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| History of Theories of Aether and Electricity | |||||||||
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Electric and magnetic Science prior to the introduction of the potentials
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supposing that there are two electric fluids, the parts of the
same fluid repelling each other according to the inverse square
of the distance, and attracting the parts of the other fluid
according to the same inverse square law." " The supposition
of two fluids," he adds, " is moreover in accord with all those
discoveries of modern chemists and physicists, which have made
known to us various pairs of gases whose elasticity is destroyed
by their admixture in certain proportions - an effect which could
not take place without something equivalent to a repulsion
between the parts of the same gas, which is the cause of its
elasticity, and an attraction between the parts of different
gases, which accounts for the loss of elasticity on combination." According, then, to the two-fluid theory, the " natural fluid " contained in all matter can be decomposed, under the influence of an electric field, into equal quantities of vitreous and resinous electricity, which, if the matter be conducting, can then fly to the surface of the body. The abeyance of the characteristic properties of the opposite electricities when in combination was sometimes further compared to the neutrality manifested by the compound of an acid and an alkali. The publication of Coulomb's views led to some controversy between the partisans of the one-fluid and two-fluid theories ; the latter was soon generally adopted in France, but was stoutly opposed in Holland by Van Marum and in Italy by Volta. The chief difference between the rival hypotheses is that, in the two-fluid theory, both the electric fluids are movable within the substance of a solid conductor ; while in the one-fluid theory the actual electric fluid is mobile, but the particles of the conductor are fixed. The dispute could therefore be settled only by a determination of the actual motion of electricity in discharges ; and this was beyond the reach of experiment. In his Fourth Memoir Coulomb showed that electricity in equilibrium is confined to the surface of conductors, and does not penetrate to their interior substance ; and in the Sixth Memoir [1] he virtually establishes the result that the electric force near a conductor is proportional to the surface-density of electrification. Since the overthrow of the doctrine of electric effluvia by Aepinus, the aim of electricians had been to establish their science upon the foundation of a law of action at a distance, resembling that which had led to such triumphs in Celestial Mechanics. When the law first stated by Priestley was at length decisively established by Coulomb, its simplicity and beauty gave rise to a general feeling of complete trust in it as the best attainable conception of electrostatic phenomena. The result was that attention was almost exclusively focused on action-at-a-distance theories, until the time, long afterwards, when Faraday led natural philosophers back to the right path. Coulomb rendered great services to magnetic theory. It was he who in 1777, by simple mechanical reasoning, completed the overthrow of the hypothesis of vortices. [2] He also, in the second of the Memoirs already quoted, [3] confirmed Michell's law, according to which the particles of the magnetic fluids attract or repel each other with forces proportional to the inverse square of the distance. Coulomb, however, went beyond this, and endeavoured to account for the fact that the two magnetic fluids, unlike the two electric fluids, cannot be obtained separately; for when a magnet is broken into two pieces, one containing its north and the other its south pole, it is found that each piece is an independent magnet possessing two poles of its own, so that it is impossible to obtain a north or south pole in a state of isolation. Coulomb explained this by supposing [4] that the magnetic fluids are permanently imprisoned within the molecules of magnetic bodies, so as to be incapable of crossing from one molecule to the next ; each molecule therefore under all circumstances contains as much of the boreal as of the austral fluid, and magnetization consists simply in a separation of the two fluids to opposite ends of each molecule. Such a hypothesis evidently accounts for the impossibility of separating the two fluids to opposite ends of a body of finite size. The same idea, here introduced for the first time, has since been applied with success in other departments of electrical philosophy. In spite of the advances which have been recounted, the mathematical development of electric and magnetic theory was scarcely begun at the close of the eighteenth century ; and many erroneous notions were still widely entertained. In a Report [5] which was presented to the French Academy in 1800, it was assumed that the mutual repulsion of the particles of electricity on the surface of a body is balanced by the resistance of the surrounding air; and for long afterwards the electric force outside a charged conductor was confused with a supposed additional pressure in the atmosphere. Electrostatical theory was, however, suddenly advanced to quite a mature state of development by Simeon Denis Poisson (b. 1781, d. 1840), in a memoir which was read to the French Academy in 1812. [6] As the opening sentences show, he accepted the conceptions of the two-fluid theory. " The theory of electricity which is most generally accepted," he says, " is that which attributes the phenomena to two different fluids, which are contained in all material bodies. It is supposed that molecules of the same fluid repel each other and attract the molecules of the other fluid ; these forces of attraction and repulsion obey the law of the inverse square of the distance ; and at the same distance the attractive power is equal to the repellent power; whence it follows that, when all the parts of a body contain equal quantities of the two fluids, the latter do not exert any influence on the fluids contained in neighbouring bodies, and consequently no electrical effects are discernible. This equal and uniform distribution of the two fluids is called the natural state ; when this state is disturbed in any body, the body is said to be electrified, and