149. Changes of form in a concentric system of lines of force, produced by bending the axis.-Let a wire, originally straight, and constituting an axis of force, be bent into the form of a circle, so that two portions of the wire which were originally far apart are brought quite close together, the direction in which they run being at the same time reversed. The lines of force embracing the various parts of the axis will thus be collected into a bundle, which passes through the nearly closed loop. Each of the lines of force retains always the form of a closed curve, whether the conductor itself be straight or curved. In the neighbourhood of the wire, where the force is greatest, and where (in accordance with our previous experiments) the tubes of force are richest in energy, the lines preserve most nearly their circular form. But since, in bending the conductor into a loop, we have collected into a bundle lines of force that were originally spread far apart, it follows that the tubes of induction of smaller energy-content are deformed in such a way that in passing through the loop these are crowded more closely together, while outside the loop they are spread further apart.

(a) Line-of-force diagram for a current flowing round a ring. To show by means of iron filings the course of the lines of force within and without the ring, bend a thick copper wire into the form of a semicircle, thrust its two extremities through corresponding holes in a horizontally placed sheet of paper, and then

FIG. 50

bend them round until they come almost, but not quite, into contact, so that the wire takes the form of a nearly complete circle (fig. 50). When a current is sent round the wire in the direction (+ to −) indicated by the arrows, the general course of the lines of force is as shown diagrammatically in fig. 50. Fig. 51 shows the actual disposition of the filings.

The direction in which the lines of force are to be positively reckoned may be easily deduced from the positive direction along the conductor (axis of force). If this latter direction is counter

FIG. 51

clockwise from our point of view, the lines of force will encircle the left limb of the wire clockwise as seen from above. These lines, then, as they thread through the ring are directed towards the observer, the same being true of the lines which encircle the right limb of the wire, and, in fact, of all lines of the system.

[Thus, a translation in the direction of the magnetic force, combined with a rotation in the sense of the current, would constitute a right-handed screw motion. Comparing this statement with the corresponding square-bracketed statement in § 147, we may say that a current and the lines of induction which it produces embrace one another right-handedly.]

The relation between the cyclic directions of currents and their lines of force may be simply exemplified as follows:-Bring the thumb and forefinger together in each hand so as to form two closed circuits, which are made to embrace one another like consecutive links of a chain. The forefinger being considered in each case as pointing along the positive direction of the circuit, the forefinger of the right hand should be pointing downward through the left-hand circuit. This being so, the right-hand circuit may be taken to represent the current, and the left-hand circuit the lines of force, or vice versa.

(b) Model of tubes of force for a current in a ring.-Over a copper wire SS, fig. 52, are threaded a series of closed rings, representing tubes of force, all of the same size, and corresponding in cyclic direction to the + to direction of the current. The wire being then bent into the form of a closed or nearly closed circular loop, the rings must be arranged so that their planes converge towards the axis of the loop, as indicated in the figure. The arrows show that the hypothetical vortical motions within the conductor are everywhere in the same sense, namely in the sense of the axis of force (sense of the current).

The model, fig. 52, with the rotatory motions which it indicates, furnishes a chart of what occurs in the field of a circular

S

FIG. 52

current, according to the kinetic hypothesis; it may also be taken to exemplify the structure of a 'vortex sponge,' such as English mathematicians have elaborated from the

kinetic theory. Although these motions may appear to be very complicated, their relations are very easily made out, for all the separate motions are determinately related to the axis of force. Moreover the model shows clearly the distribution of energy in the neighbourhood of the wire.

150. A circuital current as a store of magnetic field energy.Each tube of induction, or as we may say when the permeability of the medium is unity, each tube of force, is the seat of some definite amount of energy (compare § 119). If now, by bending a current conductor into the form of a loop, we compress into a limited compass tubes of force which were originally spread wide apart, we are thus accumulating a certain amount of energy, somewhat as we accumulate energy in a weight when we raise it against gravity.

