André-Marie Ampère

How to use this course

Every physics student meets Ampère three times: in the unit of current, in the circuital law \(\oint\mathbf B\cdot d\mathbf l=\mu_0 I\), and in the rule that parallel currents attract. Fewer know that he discovered that rule within a fortnight of hearing about Ørsted’s compass needle, that he built the first quantitative theory of forces between currents, and that he insisted magnets themselves are made of currents. James Clerk Maxwell called him “the Newton of electricity”. This course follows that work in detail and places it in a life of unusual difficulty.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements in modern SI notation, derivations and exercises with folded solutions. Ampère wrote in his own units and in the language of forces between pairs of elements; where his form differs from ours, the lesson says so.

Timeline

Lyon and Poleymieux, 1775–1793

André-Marie Ampère was born in Lyon on 20 January 1775. His father, Jean-Jacques Ampère, was a prosperous merchant and an admirer of Jean-Jacques Rousseau. Following Rousseau’s ideas on education, he never sent his son to school. From 1782 the family lived mostly at Poleymieux, a village in the Monts d’Or about ten kilometres north of Lyon, where the house is now a museum of electricity.

The boy taught himself from his father’s library. He read the articles of the Encyclopédie in no particular order and, by his own later account, remembered much of them. At thirteen he submitted a paper to the Académie de Lyon on the rectification of the circle, a construction of a straight segment equal to a circular arc. It was not published, and a reader advised him to learn the calculus first. He did. He learned Latin in order to read Euler and the Bernoullis, and he worked through the 1788 Mécanique analytique of Lagrange, redoing its calculations.

The Revolution

In 1791 his father became a justice of the peace in Lyon. In 1793 Lyon rose against the Convention, was besieged and fell. In the repression that followed, the elder Ampère was arrested for his part in the prosecution of a Jacobin leader and guillotined on 24 November 1793. His sister had died the year before. The eighteen-year-old Ampère fell into a depression that lasted more than a year. He recovered slowly through botany, Latin poetry and long walks, interests he kept all his life.

Julie, Bourg and the theory of play, 1794–1803

In 1796 Ampère met Julie Carron, from a family near Lyon. He was poor and without prospects, and her family made him wait. He supported himself by private teaching of mathematics in Lyon, and they married in 1799. Their son, Jean-Jacques, was born in 1800; he became a well-known historian and philologist.

In 1802 Ampère was appointed professor of physics and chemistry at the École centrale of the Ain department at Bourg-en-Bresse. Julie was already ill and stayed in Lyon, so the couple lived apart, writing almost daily. In Bourg he wrote his first serious work, Considérations sur la théorie mathématique du jeu (1802). It proved that a gambler who plays a fair game against an opponent with unlimited resources will be ruined with certainty (Lesson 10). He sent it to the Paris Academy. Laplace found an error in a later section, which Ampère corrected, and Jean-Baptiste Delambre, impressed by the work, recommended him for a post in mathematics at the new lycée in Lyon.

He took up the post in 1803. Julie died in July of that year. Ampère never fully recovered from the loss. In 1804 he left Lyon for Paris.

Paris, 1804–1820

In 1804 Ampère became répétiteur in analysis at the École polytechnique, the post of a tutor who drilled students on the professors’ lectures. In 1809 he was made professor of analysis and mechanics, a position he held until 1828. He wrote on the calculus of variations and on partial differential equations; his classification of such equations, presented to the Institut in 1814, supported his candidacy there.

His private life went badly. In 1806 he married Jeanne Potot, at a ceremony attended by Lagrange and Delambre. Their daughter Albine was born in 1807, but the marriage failed almost at once, and a legal separation followed; Ampère took custody of Albine. In 1808 Napoleon’s government appointed him one of the inspectors general of the new Imperial University. The post required long tours of provincial schools, which he kept up for almost three decades and which brought in the income his professorships did not.