the various phenomena of electricity begin to take place. "Material bodies do not all behave in the same way with respect to the electric fluid : some, such as the metals, do not appear to exert any influence on it, but permit it to move about freely in their substance ; for this reason they are called conductors. Others, on the contrary - very dry air, for example - oppose the passage of the electric fluid in their interior, so that they can prevent the fluid accumulated in conductors from being dissipated throughout space." When an excess of one of the electric fluids is communicated to a metallic body, this charge distributes itself over the surface of the body, forming a layer whose thickness at any point depends on the shape of the surface. The resultant force due to the repulsion of all the particles of this surface-layer must vanish at any point in the interior of the conductor, since otherwise the natural state existing there would be disturbed ; and Poisson showed that by aid of this principle it is possible in certain cases to determine the distribution of electricity in the surface-layer. For example, a well-known proposition of the theory of Attractions asserts that a hollow shell whose bounding surfaces are two similar and similarly situated ellipsoids exercises no attractive force at any point within the interior hollow; and it may thence be inferred that, if an electrified metallic conductor has the form of an ellipsoid, the charge will be distributed on it proportionally to the normal distance from the surface to an adjacent similar and similarly situated ellipsoid. Poisson went on to show that this result was by no means all that might with advantage be borrowed from the theory of Attractions. Lagrange, in a memoir on the motion of gravitating bodies, had shown [7] that the components of the attractive force at any point can be simply expressed as the derivates of the function which is obtained by adding together the masses of all the particles of an attracting system, each divided by its distance from the point; and Laplace had shown [8] that this function V satisfies the equation
\frac{\partial^2 V}{\partial x^2} + \frac{\partial^2 V}{\partial y^2} + \frac{\partial^2 V}{\partial z^2} = 0
in space free from attracting matter. Poisson himself showed
later, in 1813, [9] that when the point (x, y, z) is within the
substance of the attracting body, this equation of Laplace must
be replaced by
\frac{\partial^2 V}{\partial x^2} + \frac{\partial^2 V}{\partial y^2} + \frac{\partial^2 V}{\partial z^2} = -4\pi\rho
where \rho denotes the density of the attracting matter at the
point. In the present memoir Poisson called attention to the
utility of this function V in electrical investigations, remarking
that its value over the surface of any conductor must be
constant. The known formulae for the attractions of spheroids show that when a charged conductor is spheroidal, the repellent force acting on a small charged body immediately outside it will be directed at right angles to the surface of the spheroid, and will be proportional to the thickness of the surface-layer of electricity at this place. Poisson suspected that this theorem might be true for conductors not having the spheroidal form - a result which, as we have seen, had been already virtually given by Coulomb ; and Laplace suggested to Poisson the following proof, applicable to the general case. The force at a point immediately outside the conductor can be divided into a part s due to the part of the charged surface immediately adjacent to the point, and a part S due to the rest of the surface. At a point close to this, but just inside the conductor, the force S will still act; but the forces will evidently be reversed in direction. Since the resultant force at the latter point vanishes, we must have S=s ; so the resultant force at the exterior point is 2s. But s is proportional to the charge per unit area of the surface, as is seen by considering the case of an infinite plate ; which establishes the theorem. When several conductors are in presence of each other, the distribution of electricity on their surfaces may be determined by the principle, which Poisson took as the basis of his work, that at any point in the interior of any one of the conductors, the resultant force due to all the surface-layers must be zero. He discussed, in particular, one of the classical problems of electrostatics - namely, that of determining the surface-density on two charged conducting spheres placed at any distance from each other. The solution depends on Double Gamma Functions in the general case ; when the two spheres are in contact, it depends on ordinary Gamma Functions. Poisson gave a solution in terms of definite integrals, which is equivalent to that in terms of Gamma Functions ; and after reducing his results to numbers, compared them with Coulomb's experiments. The rapidity with which in a single memoir Poisson passed from the barest elements of the subject to such recondite problems as those just mentioned may well excite admiration. His success is, no doubt, partly explained by the high state of development to which analysis had been advanced by the great mathematicians of the eighteenth century ; but even after allowance has been made for what is due to his predecessors, Poisson' s investigation must be accounted a splendid memorial of his genius. Some years later Poisson turned his attention to magnetism ; and, in a masterly paper [10] presented to the French Academy in 1824, gave a remarkably complete theory of the subject. His starting-point is Coulomb's doctrine of two imponderable magnetic fluids, arising from the decomposition of a neutral fluid, and confined in their movements to the individual elements [1] Page 677. [2] Mem. presences par divers Savans, ix (1780), p. 165. [3] Mem de 1'Acad., 1785, p. 593. Gauss finally established the law by a much more refined method. [4] In his Seventh Memoir, Mem, de 1'Acad., 1789, p. 488. [5] On Volta's discoveries. [6] Mem. de l'Institut, 1811, Part i., p. 1, Part ii., p. 163. [7] Mem. de Berlin, 1777. The theorem was afterwards published, and ascribed to Laplace, in a memoir by Legendre on the Attractions of Spheroids, which will be found in the Mem. par divers Savans, published in 1780. [8] Mem. de 1'Acad., 1782 (published in 1785), p. 113. [9] Bull, de la Soc. Philomathique. iii. (1813,, p. 388. [10] Mem. de l'Acad., v, p. 247. |