In Search of a Newtonian Law of Electrodynamics (18201826)
By Christine Blondel and Bertrand Wolff
Translation by Andrew Butrica
Oersted's experiment reveals methodological differences
Oersted'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
Biot imagined that each "slice" of the conductor, labeled F in fig.1, underwent "a momentary magnetization of its molecules". These slices then were equivalent to the tangential magnetized needles ab, a'b', etc. Fig. 1. Elementary magnets created along the circumference of a conducting wire (JeanBaptiste Biot hypothesis) 
With Felix Savart, a physician fascinated by acoustics, Biot carried out very precise measurements to determine the force exerted by a conducting wire on the pole of a magnet and to deduce from these measurements a mathematical law for the action of a small slice of the conductor on this pole. sinω/r^{2} where r is the distance between M and the slice located at N, and ω the angle between the straight line MN and the conductor. What were the series of experimental and theoretical steps that resulted in the formulation of this law?
Fig. 2. Action of an infinitely long vertical conducting wire on a "particle of magnetism" M 
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 force exerted by the slice located at N also depends on the angle ω in accordance with a function yet to be determined. To ascertain this function, Biot undertook a second series of experiments. He measured the force exerted on the pole of a magnet M (fig. 3) by an infinitely long conductor forming a variable angle at point A. He concluded from his data that this force was proportional to the angle θ (fig.3). On 18 December, Biot announced this result. Consequently, he asserted, the elementary force was proportional to sinω/r^{2}. But we can see that there is a problem: the integration of this elementary force sinω/r^{2} along the length of the bent wire yields a total force proportional to tg(θ/2) and not simply proportional to θ. It was only in 1823, after Ampere had highlighted this contradiction, and in remaking his measurements with more care, that Biot found that the total force actually was proportional to tg(θ/2)and not to θ. So clearly then Biot derived the factor sinω not from experimentation but from intuition. If one considers that the force exerted by slice N is greatest for ω = 90°, the opposite for ω =  90° (the direction of the current reversed), and null for ω = 0, then the simplest trigonometric function that satisfies these conditions is sinω. 
Fig. 3. Action of a bent conducting wire on a "particle of magnetism" M 
or in norm :
If the factor sinω/r^{2} originated from Biot (and Laplace), the factor ids that appears in this modern form, as we shall see, marked the triumph of Ampere's current element over Biot's magnetized slice!
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:
"It remains to be found how each infinitely small molecule of the connecting wire contributes to the total action of the slice to which it belongs."
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.
Ampere's "program"
While Biot sought to reduce the action of a conducting wire on a magnet to magnetic interactions, Ampere in contrast reduced them to the interactions between currents. 
Fig. 4. Ampere employed arrows to represent his elementary currents ("suivant moi" i.e. "according to me") as well as the elementary magnets of Biot ("according to M. Biot").. 
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.
"A hypothesis . . . the simplest that one could adopt"
Shown here (fig. 5) are two current elements located at A and B. Their lengths are assumed to be infinitely small compared to the distance AB. The intensity of the force between these elements depends not only on the distance r = AB, but also on the three angles α, β and γ that define the relative positions of the two elements in threedimensional space. Ampere's first hypothesis was that the elementary force decreased "inversely with the square of this distance in accordance with what one observes for all types of action more or less analogous to this one." Like Biot, it remained for him to determine how it varied with the angles. 
Fig. 5 . The current element at point A is in plane P. The current element at B is in plane Q. 
Fig. 6. A straight wire and a "sinuous" wire. 
Several experiments suggested the idea that we call today the vector addition of current elements, that is to say, the possibility of replacing a current element with its projections along three axes. One of the experiments substantiating this property was the one showing that a sinuous wire and a straight wire exerted equal forces on a moving circuit or a compass.[See the video L'expérience du courant sinueux ]. All that was then needed was to analyze each current element individually along the three orthogonal axes and to consider the forces acting among these components in pairs. 
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:
g h (sinα sinβ cosγ + k cosα cosβ) / r^{2 }(1)
where g and h depended on "the electricity passing in equal time periods". Here is a first definition of the notion of the intensity of a current.
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.