Ampère’s interests in these years were spread across mathematics, chemistry and philosophy. He corresponded with Humphry Davy about the nature of chlorine and of the acid of fluorspar (Lesson 09), and in 1814 he published a theory of gas volumes based on molecular shapes. In November 1814 he was elected to the mathematics section of the Institut, defeating the young Augustin-Louis Cauchy. By 1820 he was a respected but not famous member of the Paris scientific world, known as an absent-minded, warm and talkative man with an encyclopaedic range. His fame came from a few weeks in the autumn of 1820.

September 1820: currents act on currents

For the general reader. In 1820 the Danish physicist Hans Christian Ørsted found that an electric current turns a compass needle. Ampère asked a bolder question: if a current acts like a magnet, does a current act on another current, with no magnet involved? Within two weeks he showed that it does. Two parallel wires pull together when their currents run the same way and push apart when they run opposite ways. For most of the twentieth century this force defined the ampere.

Three weeks

Ørsted announced his discovery in a short Latin pamphlet dated 21 July 1820. François Arago reported it to the Académie des sciences on 4 September and repeated the experiment before the members on 11 September. Ampère went home and began experimenting. On 18 September he read his first note, and on 25 September he announced that two wires carrying currents attract or repel each other. He had them built so that one wire could swing freely, and showed that the effect reverses when either current is reversed.

He also fixed the vocabulary. He separated the tension of a battery from the current in a closed circuit, and he took the direction of the current to be that in which positive electricity flows. His rule for the needle used a little man, the bonhomme d’Ampère, lying along the wire with the current entering at his feet and leaving at his head, and facing the needle. The north pole of the needle is then deflected to his left. In modern terms, the magnetic field circles a straight current according to the right-hand rule.

The parallel-wire force

A long straight wire carrying current \(I_2\) produces at distance \(d\) the field \(B=\mu_0 I_2/(2\pi d)\), tangent to circles about the wire. A parallel wire carrying \(I_1\) feels the force \(I_1\,\mathbf L\times\mathbf B\) on a length \(L\). With both currents along \(\hat{\mathbf z}\) and wire 1 at position \(d\,\hat{\mathbf x}\) relative to wire 2, \(\mathbf B=\frac{\mu_0I_2}{2\pi d}\hat{\mathbf y}\) and \(\hat{\mathbf z}\times\hat{\mathbf y}=-\hat{\mathbf x}\), so the force points back toward wire 2.

The ampere

In 1948 the General Conference on Weights and Measures defined the ampere as the constant current which, in two straight parallel conductors of infinite length and negligible cross-section placed one metre apart in vacuum, produces between them a force of \(2\times10^{-7}\) newton per metre of length. By the proposition, this fixed \(\mu_0=4\pi\times10^{-7}\ \mathrm{N\,A^{-2}}\) exactly. Since 20 May 2019 the ampere is defined instead by fixing the elementary charge, \(e=1.602176634\times10^{-19}\ \mathrm C\), so that one ampere is a definite number of elementary charges per second. The constant \(\mu_0\) is now measured; it equals \(4\pi\times10^{-7}\ \mathrm{N\,A^{-2}}\) to about one part in \(10^{10}\).

The force between current elements

For the general reader. Newton explained the planets by a force between pairs of point masses. Ampère wanted the same for electricity: a law for the force between two tiny pieces of wire, from which every effect of circuits on circuits could be computed by adding up pairs. He found it by a clever trick. Instead of measuring forces, he designed arrangements in which forces exactly cancel, and asked which laws allow those cancellations.

