needles, the frequency of the vibrations in this case offers a proportional direct measure of the numerical value of the intensity. The method of investigating the distribution of a field least open to objection, but which, at the same time, is very tedious, is that by means of an exploring coil (§§ 2, 4). From the position of maximum induction, the direction of the field may be deduced; and from the quantity of electricity set in motion, the numerical value of the intensity. For the details of this method, which is very seldom used, we may refer to the sections on ballistic methods (§§ 195, 196; see also Chapter XI., § 208). To complete this account we may mention a few instruments in this connection, viz. the declinometer, inclinometer, and local variometer. As these, however, are almost exclusively used for special work on terrestrial magnetic measurements, a detailed description need not be given here, A. Magnetometric Methods § 190. Plan of Gauss's Method.--The accurate measurement of the absolute value of the horizontal component of a uniform field was first made possible by the classical method which Gauss devised for determining the terrestrial field. The value to be measured is deduced from the deflection which the direction of the horizontal component experiences when it is combined with an auxiliary component of known value, also horizontal, but at right angles to it. To determine the direction of the resultant a magnetometer is used. The essential part of this instrument is a well-damped small system of magnets suspended by a vertical fibre, as free from torsion as possible (quartz fibres are best), and which can be turned, the azimuth being read off by a mirror.3 The above-mentioned auxiliary F. Kohlrausch, Leitfaden der prakt. Physik. 7th edition, p. 255 et sqq. Leipzig, 1892. 2 Gauss, Intensitas vis magnet. terrestris ad mensuram absolutam revocata. Werke, vol. 5, p. 89; 2 Reprint. Göttingen, 1877. See also F. Kohlrausch, loc. cit. pp. 230–236. The following are the chief points in reference to the construction of a magnetometer, of which there are many types, simple and complicated, The system of magnets must be small, so that the auxiliary component in the space occupied by it is sufficiently uniform, and its moment of inertia is small; on the other hand, its magnetic moment must be as great as possible. Perhaps the best is a thin aluminium disc on both sides of which small mag PLAN OF GAUSS'S METHOD 293 component is produced by means of the action at a distance of an auxiliary magnet, as will be described in the following paragraphs. The magnetic moment of this magnet may either be known at the outset, or its product into the horizontal intensity, which is the quantity to be measured, may be determined by one of the following methods :- A. Observation of Oscillations.-The auxiliary magnet, whose (unknown) permanent magnetic moment may be M, is suspended horizontally in the place where the horizontal component is to be determined. In this position its period 7, or its frequency 1/7, is observed, from which is obtained [by equation (8), N = § 23] The moment of inertia K of the auxiliary magnet may be either calculated, or else determined experimentally, by some dynamical method. B. Method of Weighing.'-The auxiliary magnet is fastened vertically in the middle of the scale-beam. Let the balance swing in the magnetic meridian, and let the difference in weight corresponding to half a turn about the vertical be 8 M. If D is the length of the scale-beam, g the acceleration of gravity, we obtain C. Bifilar Suspension; Torsion.-In the arrangements previously described, the auxiliary magnet oscillates about a position of equilibrium—that is, the horizontal (A) or the vertical direction (B) in the magnetic meridian. In the former case the horizontal component, and in the latter the vertical component, obviously induces in it a certain magnetisation, which is supernets are fixed. As electromagnet copper dampers are of little use in such feeble systems, air-damping, which can be regulated, is to be preferred. Although torsion may nearly always be neglected when quartz fibres are used, a torsion head should in all cases be fitted. It is, lastly, very convenient, if the mirror can be turned in reference to the system, and if the case, which must not contain the least iron, is arranged for being set upon a horizontal plane, or else suspended against a wall. Toepler, Wied. Ann. vol. 21, p. 158, 1884. posed on the existing permanent magnetisation, and therefore increases the moment M. The correction due to this is, however, only small in observations on terrestrial magnetism; nevertheless, it is always better to place the auxiliary magnet in a nearly east and west position. The transverse magnetisation then produced has scarcely any effect. For instance, the auxiliary magnet may be suspended in this position to a bifilar or torsion arrangement, the directive torque of which is known (note 3, p. 297). The latter is multiplied with the tangent of half the deflection produced by reversing the magnet to the west and east direction. The product gives then directly the value M H. § 191. Observations of Deflection.-After the auxiliary magnet has been taken away, the magnetometer M is put in its place. Its magnetic system then sets itself in the direction of the field, which we have called the magnetic meridian, and which is marked NS in fig. 61. In order now to produce a deflection of the magnetic system, the auxiliary magnet is placed in one of two different positions, its own direction in both cases being either west-easterly or east-westerly. 1. First Principal Position.-Auxiliary magnet at distance D, in W or O at right angles to the meridian. The component 5, induced by it at M is also at right angles to the meridian, and amounts to [equation (5), § 22] in which L is the geometrical (or 'virtual,' § 210) length of the auxiliary magnet. 2. Second Principal Position.-Auxiliary magnet at N or S at distance D2, at right angles to the meridian, as well as the component, due to it, which in this case has the following value [equation (6), § 22]1: The deflections a, or a, due to the second component are obviously given by the following equations: From equations (31) or (3) and (4), neglecting the factors in the brackets, we get, as a first approximation, (5) M Ditan a or M = D tan a2 After having expressed MH and M/H as functions of determinate quantities, not only the intensity 5, but also incidentally the moment M of the auxiliary magnet, may be calculated. If the latter is already known, is obviously obtained from equation (5) without further determinations. These simplified equations, however, only hold for distances which are very great in comparison with the length of the deflecting magnet (§ 22); but as the deflections obtained at such distances are usually too small, the magnet must be brought nearer the magnetometer, so that several members of the series in equation (3,) or (3) have to be brought into account. The length of the magnet is, however, usually eliminated by observing at two successive distances (Kohlrausch, loc. cit., p. 231). Gauss's method is used mostly for determining the absolute value of the earth's horizontal intensity. It may, however, in principle be applied to measuring the horizontal components of An elementary deduction of equations (3,) and (3) is found, for instance, in F. Kohlrausch, loc. cit. pp. 389, 390. uniform fields due to any agent, provided their intensity does not exceed 1 C.G.S. If the field is stronger, it is scarcely possible to produce sufficient deflection by means of magnets of the usual size. B. Electrodynamic Methods § 192. Measurement of a Dynamical Force. It was mentioned in § 1 that the magnetic field can be completely defined by the two chief forms in which it manifests itself the electrodynamic and the inductive; and for the practical methods of measurement based on this, reference was made to the present chapter. F W لس FIG. 62 مسلسل S As regards electrodynamic 1 methods, we will mention the following simple arrangement, due to Lord Kelvin.2 In the field F a metal wire hangs between two pole-pieces, supposed horizontal and at right angles to the plane of fig. 62. By means of two mercury cups C, C a current of known strength 1 (in decamperes) is passed. If then the intensity of the field is , its effective height I, and its direction is such that a force, &, expressed in dynes, is exerted, say, towards the left on the wire, this, from electro-dynamical principles, is given by the equation This force is held in equilibrium by the tension of the threads t, and t, which are fastened to the string pendulums PP or PP. It needs no explanation how that force can be By electrodynamic action all forces are understood which are exerted on conductors conveying currents in the magnetic field, no matter whether the field is due to other conductors, or depends on other sources, such as rigid magnets. 2 A. Gray, Absolute Measurements in Electricity and Magnetism, vol. 2, p. 701, London, 1893. |