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and applied by Coulomb-the torsion method, and that of oscillations. The first is applied especially to bars; the second extends also to small needles, but loses much of its precision when the magnets are of very small dimensions.
Divers physicists, among others Wiedemann*, have calculated the magnetic moments of a magnetized needle from the deviation produced by it upon a very short magnetized needle furnished with a mirror and movable about a vertical axis. This process is peculiarly suitable when the magnetic moments of one and the same needle variously magnetized are to be compared, provided that the magnetic moments to be compared are not too feeble.
In this case the needle A, whose magnetic moment we wish to compare, is fixed in a horizontal groove, the direction of which is sensibly perpendicular to the magnetic meridian. The distance from the centre of the needle A to the centre of needle B should be sufficiently great relatively to the length of A to avoid a notable alteration of the results by a slight displacement of the poles of A in conscquence of a more or less considerable magnetization. The azimuth of the line of the centres of A and B can be chosen arbitrarily; and if the needle A receive the same position in all the experiments, the deviations observed will be proportional to the moment it possesses. If the needle is regular, it may be returned to its groove with its ends reversed, and the mean of the deviations be taken.
I have frequently employed this process of measurement in the study of magnetization by currents; but mostly, as in the study of the breaking of the current, I have had to measure the ratio of the magnetic moments of needles of very small dimensions, and different in length and diameter. I have then made use of a special arrangement, which I will describe in detail,
Principle of the method.-Suppose a rigid support, movable about a vertical axis. Fix upon this support (1st) a horizontal needle the magnetic moment M of which is known, (2nd) the needle whose magnetic moment a we wish to determine. The two needles are placed one above the other in such wise that their axes form a right angle, and at a sufficient distance for their reciprocal action not to alter the distribution of the magnetism in either of them.
The system thus formed takes, under the influence of the earth's magnetism, a determinate position of equilibrium, such that the mag
netic axis of the needle M (fig. 1) makes, with the plane of the magnetic meridian N S, an angle a determined by the equation x=M tan α.
If the moment x is sufficiently small in proportion to M, the angle a can be determined by Poggendorff's optical method*. For that purpose the support of the needles carries a small vertical silvered mirror, in which, with the aid of a telescope, the image of a horizontal graduated scale is observed, which is placed beneath and very near the objective, and perpendicular to the optic axis of the telescope.
The moment M of the directing needle may be made as small as we please. It follows that it is possible by this method to measure the magnetic moment of very small needles, comparable in their dimensions to particles of file-dust. I have been able to effect measurements relative to needles of 2 millims. length and 0.2 millim. diameter.
When we wish only to compare with one another the magnetic moments x, x' of several minute needles, it is not necessary to know the moment M of the directing needle; we have, indeed, designating by a, a' the deviations they produce:
and, on account of the minuteness of the angles a and a',
where n and n' are the two readings made upon the scale, the zero of which is supposed to be in the plane of the magnetic meridian.
Apparatus.-The support of the needles is merely a small stick, thin and light, of sealing-wax, hard and not readily fusible. The directing needle A B (fig. 2) is attached to the lower extremity of the wax; and a very narrow glass tube T passes through the latter towards its upper part, in such a way that its axis is horizontal and perpendicular to the axis of the directing needle. The system is suspended by a nipper or hook of copper to a single cocoon-thread, and enclosed in a wooden box blackened on the inside, and closed in front by a plane glass. The cocoon-thread passes through an aperture in the top of the box, and is grasped by a brass nipper. And a disk of red copper is placed under and very near the directing needle, the oscillations of which it deadens.
* "Methode der Spiegelablesung " (Pogg. Ann. vol. vii. 1826), employed by Gauss (Göttingische gelehrte Anz. 1833, Nos. 205-207), to whom the invention of it is most frequently attributed.
The following are the dimensions of one of the apparatus which I have employed :-needle A B, length 6 centims., diameter 0.2 millim.; mirror M, diameter 8 millims.; tube T, length 2 centims., diameter 0.5 millim.; stick, a square prism of 4 millims. in the side, and 5 centims. length.
The construction presents no difficulty. The condition most requisite to be realized is, to render the axis of the tube T sensibly perpendicular to the axis of the needle A B. For this a graduated circle of paper is employed; in the centre of the circle the stick of wax is placed with the needle A B already adherent to its lower extremity, and the axis of the needle is made to coincide with the line 0-180°. Then on the
upper face of the stick the tube T is placed, with a long and stiff steel wire inside of it; and the axis of the tube is directed so that the eye, placed in the prolongation of the stick, sees the steel wire projected on the line joining 90° and 270°. It is sufficient then to put a lighted match near the base of the tube to slightly melt the wax and fix the tube. Finally another piece of wax, terminated by the nipper c, is fitted on.
