through its position of rest, and therefore its potential energy is zero, we get

Now if the amplitude, 0, of the vibrations is small, the chord AS may be taken as equal to the arc (AS). Then

428. Measurement of the Strength of a Magnetic Field.—We have seen in the last section that if a magnet, of which the moment is M and the moment of inertia is K, vibrates in a magnetic field of strength H, the periodic time 7 of the vibrations is given by

Hence if we measure T, and know K and M, we can calculate H. The moment of inertia K can either be calculated, if the magnet is of a simple and regular shape, or it can be determined experimentally. Hence we have only M and H to determine, so that if by any other experiment we MA

B

FIG. 415.

S

PN

can get a second relation between M and H, say their ratio, we could calculate both of them.

Now we have obtained in § 426 expressions for the couple caused by the action of one magnet on another when they are placed in certain relative positions. Suppose now a magnetised needle ns (Fig. 415) is suspended by a fine thread in the given magnetic field, then it will set

→

itself parallel to the direction, BA, of the field. If now we place the magnet, of which we have observed the period of vibration, in the position NS, it will exert a couple on the needle, which, if the distance OP is great 2MM' compared to the sizes of the magnets, is equal to 2 D3

and hence the needle will be turned into some such position as that shown in the figure, and will finally come to rest when the deflecting couple, due to NS, is

equal to the couple, tending to bring it back into the direction BA, due to the field.

If is the angle which the axis of the needle makes with the lines of force of the field when it comes to rest under the combined influence of the magnet NS and of the field, the couple acting in the clockwise direc2MM' tion due to the magnet is cos 0, while the couple acting in the D3 opposite direction due to the field is M'H sin 0. When there is equilibrium these must be equal, and hence

If then we measure the distance between the centres of the magnet and needle and the deflection, we can calculate the ratio

M
H

But we

have already seen that the vibration experiment gives us the value of the product MH, and hence by simple algebra the values of the two quantities M and H can be calculated. Therefore by measuring the periodic time of a magnet of known moment of inertia, when suspended in a given magnetic field, and then determining the angle through which a needle, suspended in the same field, is deflected by this magnet when placed at a known distance, we can obtain both the strength of the field and the magnetic moment of the magnet. Of course, when performing the deflection experiment, the magnet NS might be placed in the "B position," in

CHAPTER II

TERRESTRIAL MAGNETISM

429. The Magnetic Elements.-The most important magnetic field with which we have to do is that due to the magnetic state of the earth. In order to be able to state the condition of the magnetic field of the earth, or as we may say for short the earth's field, at any point we require to know two things, (1) the direction of the lines of force of the field, and (2) the strength of the field. That is, we want the direction in which a single unit north pole would tend to move under the influence of the field, and also the force which would act upon it. We have hitherto supposed that the directions of the lines of force of the magnetic fields with which we have been dealing were horizontal, so that a magnetised needle, which was suspended or pivoted, so as to turn about a vertical axis, was able to set itself parallel to the lines of force of the field. If a long thin unmagnetised bar of steel is suspended by a fine thread so that it hangs in a horizontal position, and is then magnetised, it will set itself in an approximately north and south position, but will no longer be horizontal. In this part of the globe the north end will dip downwards. This indicates that in these parts the lines of force of the earth's field are not horizontal, but are inclined downwards.

For most purposes it is convenient to suppose the earth's field resolved into two components, one of which is horizontal and the other vertical. Since a magnetic field is of the nature of a force, having magnitude or strength and direction, the field may be resolved into two component fields, just as a force in § 67 is resolved into two component forces.

In order to define each of these components, we require of course to know its direction and its strength. In the case of the horizontal component its strength is called the horizontal force, and is generally indicated by the letter H. Since by supposition this component is horizontal, in order to define its direction we only require to know the angle which it makes with some fixed direction. The fixed direction chosen is the geographical meridian, and the angle which the horizontal force makes with the geographical meridian is called the declination, or sometimes the variation.

The vertical component of the earth's field is called the vertical force, and is generally indicated by the letter '; its direction is along the vertical, i.e. the radius of the earth at the point considered.

Hence the

H

A

The actual strength of the earth's field, which is of course the resultant of H and V, is called the total force. The angle between the lines of force of the earth's field and the horizontal is called the dip. dip is also the angle between the direction of the horizontal component and that of the total force or actual field. The three magnetic forces, the total force and its two components, H and V, must of course lie in the same vertical plane, the angle which this plane makes with a vertical plane containing the place considered and the axis about which the earth turns, that is, the meridian plane, is equal to the declination.

If the plane of the paper is taken as the vertical plane in which the total force and its components

→

B

FIG. 416.

lie, and OA, OB, and oc (Fig. 416) represent in magnitude and direction the horizontal and vertical components and the total force, then the angle AOC or will be the dip. Hence if the total

force, OC, is called I, we have from the triangle AOC

H
= cos 0.
I

Also from the triangle BOC, since the angle BOC is 90° – 0,

These three expressions permit of our obtaining V and I if we know the horizontal component, H, and the dip, 0, or if we know V and H we can obtain / and 0.

N

Hence it is evident that if we know the declination, the horizontal component, and the dip, we can deduce the direction and strength of the earth's field. Since it is generally most convenient to measure these three quantities, they are called the magnetic elements.

W

H

It is sometimes convenient to be able to express the direction and magnitude of the earth's field by three quantities which are all of the same nature, and not, as we have done above, by means of a force and two angles. Suppose we resolve the horizontal component along a line which points to the true or geographical north, and along a line true west. If X is the northerly component and Y the

FIG. 417.

westerly component, then it is at once evident, from Fig. 417, that if 8 is the declination,

X=H cos 8,

Hence, if we know X and Y, we can calculate H and S. Thus the values of the three components of the force, X, Y, and V, are sufficient to completely define its value both in magnitude and direction.

Before proceeding to consider the general form of the earth's field as deduced from a study of the measurements which have been made of the magnetic elements at different parts of the earth's surface, it will be useful to briefly consider the methods employed to measure the magnetic elements at any given place.

430. Measurement of the Declination.-The declination is the angle between the geographical meridian and the direction of the horizontal component. Thus, since a magnet when suspended by a fine thread, so as to turn freely about a vertical axis, will set itself parallel to the direction of the lines of force of the horizontal component, the declination can be obtained by measuring the angle between the axis of such a suspended magnet and the meridian.

The practical difficulty in performing the experiment lies in the fact that the magnetic axis of a magnet does not necessarily coincide with its geometrical axis.

The magnet usually employed consists of a hollow steel cylinder, A (Fig. 418), which is fixed in a brass collar to which are attached two pegs, B and C, either of which fits into a clip attached to the end of a fine

the length of the cylinder, so that the rays of light proceeding from any point in the scale s leave the lens as a parallel pencil. The line joining the central division of the scale and the optical centre (§ 348) of the lens is taken as the geometrical axis of the magnet.

If AB (I., Fig. 419) is the plan of a magnetic needle suspended by a fine thread attached at C, and of which the magnetic axis is ns, then it will set itself with the magnetic axis in the magnetic meridian NS. In