Ampère’s assumptions and the null method

Ampère assumed that the force between two current elements \(I_1\,d\mathbf s_1\) and \(I_2\,d\mathbf s_2\) acts along the line joining them, as Newton’s third law in its strong form requires, and that its size depends only on the lengths of the elements, the currents, the distance \(r\) and the angles. Let \(\theta_1,\theta_2\) be the angles the elements make with the joining line, and \(\omega\) the angle between the two planes each element forms with that line. He wrote the attraction as

with constants \(n\) and \(k\) to be found from experiment. Between 1820 and 1825 he built four equilibrium arrangements:

1. A wire doubled back on itself exerts no force: reversing a current reverses its action.
2. A wire bent into small zigzags acts like a straight wire between the same ends: elements add like vectors.
3. A closed circuit exerts no force on an element along the element’s own direction.
4. Three similar circular circuits, with the middle one’s radius and distances scaled by a common factor, leave the middle circuit in equilibrium: the force law is homogeneous.

Case 3 gives the relation \(n+2k=1\), and case 4 gives \(n=2\), so \(k=-\tfrac12\). Ampère was unable to measure these forces directly with any precision; his law rests on cancellations, which can be observed far more accurately.

To connect the two forms, note that \(\cos\varepsilon=\sin\theta_1\sin\theta_2\cos\omega+\cos\theta_1\cos\theta_2\), where \(\varepsilon\) is the angle between the elements. With \(k=-\tfrac12\) the bracket in Ampère’s form becomes \(\cos\varepsilon-\tfrac32\cos\theta_1\cos\theta_2=\tfrac12\big(2\cos\varepsilon-3\cos\theta_1\cos\theta_2\big)\), matching the vector form up to the choice of units.

Comparison with the Biot–Savart–Lorentz form

The modern textbook gives the force on element 1 as \(I_1\,d\mathbf s_1\times d\mathbf B_2\) with the Biot–Savart field \(d\mathbf B_2=\frac{\mu_0I_2}{4\pi}\frac{d\mathbf s_2\times\hat{\mathbf r}}{r^2}\). Hermann Grassmann proposed this element law in 1845. It is not central and violates Newton’s third law for isolated elements. Yet:

Proof sketch. Expand \(d\mathbf s_1\times(d\mathbf s_2\times\hat{\mathbf r})=d\mathbf s_2\,(d\mathbf s_1\cdot\hat{\mathbf r})-\hat{\mathbf r}\,(d\mathbf s_1\cdot d\mathbf s_2)\). Moving along \(C_1\), \(d(1/r)=-\hat{\mathbf r}\cdot d\mathbf s_1/r^2\), so the first term integrates to zero around \(C_1\). This gives the displayed form for Grassmann’s law. For Ampère’s law, fix \(\mathbf a=d\mathbf s_2\) and differentiate \(\mathbf g=\hat{\mathbf r}(\hat{\mathbf r}\cdot\mathbf a)/r\) along \(C_1\):

Since \(\oint_{C_1}d\mathbf g=0\), the extra term \(\hat{\mathbf r}\big[(d\mathbf s_1\cdot d\mathbf s_2)-3(\hat{\mathbf r}\cdot d\mathbf s_1)(\hat{\mathbf r}\cdot d\mathbf s_2)\big]/r^2\) by which Ampère’s bracket exceeds Grassmann’s integrates over \(C_1\) to \(-\oint_{C_1}d\mathbf s_1(\hat{\mathbf r}\cdot d\mathbf s_2)/r^2\). Now integrate over \(C_2\): along \(C_2\), \(d(1/r)=+\hat{\mathbf r}\cdot d\mathbf s_2/r^2\), so this vanishes too. \(\square\)

Because all real steady currents flow in closed circuits, experiments with steady currents cannot decide between the two element laws. The question was revived several times, notably around Ampère’s floating “hairpin” experiment, in which a U-shaped wire floating on mercury is pushed along its legs. The modern view is that the Lorentz force on moving charges is fundamental and that electromagnetic momentum accounts for apparent violations of the third law.

Magnetic shells and the circuital law

For the general reader. A loop of current and a thin sheet of magnetism stretched across the loop are indistinguishable from outside. Ampère proved this in 1826. It means the magnetic effect of a loop depends only on how large the loop looks from where you stand. Walk once around the wire and that apparent size changes by a fixed amount, which gives the circuital law now taught under Ampère’s name.

