley discovered in 1728 that stars appear displaced towards the direction of the Earth’s motion by an angle up to \(\kappa\approx v_\oplus/c\), the constant of aberration. The value then accepted, \(\kappa=20.445''\), gives \(v_\oplus\) once \(c\) is known. The Earth’s orbital speed and the length of the year give the radius of the orbit, and the radius of the Earth then gives the solar parallax \(\pi_\odot\), the angle the Earth’s equatorial radius subtends at the Sun.

With Foucault’s \(c\) this gives \(\pi_\odot=8.86''\), against the 8.57″ derived from the transits of Venus of 1761 and 1769 in Johann Encke’s reduction. Le Verrier had argued from planetary theory that the parallax was larger than Encke’s value; Foucault’s result supported him and reduced the accepted distance to the Sun by about 3%. The modern value is \(\pi_\odot=8.794''\).

The Observatory and the Academy, 1855–1868

The pendulum made Foucault famous, but it did not bring him a seat in the Academy. In March 1851 he stood for election, as Fizeau did, and failed; his reviews in the Journal des débats and his lack of formal qualifications counted against him. His doctorate of 1853 answered the second objection. Honours came from abroad and from the state. The Royal Society awarded him the Copley Medal for 1855 for the pendulum and gyroscope. Napoleon III, who had followed the pendulum demonstrations, had a post created for him: physicist attached to the Imperial Observatory, where Urbain Le Verrier had become director in 1854. The post, held from 1855, gave him a workshop and a modest salary.

At the Observatory he built the silvered-glass telescopes and the regulators that drove the instruments at constant speed. He went with Le Verrier to Spain to observe and photograph the total solar eclipse of 18 July 1860. In 1862 he became a member of the Bureau des longitudes and an officer of the Légion d’honneur. The Royal Society elected him a Foreign Member on 9 June 1864. He was also elected to the Leopoldina. In 1865, after earlier defeats, he was elected to the mechanics section of the Académie des sciences, in the seat of Émile Clapeyron. That year he published studies of the centrifugal governor of James Watt and of the regulation of electric light.

In 1867 he began to lose sensation in his hands, and paralysis followed quickly. The illness is usually described as a rapid form of multiple sclerosis; some writers suggest that a lifetime of work with mercury contributed. Near the end of his life he returned to the Catholic faith of his childhood. He died at his home in the rue d’Assas on 11 February 1868, aged 48, and was buried in Montmartre Cemetery.

Legacy

Foucault was an instrument-maker of genius who always aimed his instruments at a question of principle. His work divides between demonstrations that made physics visible and techniques that made it precise.

• The Earth’s rotation. Foucault pendulums swing in hundreds of science museums and universities. The original Panthéon bob is in the Musée des Arts et Métiers in Paris, and a pendulum again hangs in the Panthéon.
• Gyroscopes. Gyrocompasses guided ships and submarines through the twentieth century, and gyroscopes, mechanical, optical and micro-machined, stabilize aircraft, spacecraft and phones. Gravity Probe B used spinning quartz spheres as gyroscopes to measure relativistic precession.
• The speed of light. The rotating mirror was the basis of Michelson’s measurements from 1879 to 1926. Foucault’s 1862 value gave the first accurate laboratory determination of the scale of the solar system.
• Telescopes. Silvered (and later aluminized) glass mirrors made possible the great reflectors of the twentieth century. The knife-edge test is still taught to every amateur mirror-maker, and its descendants are used to test large optics.
• Electromagnetism. Eddy currents brake trains, heat induction furnaces and cookers, and detect cracks in metal; their losses shaped the design of every transformer.

His name is one of the 72 inscribed on the Eiffel Tower. A lunar crater, the asteroid 5668 Foucault and a street in the 16th arrondissement of Paris bear his name.

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Léon Foucault, MacTutor History of Mathematics.
2. Royal Society, Obituary notice of Léon Foucault, Proceedings of the Royal Society (1868).
3. Léon Foucault and Léon Foucault (French), Wikipedia.
4. W. Tobin, The Life and Science of Léon Foucault: The Man Who Proved the Earth Rotates, Cambridge University Press, 2003.
5. J. Lequeux, Le Verrier: Magnificent and Detestable Astronomer, Springer, 2013 (for the Observatory years).
6. A. D. Aczel, Pendulum: Léon Foucault and the Triumph of Science, Atria, 2003.

Physics

1. L. Foucault, Sur les vitesses relatives de la lumière dans l’air et dans l’eau, doctoral thesis, Paris, 1853.
2. L. Foucault, Démonstration physique du mouvement de rotation de la Terre au moyen du pendule, Comptes rendus 32 (1851).
3. L. Foucault, Sur les phénomènes d’orientation des corps tournants entraînés par un axe fixe à la surface de la Terre, Comptes rendus 35 (1852).
4. L. Foucault, Mémoire sur la construction des télescopes en verre argenté, Annales de l’Observatoire impérial de Paris 5 (1859).
5. L. Foucault, Détermination expérimentale de la vitesse de la lumière, Comptes rendus 55 (1862).
6. Foucault’s measurements of the speed of light, Foucault pendulum and Foucault knife-edge test, Wikipedia.
7. Historical synthesis supporting Duhem’s demonstration of the impossibility of the experimentum crucis, arXiv:2310.17982.
8. J. Texereau, How to Make a Telescope, Willmann-Bell, 2nd ed. 1984 (the knife-edge test in practice).