A certain amount of work must be expended in the operation, for we have to overcome by a muscular effort the pressure exerted upon one another by the energy tubes in directions perpendicular to their length. [It should be remarked, however, that the more work is expended in this way, the smaller the magnetic energy becomes; the explanation being that work is meanwhile done upon the source of the galvanic current.]

That such pressure really exists in the tubes of force enclosing the current conductor, and gives rise to mechanical forces upon the conductor itself, may be easily verified experimentally.

Experiment 54.-A short, narrow, thin strip of gold has

its ends joined to wires conveying a current. When these wires are held close together, the freely-hanging loop of gold, each time that the current is made to flow, is seen to bulge out at the sides. It strives to attain to the circular form, which, as we know, has the greatest possible area corresponding to a given perimeter. The lines of force passing through a circularly bent current conductor exert an outward thrust on the conductor itself, just as a confined mass of gas exerts a pressure on the walls of the containing vessel.

A current with its system of lines of force constitutes a store of magnetic field energy. The energy is much greater (for a given value of the current) when the space, or a part of the space surrounding the conductor, is occupied by a medium of high magnetic permeability, as for example when the conductor surrounds a mass of soft iron. Later on we shall have to speak more particularly of these important relations; for the present this preliminary note must suffice.

151. A circuital current as the equivalent of a magnetic shell. When the axis of a concentric system of lines of force is bent into a circular form, we obtain something of the nature of a magnetic shell, whose surface is bounded by the current conductor. For the circuit, like the magnetic shell, emits lines of force in one direction, and these bend round, and return to it from the opposite direction. One side behaves like the north-seeking (red) face of the model described in § 23, the other like the south-seeking (blue) face. From a comparison of figs. 50 and 52 we

obtain the following rule :

When we look at a circuit in such a direction that the current appears to flow in it counter-clockwise, the lines of force will be threading through the circuit towards us, and the north surface of the equivalent magnetic shell will be facing us; if, on the other hand, the current is so regarded that it appears clockwise, the lines of force will be threading through away from us, and we shall be turned towards the south surface of the equivalent shell.

Experiment 55.-To illustrate this law a thick aluminium wire is to be bent into a circular hoop, of the same diameter as the red and blue cardboard disc mentioned in § 23, and suspended by two gold strips about a metre in length. The aluminium ring has then but little freedom to move, except directly backwards and forwards; it cannot so easily rotate, and the mode of suspension does not allow it to move sideways. The ends of the wire are to be arranged crossing one another, but not quite touching, so that if we follow the wire along its whole length we pass first along a tangent to a circle, then around the circle itself, and, finally, along a continuation of the same tangent, the distance between the extremities of the wire being equal to the diameter of the circle. In these extremities saw-cuts are made, in which the ends of the gold strips are fastened. When the ring is suspended, a pin is placed underneath to mark its position of rest. If the upper ends of the suspending strips are now joined through a commutator to a pair of terminals, the freely-movable ring becomes associated with a bundle of lines of force. In each case the cardboard disc may be arranged so as to indicate the polarity of the circuit. On bringing near the north or south pole of a bar magnet, we observe the same phenomena of attraction and repulsion which were described in § 33.

If the extremities of a circularly bent wire are furnished with points which are bent downwards, and dip into mercury cups placed one below the other, we may then connect these mercury cups to terminals, so that a current flows round the wire (AMPÈRE'S suspension). Slight impurities at the surface of the mercury, however, interfere with the free motion of the system.

For purposes of comparison, and in view of the applications which we shall afterwards have to make, let us imagine a model to be constructed representing a circuital axis of force (circular current) with a series of tubes of induction passing through it (fig. 53).

A number of model tubes of force, appropriately marked with arrows, are collected into a bundle which passes through a single loop of the copper wire SS. If the tubes of force are bent into curves so as to follow the course of the lines of force, the resulting model resembles very closely that which was made to represent a magnetic shell (§ 128). The circularly bent axis of