Sketch. Divide \(S\) into small meshes, each carrying the current \(I\) around its edge; interior edges cancel in pairs. A small loop of area \(dA\) and normal \(\hat{\mathbf n}\) has dipole moment \(I\,dA\,\hat{\mathbf n}\) and scalar potential \(I\,dA\,\hat{\mathbf n}\cdot\hat{\mathbf R}/(4\pi R^2)\), where \(\mathbf R\) runs from the mesh to \(P\). The factor \(dA\,\hat{\mathbf n}\cdot\hat{\mathbf R}/R^2\) is the solid angle of the mesh seen from \(P\). Summing gives \(I\Omega/(4\pi)\). The small-loop field itself follows from the Biot–Savart law, or from Ampère’s force law applied to closed circuits.

The circuital law

Follow a closed path \(\gamma\) that links \(C\) once. As \(P\) goes around \(\gamma\), it crosses \(S\) once, and \(\Omega\) changes continuously by \(-4\pi\) along the path before jumping back by \(4\pi\) at the crossing. Hence the line integral of \(\mathbf B=-\mu_0\nabla\Phi\) along \(\gamma\), which only sees the continuous change, is \(\mu_0 I\).

The integral form is named for Ampère but written in this way by Maxwell in the 1850s. Maxwell noticed that it cannot be right for changing currents: taking the divergence of \(\nabla\times\mathbf B=\mu_0\mathbf J\) gives \(\nabla\cdot\mathbf J=0\), which contradicts charge conservation \(\nabla\cdot\mathbf J=-\partial\rho/\partial t\) when charge accumulates. In 1861 he added the displacement current:

With Gauss’s law \(\nabla\cdot\mathbf E=\rho/\varepsilon_0\) the divergence of the right side now vanishes identically, and the corrected law led Maxwell to electromagnetic waves.

Magnets are currents

For the general reader. Ampère took the boldest step of 1820: he proposed that there is no separate magnetic substance at all. A bar magnet is a bundle of tiny electric currents circulating inside its particles, all lined up. A coil of wire, which he called a solenoid, then behaves exactly like a bar magnet, and the Earth’s magnetism should come from currents inside the Earth. Electron spin aside, this is the modern view.

From magnets to loops

Ampère first imagined currents circulating around the axis of a magnet on a large scale. His friend Augustin Fresnel pointed out in 1821 that such currents would heat the magnet, and that currents confined to individual molecules would avoid the problem. Ampère adopted the molecular picture. Magnetization then means the alignment of pre-existing molecular currents, and a magnetized body behaves like a set of current loops whose interior currents cancel, leaving an effective surface current.

Torque on a rectangle. Take sides \(a\) (along \(\hat{\mathbf x}\)) and \(b\), in a uniform field \(\mathbf B\). The forces on opposite sides are equal and opposite, so the net force vanishes. The forces on the four sides form couples whose total torque works out to \(I ab\,\hat{\mathbf n}\times\mathbf B\) for any orientation of \(\mathbf B\). Any plane loop is a sum of small rectangles with shared edges cancelling, so \(\boldsymbol\tau=\mathbf m\times\mathbf B\) holds for any plane loop in a uniform field.

The solenoid

A uniformly magnetized cylinder with magnetization \(\mathbf M\) (moment per unit volume) along its axis is equivalent to a surface current density \(\mathbf K=\mathbf M\times\hat{\mathbf n}\), of size \(M\), circulating around the side. A solenoid with \(n\) turns per unit length carrying \(I\) has exactly this surface current with \(M=nI\). Outside, the two produce identical fields; inside, the solenoid has \(B=\mu_0nI\) and the magnet has \(B=\mu_0M\) (for a long magnet). Ampère demonstrated with suspended helices that a solenoid aligns with the Earth’s field and attracts and repels magnets just as a compass needle does.

