Ampère's Work

The weakness of Ampère’s work is its assumption that the force is directed along the line joining the 2 elements.

In the analogous case of the action between two magnetic molecules, we know that the force is not directed along the line joining the molecules. It is therefore of interest to find the form of F when this restriction is removed.

For this purpose we observe that we can add to the expression already found for F any term of the form

where φ(r) denotes any arbitrary function of r; for since

this term vanishes when integrated round the circuit s; and it contains ds and ds′ linearly and homogeneously, as it should. We can also add any terms of the form

where χ(r) denotes any arbitrary function of r, and d denotes differentiation along the arc s, keeping ds′ fixed (so that dr = - ds); this differential may be written

In order that the law of Action and Reaction may not be violated, we must combine this with the former additional term so as to obtain an expression symmetrical in ds and ds′: and hence we see finally that the general value of F is given by the equation

The simplest form of this expression is obtained by taking

when we obtain

The comparatively simple expression in brackets is vector part of the quaternion product of the three vectors ds, r, ds′. [44]

From any of these values of F we can find the ponderomotive force exerted by the whole circuit s on the element ds’: it is, in fact, from the last expression,

Now this value of B is precisely the value found by Biot and Savart[45] for the magnetic intensity at ds′ due to the current i in the circuit s. Thus we see that the ponderomotive force on a current-element ds′ in a magnetic field B is i′[ds′.B].

Ampère developed to a considerable extent the theory of the equivalence of magnets with circuits carrying currents; and showed that an electric current is equivalent, in its magnetic effects, to a distribution of magnetism on any surface terminated by the circuit, the axes of the magnetic molecules being everywhere normal to this surface:[46] such a magnetized surface is called a magnetic shell. He preferred, however, to regard the current rather than the magnetic fluid as the fundamental entity, and considered magnetism to be really an electrical phenomenon: each magnetic molecule owes its properties, according to this view, to the presence within it of a small closed circuit in which an electric current is perpetually flowing.

The impression produced by Ampère’s memoir was great and lasting.

Writing half a century afterwards, Maxwell speaks of it as “one of the most brilliant achievements in science.” “The whole,” he says, “theory and experiment, seems as if it had leaped, full-grown and full-armed, from the brain of the ‘Newton of electricity.’ It is perfect in form and unassailable in accuracy; and it is summed up in a formula from which all the phenomena may be deduced, and which must always remain the cardinal formula of electrodynamics.”

Not long after the discovery by Oersted of the connexion between galvanism and magnetism, a connexion was discovered between galvanism and heat. In 1822 Thomas Johann Seebeck (b. 1770, d. 1831), of Berlin discovered[47] that an electric current can be set up in a circuit of metals, without the interposition of any liquid, merely by disturbing the equilibrium of temperature Let a ring be formed of copper and bismuth soldered together at the two extremities; to establish a current it is only necessary to heat the ring at one of these junctions. To this new class of circuits the name thermo-electric was given.

It was found that the metals can be arranged as thermo-electric series, in the order of their power of generating currents when thus paired, and that this order is quite different from Volta’s order of electromotive potency. Indeed antimony and bismuth, which are near each other in the latter series, are at opposite extremities of the former.

The currents generated by thermo-electric means generally feeble: and the mention of this fact brings us to the question, which was about this time engaging attention, of the efficacy of different voltaic arrangements.

Comparisons of a rough kind had been instituted soon after the discovery of the pile. The French chemists Antoine François de Fourcroy (b. 1755, d. 1809), Louis Nicolas Vauquelin (b. 1763, d. 1829), and Louis Jacques Thénard (b. 1777, d. 1857) found[48] in 1801, on varying the size of the metallic disks constituting the pile, that the sensations produced on the human frame were unaffected so long as the number of disks remained the same, but that the power of burning finely drawn wire was altered; and that the latter power was proportional to the total surface of the disks employed, whether this were distributed among a small number of large disks, or a large number of small ones. This was explained by supposing that small plates give a small quantity of the electric fluid with a high velocity, while large plates give a larger quantity with 10 greater velocity. Shocks, which were supposed to depend on the velocity of the fluid alone, would therefore not be intensified by increasing the size of the plates.

The effect of varying the conductors which connect the terminals of the pile was also studied. Nicolas Gautherot (b. 1753, d. 1803) observed[49] that water contained in tubes which have a narrow opening does not conduct voltaic currents so well as when the opening is more considerable. This experiment is evidently very similar to that which Beccaria had performed half a century previously[50] with electrostatic discharges.

As we have already seen, Cavendish investigated very completely the power of metals to conduct electrostatic discharges; their power of conducting voltaic currents was now examined by Davy.[51] His method was to connect the terminals of a voltaic battery by a path containing water (which it decomposed), and also by an alternative path consisting of the metallic wire under examination. When the length of the wire was less than a certain quantity, tho water ceased to be decomposed; Davy measured the lengths and weights of wires of different materials and cross-sections under these limiting circumstances; and, by comparing them, showed that the conducting power of a wire formed of any one metal His inversely proportional to its length and directly proportional to its sectional area, but independent of the shape of the crosssection[52]. The latter fact, as he remarked, showed that voltaic currents pass through the substance of the conductor and not along its surface.

