the same strength. The axis of the diagram is of course drawn through the positions occupied by the two poles. A unipolar diagram is then constructed for each pole separately, and on combining the two we obtain the diagram of the resultant field. (a) Diagram of two unlike poles.-A north pole 20 whose strength is + = 1.592 unit, and a south 4π pole of smaller strength 5 4π = −0·3979 unit are placed at a determinate distance apart. The flux of force from the first is twenty units, while the flux inwards to the second is five units. Fig. 28, plate I, shows the separate diagram for each pole. The resultant lines of force in the combined field are determined as follows:-Consider any point of intersection of two lines of force, one belonging to each system, and imagine a positive unit pole to be placed there; it will be repelled by N and attracted by S, and this sufficiently indicates from which corner of the parallelogram the diagonal is to be drawn. Through the two conical regions which lie closest to the axis, between the two poles, the flux of force passes directly from one pole to the other, since the field-intensities due to N and s are there in the same general direction, namely, from N towards s. Hence the innermost lines of force are bowed out, curving somewhat away from their original directions, and the same is true of all the remaining lines. In the diagram thus obtained we may observe the following features: The lines of force which proceed from N become bowed out more and more widely, and finally are gathered together at s, where they terminate. On rotating the figure, five coaxal shells are mapped out, through each of which the unit flux of force passes over from N to s. This constitutes the entire flux of force which s is capable of receiving, so that the remaining lines from N do not terminate on s, but pass by it. There are thus 15 unit solenoids starting from N, which do not terminate within the diagram, but carry their flux of force to remoter parts of the field. The entire distribution of the flux of force originating at N is deflected somewhat towards the right by the presence of the unlike pole s. Owing to the tension along the lines. of force and the pressure across them, the two poles will be urged towards one another (attraction). If the two poles are free to move, they will mutually approach one another. [The forces which they exert upon one another will be equal and opposite, in accordance with the third law of motion, and it should be noticed that though all the lines of force attached to and pulling s are pulling N more directly along the axis, there are also a considerable number of lines pulling N away from 8.] Here as before we can calculate at once the field-intensity at any place when we know the breadth b between consecutive lines and the distance r from the axis. (b) Diagram of two like poles.—Let the strength of one pole be as before = 1.592 unit, that of the other pole being 5 4π 20 4π = 0·3979, so that the two poles in this case are of the same name, a north pole N being placed at a determinate distance from a weaker north pole n. At the points of intersection of the lines corresponding to the two unipolar fields, the forces to be compounded are both directed away from their poles. We have a network of intersecting lines precisely like that of fig. 28, but through each of the meshes we have now to draw the other diagonal, so that we obtain the result shown in fig. 29. The lines of force, as they approach one another from the two poles, bend round so as to avoid meeting. There is one line which separates the two systems from one another, and does not pass through either pole. Where this line meets the axis of the diagram, there is an indifferent point, or point of zero magnetic force, J. The 20 units of flux of force proceeding from the pole N all pass outwards into the surrounding space, and so likewise do the 5 units from the pole n, no intermixture of the two taking place. Owing to the pressure across the direction of the lines of force, there will be a force on each pole urging it away from the other; that is, the two poles will repel one another. If we suppose pressure and tension to be interchanged, so that there is pressure along the lines of force, and a tension across them, the diagram may be taken to represent a double star, whose members attract one another gravitationally. 91. Diagrams of homogeneous fields. In a uniform field the lines of force are all straight and parallel, and distributed throughout with the same density. If we intend to apply to this case the same principle of construction that was used for systems having an axis of symmetry, we must choose as axis some line parallel to the direction of the field, and divide the space surrounding it into cylindrical shells, each of such cross-sectional area that the flux of force through it is unity. In the plane of the diagram, the limits of these cylindrical shells appear as straight lines parallel to the axis. The distances r1, 72, 73 . . . of these lines from the axis must satisfy the simple relation When the diagram has been prepared in accordance with these conditions, it may be combined with the diagram for a single pole by drawing the diagonals of the small quadrilateral figures produced by the crossing of the two systems of lines. With the resultant lines we may then use the formula 1 . In fig. 30 a uniform field is represented, having the absolute strength = 0.00771 cm' gr sec1. The radii of the sucH cessive cylindrical surfaces must accordingly be The positive direction of the lines of force is from right to left. We must suppose, therefore, that there are two strong magnet-poles at distances far beyond the limits of the diagram; a north pole to the right and a south pole on the left. Combined with this field is that of a north-seeking pole 10 of strength + m = + = 0.7958 cm. gr. sec.1. If we 4π suppose a positive unit pole placed at a point where lines of the two systems intersect one another, it will be urged towards the left in the homogeneous field, and at the same time repelled by the pole N. This consideration shows us along which diagonals the resultant lines of force are to be drawn. The 10 units of flux of force which proceed from the pole N are so deflected by the influence of the uniform field, that they are ultimately all passing towards the left. The innermost conical surface has become narrowed to the tubular form, 1, 1, and all the remaining surfaces are correspondingly distorted from their original form. The shells numbered seven to ten, which start from the pole towards the right hand of the diagram, are seen to be entirely bent round, and all the ten shells have so completely acquired the direction of the field that further to the left the uniformity is completely restored. The originally cylindrical shells of the uniform field spread out as they approach N, and bend round it as if it were an obstacle in their path, or more strictly, as if the uniform field were a uniform flow of fluid, and the pole N a source from which fluid was emanating. The surfaces more remote from N maintain their course with but little deviation. The present example shows very clearly the insight to be obtained from the construction of such diagrams. If we consider the figure from the point of view of the tensions and pressures accompanying the lines of force, it is clear that N must experience a force urging it through the field towards the left. This is in accordance with the tendency of a north pole to move along the positive direction of the lines of force (compare § 28). The reverse would be the case if we replaced the north pole by a south pole, which behaves as a sink towards the lines of force. 92. Combination of more than two fields.-A resultant field mapped out by this method may in turn be combined with another field, and so on; so that by successive applications of our construction we may obtain the diagram for even a very complicated case. The separate diagrams may be conveniently drawn on tracing paper, so that when two are superposed they are both easily visible. They must of course be drawn to the same scale. A very instructive case is that of the bipolar field of a bar magnet, combined with the uniform field of the earth, but this is left as an exercise for the reader. |