Experiment 39.-Magnetise a considerable number of equal and similar needles, and select three that are equally magnetised, testing the equality in the following way. A short piece of straw is fastened with wax to one end of a long cocoon fibre, from which it hangs suspended. The needles are taken in turn, and each one is stuck to the straw, so as to be free to oscillate in a horizontal plane, the number of oscillations in some definite time being observed. If there are three of the needles whose magnetic moments (M) are all equal, the same number will be obtained in the three cases. Now take two of these needles at the same time and afterwards all three, fastening them side by side to the straw, with their north-seeking poles in the same sense, and then allowing them to oscillate. The moment of inertia of the straw and wax may be neglected in comparison with that of the needles. When a single needle is thus replaced by two or three, the moment of inertia of the mass to be set in motion is doubled or trebled, but the magnetic moment of the system is likewise increased in the same proportion, there being two or three poles of strength m to be acted upon by the earth's field whose horizontal component is H. The moment of the forces acting on the vibrating system has thus been doubled or trebled.

Our experiment shows that the frequency of vibration is unaltered; so that it can only depend on the ratio of MH to I.

A more rigorous investigation shows that the frequency of vibration n is given by the expression

77. Determination of field intensities in absolute measure by oscillations. The strength m of the poles of a thin bar magnet may be determined by means of the balance, and the magnetic moment M=ma calculated. If the bar is cylindrical, of length 7 cm. and of inertia can be calculated and the known mass P (§76). n gives the strength of field tion (4).

radius p cm., the moment from these measurements, The frequency of oscillation in accordance with equa

Experiment 40.-Determination of the intensity of the earth's magnetic field in absolute measure.-Let a needle of known

magnetic moment M be fastened at its middle to a fine fibre; it will set itself in the direction of the earth's magnetic lines of Now disturb the needle from its position of rest and allow it to oscillate. From the frequency of oscillation we can calculate T, the resultant intensity of the earth's magnetic field, or total force'; and we shall find in Germany approximately

T= cm. gr. sec.-1;

that is, there is half a line of force to each square centimetre of cross-section, or one line of force to each two square centimetres. The model, fig. 19, in which 35 lines of force pass through an area of 70 sq. cm., thus represents the magnetic field quantitatively, provided we give to it the proper orientation.

78. Determination of magnetic moments in absolute measure by oscillations.-We have calculated the total intensity T of the earth's magnetic field. On multiplying this by the cosine of the angle of dip, we obtain (§ 41) the horizontal component of the field, which is approximately

H = 0·18 cm.-1 gr.' sec.-1

for central Germany [and also for London]. Knowing this value, we can now find conversely (§ 76) the magnetic moments of given bar magnets, and hence the strengths of their poles. We have only to determine the frequency of oscillation when the magnets are allowed to oscillate in a horizontal plane under the influence of the earth's horizontal magnetic component.

Example. Each of the rectangular prismatic bar magnets used in producing the line-of-force figures 10, 11, 15, 16, 17, 18, has a length =7·8 cm., a breadth b=0·4 cm., and a mass P= 22.05 grams. When any one of them is suspended by a fine cocoon fibre, it makes six complete oscillations in a minute under the earth's horizontal component H=018. The oscillation frequency is in this case n=0·1. From the formula (4) we have M=472n2/IH =246. Since the distance between the poles a=l=6.5 cm., and Mma, the strength of either pole of this magnet is m=38 units (cm. gr. sec. 1).

79. Determination of the horizontal component of the earth's magnetic field in absolute measure. (Method of Gauss and Weber.) The method given in § 77 for determining the

H

intensity of the earth's magnetic field has the disadvantage that the moment of the bar magnet employed is involved in the result. But the magnetisation of steel magnets changes considerably with the temperature, and it is therefore important to devise a method by which the magnetic moment is finally eliminated. Such a method is that of Gauss and Weber. It combines the oscillation method, giving the value A of the product MH, with the deflection method of § 73, giving the value B of the quotient M/H; the magnet used for deflecting the declination-needle being that which has been made to oscillate freely in the earth's magnetic field. On division we obtain

the magnetic moment M disappearing from the result.

The expressions A and B only involve such quantities as may be expressed directly in terms of the fundamental units of length, mass, and time. Thus we have the value of the important magnitude H expressed in absolute measure.

99

CHAPTER V

THE REPRESENTATION OF FIELDS OF FORCE

In the last chapter it was shown how magnetic fields could be measured in absolute units. We have next to consider how they may be represented geometrically, the method which we shall follow being founded on a series of important conceptions which were introduced by FARADAY and MAXWELL, and to which we must now turn our attention. The methods for mapping out a field in a plane are of great importance, especially in technical applications, and with these we shall deal first, confining our attention to a few simple but typical problems.

A.-Conceptions of Faraday and Maxwell

In all modern representations of magnetic and electrical phenomena the conceptions of bundles of lines of force, tubes of force and flux of force play a prominent part. But when we here introduce these conceptions, it must be remembered that for the present their significance is purely geometrical. They merely provide us with a construction for mapping out a field in a more convenient way. But they have also a physical significance, for they are an expression of that doctrine of continuity of the magnetic condition which is the foundation of all modern theories, so that they are of the greatest assistance in making clear our physical conceptions.

80. The bundle of lines of force. So far we have only spoken of the separate lines of force; but we must now consider collectively a group of neighbouring lines of force

running all in the same direction. If the direction of the lines of force were everywhere strictly the same, a bundle of such lines would not spread out sideways as it passed through the field. But this disposition of lines of force exemplifies most simply the transmission through the medium of the tensions and pressures which constitute the magnetic field, though the uniformity which we assume is only possible through a limited region of space. If we consider collectively a number of lines of force, we have then a kind of rope or bundle of lines, of finite cross-section. This consists of lines of force which travel through neighbouring parts of space, so that if the cross-section of the bundle is small, we may assume that all the lines composing it lie appreciably in the same direction. Any given bundle of lines of force is, throughout its entire length, made up of the same set of lines. It is only where magnet-poles exist that lines of force can leave a bundle, or new lines enter it.

We have no direct evidence to show that the magnetic lines of force have any definite cross-section, even of very small extent. But in the case of electrical lines of force, which are in many respects analogous, we find that at each atom, or rather at each unit of chemical valency, a certain number of the lines of force terminate, no more and no less. This fact is of the very greatest importance, and well worthy of attention from the point of view of the analogy between electrostatic and magnetic phenomena.

81. Tubes of force. From the bundle of lines of force we pass on to the conception of a tube of force,' which is of the highest importance throughout the whole theory of the subject. Imagine a closed curve drawn anywhere within the magnetic field, and not itself in the direction of the lines of force (fig. 24). In the figure the lines of force l1, l1⁄2, l, which pass through the various points P1, P2, P3 of this curve are prominently shown. These lines taken collectively constitute an infinitely thin-walled tube, which encloses a bundle of lines of force. On the outer wall of the tube lie the points P1, P2, P3, and consequently the closed curve from which we started. The generating lines