Induction: the near miss and the sequel

For the general reader. If currents make magnetism, can magnetism make currents? Ampère looked for this and, in 1822, saw a hint of it without recognizing what he had seen. Michael Faraday found the answer in 1831: a changing magnetic field produces a current. Yet Ampère’s mathematics had already contained much of the theory that would describe it.

Geneva, 1822

In Geneva in 1822, Ampère and the physicist Auguste de la Rive suspended a light copper ring inside a coil and brought a strong magnet near. When the current in the coil was switched, the ring moved slightly. The effect was a current induced in the ring by the changing field. Ampère was looking for a steady effect, consistent with his theory of permanent molecular currents, and did not pursue a transient one. Arago’s rotating disc of 1824, another induction effect, also went unexplained until Faraday.

Neumann’s formula

In 1845 Franz Neumann showed that induction between circuits can be derived from a single function built on Ampère’s work. Define the mutual inductance of two closed circuits

Check. Write \(\mathbf B_2=\nabla\times\mathbf A_2\) with \(\mathbf A_2(\mathbf x)=\frac{\mu_0I_2}{4\pi}\oint_{C_2}\frac{d\mathbf s_2}{|\mathbf x-\mathbf r_2|}\). By Stokes, \(\Phi_{12}=\oint_{C_1}\mathbf A_2\cdot d\mathbf s_1=M_{12}I_2\). For the force, \(\nabla_1(1/r)=-\hat{\mathbf r}/r^2\), so \(I_1I_2\nabla_1M_{12}=-\frac{\mu_0I_1I_2}{4\pi}\oint\oint\frac{\hat{\mathbf r}}{r^2}(d\mathbf s_1\cdot d\mathbf s_2)\), exactly the closed-circuit force of Lesson 05. Ampère’s force law and the inductance that governs Faraday’s effect are two faces of the same double integral.

Gases, molecules and fluorine

For the general reader. Before electricity, Ampère’s main scientific passion was chemistry. In 1814, independently of Amedeo Avogadro, he proposed that equal volumes of gases contain equal numbers of molecules. Chemists ignored the idea for almost fifty years. He also suggested that a still-unisolated element lay hidden in fluorspar and gave it its name, fluorine.

Combining volumes

In 1808 Joseph Louis Gay-Lussac found that gases react in simple ratios of volume: two volumes of hydrogen with one of oxygen give two volumes of water vapour. Avogadro (1811) and Ampère (1814, in a long letter to Claude-Louis Berthollet) both explained this by a hypothesis.

Deduction. If two volumes of hydrogen and one of oxygen give two volumes of water vapour, then \(2N\) hydrogen molecules and \(N\) oxygen molecules give \(2N\) water molecules. Each oxygen molecule must therefore split between two water molecules, so it contains an even number of oxygen atoms, at least two. Ampère reasoned with the shapes of molecules: he pictured the particles of compounds as polyhedra built from atoms at their vertices, and used the combining volumes to assign them. His geometry was speculative, but the conclusion that elementary gases such as hydrogen, oxygen and nitrogen are made of compound molecules was right. It became accepted only after Stanislao Cannizzaro’s presentation at Karlsruhe in 1860.

Chlorine and fluorine

Ampère followed closely the debate over whether “oxymuriatic acid” contained oxygen. He agreed with Humphry Davy that chlorine is an element, and he argued by analogy that the acid obtained from fluorspar contained another unknown element of the same family. In letters to Davy he proposed the name fluor, which Davy adopted as fluorine. The element was isolated by Henri Moissan in 1886.

Mathematics: ruin and the Monge–Ampère equation

For the general reader. Ampère first made his name as a mathematician. His first paper proved that a gambler who keeps playing a fair game against a casino with unlimited funds is sure to lose everything in the end. Later he studied a class of nonlinear partial differential equations, first met by Gaspard Monge, which now carry both names and sit at the centre of modern geometry and optimal transport.

