B. Electrodynamic flux-of-induction diagrams

The diagrams of uniaxal magnetic fields described in § 172 may be combined with other diagrams of the same kind, just as the diagrams of two or more magnetic fields were combined to represent the resultant of those fields. Thus we obtain a representation of multiaxal or electrodynamic fields.

204. Combination of the diagrams of two axial fields of force. Let the diagram for each current be prepared as in § 172, and then let the two diagrams be superposed. We thus obtain the distribution of lines of force in the field of two currents, whose directions may be either the same or opposite. The considerations involved in this case are quite. similar to those already mentioned in connection with the production of a bipolar magnetic field from two unipolar fields. They lead to the following rule: In a plane at right angles to the axes of force, construct the line-of-force diagrams for the two currents (the permeability of the surrounding medium is assumed equal to unity). Thus the diagram is divided up into quadrilateral figures, through each of which a diagonal has to be drawn. In the case of currents whose direction is the same, the lines of force of the two systems encircle their axes in the same sense, and the diagonals are to be drawn as in figs. 89 and 90, Plate II. When the currents are in opposite directions, the lines encircle their respective axes in contrary senses, and the diagonals are to be drawn as in figs. 91 and 92.

In the former case the lines of force in the immediate neighbourhood of the axes A, and A, are in the form of separate rings, and as we follow the system of lines further out, we see that they pass through an intermediate form resembling a lemniscate to curves which are more and more nearly circular. This case is represented in the iron-filing diagram, fig. 86. In the second case the lines of force surrounding the two conductors are distinct, and never coalesce, but press upon one another so as to be displaced

further from the middle of the field, the resulting diagram being as shown in fig. 87.

The correctness of the diagonal rule will be evident if we remember a property characteristic of the diagram for a uniaxal field; namely, that at each point the field-intensity is inversely proportional to the distance between the nearest lines of force (compare § 173). In fig. 88, let r', r" be two consecutive lines of force of the system A1, distant a, from one another; r', r2" two consecutive lines of the system A2, whose distance apart is a2. Suppose also that we have to deal with currents in the same sense (both passing for example from above downwards through the plane of the diagram), so that the annular lines of force run in the same cyclic (clockwise) direction. If the lines are drawn at sufficiently small intervals, we may regard the sides of the resulting quadrilateral figures as straight, and these figures themselves as parallelograms. At a point P in the field, let the magnetic forces due to the two separate systems be denoted by H1, H2, their direc

FIG. 88

2

tions being tangential to the circles r', r" at the point P. If the quadrilateral is small, its sides lie very nearly along the lines of action of H, and H2.

1

In accordance with § 173 these magnetic forces are inversely proportional to the distances between the respective pairs of lines of force, so that

Again, from the similarity of the corresponding triangles the ratio between these distances is equal to the ratio between the sides of the parallelogram, that is a,: a=S, S1. Hence th resultant of the two magnetic forces is along the line joining P to the opposite corner of the parallelogram.

Here again the distance between consecutive lines of force in the resultant field is at each point inversely proportional to the resultant field-intensity, as may be seen from considerations quite similar to those which we encountered when combining the fields of two magnets. When a field is to be formed by combining two uniaxal fields, the representation of the lines of force may also be accomplished by means of the following rule:

In each of the two original simple fields let the lines of force be numbered, the innermost circle of radius ro having the index p=0, the next in order the index p=1, and so on; we have then to combine any line of force of the one system-say that whose index is m—with some line of the other system-for example that line whose index is n. In accordance with § 172 rm=em/2i, r„=e" 2. If the two fields are in the same (cyclic) sense, we pass from the point P (fig. 88) where the mth and nth circles of the respective systems intersect, to the point of intersection of the (m+1) circle of the first system and the (n-1) circle of the second. That is to say, we join together those two corners of the parallelogram for which the sum of the indices has the same value m+n. Thus each of the resultant lines of force, in the case of same-way directed uniaxal fields, corresponds to some determinate value m+n of the sum of the indices. To obtain the lines of force of the resultant field when the original fields are oppositely directed, we must pass from a point of intersection, such as (7, 7) to intersections where both indices are higher or both lower, the difference m-n having a constant value along each of the resultant lines of force. The separate lines of the resultant system are in this case distinguished by different values of the number m―n.

