170. Energy of the field of a current loop.-In § 150 we saw that the space within a current loop, where the lines of force are most closely crowded together, is the seat of a certain quantity of energy. To evaluate this we may proceed as in § 119, where we found a measure for the energy existing throughout a magnetic field. We shall here confine our attention to the calculation of an expression for the variation of the magnetic energy around a given axis of force, i.e. for the energy differential dE.

In the first place it is evident that the energy of the field depends upon the current-strength i. For, when all other circumstances are given, a current-strength of 1, 2, 3.... units produces 1, 2, 3 .... times as many lines of force per unit area at each part of the field. If, then, i is the strength of the current around the given circuit, then i will furnish a measure of the extent to which the surrounding region is filled with lines of force (compare equation (30)). If more lines of force are introduced, the current-strength remaining the same, work will have to be done, as in introducing a unit pole into a given field. This work will be greater in proportion to the closeness with which the field is already filled with lines of force-that is, it will be proportional to i. It will also be the more considerable the greater the number of new lines of force introduced into the field. Let N denote the number of lines of force originally embraced by the conductor, and dN the increment of this number, so that the work expended will be proportional to dN. If no work is lost in the process, all the work expended in increasing the number of lines of force is transformed into energy of the magnetic field; and it may be shown that

dE=idN

(33)

What is here stated concerning a single current loop holds good also for any closed circuit in which a current is flowing; the increment of the energy of the surrounding field due to the introduction of dN new lines of force is always idN.

171. The field of a current as a monocyclic mechanism.— Since the field-intensity at each point of the field of a current is proportional to the current-strength i, while in the kinetic theory is proportional to the cyclic velocity, it follows that the whole distribution of cyclic motions in the field of a current has the character of a monocycle (§ 143) whose cyclic velocity q is equal to the currentstrength i.

A large number of very suggestive models have been constructed (by O. J. LODGE and others) with the object of illustrating the motion of the smallest parts of the field of a current. For the most part, however, these models do not express the falling of angular velocity with increase of distance from the path of the current. Fig. 67 represents a section through the upper part of the field of a straight conductor S, the relative values of the constituent angular velocities being secured by a monocyclic connection of the moving parts (compare §§ 136–143).

BB, fig. 67, is a board 50 cm. long and 43 cm. broad, and along one of its longer sides a wooden lath 3 cm. broad is fastened to represent the current-conductor S. The further side

K

B

FIG. 67

of this beam (at the back of the figure) is accordingly covered with a strip of red paper of the same width, with red arrows attached to it. The remainder of the surface of the board,

measuring 40 cm. x 50 cm., is pierced with four rows of five holes each, at distances of 10 cm. apart. Immediately above the five vertical series of these holes, supports t are fixed to the edge of the board BB, and serve as an attachment for five wooden laths whose lower ends are to be fastened to S, and which are pierced with holes corresponding to those in the board. Between the board and the laths are twenty grooved wooden pulleys, freely movable about axles which are in the form of short rods passing through the holes already mentioned. The pulleys in the lowest row r1 are 2 cm. in diameter (the measurement corresponding to the middle of the depth of the groove); in the second row r2, 4 cm.; in the third row r3, 6 cm.; in the fourth row r4, 8 cm. The laths having been fastened to S and to the supports t, rubber connecting bands are passed round the grooved pulleys, so that each pulley is coupled to all its neighbours. The axles of the pulleys project a little in front, and attached to them are paper discs on which arrows are marked (as shown in the figure). To the roller most to the left in the lowest row, a crank K is attached, and serves to set the system in motion. If at some given instant all the arrows are pointing vertically upward (dotted positions), and if K is turned through one complete revolution in the sense of the current S, the pulleys in the row r1 will have turned through of a revolution, in the row r, through, in the row r1⁄2 through, as indicated by the arrows in the figure. The angular velocity (and consequently the magnetic force which it represents) is thus inversely proportional to the distance from the current conductor. At r, the number of lines of force per unit area is four times as great as at r2, and this is indicated by pegs stuck into the board, which is divided up into squares.

The motion in the entire field is determined by a single quantity, the angular velocity of the crank K (cyclic velocity current-strength).

C.-Electro-magnetic flux-of-force diagrams

Just as in Chapter V. we represented graphically the field of a magnet-pole, and constructed bipolar fields by combining two unipolar diagrams, so now we will consider how to represent the fields of rectilinear portions of a current. To this end we must agree upon a certain

principle of diagrammatic representation, which, like that employed for the purely magnetic case, involves something of a conventional nature. On combining these currentfields with magnetic diagrams, we obtain representations of the action of electro-magnetic forces.

172. Graphic representation of the system of lines of force due to a rectilinear current. The plane of the diagram is taken perpendicular to the current-conductor, for it is only then that the lines of force appear in their natural circular form. As before, we assume that the medium surrounding the conductor has everywhere the permeability μ=1. Thus the flux of induction, with the observation and representation of which we are really concerned, coincides with the flux of force, the lines of induction being everywhere identical with the lines of force. Moreover, out of the infinitely numerous lines of force, we shall only choose such as form the boundaries of regions through which the flux of force has the same value, namely unity, these bounding lines of the unit solenoids (§ 84) being alone represented in our diagrams. In our present inquiry we have an advantage which was lacking when we were preparing diagrams of the fields of permanent magnets, for we are able to adopt very simple conventions. In the former case, figures had to be constructed whose rotation about an axis divided the field into regions of equal flux of force, while in the case of coaxal fields we may proceed according to the following rule: Imagine a second plane to be drawn parallel to the plane of the diagram, and at a distance of 1 cm. from it. Then through every line of force represented in the diagram construct a right circular cylinder, whose axis, of course, coincides with that of the conductor. Any consecutive two of these cylinders, in combination with the two planes, enclose a canal of rectangular section, encircling the currentconductor. The lines of force represented are to be so chosen that through each such canal the flux of force is the same, and equal to one absolute unit. Since the fieldintensity is inversely proportional to the distance from the conductor, the radius of the pth circular line of force must

bere, when that of the innermost line is 1 cm. ; e being the base of the natural logarithms (=2-71828 . . .) and i the constant of the coaxal system of lines of force (the socalled current-strength).

Proof of the formula.-The field-intensity at the distance r is = SO that if we consider a narrow plane strip of surface

2i

perpendicular to the direction of the lines, of (radial) breadth = dr and of height (measured parallel to the conductor) = 1 cm., the number of lines of force intersecting it will be §. dr. 1, that dr is the flux of force through this element of surface will be 2i Hence we have for the flux of force through a plane area, whose plane passes through the axis of the conductor (and therefore intersects the lines of force perpendicularly), whose height is 1 cm. (compare fig. 68) and which lies between the radii r。, ~1,

If corresponding to a given value of r。 we wish to find the value of r, which makes the flux of force equal to unity, we shall have to determine r1 from the equation

For the next ring r2, between which and r1 (still with height = 1 cm.) lies a surface. through which the flux of force is unity, we have

1cm

FIG. 68