The gambler’s ruin

Two players start with \(a\) and \(b\) units and play a sequence of independent rounds, each won by the first player with probability \(p\), with one unit changing hands each time, until one is ruined. Let \(q=1-p\).

Proof. Conditioning on the first round, \(P_k=pP_{k+1}+qP_{k-1}\) for \(0<k<N\), with \(P_0=1\), \(P_N=0\). For \(p=\tfrac12\) the solutions are linear, \(P_k=1-k/N\). Otherwise the characteristic equation \(p\lambda^2-\lambda+q=0\) has roots \(1\) and \(q/p\), so \(P_k=\alpha+\beta(q/p)^k\), and the boundary conditions give the formula. Ampère treated the fair game in 1802; the difference equation above is the standard modern proof.

The Monge–Ampère equation

Ampère’s memoirs on partial differential equations, from his 1814 classification to his work of 1815–1820, developed methods for second-order equations in two variables. Among them is the class Monge had met in 1784, now called Monge–Ampère equations. The basic case is

The equation returned in the twentieth century in the Minkowski problem, in the Calabi conjecture solved by Shing-Tung Yau, and in optimal transport, where the transport map between two densities \(\rho_0,\rho_1\) for quadratic cost is \(\nabla u\) with \(\rho_1(\nabla u)\det D^2u=\rho_0\).

The theory and its rivals, 1820–1827

The weeks after September 1820 were a race. On 30 October 1820, Jean-Baptiste Biot and Félix Savart reported measurements of the force of a long wire on a magnetic pole, falling off as the inverse of the distance; Laplace supplied the element law that underlies the Biot–Savart formula. Biot treated the wire as a magnet; Ampère treated the magnet as a current. The two men disagreed in public for years, and Ampère’s view, that electrodynamics is the fundamental science and magnetism a consequence, eventually prevailed.

Ampère experimented with improvised instruments in his apartment, and in 1820 he suggested an electric telegraph with one needle and one circuit for each letter. In the early 1820s the young Félix Savary helped him with calculations and drafting. In 1821 Michael Faraday showed continuous electromagnetic rotations of a wire about a magnet. Faraday and Ampère corresponded, and although Faraday distrusted mathematical theory, he respected Ampère’s experiments. From 1822 Augustin Fresnel lodged in Ampère’s house, and the two friends discussed the molecular currents and the wave theory of light at length.

In 1824 Ampère was elected to the chair of physics at the Collège de France, where he taught electrodynamics. In 1826 he completed his masterpiece, Théorie mathématique des phénomènes électro-dynamiques, uniquement déduite de l’expérience, printed in the Académie’s Mémoires in 1827. Its title stated a method: derive the law from experiments, as Newton derived gravitation from Kepler’s laws, and make no hypothesis about the nature of electricity beyond what the equilibria require. The Royal Society elected him a foreign member in 1827, and the Royal Swedish Academy of Sciences in 1828. Fresnel died in 1827, a heavy blow.

Last years, death, legacy

Ampère gave up his professorship at the École polytechnique in 1828 but kept his chair at the Collège de France and his inspectorate. His last decade was absorbed by a vast classification of all human knowledge, published as Essai sur la philosophie des sciences (first volume 1834; the second appeared after his death, edited by his son). His family life remained troubled: his daughter Albine married an officer in 1827 whose drinking drove her back to her father’s house in 1830. His relations with his son, a respected scholar, were affectionate but often strained.

He continued the exhausting tours of inspection to the end. On one of them he fell ill and died in Marseille on 10 June 1836, aged 61, in the infirmary of the lycée. He was buried in Marseille. In 1869 his remains were brought to Paris and laid with those of his son in the cimetière de Montmartre.