The mirror is set perpendicular to AB by placing the image of A B on the prolongation of its direction.
Placing. The placing of the telescope and the scale is done in the usual way*. The distance chosen, of the mirror from the scale, was equal to 4 metres in most of the experiments, and 7 metres in some of them.
For the regulation of the apparatus, a few drops of wax are added, according to need, before or behind A B, so as to bring the image of the scale into the field. The scale is suspended by a system of strings and counterpoises, so that it can be raised or lowered at will by a simple movement of the hand, without the eye needing to leave the ocular of the telescope. It thus becomes very easy, after a few trials, to bring the mirror M into a rigorously vertical position; and consequently the bar AB, perpendicular to the mirror, is itself horizontal.
• Vide the writings of Gauss and Weber, or Verdet, Conférences de physique faites à l'Ecole Normale.
The optic axis of the telescope will be normal to the mirror when the image of the objective occupies the centre of the field. The other conditions, relative to the scale, may be fulfilled approximately; and we shall see by-and-by how this is accomplished.
It is to be remarked that, when a needle is introduced into the axis of the tube T, the centre of gravity of the suspended system is in general displaced slightly; but it is easy to see that this displacement has no influence on the readings. In fact we can produce the corresponding displacement of the suspended system by the combination of two rotations:-one about an axis perpendicular to the mirror, without influence on the phenomenon of reflection; the second about a horizontal axis parallel to the mirror, the effect of which is to raise or lower the normal to the mirror in a vertical plane-that is, to raise or depress the image of the scale. This perturbing effect is, besides, reduced to its minimum by disposing the mirror parallel to the axis of the tube in which the needles are placed. The rotation about the axis of suspension, produced by the introduction of a magnetized needle, is due solely to the earth's action, as we have above supposed.
Corrections and Mode of Observation.-The axis of the tube T is never quite horizontal; but when we confine ourselves to relative measurements (we shall see how absolute measures are reduced to relative ones), it is enough if the axis has always the same inclination. For this the weight of the needles introduced into the tube must be very little in proportion to the total weight of the apparatus.
We have supposed that the vertical planes which contain the axis of the tube and the axis of the directing needle form a right
angle. In general they make with each other an angle -B: the precise formula which would replace formula (1) on this hypothesis would be
in which, to quantities of the second order relative to ß,
If the needle be turned in the tube end for end, the angle of the
from which, adding (3) and (4) term to term, and observing
that tan a-tan a' is extremely small, we have, excepting quantities of the second order :
tan a tan al
It will therefore be sufficient to take the mean of the two observations to correct the result of the imperfection of the apparatus*.
Again, in the preceding we have supposed that the magnetic meridian is invariable, which is not rigorously true; besides, our apparatus realizes a veritable compass of variations; consequently the error resulting from the variations of the declination is an appreciable quantity in the conditions in which we are placed.
Except the case of abrupt and irregular variations, this cause of error will be eliminated by making a third measurement after restoring the needle to its first position, turning it end for end. If the observations have been made at nearly equal intervals, the mean of the first and third measurements may be taken, and the mean of this and measurement 2 will not be sensibly affected by the variation of the declination. In all cases the equality of Nos. 1 and 3 will be a guarantee of the accuracy of the mea
For the purpose of making an observation, the oscillations of the apparatus are first eased by hand, in order to abridge the duration of an experiment. Besides, as soon as the amplitude of the oscillations is sufficiently small, the divisions n, and no of the scale, corresponding to the commencement and the end of an oscillation, are noted, and the division ng, corresponding to n+nz .+ ջ.
the end of the following oscillation. The mean, N is taken several times; and thus the position of equilibrium is determined with great exactness.
The number N also has to be corrected when the deviation is rather considerable, in such a manner that the arc cannot be confounded with its tangent. To effect the reduction a Table is employed, giving the values of tan a when tan 2a is known†. The distance of the scale from the mirror being known approximately, it is easy to draw up a Table giving the reduced values
* The two readings n and n', corresponding to the deviations and a', will differ very little from one another, if the scale be exactly perpendicular to the optic axis of the telescope. This condition will be realized by rotating the scale, in a horizontal plane, about its centre until the two readings n and n', obtained with one and the same needle direct and reversed, differ the least possible one from the other.
† See a Table of this sort in Wiedemann's Galvanismus, vol. ii. p. 207 (2nd edition).