ELECTROMAGNETISM.] E L E C T K I C I T Y 67 covery formed the starting-point of that division of electrical science with which we are now to deal. It was natural, once the action of a current 1 on a magnet was observed, to look for the reaction of the magnet on the current, and after seeing two currents act on the same magnet, it was reason able to expect that the currents would act on each other. Yet it may be doubted whether the first of these results is a legitimate deduction from the discovery of Oersted, and the second certainly is not so. Before we can apply the prin ciple of the equality of action and reaction we must be quite certain of the source of the whole of any action to which the principle is to be applied. Again, two bodies A and C may act on B owing to properties acquired by virtue of B s presence, so that in the absence of B they need not neces sarily act on each other. A good example is the case of two pieces of perfectly soft iron, each of which will act on and be acted on by a magnet, but which will not act on each other when the magnet is not near them. The questions thus raised by Oersted s discovery were experimentally settled by Ampere. He found that a magnet or the earth (which behaves as if it were a magnet) acts on the current, and the direction of these actions is found to be consistent with the principle of equality of action and reac tion. As no experimental fact has yet been quoted against the application of this principle in such cases, we shall assume it henceforth. Ampere also discovered the action of one electric current on another, and thereby settled the second question. We may conclude, therefore, that the space surrounding an electric current is a field of magnetic force just as much as the space around a magnetized body. The next step is to determine the distribution of magnetic force, or what amounts to the same thing, to find a distri bution of magnetism which shall be equivalent in its mag netic action to the electric current. This also was com pletely accomplished by Ampere. In expounding his results we shall follow the order of ideas given by Maxwell, 2 which we think affords the simplest view of the matter, and is the best practical guide that we know of through the somewhat complicated relations to which the subject intro duces us. We shall in addition give a sketch of the actual course which was followed by Ampere, and which is adhered to by the Continental writers of the present day. It results alike from the fundamental experiments of Ampere and the elaborate researches of Weber, to both of which we shall afterwards allude, that an electric current circulating in a small plane closed circuit, acts and is acted upon magnetically exactly like a small magnet placed per pendicular to its plane at some point within it, 3 provided the moment of the magnet be equal to the strength of the current multiplied by the area of the circuit, 4 and its north pole be so placed that the direction of the axis of the magnet (from S-pole to N-pole), and the direction in which the current circulates are those of the translation and rotation of a right-handed (ordinary) screw which is being screwed in the direction of the axis. In this statement we have spoken of a small closed circuit. The word " small " means that the largest dimensions of the circuit must be infinitely smaller than its distance from the nearest magnet or electric current on which it acts, or by which it is acted on. We may break up our small magnet into a number of similar magnets, and distribute them over the area of the small circuit, so that the sum of the moments of all the mag nets on any portion w of the area is un, where i is constant. We thus replace fehe circuit by a " magnetic shell " of strength " Current " is used here and in corresponding cases as an abbre viation for the " the linear conductor conveying a current." Electricity and Magnetism, vol. ii. 475, &c. 3 Naturally the centre ol the area ii it is symmetrical. 4 We shall see directly what system oi units this statement pre supposes. magnetic. shell. i, which, if we choose, may be represented by two layers parallel to the area, one of north the other of south mag netism, the surface density of which is i -i- 0, where # is the distance between the layers. 5 Starting from the principle thus laid down we cau derive Finite all the laws of the mutual action of magnets and electric currents - Consider any finite circuit ABC (fig. 29). Imagine it filled with a surface of any form, and a network of lines drawn on the surface as in the figure, di viding it up into por tions, such as abed, so small that they may be regarded as plane. It is obvious that any current of strength i circulating in ABC may be replaced by a series of closed cur rents, each of strength i circulating in the meshes (such as abcd of the network on the surface ; for in each line such as be we have two equal and opposite currents circulating whose action must be nil. Now, we may replace each of the small circuits by a magnet as above, or by a magnetic shell of strength i. The assemblage will constitute a magnetic shellof strength* filling up the circuit, whose magnetic action, at every point external* to the shell will be the same as that of the current. The north side of the shell is derived from the direction of the current by the right-handed screw relation given above. If dS be an elemeu u of tlie surface of a magnetic shell t-f strength i, D its distance from P, and the angle which the positive direction of magnetization (which is normal to f/S) makes with D, then the magnetic potential 7 at P is given by the integration extending all over fcj. When properly interpreted this double integral is found to represent the " solid angle " subtended at P by the surface S, or, as it may also be put, by the circuit ABC which bounds it. Hence, solid angles subtended by the north side being taken as positive, and the usual conventions as to sign adhered to, we may write V-i, (2), where w is the solid angle in question. We see, therefore, that the potential of a magnetic shell Poten at any point P is equal to the product of the strength of Wo the shell into the solid angle subtended by its boundary at P. Now the potential of such a shell is continuous^ and single-valued at all points without it. (With points within it we are not now concerned, since the action of the current at such points is not the same as that of the shell.) If, therefore, a unit north pole start from any point P and return to the same, after describing any path which does not cut through the shell, i.e., does not embrace the current, the work done by it will be nil. Let us now examine what happens if the path cuts through the shell S. Take two points P and Q, infinitely near each other, but the one P on the positive side, the other Q on the negative side of the 5 The reader who finds difficulty with the magnetic shell may adhere to the small magnet; it will be found sufficient for most practical purposes. 6 This limitation is the equivalent of the limitation small applied to the elementary plane circuit, and follows therefrom. 7 We need scarcely remind the reader that all the definitions of potential, &c., in the theory of electrostatics apply here if we substi tute + and - magnetism for + and - electricity. The unit of +
magnetism is sometimes called a unit north pole.