tion, is without action at external points. The magnetisation would only manifest itself if the ring were again opened, or if it were broken into pieces.

Let us now suppose that an infinitely thin layer, NS, is cut out at right angles to the magnetisation. One free face of the slit will be covered with a uniform layer of density +A, and the other with a layer of density A. A unit positive pole placed at P in the empty space will be repelled with a force, 2πA, by the positive face, and attracted with a force, 2πА, by the negative face. The magnetic force in the slit is, therefore, equal to 4πA. The flux, or the number of lines of force which represent it, is equal to 4πA per unit surface. It may be assumed that the lines which traverse the slit are only the continuation of the lines existing in the magnet; we shall call them lines of magnetisation.

If there were other centres of force, for instance, magnetic poles situate in the field, their action on the pole would be added to the action of the two faces, and would have to be compounded with the force 4πА.

Let the ring be cut at any part, O, for instance (Fig. 166), and let it be stretched out so that it has again the shape of a cylinder (Fig. 167). Nothing is changed in the magnetic condition, except that the pole P, besides the force 4πA, is also subject

FIG. 167.

to the force due to the layers of density ± A, which cover the two ends, a force which is obviously opposed to the former.

The pole P, situate in the infinitely narrow slit perpendicular to the magnetisation, is then under the action of two systems of forces, one depending on the intensity of magnetisation, which is equal to 4πA; the other, which is due to the surface magnetisation and to the field, may be represented by F. These two systems produce a resultant which, as we shall afterwards see, can be directly determined by experiment. We shall call it the magnetic induction, and shall represent it by B.

If the two forces are in the same direction, we have

B = F + 4πA,

If the force F results solely from the surface magnetism, it is always in the opposite direction to 4πA.

The pole which, if placed in a slit perpendicular to the magnetisation, is subject to the action of B, is only under the action of F if it is placed in a cavity which has the shape of an infinitely narrow cylinder with its axis parallel to the magnetisation (Fig. 167); in this case, in fact, the magnetisation of the medium only comes into play in so far as it is represented by two infinitely small magnetised surfaces forming the ends of the cavity, the action of which may be neglected.

188. Magnet of Uniform Induction.—In order that the induction B may be constant throughout the whole extent of the body, not only must the magnetisation A be uniform, but apart from the external field the action of the surface layer must be the same on all points of the interior.

This condition is realised only in very special cases; it is so in that of a sphere magnetised uniformly (§ 26); for an intensity of magnetisation A, the force due to the layers of displacement has the constant value

F = − ̃А.

It is also realised with an ellipsoid of revolution magnetised uniformly in the direction of its axis (Fig. 168). If 2a is the axis of revolution, 26 that of the equator, the total mass of each layer is πb2A (§ 185). The value of that which is comprised between the

equator and a parallel plane situated at the distance r is πA(b2 - y2) or #A6223 πAb2x2 it is proportional to x2, and may thence be reprea2 ;; sented by the surface of a triangle such as OAP. The centre of gravity is at two-thirds of the base, that is to say, at two-thirds of the axis counting from the centre. The moment is therefore GG'× πb2 A Tab2A; which is otherwise evident, seeing that Amab is the volume of the ellipsoid,

=

Calculation shows that the force due to the layer at a point in the interior may be expressed by

N being a coefficient which only depends on the ratio of the axes, and which is smaller as the ellipsoid is more elongated. We have seen that in the case of a sphere it is equal to . For a = 100 b, and a = 50ɔ b, calculation gives respectively N = 0.0054 and 0.0003.

Finally, the force F vanishes in the case of an annular magnet. This may also be assumed to be the case very approximately with a cylinder whose length is very great in proportion to its diameter. The action of the terminal masses may then be considered as negligable throughout the greater part of the length.

CHAPTER XVIII.

MAGNETIC INDUCTION.

189. Magnetisation by Induction.-Any piece of iron placed in a magnetic field becomes a magnet. The direction of the magnetisation is that of the lines of force, the south pole being at the part where they pass in, and the north pole where they pass out. This phenomenon is known as magnetisation by induction.

The phenomenon is molecular, and is analogous to the inductive electrification of an insulating body. It always precedes the attraction of iron by a magnet, so that attraction is, in reality, always exerted between two magnets which present to each other their opposite poles.

A filament of iron free to move, and not subject to any other action than that of the field, tends to take at each point the direction of the line of force. The figures of iron-filings are formed in this way; each particle tends to place itself tangentially to the line of force and, the opposite poles of adjacent filings attracting each other, they arrange themselves in lines which mark out the lines of force.

190. Residual Magnetism-Coercive Force.—In a short piece of pure and unhardened iron, what is called soft iron, magnetisation is produced at once, and disappears when the inductive action is withdrawn. With iron which has been hardened, with impure iron, cast iron, steel, and particularly with hardened steel, the magnetisation produced by a given magnetising field is less intense, but in general it persists in greater or less degree after the cessation of the magnetising force. This is the origin of artificial magnets. The magnetisation, which lasts only as long as the inductive action, is called temporary magnetism, while that which continues after the inductive action has ceased is called permanent, or remanent, or residual magnetism.

The name coercive force has been applied to the property which certain materials thus possess of retaining, after the inductive action

has ceased, a portion of the magnetism which this action had developed. Coercive force is, to a certain extent, a property analogous to friction, and constitutes a resistance to any change of magnetic condition. A method of estimating its numerical value will be given later (§ 197).

191. Magnetic and Diamagnetic Bodies.-Iron, including its varieties, is not the only body which is magnetised by induction. Nickel and cobalt have similar properties. These properties are also possessed, though to a less degree, by various compounds of iron, such as magnetic oxide of iron, Fe3O4, which constitutes loadstone, iron perchloride, and iron sulphate, whether solid or in solution. Indeed, when a more and more intense field and delicate means of observation are employed, it is found that there is probably no body, whether solid, liquid, or gaseous, which is not susceptible to the action of magnetism, and does not acquire by induction magnetic polarity. But in this respect bodies fall into two distinct categories: those of one class are acted on in the magnetic field like iron, their axis of magnetisation is along the lines of force (Fe, Fig. 169); those of the other class acquire an inverse magnetisation, that is, parallel, but in the opposite direction to the lines of force (Bi, Fig. 169). The term magnetic is applied to the former, and the term diamagnetic to the latter class of bodies. Diamagnetic properties are only shown to a very slight extent; the most strongly diamagnetic body is bismuth.

Fe

N

S

FIG. 169.

Bi

Magnetism is thus seen to be a general property of bodies, but it is extremely remarkable that while three substances-iron, cobalt, and nickel-possess it in a very high degree, it may be considered as in comparison non-existent in the others.

192. Coefficient of Magnetisation-Magnetic Susceptibility and Permeability. The fundamental problem of magnetisation by induction is this: the magnetising force acting at a point of a piece of iron being given, to determine the intensity of the resulting magnetisation, or the value of magnetic induction at this point.

The magnetic force which is here to be considered is that resulting from the field and the reaction of the surface layer; it is that which would act on a pole placed at the point in question in a