"It is on these general considerations that I constructed an expression for the attraction of two infinitely small currents, which in truth was a hypothesis, but the simplest that one could adopt, and consequently the one that should be tested first."
he wrote in September 1820 in a manuscript proposing a preliminary formula. The next step would be to deduce mathematically the action between finite currents, and then to compare the results of these calculations with measurements. Ampere described complex apparatus such as the one shown in fig. 7. Did he use them? We find no numerical results in his rough drafts. His ingenuity in conceiving such new devices, combined with an almost total lack of measurements, is a rather general characteristic of Ampere's approach in physics. In this first phase of his research, in the absence of exact measurements, qualitative experiments strongly influenced his theoretical thinking. 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. 
Fig. 7. Apparatus imagined for measuring the force between the moving (vertical) conductor BC and the fixed conductor RS capable of being positioned at various angles. 
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 repercussions
Ampere'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, Ampere, and the mystery of continuous rotations ].
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.
Towards a new method: the cases of equilibrium
Little by little Ampere abandoned all plans to realize absolute measurements of electrodynamic forces. Starting in 1822 a new method was underway: the use of "cases of equilibrium". As a result, Ampere forged what has since come to be called the "zero method". The idea was to make two circuits traversed by the same current act simultaneously on a movable conductor in such a way that the forces created by the two circuits on the mobile conductor counterbalanced each other exactly. To eliminate the parasitic action of terrestrial magnetism on this mobile circuit, Ampere imagined "astatic" circuits such as those represented in fig. 8. 
Fig. 8. Two examples of circuits insensitive to terrestrial magnetism To make a mobile circuit insensitive to terrestrial magnetism, Ampere added to it a second circuit that was contiguous on one side. He then called the ensemble "astatic". 
Fig. 9. The actions exerted simultaneously by a curving current and a rectilinear current on the side cd of an astatic mobile circuit compensate for each other (the frame EFKL serves to support the connecting wires between the mobile circuit and the battery). 
Ampere was led to this method by two experiments that already constituted "cases of equilibrium" : The discovery of a third case of equilibrium from the experiments with continuous rotations led Ampere to two important conclusions: 
A mutual nonzero action between two collinear elements?
We saw above that a nonzero value for k implied the existence of a force between collinear elements. Ampere thought to confirm this conclusion with an experiment conducted in 1822 in Geneva. The two branches of a metal wire in the form of a hairpin floated on a mercury surface in two independent compartments of a circular vessel (fig. 10). The end of each branch was in contact with a metal wire connected to a battery. When the current was started, the branches moved away from the contacts regardless of the direction of the current. [See the video L'expérience du conducteur flottant ]. For Ampere, this indicated a repulsion for each wire between the current established in the mercury and its extension in the wire itself, and he concluded: this "is a great proof in favor of [my] formulas.". 
Fig. 10. The experiment with the floating conductor intended to show the repulsion of collinear currents. 
A "définitive" formulation
Ampere 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:
i i' ds ds' (sinα sinβ cosγ  ½ cosα cosβ) / r^{2 } (2)
The mathematical consequences of the elementary law
As 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".
These successes motivated Ampere to carry out new research on the action of whatsoever kind of closed circuit on a current element. In the course of his calculations, Ampere introduced an important calculating aid: a line that he called the "directrice". At each point in space this line has a specific direction that depends only on the closed circuit (and not the orientation of the current element). Ampere showed that the force experienced by current element ds' located at a given point is perpendicular to the directrice at that same point. The force equally being perpendicular to the current element, it is therefore perpendicular to the plane defined by the directrice and the element. Finally, if the angle between the directrice and current element ds' was ε, Ampere showed that one could write the intensity of the force in the form: Fig. 11. Ampère manuscript showing the "directrice" and the expression ½ D i i' ds' sinε 
dF = i' ds' . B sinε or in vector form:
(4) where the intensity B of the magnetic induction vector is, by nearly a factor, the product Di and its direction that of the directrice.
But the similarity of mathematical formulas should not make us forget that the present day formulation in terms of magnetic induction implicitly supposes the notion of a field, that is to say, the idea that the space around a circuit undergoes a modification characterized at each point by a physical quantity. Furthermore, the mathematical definition of a vector did not come along until after Ampere.
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.
