@. Ampère et l'histoire de l'électricité 
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Parcours historique > Des lois pour le courant : Ampère, Ohm et quelques autres...  
In Search of a Newtonian Law of Electrodynamics (18201826)FrançaisBy Christine Blondel and Bertrand Wolff Oersted's experiment reveals methodological differencesOersted's discovery of the magnetic effects of electric currents [See the page L'expérience de H.C. Oersted] stimulated a proliferation of experimental research and fresh discoveries throughout Europe. Given the complexity of the new phenomena, most physicists and chemists made a greater number of purely qualitative observations. From the outset, only Ampere and Biot  in competition with each other  attached great importance to the quest for a mathematical law that would express the magnetic action of a current and that would be capable of predicting new experimental results. JeanBaptiste Biot, author of an acclaimed Treatise on Experimental and Mathematical Physics (Traité de physique expérimentale et mathématique), personally had experimented with Coulomb and was an ardent Newtonian. Rejecting the wealth of experimental qualitative results, Biot sought to find a single law that would express the amount of magnetic force exerted by an infinitely long wire on a magnetic pole as a function of its distance to the wire. For the mathematician Ampere, his own search for an "electrodynamic formula", which he launched less than a month after the announcement of Oersted's discovery, fell within the broader framework of his "grand theory". Indeed, his lectures before the Academy of Sciences gave a large place to hypotheses and to the first experimental verifications of this theory according to which all magnetic phenomena could be reduced to interactions between currents [See the page Ampere lays the basis for electrodynamics...] The law of Biot and Savart
To study the action of a very long vertical conductor on the pole of a magnet through experimentation, Biot placed a horizontal magnetized needle at various distances from the wire. The influence of terrestrial magnetism on the needle was neutralized by an auxiliary magnet. Following Coulomb's method, he measured the period of the small oscillations of the needle around its point of equilibrium: the force that each pole is subjected to is proportional to the square of this period. This experimental technique was familiar to Biot [On Coulomb's use of this oscillation method, see the page Les lois fondamentales de l'électricité et du magnétisme]. The experiment, however, raised new problems. Specifically, the electric current decreased quickly because of the rapid polarization of the batteries in use at the time. On 30 October, Biot announced his experimental results: the force exerted by an infinite conductor on the pole of a magnet is inversely proportional to the distance MH of the pole to the wire. As Laplace remarked to Biot shortly thereafter, one could deduce that the elementary force exerted by an infinitesimal wire slice located at a distance r from the pole was proportional to 1/r^{2}.
The elementary force being proportional to 1/r^{2} followed closely in the tradition of Newton and Coulomb. However, it did not obey the Newtonian principle of action and reaction, because it was not directed along the line MN (fig. 2). It was, to use the terminology of those times, a "transversal" force perpendicular to the plane defined by MN and the wire. Moreover, for Biot, the action of the slice was a composite action. For him the problem of the action of a conducting wire on a magnet was far from being resolved:
In this way the action of a current on a magnet could be reduced to simple magnetic interactions. Biot maintained that it was possible to conceive of an assemblage of miniscule magnetized needles along the circumference of the wire and from which one could deduce his experimental law. But he acknowledged "great difficulty" in achieving that goal.
The portion of the current that he considered was not the infinitely thin slice imagined by Biot, but rather an element of infinitesimal length ds. One could derive the force between two finite currents, at least theoretically, by two successive integrations of the elemental force between two current elements ds and ds'. For the action between a magnet and a current a triple integration would be necessary, each slice of the magnet containing in effect, according to Ampere, an infinity of coaxial circular currents. But, while one can go from the elemental force to the total force through integration, the reverse is not possible. Biot collided with the same problem and, as we saw, determined its angular factor by intuition, not experiment. How then can one determine the elemental force between two current elements? Ampère adopted a very original strategy. He proposed to start from the most general interactive force possible and to determine its expression by qualitative experiments on finite circuits. In good Newtonian style, Ampere assumed that this elementary force should obey the principle of action and reaction, and therefore it would be directed along the straight line that joined the two current elements.
The circuitous path of Ampere's thinking as it meandered between experiments, calculations, and theoretical arguments is too complex to be presented here in detail. We therefore will stay within the broad outlines at the risk of giving an incomplete picture of the breadth of his work that was interrupted and restarted several times over many years. To appreciate the full measure of Ampere's intellectual audacity, his experimental ingenuity, and the sophistication of his lengthy mathematical reasoning, please consult the Publications d'Ampère.
