Siméon Denis Poisson

Electrostatical theory was, however, suddenly advanced to quite a mature state of development by Siméon Denis Poisson (b. 1781, d. 1840), in a memoir which was read to the French Academy in 1812.[76]

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.

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.

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;

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 2 similar and similarly situated ellipsoids exercises no attractive force at any point within the interior hollow.

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[77] 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[78] that this function V satisfies the equation

in space free from attracting matter. Poisson himself showed later, in 1813,[79] that when the point (x, y, z) is within the substance of the attracting body, this equation of Laplace must be replaced by

where ρ 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.

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 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 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[80] 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 of the magnetic body, so as to be incapable of passing from one element to the next.

Suppose that an amount m of the positive magnetic fluid is located at a point (x, y, z); the components of the magnetic intensity, or force exerted on unit magnetic pole, at a point (ξ, η, ζ) will evidently be

where r denotes |(ξ-x)2 + (η-y)2 + (ζ-z)2|

Hence if we consider next a magnetic element in which equal quantities of the two magnetic fluids are displaced from each other parallel to the x-axis, the components of the magnetic intensity at (ξ, η, ζ) will be the negative derivates, with respect to ξ, η, ζ respectively, of the function

where the quantity A, which does not involve (ξ, η, ζ), may be called the magnetic moment of the element: it may be measured by the couple required to maintain the element in equilibrium at a definite angular distance from the magnetic meridian.

If the displacement of the two fluids from each other in the element is not parallel to the axis of s, it is easily seen that the expression corresponding to the last is

where the vector (A, B, C) now denotes the magnetic moment of the element.

Thus the magnetic intensity at an external point (ξ, η, ζ) due to any magnetic body has the components

where

integrated throughout the substance of the magnetic body, and where the vector (A, B, C) or I represents the magnetic moment per unit-volume, or, as it is generally called, the magnetization. The function V was afterwards named by Green the magnetic potential.

Poisson, by integrating by parts the preceding expression for the magnetic potential, obtained it in the form

the first integral being taken over the surface S of the magnetic body, and the second integral being taken throughout its volume. This formula shows that the magnetic intensity produced by the body in external space is the same as would be produced by a fictitious distribution of magnetic fluid, consisting of a layer over its surface, of surface-charge (I.ds) per element dS, together with a volume-distribution of density — div I throughout its substance. These fictitious magnetizations are generally known as Poisson’s equivalent surface- and volume-distributions of magnetism.

Poisson, moreover, perceived that at a point in a very small cavity excavated within the magnetic body, the magnetic potential has a limiting value which is independent of the shape of the cavity as the dimensions of the cavity tend to zero; but that this is not true of the magnetic intensity, which in such a small cavity depends on the shape of the cavity. Taking the cavity to be spherical, he showed that the magnetic intensity within it is

where I denotes the magnetization at the place.

This memoir also contains a discussion of the magnetism temporarily induced in soft iron and other magnetizable metals by the approach of a permanent magnet.

Poisson accounted for the properties of temporary magnets by assuming that they contain embedded in their substance a great number of small spheres, which are perfect conductors for the magnetic fluids; 80 that the resultant magnetic intensity in the interior of one of these small spheres must be zero. He showed that such a sphere, when placed in a field of magnetic intensity F,[83] must acquire a magnetic moment of amount

the volume of the sphere, in order to counteract within the sphere the force F. Thus if kp denote the total volume of these spheres contained within a unit volume of the temporary magnet, the magnetization will be I, where

and F denotes the magnetic intensity within a spherical cavity excavated in the body. This is Poisson’s law of induced magnetism.

It is known that some substances acquire a greater degree of temporary magnetization than others when placed in the same circumstances: Poisson accounted for this by supposing that the quantity kp varies from one substance to another. But the experimental data show that for soft iron kp must have a value very near unity, which would obviously be impossible if kp is to mean the ratio of the volume of spheres contained within a region to the total volume of the region.[84] The physical interpretation assigned by Poisson to his formulae must therefore be rejected, although the formulae themselves retain their value.

Poisson’s electrical and magnetical investigations were generalized and extended in 1828 by George Green[85] (b. 1793, d. 1841). Green’s treatment is based on the properties of the function already used by Lagrange, Laplace, and Poisson, which represents the sum of all the electric or magnetic charges in the field, divided by their respective distances from some given point: to this function Green gave the name potential, by which it has always since been known.[86]

Near the beginning of the memoir is established the celebrated formula connecting surface and volume integrals, which is now generally called Green’s Theorem, and of which Poisson’s result on the equivalent surface- and volume-distributions of magnetization is a particular application.

Green used this theorem to investigate the properties of the potential and arrived at many results of remarkable beauty and interest.

Suppose that there is a hollow conducting shell, bounded by two closed surfaces, and that a number of electrified bodies are placed, some within and some without it, and let the inner surface and interior bodies be called the interior system, and the outer surface and exterior bodies be called the exterior system.

Then all the electrical phenomena of the interior system, relative to attractions, repulsions, and densities, will be the same as if there were no exterior system, and the inner surface were a perfect conductor, put in communication with the earth.

All those of the exterior system will be the same as if the interior system did not exist, and the outer surface were a perfect conductor, containing a quantity of electricity equal to the whole of that originally contained in the shell itself and in all the interior bodies.

Electrostatics had by this time attained a state of development in which further progress could be hoped for only in the mathematical superstructure, unless experiment should unexpectedly bring to light phenomena of an entirely new character. This will therefore be a convenient place to pause and consider the rise of another branch of electrical philosophy.