804.] ESTABLISHment of the diSTRIBUTION OF Force.

397

803.] We have to remark in the first place, that in this problem the thermal conductivity of Fourier's medium is to be taken inversely proportional to the electric conductivity of our medium, so that the time required in order to reach an assigned stage in the process of diffusion is greater the higher the electric conductivity. This statement will not appear paradoxical if we remember the result of Art. 655, that a medium of infinite conductivity forms a complete barrier to the process of diffusion of magnetic force.

In the next place, the time requisite for the production of an assigned stage in the process of diffusion is proportional to the square of the linear dimensions of the system.

There is no determinate velocity which can be defined as the velocity of diffusion. If we attempt to measure this velocity by ascertaining the time requisite for the production of a given amount of disturbance at a given distance from the origin of disturbance, we find that the smaller the selected value of the disturbance the greater the velocity will appear to be, for however great the distance, and however small the time, the value of the disturbance will differ mathematically from zero.

This peculiarity of diffusion distinguishes it from wave-propagation, which takes place with a definite velocity. No disturbance takes place at a given point till the wave reaches that point, and when the wave has passed, the disturbance ceases for ever.

804.] Let us now investigate the process which takes place when an electric current begins and continues to flow through a linear circuit, the medium surrounding the circuit being of finite electric conductivity. (Compare with Art. 660).

When the current begins, its first effect is to produce a current of induction in the parts of the medium close to the wire. The direction of this current is opposite to that of the original current, and in the first instant its total quantity is equal to that of the original current, so that the electromagnetic effect on more distant parts of the medium is initially zero, and only rises to its final value as the induction-current dies away on account of the electric resistance of the medium.

But as the induction-current close to the wire dies away, a new induction-current is generated in the medium beyond, so that the space occupied by the induction-current is continually becoming wider, while its intensity is continually diminishing.

This diffusion and decay of the induction-current is a phenomenon precisely analogous to the diffusion of heat from a part of

the medium initially hotter or colder than the rest. We must remember, however, that since the current is a vector quantity, and since in a circuit the current is in opposite directions at opposite points of the circuit, we must, in calculating any given component of the induction-current, compare the problem with one in which equal quantities of heat and of cold are diffused from neighbouring places, in which case the effect on distant points will be of a smaller order of magnitude.

805.] If the current in the linear circuit is maintained constant, the induction currents, which depend on the initial change of state, will gradually be diffused and die away, leaving the medium in its permanent state, which is analogous to the permanent state of the flow of heat. In this state we have

▼2F = ▼2G = v2 H = 0

(2)

throughout the medium, except at the part occupied by the circuit,

These equations are sufficient to determine the values of F, G, H throughout the medium. They indicate that there are no currents except in the circuit, and that the magnetic forces are simply those due to the current in the circuit according to the ordinary theory. The rapidity with which this permanent state is established is so great that it could not be measured by our experimental methods, except perhaps in the case of a very large mass of a highly conducting medium such as copper.

NOTE. In a paper published in Poggendorff's Annalen, June 1867, M. Lorenz has deduced from Kirchhoff's equations of electric currents (Pogg. Ann. cii. 1856), by the addition of certain terms which do not affect any experimental result, a new set of equations, indicating that the distribution of force in the electromagnetic field may be conceived as arising from the mutual action of contiguous elements, and that waves, consisting of transverse electric currents, may be propagated, with a velocity comparable to that of light, in non-conducting media. He therefore regards the disturbance which constitutes light as identical with these electric currents, and he shews that conducting media must be opaque to such radiations.

These conclusions are similar to those of this chapter, though obtained by an entirely different method. The theory given in this chapter was first published in the Phil. Trans. for 1865.

CHAPTER XXI.

MAGNETIC ACTION ON LIGHT.

806.] THE most important step in establishing a relation between electric and magnetic phenomena and those of light must be the discovery of some instance in which the one set of phenomena is affected by the other. In the search for such phenomena we must be guided by any knowledge we may have already obtained with respect to the mathematical or geometrical form of the quantities which we wish to compare. Thus, if we endeavour, as Mrs. Somerville did, to magnetize a needle by means of light, we must remember that the distinction between magnetic north and south is a mere matter of direction, and would be at once reversed if we reverse certain conventions about the use of mathematical signs. There is nothing in magnetism analogous to those phenomena of electrolysis which enable us to distinguish positive from negative electricity, by observing that oxygen appears at one pole of a cell and hydrogen at the other.

