This result reduces to the formula given above in the special case where ro=1 cm.

Fig. 69, Plate II., was constructed in this way. It represents, to a scale of one-half, the field of a long straight conductor, carrying a current of 50 ampères 5 deca-ampères 5 electro-magnetic units of current; 1/2i being consequently equal to .

=

173. Deduction of the field-intensity at any point of an axially symmetrical field from the diagram.-We may use the diagram, fig. 69, as we have already done with those representing the fields of magnets, to determine the field-intensity at any given point in absolute measure. From the nature of the construction this may evidently be obtained by measuring with a rule the distance b between the two nearest lines of force on either side of the point, and dividing 1 by b. In this operation the scale to which the diagram is constructed must of course be taken into account.

That the process is justified we see at once from the relation xb.1=1. As an example we will calculate the fieldintensity at a place marked x in fig. 69, and lying between the 21st and 22nd lines of force. Measuring the distance between these two lines, we find b=4.1 mm. But the scale of the diagram is 1: 2, so that in reality b=0·82 cm., whence =1.2. To check this result we may make an independent calculation from the formula =2i/r; the radius of the inner line of force (number 21) is in the diagram 4.1 cm., and therefore in reality 8.2 cm. For points lying on this line, since the diagram was constructed for i=5 deca-ampères, the field-intensity =1.2 cm. gr. sec.-1. And for points lying near to this line the field-intensity will have approximately the same value.

174. Diagram of the fundamental electro-magnetic experiment. The changes produced by the introduction of a magnetic pole in the disposition of the lines of force in the field of a current may be graphically represented by superposing the diagrams corresponding to the two fields. But we must now represent the unipolar magnetic field in a

manner somewhat different from that used in Chapter V., since we have represented the field of the current by an orthogonal plane projection of a flat region one centimetre. thick, and not by a figure of rotation about an axis. We assume, then, that the effective pole of the long thin bar magnet which we introduce into the field is confined to a length of about 1 cm. at its extreme end (§ 21). This will enable us to arrive at a construction giving the variation produced by the pole in the flux of force between the two planes 1 cm. apart. The lower of these planes we suppose to be the plane of the diagram, fig. 70, Plate II., in which, for both fields, we map out the boundary lines of the strips corresponding to unit flux of force. In the case of the current these bounding lines of force are the concentric circles already referred to in § 172, on the tenth of which (reckoning zero at the innermost) the pole S is situated. Here again the scale of the diagram is 1: 2, so that the actual distance from the current to the pole is 2-4 cm. With a current of five deca-ampères, the field-intensity at the place where the pole S is situated will accordingly be

4.2 cm. gr. sec.-1

The current passes from above downwards through the plane of the paper, so that the lines of force in the field of the current pass clockwise around A, as the arrows above A indicate. For the pole, the diagram consists of a fanshaped system of compartments, diverging from the last cm. of the length of the magnet and bounded above and below by the two parallel planes already mentioned; the space between these planes being divided up by a concurrent system of planes, which are perpendicular to them, and have equal angles between nearest neighbours. The number of compartments in the diagram of the polar field is equal to the number of units of flux of force which pass into the pole (sink) from the space between the parallel planes. We shall suppose that there are eight such units.

The strength of the pole is thus greater than 8, since lines of force also pass to it from above and below.

In the plane of the diagram the eight partition-planes appear in orthogonal projection as eight straight lines, passing through S and ranged round at equal angular intervals. That line of force of the current-field on which S itself is situated passes without deviation directly into this sink. A line of force belonging to S must therefore be tangential to it, and thus we arrive at the relative disposition of the two diagrams shown in fig. 70.

On either side of this tangential line, four units of flux of force pass from the surrounding field into the pole, where they appear to end, disappearing from the field, and passing into the magnet. The direction of the lines of force at each point is to be found by combining the two diagrams together. The resultant course of the bounding lines of force is obtained by drawing the proper diagonals across each of the four-sided figures which are formed by the crossing of pairs of neighbouring lines of the two systems-those diagonals, namely, which start from the same corner of the quadrilateral as do the lines of force. At the sink S itself, the resultant lines are all tangential to the straight lines of the polar system. Fig. 70, which was constructed in this way, should be compared with the iron-filing diagram fig. 57, which corresponds approximately with the lower of the two parallel planes here considered, except that the current and its lines of force are reversed in direction.

Eight of the flux of force channels surrounding the current are collected together at S like the tail of a comet ; and the lines of force, which were originally in the form of closed curves, have become opened out into infinite spirals.

If the south pole S were free to follow the pull exerted by the lines of force, it would tend to move round the currentconductor A in the counter-clockwise direction, since it strives to move along the lines of force which terminate upon it. It might thus appear that a rotation of the pole around the

conductor could be maintained indefinitely by electro-magnetic forces, thus providing us with a veritable 'perpetuum mobile.' It must not be forgotten, however, that we have here assumed the fulfilment of impossible conditions, since a free isolated magnetic pole cannot exist, not to mention that energy must be continuously expended to maintain the magnetic field of the current, as we shall see later.

CHAPTER X

COILS, SOLENOIDS AND ELECTRO-MAGNETS

HAVING now become acquainted with the simple current, its lines of force, and their action on magnetic systems, we proceed to consider some forms of current-conductors which are of great theoretical interest as well as of practical importance. They are all developed from the concentric axis of force system in such a way as to collect together a great number of lines of force through some definite portion of the field of the current. There are two ways of arriving at this result; in the first place, we may collect into a small compass some considerable length of the conductor itself, by winding it in the form of a coil or bobbin ; and in the second place, by the introduction of masses of iron we may greatly increase the permeability of the medium through which the lines of force have to pass. By the latter method we may convert a coiled current-conductor into an electro-magnet, of far greater strength than any natural magnet, or even than any of the compound permanent magnets, which we have been so far able to prepare. Such electro-magnets enable us to study the magnetic properties of various substances much more accurately than would be possible with weaker magnets of steel.

175. Coils. We have already (§ 149) described how the lines of force in the field of a current become collected together into a bundle, when the conductor is bent into the form of a loop. We may further increase the number of lines of force by adding more turns of wire to the loop. We shall cause the more lines of force to thread through the loop, the greater the length of the conducting wire we wind on, i.e. the