MAGNETISM 227 Elemen tary law of action it a listance. Unit of nagnet- ism. Strength >f mag- letic ield. Re- -ultant nagnetic
- orce.
Volume and surface lensity
- >f mag-
aetism. Total nagnet- sm zero. of the same sign and attracts magnetism of the opposite sign. Magnetism is supposed to be so associated with the matter of the body that magnetic force exerted on the magnetism is ponderomotive force exerted on the matter. On the other hand, magnetic force is always supposed to be exerted by magnetism upon magnetism, and never directly by or upon matter. Into the nature of this association of magnetism with matter there is no pretence, indeed no need, to enter. The elementary law of action assumed is that the attrac tion or repulsion (as the case may be) between two quantities m and m of magnetism siipposed concentrated in two points at a distance r apart is %-> and is in the line joining the two points. This supposes that the unit quantity of magnetism is so chosen that two units of positive magnetism at unit distance apart repel each other with unit force. This definition, which is fundamental in the electromagnetic system of units, gives for the dimensions of a quantity of magnetism [iJM T" 1 ]. If the electrostatic system be adopted the result would of course be different. An accurate meaning can now be given to the phrase " strength of a magnetic field," or its equivalent " resultant magnetic force at a point in the field;" it is defined to be the force exerted upon a unit of positive magnetism supposed concentrated at the point. The force exerted on a unit of negative magnetism would of course be equal in magnitude, but oppositely directed ; and in general, if R denote the resultant magnetic force at the point, the magnetic force exerted on a quantity K of magnetism concentrated there is K~R. We may, as in the corresponding theory of electricity, introduce the ideas of volume density (p) and surface density (cr), so that pdv and o~dS denote the quantities of magnetism in an element of volume and on an element of surface respectively ; p and cr may of course be positive or negative according to circumstances. It will now be seen that, mathematically speaking, the theories of action at a distance for electricity and magnetism are identical, and every conclusion drawn will have, so far as the physical diversity of the two cases may allow, a double application. 1 In particular it will be found that the theory of magnetism, when properly interpreted, gives the theory of dielectrics polarized in the way imagined by Faraday. The fact of magnetic polarity requires the conception of negative as well as positive magnetism ; the fact that the properties of the smallest parts of a magnet are similar to those of the whole requires that in every element of the body there shall be both negative and positive magnetism. From the fact that in a uniform field, i.e., one in which the resultant magnetic force has at every point the same magnitude and direction, the force of translation upon a magnet is nil, it follows that the algebraic sum of all the magnetism in any magnet must be zero ; for, if R denote the strength of the field, by the theory of parallel forces the whole force on the magnet will be 2(*R), = RS* ; hence 2* = 0. In other words, in every magnet there must be as much negative as positive magnetism ; and this con clusion also must be extended to the smallest parts of every magnet, so long as we do not go behind the mere facts of observation. The positive and negative magnetism cannot be coincident throughout, otherwise there would be no external magnetic action, but the separation is in the elements of the body. Thus, although there is no force of translation in a uniform field, there will in general be a couple. Consider the positive and negative magnetism 1 To prevent needless repetition, we shall adopt henceforth, without further explanation, the definitions, terminology, and results given in the article ELECTRICITY, vol. viii. p. 24 sq. separately, and let K denote any element of the former and any element of the latter. Let 1ST be the centre of mass of the positive, S the centre of mass of the negative magnetism ; so that, if the magnet be referred to a set of rectangular axes, the coordinates of N and S are and 2/cY 2/c (1). Let the distance NS = I, and let K = 12,K, = I^K. ; this Magnetic quantity K is called the "magnetic moment." By the theory moment, of parallel forces, if we suspend the magnet in a uniform field of strength R, the action upon it reduces to two forces R2x and - R2, each parallel to the direction of the field, acting respectively at N and S, in other words to a couple Magnetic whose moment is R^sin^ or KRsin^, where x is the couple ir ^_ > uniform angle between SN and the direction of the field. Hence, field, if the magnet be perfectly free to follow the magnetic action -> -* of the field, it will set so that the line SN or the line NS is parallel to the direction of the field, the equilibrium being stable in the former case, but unstable in the latter. The line NS is therefore parallel to what we have already defined on experimental grounds as the axial direction in the magnet. N, S, and NS are sometimes called par excellence the poles and the axis of the magnet ; we have adopted the looser definition given above because it is more convenient and nearer the popular usage. The above results maybe applied to some cases very important in Theory practice. Let the magnet whose centres of positive and negative of magnetism are N and S be suspended by the middle point of NS, dipping which, for simplicity, may be assumed to be also its centre of needle, gravity. Let OX (fig. 20) be a horizontal line drawn northwards, OZ X a vertical drawn downwards, both in the magnetic meridian. Let the vertical plane through NS make an angle with the magnetic meridian, and let ON make an angle <p with the horizon. If R be the strength of the earth s magnetic field, and t the angle of dip, then the horizontal and vertical components of the earth s force are H = R cost and Z = R sim. First, suppose the angle 4> fixed, and the magnet free to rotate about OZ only ; then the couple tending to diminish the angle 6 is 2 . 2* . R cost cos</> sine, or KR cost cos< sin0. G In other words the directive couple varies as the sine of the angle of deviation from the magnetic meridian. This conclusion was verified experimentally by Lambert, and also by Coulomb 2 by means of his torsion balance. It will be seen that, cset-eris paribiis, the directive couple is greatest when the magnetic axis is horizontal.
2 Mem. de I Acad., 1785.