Legacy

• Electrodynamics: Ampère made the action of currents on currents the foundation of electromagnetism and gave the first mathematical law for it. Wilhelm Weber extended his force law to moving charges in 1846. Maxwell, who called him “the Newton of electricity”, built on his work and gave the circuital law its modern form.
• Words: he introduced the terms electrodynamics, electric current in its modern sense, and solenoid. His classification of the sciences also coined cinématique, kinematics.
• Units and honours: the International Electrical Congress of 1881 named the unit of current after him, and the ampere was one of the base units of the SI when it was established in 1960. His name is one of the 72 engraved on the Eiffel Tower. The house at Poleymieux is the Musée Ampère.
• Mathematics: the Monge–Ampère equation and the gambler’s ruin remain standard in geometry and probability.

Connections

People linked to André-Marie Ampère across this collection. A name without a link has no page here yet. See André-Marie Ampère in the network of all 337 lives →

Students

• Augustin-Louis Cauchy
• Jean-Victor Poncelet
• Sadi Carnot 1812-1814

Influenced by

• Alessandro Volta
• Gaspard Monge
• Hans Christian Ørsted 1820
• Joseph-Louis Lagrange

Influenced

• Hermann Grassmann
• James Clerk Maxwell
• Joseph Henry
• Joseph Liouville
• Wilhelm Weber

Colleagues and collaborators

• Amedeo Avogadro
• Augustin-Jean Fresnel
• François Arago 1820
• Humphry Davy
• Jean-Baptiste Biot
• Michael Faraday
• Pierre-Simon Laplace
• Auguste de la Rive 1822
• Félix Savary 1822-1823
• Jean-Baptiste Delambre

Rivals

• Augustin-Louis Cauchy 1814
• Jean-Baptiste Biot 1820-1827

Friends

• Augustin-Jean Fresnel 1822-1827

Family

• Jean-Jacques Ampère

Also mentioned in this life

• Niels Bohr
• Stanislao Cannizzaro
• Leonhard Euler
• Carl Friedrich Gauss
• Joseph Louis Gay-Lussac
• Johannes Kepler
• Hendrik Antoon Lorentz
• Hermann Minkowski
• Isaac Newton
• George Gabriel Stokes

Fields

• electrodynamics 3 lives
• mathematics 15 lives
• chemistry 21 lives
• philosophy of science 7 lives

On CoursesHub

• Classical Electrodynamics
• Philosophy of Science

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, André-Marie Ampère, MacTutor History of Mathematics.
2. André-Marie Ampère, Wikipedia; and the French article.
3. Linda Hall Library, Scientist of the Day: André-Marie Ampère.
4. École polytechnique, The bonhomme d’Ampère celebrates its bicentenary (2020).
5. J. R. Hofmann, André-Marie Ampère: Enlightenment and Electrodynamics, Cambridge University Press, 1996.
6. L. Pearce Williams, Ampère, André-Marie, in Dictionary of Scientific Biography, vol. 1, Scribner, 1970.
7. CNRS, Ampère et l’histoire de l’électricité (edition of his correspondence and papers).

Physics and mathematics

1. A.-M. Ampère, Théorie mathématique des phénomènes électro-dynamiques, uniquement déduite de l’expérience, Mémoires de l’Académie royale des sciences 6 (1827).
2. A. K. T. Assis and J. P. M. C. Chaib, Ampère’s Electrodynamics, Apeiron, 2015 (English translation and analysis of the 1826 memoir).
3. J. C. Maxwell, A Treatise on Electricity and Magnetism, Oxford, 1873, vol. 2, chapters on electromagnetic force.
4. Ampère’s force law, Wikipedia.
5. A.-M. Ampère, Considérations sur la théorie mathématique du jeu, Lyon, 1802.
6. A.-M. Ampère, Lettre de M. Ampère à M. le comte Berthollet sur la détermination des proportions dans lesquelles les corps se combinent, Annales de chimie 90 (1814).
7. BIPM, The International System of Units (definition of the ampere).
8. J. D. Jackson, Classical Electrodynamics, 3rd ed., Wiley, 1999, chapter 5.