CHAPTER X EXPERIMENTAL DETERMINATION OF FIELD-INTENSITY § 188. General Introduction.-The complete determination of a magnetic field embraces on the one hand a topographical survey of its distribution (Chapter III.) within the region in question, and on the other hand the numerical evaluation of the intensity at a given place, which then also gives that in other parts of the field; the latter problem is by far the more important, as, in general, we are dealing with fields which are either quite uniformly distributed throughout a certain region, or may be approximately considered so. In the present chapter we shall review the various methods of measurement; in so doing we shall give only the chief features of the older, more or less classical methods, which may be assumed to be known,' while several newer methods, less generally understood, will be discussed more in detail. In so far as electromagnetic' fields are concerned that is, such as are exclusively produced by electrical currents-it will of course, if possible, be simpler to calculate the distribution, and the absolute value of the field from the dimensions of the conductors, and the easily measurable current; the corresponding formulæ for the cases of most frequent occurrence are given in §§ 5, 6. In many cases, however, the calculation meets with insuperable analytical difficulties; any such field is, however, subject to the laws of distribution in §§ 44 and 45. The same holds also for fields due to rigid magnets, the distribution of which was discussed in §§ 47-19. When the calculation is seen to be impracticable, as is See, among others, F. Kohlrausch, Leitfaden der prakt. Physik, 7th edition, Leipzig, 1892; Heydweiller, Hülfsbuch für elektr. Messung., $$ 62-77, Leipzig, 1892;; Mascart and Joubert, Electricity and Magnetism, vol. 2, §§ 1139-1188, London, 1888. U almost always the case in considering ferromagnetic bodies, recourse must be had to experimental methods; which of these is to be preferred depends on the special circumstances of the case; that is on the accessibility, the extent, and the order of magnitude of the field; and also whether its direction is horizontal or not; whether an absolute or a relative, an approximate, or an exact measurement is intended. The elements necessary for forming a judgment will appear in what follows. § 189. Distribution of Magnetic Fields.-The distribution of a magnetic field in two dimensions may, as already stated (§ 4), be represented by means of iron filings. The magnetic diagram, as it is called, obtained in this way not only gives a clear representation of the course of the lines in the plane chosen, but it also gives an approximate idea as to the relative value of the intensity in various places; as the lines are closer, the higher is that value. Lindeck' has represented a number of such figures, one of which is reproduced in fig. 60. This is the case of a field in the meridional plane of a circular conductor conveying a current, the position of which appears from the blank places at the top and bottom. The lines of intensity close to the conductors form circles around them, while the field in the middle of the circular conductor is pretty uniform. This figure agrees with the distribution theoretically calculated for this case (§ 6, C). A method, which is more accurate, though, at the same time, more troublesome, than the use of iron filings, and which gives the lines of intensity in three instead of in two dimensions, consists in following them with a small movable magnetic needle, proceeding from point to point always in its own direction. A small cross-piece EW fixed at right angles to the needle gives, according to Searle, the direction of the corresponding equipotential surface. The value of the intensity may often be deduced from the frequency of the vibrations, since the square of the frequency is proportional to that value [see equation (I) in the next article]. When using permanent magnetic needles the superposed induced. magnetisation constitutes 1 Lindeck, Zeitschrift für Instrumenten Kunde, vol. 9, p. 352, 1889. 2 C. Hering, Electrical Engineer, vol. 6, p. 292, 1887. DISTRIBUTION OF MAGNETIC FIELDS 291 a considerable source of error, especially in intense fields. Hence, in many cases permanent magnetism is dispensed with, and a soft iron needle is used. In so far as its magnetisation, in the first stage, is approximately proportional to the intensity of the field, which, from § 33, is nearly the case for not too long 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 1 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 N = 1/7, is observed, from which is obtained [by equation (8), § 23] (1) M H = 42 K = 472 N2 K 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 (4) 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. |