Davy, in the same memoir, compared the conductivities of various metals, and studied the effect of temperature: he found that the conductivity varied with the temperature, being “lower in some inverse ratio as the temperature was higher.”

He also observed that the same magnetic power is exhibited by every part of the same circuit, even though it be formed of wires of different conducting powers pieced into a chain, so that “the magnetism seems directly as the quantity of electricity which they transmit.”

The current which flows in a given voltaic circuit evidently depends not only on the conductors which form the circuit, but also on the driving-power of the battery. In order to form a complete theory of voltaic circuits, it was therefore necessary to extend Davy’s laws by taking the driving-power into account. This advance was effected in 1826 by Georg Simon Ohm[53] (b. 1787, d. 1854).

Ohm had already carried out a considerable amount of experimental work on the subject, and had, e.g., discovered that if a number of voltaic cells are placed in series in a circuit, the current is proportional to their number if the external resistance is very large, but is independent of their number if the external resistance is small. He now essayed the task of combining all the known results into a consistent theory.

For this purpose he adopted the idea of comparing the flow of electricity in a current to the flow of heat along a wire, the theory of which had been familiar to all physicists since the publication of Fourier’s Théorie analytique de la chaleur in 1822. “I have proceeded,” he says, “from the supposition that the communication of the electricity from one particle takes place directly only to the one next to it, so that no immediate transition from that particle to any other situate at a greater distance occurs. The magnitude of the flow between two adjacent particles, under otherwise exactly similar circumstances, I have assumed to be proportional to the difference of the electric forces existing in the two particles; just as, in the theory of heat, the flow of caloric between two particles is regarded as proportional to the difference of their temperatures.”

The comparison between the flow of electricity and the flow of heat suggested the propriety of introducing a quantity whose behaviour in electrical problems should resemble that of temperature in the theory of heat. The differences in the values of such a quantity at two points of a circuit would provide what was so much needed, namely, a measure of the “driving-power” acting on the electricity between these points.

To carry out this idea, Ohm recurred to Volta’s theory of the electrostatic condition of the open pile. It was customary to measure the “tension” of a pile by connecting one terminal to earth and testing the other terminal by an electroscope.

Accordingly, Ohm says:

“In order to investigate the changes which occur in the electric condition of a body A in a perfectly definite manner, the body is each time brought, under similar circumstances, into relation with a second moveable body of invariable electrical condition, called the electroscope; and the force with which the electroscope is repelled or attracted by the body is determined. This force is termed the electroscopic force of the body A.”

“The same body A may also serve to determine the electroscopic force in various parts of the same body. For this purpose take the body A of very small dimensions, so that when we bring it into contact with the part to be tested of any third body, it may from its smallness be regarded as a substitute for this part: then its electroscopic force, measured in the way described, will, when it happens to be different at the various places, make known the relative differences with regard to electricity between these places.”

Ohm assumed, as was customary at that period, that when two metals are placed in contact, “they constantly maintain at the point of contact the same difference between their electroscopic forces.” He accordingly supposed that each voltaic cell possesses a definite tension, or discontinuity of electroscopic force, which is to be regarded as its contribution to the driving-force of any circuit in which it may be placed.

This assumption confers a definite meaning on his use of the term “electroscopic force”; the force in question is identical with the electrostatic potential. But Ohm and his contemporaries did not correctly understand the relation of galvanic conceptions to the electrostatic functions of Poisson.

The electroscopic force in the open pile was generally identified with the thickness of the electrical stratum at the place tested; while Ohm, recognizing that electric currents are not confined to the surface of the conductors, but penetrate their substance, seems to have thought of the electroscopic force at a place in a circuit as being proportional to the volume-density of electricity there—an idea in which he was confirmed by the relation which, in an analogous case, exists between the temperature of a body and the volume-density of heat supposed to be contained in it.

Denoting, then, by S the current which flows in a wire of conductivity γ, when the difference of the electroscopic forces at the terminals is E, Ohm writes

From this formula it is easy to deduce the laws already given by Davy. Thus, if the area of the cross-section of a wire is A, we can by placing a such wires side by side construct a wire of cross-section nA. If the quantity E is the same for each, equal currents will flow in the wires; and therefore the current in the compound wire will be n times that in the single wire; so when the quantity E is unchanged, the current is proportional to the cross-section; that is, the conductivity of a wire is directly proportional to its cross-section, which is one of Davy’s laws.

In spite of the confusion which was attached to the idea of electroscopic force, and which was not dispelled for some years, the publication of Ohm’s memoir marked a great advance in electrical philosophy. It was now clearly understood that the current flowing in any conductor depends only on the conductivity inherent in the conductor and on another variable which bears to electricity the same relation that temperature bears to heat; and, moreover, it was realized that this latter variable is the link connecting the theory of currents with the older theory of electrostatics.

These principles were a sufficient foundation for future progress; and much of the work which was published in the second quarter of the century was no more than the natural development of the principles laid down by Ohm.[54]

It is painful to relate that the discoverer had long to wait before the merits of his great achievement were officially recognized. Twenty-two years after the publication of the memoir on the galvanic circuit, he was promoted to a university professorship; this he held for the five years which remained until his death in 1854.