30. Exploration of the course of the lines of force.-Short magnets with one degree of freedom, such as needles freely turning on points, will serve to determine the course of the lines of force in a plane.

Experiment 23.-Exploration of the lines of force of a large bar magnet. Let a strongly magnetised steel bar or a compound bar magnet be laid on a roll of drawing paper spread out horizontally, and then bring into various parts of the field a small magnetic needle, mounted on a low stand, so that its height above the paper is about equal to half the thickness of the bar magnet. The needle will everywhere set itself in a determinate direction, which may be recorded by marking upon the paper the points vertically beneath the two ends of the needle, joining them by a line, and affixing an arrow-head to show in which sense (s, n) the line is to be taken. If this operation is performed at many points the system of arrows will give a very clear idea of the course of the lines of force in the field. The results may afterwards be verified by means of iron filings.

This method of recording isolated positions of the needle is in some measure incomplete, since it does not give directly the actual course of any single line. But if to the upper end of a lead pencil we attach a metal point, upon which a magnetic needle can turn freely in a horizontal plane, the course of a line of force may be followed through the field by so guiding the pencil that the projection of the needle on the paper is always tangential to the last drawn element of the curve.

A very convenient and simple arrangement for this purpose is shown in fig. 7. A round brass foot A, whose under surface is plane, has a hole in the middle into which is inserted a brass tube B, five centimetres long, and provided at its upper end with a needle-point c. Inside the tube is a piece of lead pencil D, which is pressed downwards by the spiral spring E, but is kept from falling out by a constriction of the lower end of the tube, only the point F being able to protrude. When the pencil is to be renewed, the top c of the tube can be removed. On the pointed end of c, the small bar-magnet G, with poles marked N and S, can turn freely in a horizontal plane. When not in use, the

instrument stands on a cork base, which has a small depression to receive the point F.

с

If we desire to map out the lines of force of a magnet (for example, of a horse-shoe magnet), we place it over a sheet of paper, supporting it on a wooden block so as to raise its 'median plane' to the same height as the magnetic needle G. By looking vertically down upon G, it is easy so to move the foot A, that the tangent to its path is always parallel to G. We have thus a very fairly accurate graphic construction for the course of a line of force, so far as it corresponds to the median plane of the magnet under examination.

One of the most comprehensive collections of such systems of curves, for magnets of the most various forms (bar-, horse-shoeand ring-magnets) was brought out in 1844, by J. E. HERGER, with a preface by ERMAN. It was published by E. PÖNICKE and Son, of Leipzig, and contains thirty-seven large diagrams of lines of force, in thirty-one folio plates.

A

A

F

FIG. 7

Specification of the magnetic force as a directed quantity.— Two data are necessary for the complete specification of the effect produced upon a movable pole when placed in a magnetic field.

(1) A determinate direction.-This will be at each point coincident with that of the lines of force passing through the point, and is to be determined by drawing the corresponding tangent to the line of force, or more simply by taking an element of the curve which includes the point itself and is short enough to be regarded as straight. The complete specification of a direction includes also a determinate sense. Lines of force at a given position in space may lie along the same curve and yet differ from one another in direction, a north-seeking pole being urged one way along the one line of force, and the contrary way along the other. It is therefore essential to know the sense in which a line of force proceeds, as well as the form of the curve which it follows.

(2) A determinate strength.--We have already seen that the magnetic effects in the more remote parts of the field are weaker than those in the neighbourhood of the poles. We shall learn later from quantitative experiments that the same is true of the mechanical force exerted upon a movable pole. Thus the effects observed at different places in the same field differ also in intensity.

Quantities which have direction (including sense) and magnitude are called 'vectors'; they may be represented geometrically by a straight line of definite length drawn from a point in a definite direction. The magnetic force is a vector, or directed quantity.

The simplest vector quantity is the displacement of a point from one position to another along a straight line of determinate length drawn in a determinate direction; hence the name (vehere to carry). Velocity and acceleration are also vectors. The part played by vector quantities in physics is very important. We shall find that rotations about axes are also included in the same category.

32. Faraday's representation of the distribution of magnetic force by means of the lines of force.-The lines of force furnish us with sufficient data for the complete determination of the magnetic force at each point of the field. The course followed by the lines shows immediately the direction of the force; while by introducing a movable north pole into the field, we can discover which is the positive sense along any line of force. It would seem, perhaps, more difficult to express the magnitude of the magnetic force by means of the distribution of lines. But the density or closeness with which these lines are packed together in any given part of the field may be made to furnish a measure of the intensity of the corresponding force. This density is most simply measured by the number of lines of force intersecting some area whose plane is at right angles to their course, for example, the number of lines per square centimetre; this last-named quantity being called the field intensity at the corresponding point.

It will be shown later on that this mode of representation

leads to an absolute and not merely to a relative measure of the magnetic force. As the principle of the method is somewhat unfamiliar, we shall here introduce a simple example which may help to make it clearer.

Experiment 24.-Examination of the distribution of the lines of force in three dimensions.-Cut down upon one of the gelatine blocks described in § 25, and count the lines of force which intersect the different sections.

In the neighbourhood of the pole of the magnet, a square centimetre of surface, taken perpendicularly to the direction of the lines of force, will be marked with many points where these lines intersect it (considerable strength of field); while in places more remote from the pole the intersections per square centimetre will be few (small strength of field). The numbers thus obtained are not directly comparable, because the formation of the chains of iron filings is influenced by numerous accidental circumstances; the method will only become exact when we suppose the lines of force to be built up and distributed according to a definite law. The example is merely intended to illustrate the possibility of using the method to show graphically the magnitude of the magnetic force.

B.-Mechanical action between two movable magnets.

We shall next consider the case where two magnets, subject to one another's influence, are both movable. One of them (M,) we shall suppose to have one degree of freedom, the constraints imposed upon it being such that it can only move to and fro in the direction of its axis, or at right angles to it; or we shall suppose it suspended by a thread or mounted on a needle-point, so that it can turn freely in a horizontal plane. In its position of equilibrium in the last-mentioned case, it will point north and south. The second magnet is to be entirely free; it is to be held in the hand, so that it can be approached to the first magnet from every possible side.

We shall now dispense with the condition that the distance from pole to pole of a magnet is to be very great; on the contrary we shall suppose both pairs of poles to take part in the effects observed, and shall seek to determine how

the lines of force of the one magnet are influenced by those of the other.

33. Reciprocal action of the fields of two neighbouring coaxal bar magnets. To render a bar magnet freely movable in the direction of its own axis, it is placed in a little boat made of sheet-brass (fig. 8) and suspended by threads from the ceiling.

A rectangular piece of sheet brass about 12 cm. long is bent round the bar magnet m1, which is about 25 cm. in length and of circular section. At the four corners S-shaped hooks are inserted into holes drilled in the brass. The hooks are fastened to long thin threads, which are passed in pairs over other hooks fixed in the ceiling. Beneath one end of the bar magnet is a block of wood K, which carries a pointer Z.

Experiment 25.-Let the pole n of the magnet m, be brought axially towards the pole n of m1; m, recedes from

m2, as indicated in the figure by the arrows.

If the pole s of ma M2 is approached to the pole s of m1, a similar repulsion is observed. The receding of the magnet m, causes it to rise somewhat above its original height, so that a certain amount of work is necessarily done against gravity.

The field due to a magnet-pole is in some degree impenetrable by the like pole of another magnet.

When we stretch out our hand, and suddenly feel a resistance to its movement, we say that there is something there.'