240 MAGNETISM PTrf 2K i,2
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- " (63). The deflecting magnet is reversed in its carriage, and the whole operation repeated. If the deflexion now be 2 , irrespective of sign, then -i=- sin 0.1 + + -!+ (64). The mean of these gives The magnet is finally removed to a distance r 1 west, and the previous observations repeated ; we thus get - sin0-) = lH -.+ (66). 4/ s The mean of (65) and (66) is then taken, and we get <r 3 TT P M n a _ I . | 2K ^ 1+ 7? where S x = | (sin 0j + sin 2 + sin 3 + sin 4 ), or, what is practically the same, the sine of the mean of 2 , 2 , 3 , and 6 4 . The object in taking the mean of (65) and (66) is to eliminate any error arising from the non-coincidence of the middle point of the cross bar with the axis of suspension. In order to eliminate P 2 , another set of observations are made with a new distance r. 2 (26 cm. or so), giving the equation r 3 H P From (67) and (68) we have finally H p _ Statical instru ments. Measure ments by electro magnetic induc tion. When great accuracy is required, several corrections have to be applied: (1) the moment of the deflector must be corrected for induction ; (2) the moment of the deflector must be corrected for temperature ; (3) the lengths r and r 2 on the cross bar must be corrected for temperature. Statical Method. There is another method by which we may determine the product KH, viz., we may oppose a statical couple to the couple exerted by the earth on the magnet in a given position, so that there may be equili brium ; the statical couple, which may arise from the torsion of a fibre, from a bifilar suspension, or other gravitational force, thus becomes the measure of the magnetic couple; and hence KH can be determined in absolute measure. Coulomb s torsion balance experiments are an example of this method. It finds numerous ap plications in the variation instruments of fixed magnetical observatories, and also in instruments for magnetic ob servations at sea, but it is very little used in the ordinary rork of a physical laboratory. Magnetic Measurement by Electromagnetic Induction. It has been explained in the article ELECTRICITY that, if, either owing to the variation of the magnetic field, or owing to the motion of a closed linear conductor in it, the number of lines of magnetic force N passing in the positive direction through the conductor vary, this variation will cause an electromotive force - dNjdt in the positive direction round the circuit. Let us suppose, to take a simple case, that we have a coil of wire made up of a number of parallel plane circular windings, and that the sum of all the areas of the separate windings is A. If we place this in a field of uniform intensity R, so that the normal to the windings makes an angle with R, the number of lines of force passing through the coil will be N 1 = ARcos^. If we now suddenly reverse the coil, by turning it through 180 about an axis perpendicular to its normal, the value of N in the new position is N 2 = ARcos0. Hence the integral electromotive force during the motion is - fdtdN/dt = N 2 - N^ = - 2ARcos0, and the whole quantity Q of electricity which passes will be Q= - 2AIlcos0/S, where S is the resistance of the coil. If Q be found in absolute measure, 1 and A and S be known, we thus obtain the value of Rcos0. This is the principle of Weber s "earth inductor," 2 by means of Weber s which the horizontal and vertical components of the earth s earth in- force can be measured, and in consequence the declination ductor - and inclination determined. If the test coil be made very small, so that the portion Verdet s of the field which it occupies may be supposed uniform, explorin this method may be applied to measure the intensity at 01 different parts of a non-uniform field. 3 The small coil is placed with its windings perpendicular to the lines of force, and then suddenly reversed, or, if that be impossible, suddenly removed to a part of the field where the number of lines of force passing through it is zero. The integral electromotive force is of course in the latter case only half what it is in the former. This method is often of use where, owing to the great strength of the field and the consequent disturbances arising from induction, any other method would be utterly useless. The method of electromagnetic induction may also be Magnet: applied to measure the component of the magnetic moment momeu1 of any body parallel to a given line. mined t Let aa bb (fig. 30) be the section of a uniform cylindrical coil of electro- length 21, made up of a single layer of flat circular windings of niagneti radius b, n to the centimetre. Let the axis of the coil be taken inductic for a:-axis, and let K be any magnet within the coil, placed with the given line parallel to the axis of the coil. Let pq be any single winding of the coil, then the surface integral of the magnetic induction for pq is given by ffadydz ; hence the whole number of lines of force through the coil is given by 1ST = Jndxjfadydz , = nJJJ~adxdyd~ , the integration being extended all over the cylindrical space abb a . Now, since a = a + 4irA = dV/dx + 4TrA, we get K = - n/// -r-dxdydz + 4irn///Adxdydz , = - n(ffVdydz - 9) where K is the component parallel to the axis of the coil of the moment of the magnet, and S and S the values of the surface integral of the potential of the magnet (derived from Poisson s distribution) over the two ends of the coil. When there are more layers than one, we must of course sum the different parts of iST arising from the different layers. The formulas are quite general, and some applications will be given later. Meantime we see that, if the coil be 1 See arts. ELECTRICITY and GALVANOMETER. 2 Pogg. Ann., xc., 1853.
3 Cf. Verdet, Ann. d. Chim. et d. Phys., xli., 1854.