Experiments showed Ampere that two parallel wires attracted each other when they were carrying currents flowing in the same direction, and they repelled each other if the currents ran in opposite directions. He maintained that it was the same for two current elements that were infinitely small and parallel. Furthermore, Ampere was led to suppose that the force between two elements was zero, if one of them were situated in a plane perpendicular to the other element in its middle. He then arrived at an expression for the elementary force being proportional to: The first term corresponded to the force between the two parallel components and the second to the force between the two collinear components. In the autumn of 1820, Ampere assumed, with some hesitation, that the force between two collinear elements was nil, that is, that k = 0.
On 4 December, he had his first notable success: Ampere announced that he came upon  with the aid of this preliminary formula  the experimental results of Biot that gave a force of 1/r for an infinite wire acting on a magnetic pole (fig. 2). To that end, he conceived any bar magnet as a set of currents in planes perpendicular to the axis of the magnet. Ampere's hypothesis thus led to results consistent with experiment. Additionally, it had the advantage of avoiding the “gratuitous assumption” of the magnetization of the wire. On the other hand, in the eyes of Biot and his partisans, it was the currents within the magnets that were the gratuitous assumption! [See the page Des théories mathématiquement équivalentes, physiquement différentes]. This phase of research, interrupted by illness, ended in January 1821. A cold reception and surprising repercussionsAmpere's theory was far from receiving unanimous support. The relative confusion of memoirs written in haste, the difficulty of other physicists trying to repeat his radically new experiments, the reluctance toward what seemed to some people an abuse of the infinitesimally small, all of these contributed to misunderstanding, even distrust. More fundamentally, the hypothesis of the existence of electric currents within magnets aroused skepticism. In England, Faraday praised Ampere's experiments and the ingenuity of his theory, but he doubted the reality of currents not proven directly by experiment It was a discovery by Faraday that relaunched Ampere's research in the autumn of 1821. Faraday announced that he had achieved the continuous rotation of a magnet under the action of a conductor and vice versa. These continuous rotations astonished Ampere a great deal. [See the page Faraday, Ampère et les rotations continues]. Replacing the magnet with a solenoid, he realized continuous rotations solely with circuits, which further supported his theory. On the other hand, it was impossible for him to obtain these rotations with only magnets. For Ampere, that fact dealt a fatal blow to Biot's theory: one could not reduce electromagnetism to interactions between magnets.
A mutual nonzero action between two collinear elements?
A "définitive" formulationAmpere then announced his definitive formula. He specified the factors g and h in formula (1) as the intensities of the currents and defined the way to measure these intensities by the force that a current exerts on a parallel current taken as a reference.
By replacing k with the value  1/2, the mutual action between two infinitely small current elements with lengths ds and ds' became: The mathematical consequences of the elementary lawAs such Ampere did not consider his task complete. He wanted to show that his formula predicted all possible cases of interactions between currents and magnets. In this effort Ampere benefited from the valuable collaboration of two young men, Félix Savary and JeanFirmin Demonferrand. The Manuel d'électricité dynamique of the latter, published in 1823, had a wider circulation than Ampere's own collected electrodynamic observations Recueil d'observations électrodynamiques published somewhat earlier. Savary deduced the law of Biot and Savart from Ampere's elementary formula and in the process pointed out the inconsistencies of Biot in his determination of that law. Coulomb's law for magnetic poles equally became a mathematical consequence of Ampere's formula. In February 1823, Ampere declared that, thanks to Savary and Demonferrand, "all the facts not yet explained completely [...] are necessary consequences of his formula".
Beginning in the winter of 18231824, Ampere was overwhelmed by personal problems and his teaching load. It was only in August 1825 that he returned to electrodynamics.
A fourth and last case of equilibrium consisted of making two fixed circular circuits, one p times smaller, the other p times larger(fig. 13). The two fixed circles were placed on either side of the movable circle, the distances between the centers of the circles being in the same relationship p as their radii.