Two new cases of equilibrium
Striving to establish his formula in a more general way, Ampere assumed the elementary force to be proportional to 1/r^{n} with n an integer, and no longer a priori 1/r^{2}. He also was dissatisfied with the third case of equilibrium that allowed him to determine the value of the constant k =  1/2. So, he started from the most general expression of the elementary force deduced from the first two cases of equilibrium: i ds i' ds' (sinα sinβ cosγ + k cosα cosβ)/r^{n}. A new experiment, which became his third case of equilibrium, intended to show that a closed circuit placed in the vicinity of a small portion of a conductor can make it move only in a direction perpendicular to the conductor (fig. 12). But the experiment was, by Ampere's own admission, "hardly prone to accuracy" mainly because of friction. The expected result of this experiment provided, at the cost of long calculations, a relationship between n and k: 

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.
Fig. 13. The fourth equilibrium case 
A rather simple argument shows that, if the movable circle remains in equilibrium under the opposing actions of the two fixed circles, the exponent n takes the value 2. Ampere thus again found that the value Ampere now had at his disposal all the elements that he deemed necessary for the definitive account of his theory. The synthesis of his many publications scattered in various journals became his Théorie des phénomènes électrodynamiques, uniquement déduite de l'expérience [Theory of Electrodynamic Phenomena Deduced Solely from Experiment], published in 1826. 
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:
i ds i' ds' (sinα sinβ cosγ + k cosα cosβ) / r^{n}
The two last cases of equilibrium determined that n = 2 and k = 1/2.
The derivation of this elementary force occupied barely the first quarter of the Théorie [mathématique] des phénomènes électrodynamiques. The rest was devoted to very long calculations extending this work into specific problems. These deal first with obtaining the laws of interactions among all types of circuits by integrating the formula. One of the calculations concerned the mutual action of two parallel rectilinear conductors carrying currents of intensities I and I'. Ampere showed that the force exerted by one of the conductors, considered as being infinitely long, on a portion of the other conductor whose length is L, is proportional to I I'L / a, where a is the distance between the two wires. It is from this force between two parallel currents that we define the unit of intensity in the present international system of units and which in 1881 received the name Ampere. [See the page Le système international d'unités]
By means of even longer calculations, Ampere established the equivalence between a bar magnet and a solenoid. He also showed that any given closed circuit was equivalent to a set of two surfaces infinitesimally close to the circuit and over which were spread the two opposite magnetic fluids (a "magnetic leaf"). From that moment forward electrodynamics encompassed all magnetic phenomena. Fig. 14. Extract from Ampere's calculations concerning the interactions between circuits 
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:
"Observe first the facts, vary the circumstances as much as possible, carry out precise measurement in order to deduce general laws founded solely on experimentation, and deduce from these laws  independently of all hypotheses about the nature of the forces that produce the phenomena  the mathematical value of these forces, that is to say, the formula that represents them, such is the path taken by Newton. It has been adopted generally in France by savants to whom physics owes the tremendous progress that it has achieved in recent times, and it was what has served me as a guide in all my research on electrodynamic phenomena."
As we have seen, this was not so simple.
What about the four experiments intended as the basis of the theory? Experimental difficulties, Ampere recognized, made the results of his third experiment very doubtful. Regarding the fourth experiment, he wrote:
" I believe I must admit, in finishing this memoir, that I have not had the time to build the instruments [...] The experiments for which they were intended have not been carried out yet; but, as the purpose of these experiments was only to verify a series of results obtained otherwise, moreover as it would be useful to perform them as a counterproof to those that furnished these results, I have not thought it necessary to delete their description."
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 Ampère's Force Law: An Obsolete Formula? ]. 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 reading
AMPERE, 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.
AMPERE, AndréMarie. Théorie des phénomènes électrodynamiques, uniquement déduite de l'expérience, in Société française de physique. Collection de mémoires relatifs à la physique. t. 2 : Mémoires sur l'électrodynamique. 1ère partie. Paris : GauthierVillars, 1885, p. 1193. [See pdf]
ASSIS, A. K.T. & CHAIB, J.P.M.C. Ampère’s Electrodynamics Analysis of the Meaning and Evolution of Ampère’s Force between Current Elements, together with a Complete Translation of his Masterpiece: Theory of Electrodynamic Phenomena, Uniquely Deduced from Experience. Montreal: Apeiron, 2015. [see PDF]
[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