the angle through which it turns, then clearly ′ = 90—4. The moment of the earth's force on the magnet is M'H sin o, that of the couple due to the other magnet is

Next place the first magnet with its north pole west and its centre exactly to the south of the second; the north pole of the second will move to the east through an angle 4, say, and in this case we shall have 0' = = 4.

The moment of the couple due to the earth will be as before M'H sin; that due to the first magnet is

We shall see shortly how these formulæ may be used to measure M and H.

On the Measurement of Magnetic Force.

The theoretical magnets we have been considering are all supposed to be, in strictness, simply solenoidal rods without thickness, mere mathematical lines in fact.

The formulæ may be applied as a first approximation, however, to actual magnets, and we shall use them in the experiments to be described.

There remains, however, for consideration the theory of an experiment which will enable us to compare the magnetic moments of a magnet of any form under different

conditions of magnetisation, or of two magnets of known form, or to compare the strengths of two approximately uniform magnetic fields, or, finally, in conjunction with the formulæ already obtained, to measure the moment of the magnet and the strength of the field in which it is.

We have seen (p. 166) that, if a body, whose moment of inertia about a given axis is K, be capable of vibrating about that axis, and if the force which acts on the body after it has been turned through an angle from its position of equilibrium, tending to bring it back to that position, be μ 0, then the body will oscillate isochronously about this position; also if the time of a complete oscillation be T, then T is given by the formula

T= 2π

μ

We shall apply this formula to the case of a magnet. We have seen already that, if a magnet be free to oscillate about a vertical axis through its centre of gravity, it will take up a position of equilibrium with its magnetic axis in the magnetic meridian. The force which keeps it in. the meridian arises from the horizontal component of the earth's magnetic force; and if the magnet be disturbed. from this position through an angle 0, the moment of the couple tending to bring it back is м H sin 0, м being the magnetic moment. Moreover, if @ be the circular measure of a small angle, we know that the difference between 0 and sin depends on 03 and may safely be neglected; we may put, therefore, with very high accuracy, if the magnet be made to oscillate only through a small angle, the value @ for sin in the above expression for the moment of the couple acting on the magnet, which thus becomes MH0; so that, if K be the moment of inertia of the magnet about the vertical axis, the time of a small oscillation T is given by the equation

T = 27

K

T can be observed experimentally, and hence we get an equation to find м H, viz.

If we have in addition a relation which gives the ratio of M/H from the two we can find м and H. Such a relation has been obtained above (p. 450), and with the notation there employed we have

There are some substances in which the action of magnetic forces produces a magnetic state which lasts only as long as the magnetic forces are acting. Such substances, of which iron is the most marked example, become themselves temporary magnets when placed in a magnetic field. They are said to be magnetised by induction. They lose nearly all their magnetic property when the magnetising forces cease to act. In most specimens of iron a certain amount of this remains as permanent magnetism after the cessation of the magnetising forces. In very soft iron the amount is very small; in steel, on the other hand, the greater portion remains permanently. We shall call such substances magnetic.

The attraction between a magnet and a magnetic substance is due to this induction.

Wherever a line of force from a magnet enters a magnetic substance it produces by its action a south pole. Where it leaves the substance it produces a north pole. Thus, if a magnetic body be brought near a north pole, those portions of the surface of the body which are turned towards the pole become endued generally with south

polar properties; those parts of the surface which are away from the north pole acquire north polar properties. An attraction is set up between the north pole of the magnet and the south polar side of the induced magnet, a repulsion of weaker amount between the north pole and the north polar side, so that on the whole the magnetic body is attracted to the north pole. This may even be the case sometimes when the magnetic body is itself a somewhat weak magnet, with its north pole turned to the given north pole. These two north poles would naturally repel each other; but, under the circumstances, the given pole will induce south polar properties in the north end of the weak magnet, and this south polarity may be greater than the original north polarity of the magnet, so that the two, the given north pole and the north end of the given magnet, may actually attract each other.

69. Experiments with Magnets.

(a) To magnetise a Steel Bar.

We shall suppose the magnet to be a piece of steel bar about 10 cm. in length and o'5 cm. in diameter, which has been tempered to a straw colour. The section of the bar should be either circular or rectangular.

We proceed first to shew how to determine if the bar be already a magnet. We may employ either of two methods. Take another delicately-suspended magnet-a well-made compass needle will do-but if great delicacy be required, a very small light magnet suspended by a silk fibre. A small mirror is attached to the magnet, and a beam of light, which is allowed to fall on it, is reflected on to a screen; the motions of the magnet are indicated by those of the spot of light on the screen, as in the Thomson reflecting galvanometer. Bring the bar into the neighbourhood of the suspended magnet, placing it with its axis east and west and its length directed towards the centre of the magnet, at a distance of about 25 cm. away. Then, if N s be the suspended

magnet, N's' the bar, and if N' be a north

end, s' a south end, N s will be deflected as in fig. 50 (1). On reversing

deflected in the opposite

direction. If the action N/between the two be too small to produce a visible permanent deflexion of the magnet N S, yet, by continually reversing the bar at intervals equal to the time of oscillation of the needle, the effects may be magnified, and a swing of

considerable amplitude given to the latter. The swing can be gradually destroyed by presenting the reverse poles in a similar way.

F:G. 51.

This is a most delicate method of detecting the magnetism of a bar, and there are few pieces of steel which will not shew some traces of magnetic action when treated thus. The following is the second method. Twist a piece of copper wire to form a stirrup (fig. 51) in which the magnet can be hung, and suspend it under a bell-jar by a silk fibre, which may either pass through a hole at the top of the jar and be secured above, or be fixed to the jar with wax or cement. If the magnet to be used be rectangular in section, the stirrup should be made so that one pair of faces may be horizontal, the other vertical when swinging. For very delicate experiments this fibre must be freed from torsion. To do this take a bar of brass, or other non-magnetic material, of the saine weight as the magnet, and hang it in