The Théorie [mathématique] des phénomènes électrodynamiques..."Ampere's theory of the mutual action of electric currents is founded on four experimental facts and one assumption", wrote Maxwell in the final chapter of his Treatise on Electricity and Magnetism (1873), that he dedicated to Ampere. The four experimental facts were the four cases of equilibrium. The hypothesis was that of the instantaneous action at a distance between current elements obeying the principle of action and reaction and, therefore, directed along the line that joined these two elements. The first two cases of equilibrium  the reversal of the direction of the force with the reversal of the direction of the current and the experiment with the sinuous current  legitimated, as we have seen, the analysis of current elements along three axes from which one can deduce that the mutual action necessarily has the expression: The two last cases of equilibrium determined that n = 2 and k = 1/2.
Ampere's formula: a formula with an ambiguous appraisal...The title of Ampere's treatise asserts that the theory was "derived solely from experiments". The treatise's first page presents a bubbly tribute to Newton:
As we have seen, this was not so simple.
This admission may be surprising. It illuminates the state of experimentation in this final work in which Ampere aimed to present the ideal development of his formula as it could be derived from a minimal number of experiments. The mathematical consequences for Ampere had the same status as Kepler's empirical laws for Newton's determination of the universal law of gravitation. The absolute confidence that he expressed regarding the result of his fourth experiment  which he quite probably never carried out  rests on "results obtained otherwise," that is to say, by a subtle back and forth between simpler experiments, intuition, and sophisticated computations. On the other hand, as Maxwell indicated a halfcentury later, an infinity of differential formulas can yield, by integration, the same expression for the force between two finite circuits. However, if one accepts Newton's principle of action and reaction as applying to the infinitely small current elements, only one elementary formula is possible: that of Ampere, and for Maxwell that, therefore, is the best. We understand why Maxwell called Ampere the "Newton of electricity" (an ambiguous compliment, as Maxwell's own theory broke radically with Newtonian philosophy). Moreover, if Ampere introduced the Théorie [mathématique] des phénomènes électrodynamiques with his profession of Newtonian faith and strongly fought the ideas of Biot as being antiNewtonian, he showed in other texts his repugnance toward the idea of instantaneous action at a distance and his private conviction of a gradual propagation through the ether, the hypothetical medium supposedly filling space ... [See the page Des théories mathématiquement équivalentes, physiquement différentes] A final question: as Ampere's formula was undoubtedly effective, why, after having given rise to so much debate, did it disappear from modern science? [See the page La force d'Ampère, une formule obsolète ?]. In 1888 the English physicist Oliver Heaviside, a disciple of Maxwell, recognized that this formula had been considered by Maxwell himself as the cardinal formula of electrodynamics. But, he added: "If so, should we not be always using it ? Do we ever use it ? Did Maxwell, in his treatise ? Surely there is some mistake." He suggested transferring Ampere's name to the formula that expresses the force undergone by a section of conductor placed in a magnetic field, "it is fundamental and, as everybody knows, it is in continual use, both by theorists and practicians." But the attribution of a savant's name to this physics formula succumbed to the accidents of history and, paradoxically, at least in France, this force is known as Laplace's force! Further readingAMPERE, AndréMarie. Mémoire [...] sur les effets des courants électriques, Annales de chimie et de physique, 1820, vol. 15, p. 5975, p. 170218. AMPERE, AndréMarie. Recueil d'observations électrodynamiques, Paris, 1822. AMPERE, AndréMarie. Théorie des phénomènes électrodynamiques uniquement déduite de l'expérience, Paris, 1826. [Recherches de JeanBaptiste BIOT in] Société française de physique (Ed.). Collection de mémoires relatifs à la physique. t. 2, Mémoires sur l'électrodynamique [publiés par les soins de J. Joubert] 1ère Partie. Paris : GauthierVillars, 1885, p. 80127. BLONDEL, Christine. Avec Ampère le courant passe, Les mathématiques expliquent les lois de la nature. Le cas du champ électromagnétique. Les Cahiers de Science & Vie, 67, 2002, 2027. LOCQUENEUX, Robert. Ampère, encyclopédiste et métaphysicien. Les Ulis : EDP sciences, 2008. HOFMANN, James R. AndréMarie Ampère. Cambridge : Cambridge University Press, 1996. BLONDEL, Christine. Ampère et la création de l'électrodynamique, 18201827. Paris : Bibliothèque nationale, 1982. A bibliography of "secondary sources " about the History of Electricity French version: May 2009 (English translation: March 2013)


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