205. Properties of biaxal flux-of-force diagrams. From our construction it follows that in each of the diagrams in figs. 89 to 92, Plate II., the flux of force between each pair of consecutive lines is unity. The canals or tubes of force are supposed to be limited above by a second plane parallel to the plane of the diagram and at a distance of 1 cm. from it. Here again, as in the case where we combined the fields of two magnets, the field-intensity at any place may be immediately deduced from the distance between two consecutive lines at that place. If the deca-ampère is taken as the unit for the measurement of current-strength, and the centimetre as the unit of length, the field-intensity is equal to the reciprocal of the number of centimetres in the distance measured.

(a) Diagram for same-way-directed parallel currents.— Figs. 89 and 90. The latter of these figures represents a different but similar system of lines of force on a larger scale. The arrows by the conductors A, A, are intended to

indicate that the constituent fields follow the same cyclic direction. The special features which characterise the disposition of the lines of force are much more clearly shown. in these diagrams than in the direct iron-filing figure (fig. 86), which is not altogether easy to follow, especially in its more central part. It will be seen how the inner lines of force are elongated to the form of ovals, whose narrower ends are directed towards one another from either side of the field, and approach one another more and more nearly as we pass to the larger ovals. The characteristic features. of the diagram are the indentation of the lines which have become fused together, and the way in which these lines continually approximate to the circular form as the distance from the conductors increases.

m+n

Along each line, in accordance with the investigation in the last paragraph, TXT,e constant. Lines whose equations are of this form are called lemniscates. It follows from our construction, and from the equation obtained, that for all points on any one of these lines the product of the distances from the two fixed points A, and A, has the same value.

(b) Diagram for oppositely-directed parallel currents.Fig. 91 shows the course of the lines of force through a somewhat wider range than fig. 92, which represents the central portion of the field on a larger scale. As in the component fields, the lines in the resultant field are all circles, their centres, however, being displaced further and further out along the continuations of the line A,A,. The central line of force, which is straight, and divides the field symmetrically, may be regarded as belonging to the circumference of a circle of infinite radius. Every other line cuts. the line A,A, in two points, one nearer to the conductor, and lying between A, and A,, the other more remote from the conductor and lying in the outer part of the field.

In the middle, between the two wires, the lines of force are much more closely packed together. In the figure, for example, the field-intensity is increased from 1/0-26 (reciprocal of the distance in cm. between two lines of force

in the simple field) to 1/0·13 (corresponding quantity for the resultant field); that is, in the ratio 2: 1.

Along each of these lines we have

The system of lines represented by equations of this form being a family of circles. The straight line A, A, is divided har monically by each of the circles.

It will be well also to construct line-of-force diagrams for currents of unequal strength, the process following the same rules as in the case of equal currents. MAXWELL, to whom we owe these and other similar constructions, has also mapped the lines of force for a grating consisting of many parallel wires.

206. Diagram for a circular current with a central magnetic needle. The diagrams of multiaxal fields, like those of uniaxal fields, may be combined with the diagrams of the fields of magnets, thus furnishing an instructive representation in cases which would be difficult to follow analytically. As an example we shall take an arrangement which is of especial practical importance; a magnetic needle being placed in the field between two conductors, and being free to turn in a plane perpendicular to those conductors. We shall suppose the conductors to be rectilinear and parallel, and traversed by currents in opposite senses. The resulting diagram does not differ essentially from that which would be obtained if the same current were made to pass round a circular conductor, with the magnetic needle at its centre. This latter arrangement is the one adopted in the tangent galvanometer.

Suppose that at A', fig. 93, a current is flowing upwards through the plane of the diagram, the conductor bending across the paper from left to right, and passing down again through A. Round A' and A draw the concentric circles representing the lines of force; then draw diagonal lines joining those points of intersection for which the