LAW OF FORCE.] ELECTRICITY 21 assume the two bails in the balance to be, repel each other, as if their electricity were collected at their centres. Let be the angle of equilibrium in any case, T the angle of tor sion. (fig. 10) is the centre of the beam, F and M the centres of the fixed and mov able ball (we suppose OF = OM); OK is perpendicular to FAT. Then FM 2 oc sin* |L 2t Hence moment of the force on M about oc _ , and the torsonial couple oc T + . Fi Hence in the three cases the value of (r -f e) sin tan fl . = A (say) must be the same, if the law of the inverse square agree with the experiments. Coulomb made many experiments of the kind we have described. The following is the result which he has <nven of one such : Observed. Calculated. Difference. T
E
36" 36 126 18 18 6 6 567 8 30 .) 4 34 The third column is obtained from the two preceding. A is calculated by putting T = and e 36 in the formula (T + e) siu ^ tan x - A . Then using this value of A and the observed value of T, the formula is employed to find e in the two second cases. The agreement between the observed and calculated values of e is the test of the truth of the law we have assumed. The agreement in the second line is as good as can be expected when possible errors of experiment are considered. It will be seen, moreover, that the calculated is in excess of the observed value, which is what we ought to expect, owing to the loss of electricity which goes on during the time con sumed in the experiment. That there is such a loss may be proved experimentally by simply leaving the movable ball to itself after any of the three operations ; it will be seen to move slowly towards the fixed ball. We shall re turn hereafter to this loss of electricity, with regard to the exact nature of which authorities aro not quite agreed. 1 In the third line the agreement is less good, but here the proximity of the balls renders the supposition of unifor mity no longer even approximately allowable. The mutual repulsion tends to drive the electricity on each ball farther from the other ball, and thus the action between the balls is as if the electricity on each were collected at points be yond the centre, so that the observed repulsion must be less than that calculated on the supposition of uniformity of distribution. Coulomb also made experiments with the torsion balance to test whether the law of the inverse square applies to the attraction as well as to the repulsion of electrified bodies. His experiments confirmed the law ; but the difficulty of operating is much greater in this case than in the former. He therefore adopted another method of experimenting. A small conducting disc was fixed nor- 1 This is only one of the many experimental difficulties which beset the use of the torsion balance, one of the most difficult of all instru ments to use successfully. To appreciate the skill and sagacity of Coulomb in this and other matters, the student must read more de tailed accounts (Riess and Mascart, or Memoires de VAcad., about 1785) of his labours than we can give here. He will be richly re paid for his trouble. Nothing is better calculated to rouse the failing enthusiasm of the tyro in experimental electricity than a perusal of the works of Coulomb, unless it be to read the Experimental Rts .rtrchc.s of Faraday. mally on the end of a small shellac needle, which was hung Law of up, so as to be horizontal, ou a fibre of raw silk attached ttrac- to a horizontal scale. An insulated conducting globe was tion set up with its centre in the same vertical plane as the n " t ^ l < ^ I scale, and in the same horizontal plane as the centre of the O f oscil small disc. The globe and disc were oppositely electrified, lations. and the period of oscillation of the needle was found by observing the duration of 15 swings. The time of oscilla tion follows the pendulum law, and varies inversely as the square root of the force acting on the needle, hence the duration of 15 oscillations will vary inversely as the square root of the force, i.e. directly as the distance between the centres of the globe and disc, if the law of the inverse square hold. Coulomb s experiment gave the following results : Distance of centres of globe and disc. Duration of 1") oscillations. Ratio of distance to duration. 9 18 24 20 41 60 2 22 2-28 2-50 The numbers in the third column ought to be all equal. The deviation from equality are not greater than can fairly be explained by loss of electricity and errors of observation. Coulomb also investigated, both by means of the torsion balance and by the method of oscillations, the relation be tween electric force and quantity. He electrified the two balls of the torsion balance by simultaneous contact vitli another ball, and observed the angle of equilibrium : he then halved the quantity on the fixed ball by touching it with an equal neutral ball ; and reduced the torsion till the angle of equilibrium, and, in consequence, the distance between the balls was the same as before ; he found the torsional couple in the second case to be somewhat less than half what it was in the first. He therefore concluded that the force between two elements of electricity varies as the product of the quantities. Coulomb s experiments were repeated, and his results confirmed by Biess, 2 and by Marie-Davy. 3 Experiments which, when properly interpreted, lead to the same results, were made by Snow Harris, 4 and by Egen. 5 We have then arrived at this general law of electric force : If two quantities q, q of electricity be supposed collected State- at two points, whose distance is d, the force beticecn them ment - of law. acts in the straight line joining the points and oc -^| So far, this law might be merely an approximation to the truth. Later on, however, it will be seen to be logically deducible from experiments which in delicacy infinitely surpass those just described. The law of Coulomb is in fact established as certainly as the law of gravitation itself. 6 By means of the law now given the unit of electrical quan- Defini- tity can be defined in a satisfactory and practical manner, tion of This unit we now state to be that quantity of positive elec- absolute tricity which, when collected into a point, repels with unit of force an equal quantity similarly collected into a point at unit distance from the former. If we take centimetre, gramme, and second as our units of length, mass, and time, tho unit force will be that force which in a second generates in a gramme of mutter a velocity of a centimetre per second. 8 Rtibungselectricitat, Bd. i. p. 94. * Mascart, 1. p. 67. 4 Phil. Trans., 1834 and 1836. In connection with which we call the attention of the student to the classical paper of Sir W. Thomsoi , Reprint of Papers on Electrostatics and Magnetism,, p. ] 5 sqq 5 Riess, Bd. i. p. 94.
- We suppose, of course, that we are dealing always with one and
