quick movements. If they diverge so widely as to touch the sides of the bell-glass, it is often difficult to detach them from the glass without tearing. To prevent this contact, two metallic columns are interposed, communicating with the ground. If the leaves diverge too widely, they touch these columns and lose their electricity.

CHAPTER XLIII.

MEASUREMENT OF ELECTRICAL FORCES.

566. Coulomb's Torsion-balance.-Coulomb, who was the first to make electricity an accurate science, employed in his researches an instrument which is often called after his name, and which is still extensively employed. It depends on the principle that the torsion of a wire is simply proportional to the twisting couple. We shall first describe it, and then point out some of its applications.

It consists of a cylindrical glass

case AA (Fig. 339), from the upper end B of which rises another glass cylinder DD of much smaller diameter. This small cylinder is fitted at the top with a brass cap a, carrying an index C. Outside of this, and capable of turning round it, is another cap b, the top of which is divided into 360 equal parts. In the centre of the cap b is an opening through which passes a small metal cylinder d, capable of turning in the opening with moderate friction, and having at its lower end a notch or slit. When the cap b is turned, the cylinder d turns with it; but the latter can also be turned separately, so as not to change the reading. These parts compose the torsion-head. A very fine metallic wire is held by the notch, and supports a small piece of metal, through which passes a light needle of shellac f, carrying at one end a small gilt ball g. A circular

Fig. 339.-Coulomb's Torsion-balance.

scale runs round the outside of the large cylinder in the plane of the needle. Finally, opposite the zero of this scale, there is a fixed ball g' of some conducting material, supported by a rod f' of shellac, which passes through a hole in the cover of the cylindrical case.

567. Laws of Electric Repulsion.-To illustrate the mode of employing this apparatus for electrical measurements, we shall explain the course followed by Coulomb in investigating the law according to which electrical repulsions and attractions vary with the distance. The index is set to the zero of the scale. The inner cylinder d is then turned, until the movable ball just touches the fixed ball without any torsion of the wire. The fixed ball is then taken out, placed in communication with an electrified body, and replaced in the apparatus. The electricity with which it is charged is communicated to the movable ball, and causes the repulsion of this latter through a number of degrees indicated by the scale which surrounds the case. In this position the force of repulsion is in equilibrium with the force of torsion tending to bring back the ball to its original position. The graduated cap b is then turned so as to oppose the repulsion. The movable ball is thus brought nearer to the fixed ball, and at the same time the amount of torsion in the wire is increased. By repeating this process, we obtain a number of different positions in which repulsion is balanced by torsion. But we know, from the laws of elasticity, that the force (strictly the couple1) of torsion is proportional to the angle of torsion. Hence we have only to compare the total amounts of torsion with the distances of the two balls. By such comparisons Coulomb found that the force of electrical repulsion varies inversely as the square of the distance.

The following are the actual numbers obtained in one of the experiments. The original deviation of the movable ball being 36°, it was found that, in order to reduce this distance to 18°, it was necessary to turn the head through 126°, and, for a farther reduction of the deviation to 8°5, an additional rotation through 441° was required. It will thus be perceived that at the distances of 36°, 18°, and 8°5, which may be practically considered as in the ratio of 1, §1⁄2, and, the forces of repulsion were equilibrated by torsions of 36°,

1 The repulsive force on the movable ball is equivalent to an equal and parallel force acting at the centre of the needle (the point of attachment of the wire), and a couple whose arm is the perpendicular from this centre on the line joining the balls. This couple must be equal to the couple of torsion. The other component produces a small deviation of the suspending wire from the vertical.

t

EQUATION OF EQUILIBRIUM.

561

126°+18°=144°, and 441+126+8.5=575°5 respectively. Now 144 is 36 x 4, and 575 5 may be considered as 576, or 36 x 16. Hence we perceive that, as the distance is divided by 2, or by 4, the force of repulsion is multiplied by 4 or by 16, which precisely agrees with the law enunciated above.

α

568. Equation of Equilibrium.-We must, however, observe that in this mode of reducing the observations two inaccurate assumptions are made. First, the distance between the balls is regarded as being equal to the arc which lies between them, whereas it is really the chord of that arc. Secondly, the force of repulsion is regarded as acting always at the same arm, whereas its arm, being the perpendicular from the centre on the chord, dimi

A

K

B

Fig. 340.

T'

nishes as the distance increases. The following investigation is more rigorous.

Let AOB (Fig. 340), the angular distance of the balls, be denoted by a, and let be the length of the radius OA. Then the chord AB is 21 sina, and the arm OK is l cosa. Let f denote the force of repulsion at unit distance, and n the couple of torsion for 1°. f Then the force of repulsion in the given position is in2 if the law of inverse squares be true, and the moment of this about the centre is fcosa, which must be equal to n A, if A be the number of degrees of torsion. Hence we have

4 l sin a'

4 l2 a

and as the first member of this equation is constant, the second member must be constant also for different values of A and a, if the law of inverse squares be true. The degree of constancy is shown by the following table:

The difference between the first and second numbers of the last

column is insignificant. That between the second and third is more considerable,1 but in reality only corresponds to an error of half a degree in the measurement of the arc.

569. Case of Attraction. The law of attractions may be investigated by a similar method. The index is set to zero, and the central piece is turned so as to place the movable ball at a known distance from the fixed ball. The two balls are then charged with electricity of different kinds. The movable ball is accordingly attracted towards the other, and settles in a position in which attraction is balanced by torsion. By altering the amount of torsion, different positions of the ball can be obtained. On comparing the distances with the corresponding torsions, it is,found that the same law holds as in the case of repulsion. The experiment, however, is difficult, and is only possible when the balls are very feebly electrified. To prevent the contact of the two balls, Coulomb fixed a silk thread in the instrument, so as to stop the course of the movable ball.

570. Law of Attraction and Repulsion as depending on Amount of Charge. We may assume as evident, that when an electrified ball is placed in contact with a precisely equal and similar ball, the charge will be divided equally between them, so that the first will retain only half the charge which it had before contact.

Suppose that an observation on repulsion has just been made with the torsion-balance, and that we touch the fixed ball with another exactly equal insulated ball, which we then remove. It will be found that the amount of torsion requisite for keeping the movable ball in its observed position is just half what it was before. The

1 We have already seen that the mutual induction of two conductors tends to diminish their mutual repulsion, and that this inductive action becomes more important as the distance is diminished. Hence the repulsion at distance 9 should be less than a quarter of that at distance 18. The apparent error thus confirms the law.

Many persons have adduced, as tending to overthrow Coulomb's law of inverse squares, experimental results which really confirm it. Except when the dimensions of the charged bodies are very small in comparison with the distance, the observed attraction or repulsion is the resultant of an infinite number of forces acting along lines drawn from the different points of the one body to the different points of the other. The law of inverse squares applies directly to these several components, and not to the resultant which they yield. The latter can only be computed by elaborate mathematical processes.

It is incorrectly assumed in the text that the law ought to apply directly to two spheres, when by their distance we understand the distance between their nearest points. It is not obvious that the distance of the nearest points should give a better result than the distance between the centres.

The strongest evidence for the rigorous exactness of the law of inverse squares is indirect; see § 574.