Coulomb’s Torsion Balance

202. The fundamental laws of electricity were established by Coulomb through experiments that measured the force between 2 small spheres charged with electricity.

• One was fixed while
• The other was held in equilibrium by 2 forces:
  1. The electrical action between the spheres
  2. The torsional elasticity of a glass fibre or metal wire

The balance of torsion consists of a horizontal arm of gum-lac. It is suspended by a fine wire or glass fibre, carrying at one end a little sphere of elder pith, smoothly gilt.

The suspension wire is fastened above to the vertical axis of an arm which can be moved round a horizontal graduated circle, so as to twist the upper end of the wire about its own axis any number of degrees.

This apparatus is enclosed in a case.

Another little sphere is mounted on an insulating stem.

• It can be charged and introduced into the case through a hole, and brought so that its centre coincides with a definite point in the horizontal circle described by the suspended sphere.

The position of the suspended sphere is ascertained by means of a graduated circle engraved on the cylindrical glass case of the instrument.

With both spheres charged, the suspended sphere in equilibrium in a known position such that the torsion-arm makes an angle θ with the radius through the centre of the fixed sphere.

The distance of the centres is then 2a sin 12 θ, where a is the radius of the torsion-arm, and if F is the force between the spheres the moment of this force about the axis of torsion is F a cos 12 θ.

Let both spheres be completely discharged, and let the torsion-arm now be in equilibrium at an angle φ with the radius through the fixed sphere.

Then the angle through which the electrical force twisted the torsion-arm must have been θ − φ, and if M is the moment of the torsional elasticity of the fibre, we shall have the equation F a cos 12 θ = M(θ − φ).

Hence, if we can ascertain M, we can determine F, the actual force beween the spheres at the distance 2a sin 12 θ.

To find M, the moment of torsion, let I be the moment of inertia of the torsion-arm, and T the time of a double vibration of the arm under the action of the torsional elasticity, then M= 4π 2 I

In all electrometers it is of the greatest importance to know what force we are measuring. The force acting on the suspended sphere is due partly to the direct action of the fixed sphere, but partly also to the electrification, if any, of the sides of the case.

If the case is made of glass it is impossible to determine the electrification of its surface otherwise than by very difficult measurements at every point.

If, however, either the case is made of metal, or if a metallic case which almost completely encloses the apparatus is placed as a screen between the spheres and the glass case, the electrification of the inside of the metal screen will depend entirely on that of the spheres, and the electrification of the glass case will have no influence on the spheres. In this way we may avoid any indefiniteness due to the action of the case.

To illustrate this by an example in which we can calculate all the effects, let us suppose that the case is a sphere of radius b, that the centre of motion of the torsion-arm coincides with the centre of the sphere and that its radius is a; that the charges on the two spheres are E1 and E, and that the angle between their positions is θ; that the fixed sphere is at a distance a1 from the centre, and that r is the distance between the two small spheres.

Neglecting for the present the effect of induction on the distribution of electricity on the small spheres, the force between them will be a repulsion

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and the moment of this force round a vertical axis through the centre will be EE1 aa1 sin θ

The image of E1 due to the spherical surface of the case is a point in the

same radius at a distance with a charge −E1 , and the moment of the

attraction between E and this image about the axis of suspension is

…

If b, the radius of the spherical case, is large compared with a and a1 , the distances of the spheres from the centre, we may neglect the second and third terms of the factor in the denominator. The whole moment tending to turn the torsion-arm may then be written

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