Hence we must not expect that if we make light fall on one end of a needle, that end will become a pole of a certain name, for the two poles do not differ as light does from darkness.

We might expect a better result if we caused circularly polarized light to fall on the needle, right-handed light falling on one end and left-handed on the other, for in some respects these kinds of light may be said to be related to each other in the same way as the poles of a magnet. The analogy, however, is faulty even here, for the two rays when combined do not neutralize each other, but produce a plane polarized ray.

Faraday, who was acquainted with the method of studying the strains produced in transparent solids by means of polarized light, made many experiments in hopes of detecting some action on polarized light while passing through a medium in which electrolytic conduction or dielectric induction exists *. He was not, however, * Experimental Researches, 951-954 and 2216-2220.

able to detect any action of this kind, though the experiments were arranged in the way best adapted to discover effects of tension, the electric force or current being at right angles to the direction of the ray, and at an angle of forty-five degrees to the plane of polarization. Faraday varied these experiments in many ways without discovering any action on light due to electrolytic currents or to static electric induction.

He succeeded, however, in establishing a relation between light and magnetism, and the experiments by which he did so are described in the nineteenth series of his Experimental Researches. We shall take Faraday's discovery as our starting point for further investigation into the nature of magnetism, and we shall therefore describe the phenomenon which he observed.

807.] A ray of plane-polarized light is transmitted through a transparent diamagnetic medium, and the plane of its polarization, when it emerges from the medium, is ascertained by observing the position of an analyser when it cuts off the ray. A magnetic force is then made to act so that the direction of the force within the transparent medium coincides with the direction of the ray. The light at once reappears, but if the analyser is turned round through a certain angle, the light is again cut off. This shews that the effect of the magnetic force is to turn the plane of polarization, round the direction of the ray as an axis, through a certain angle, measured by the angle through which the analyser must be turned in order to cut off the light.

808.] The angle through which the plane of polarization is turned is proportional

(1) To the distance which the ray travels within the medium. Hence the plane of polarization changes continuously from its position at incidence to its position at emergence.

(2) To the intensity of the resolved part of the magnetic force in the direction of the ray.

(3) The amount of the rotation depends on the nature of the medium. No rotation has yet been observed when the medium is air or any other

gas.

These three statements are included in the more general one, that the angular rotation is numerically equal to the amount by which the magnetic potential increases, from the point at which the ray enters the medium to that at which it leaves it, multiplied by a coefficient, which, for diamagnetic media, is generally positive. 809.] In diamagnetic substances, the direction in which the plane

810.]

FARADAY'S DISCOVERY.

401

of polarization is made to rotate is the same as the direction in which a positive current must circulate round the ray in order to produce a magnetic force in the same direction as that which actually exists in the medium.

Verdet, however, discovered that in certain ferromagnetic media, as, for instance, a strong solution of perchloride of iron in woodspirit or ether, the rotation is in the opposite direction to the current which would produce the magnetic force.

This shews that the difference between ferromagnetic and diamagnetic substances does not arise merely from the magnetic permeability' being in the first case greater, and in the second less, than that of air, but that the properties of the two classes of bodies. are really opposite.

The power acquired by a substance under the action of magnetic force of rotating the plane of polarization of light is not exactly proportional to its diamagnetic or ferromagnetic magnetizability. Indeed there are exceptions to the rule that the rotation is positive for diamagnetic and negative for ferromagnetic substances, for neutral chromate of potash is diamagnetic, but produces a negative rotation.

810.] There are other substances, which, independently of the application of magnetic force, cause the plane of polarization to turn to the right or to the left, as the ray travels through the substance. In some of these the property is related to an axis, as in the case of quartz. In others, the property is independent of the direction of the ray within the medium, as in turpentine, solution of sugar, &c. In all these substances, however, if the plane of polarization of any ray is twisted within the medium like a righthanded screw, it will still be twisted like a right-handed screw if the ray is transmitted through the medium in the opposite direction. The direction in which the observer has to turn his analyser in order to extinguish the ray after introducing the medium into its path, is the same with reference to the observer whether the ray comes to him from the north or from the south. The direction of the rotation in space is of course reversed when the direction of the ray is reversed. But when the rotation is produced by magnetic action, its direction in space is the same whether the ray be travelling north or south. The rotation is always in the same direction as that of the electric current which produces, or would produce, the actual magnetic state of the field, if the medium belongs to the positive class, or in the opposite direction if the medium belongs to the negative class.