PARTICULAR CASES OF ELECTRIFICATION

81.] A spherical conductor is electrified and insulated within the concentric spherical internal surface of a conducting vessel. On account of the perfect symmetry of this system in all directions, it is manifest that the distribution of electricity will be uniform over each of the opposed spherical surfaces, that the lines of force will be in the directions passing through the common centre of the spheres, and that the equipotential surfaces will be spheres having this point for their centre. If e is the quantity of electricity on the inner sphere and E that on the internal surface of the outer sphere, then by Experiment VIII

E = −e. (1) If r and R are the radii of the spheres, s and S their surfaces, and σ and Σ the surface-densities of the electricity on these surfaces, then by geometry, s = 4πr2 , S = 4πR2 , (2) where π denotes the ratio of the circumference of a circle to its diameter. The whole charge on either sphere is found by multiplying the surface into the surface-density, or e = sσ, E = SΣ. (3) Hence, σ= e , 4πr2 Σ=E , 4πR2(4) Σ=−e . 4πR2(5) and by (1), It appears, therefore, that when the charge, e, of the inner sphere is given, the surface-density, Σ, on the internal surface of the vessel is inversely as the square of the distance of that surface from the centre of the electrified sphere.

ELECTROSTATIC UNIT OF ELECTRICITY

Hence by Coulomb’s law (Experiment XIII, Art. 47) the electromotive force at the outer spherical surface is inversely as the square of the distance from the centre of the sphere.

This is the law according to which the electric force varies at different distances from a sphere uniformly electrified. The amount of the force is independent of the radius of the inner electrified sphere, and depends only on the whole charge upon it. If we suppose the radius of the inner sphere to become very small till at last the sphere cannot be distinguished from a point, we may imagine the whole charge concentrated at this point, and we may then express our result by saying that the electric action of a uniformly electrified sphere at any point outside the sphere is the same as that of the whole charge of the sphere would be if concentrated at the centre of the sphere. We must bear in mind, however, that it is physically impossible to charge the small sphere with more than a certain quantity of electricity on each unit of area of its surface. If the surface-density exceed this limit, electricity will fly off in the form of the brush discharge. Hence the idea of an electrified point is a mere mathematical fiction which can never be realised in nature. The imaginary charge concentrated at the centre of the sphere, which pro- duces an effect outside the sphere equivalent to that of the actual distribution of electricity on the surface, is called the Electrical Image of that distribution. See Art. 100.

Measurement of Electricity.

82.] We have already described methods of comparing the quantity of elec- trification on different bodies, but in each case we have only compared one quantity of electricity with another, without determining the absolute value of either. To determine the absolute value of an electric charge we must com- pare it with some definite quantity of electricity, which we assume as a unit. The unit of electricity adopted in electrostatics is that quantity of positive or vitreous electricity which, if concentrated in a point, and placed at the unit of distance from an equal charge, also concentrated in a point, would repel it with the unit of mechanical force. The dielectric medium between the two charged points is supposed to be air.

ELECTROMOTIVE FORCE AT A POINT.

83.] Let us now suppose two bodies, whose dimensions are small com- pared with the distance between them, to be charged with electricity. Let the charge of the first body be e units of electricity and that of the second e′ units, and let the distance between the bodies be r. Then, since the force varies inversely as the square of the distance, the force with which each unit of electricity in the first body repels each unit of 1 electricity in the second body will be 2 , and since the number of pairs of r units, one in each body, is ee′ , the whole repulsion between the bodies will be ee′ f= 2 r If the charge of the first or the second body is negative we must consider e or e′ negative. If the one charge is positive and the other negative, f will be negative, or the force between the bodies will be an attraction instead of a repulsion. If the charges are both positive or both negative, the force between the bodies will be a repulsion.