Page:Elementary Text-book of Physics (Anthony, 1897).djvu/301

§ 254] of the walls enters, and if the balls be small and so far apart that their inductive action on one another may be neglected, the repulsion in the second case is found to be one half that in the first case. In general the problem is a far more difficult one, for the distribution on the two spheres is not uniform. That portion of the distribution dependent on the induction of the balls can be calculated, but the irregularities of distribution due to the action of the walls of the case and other disturbing elements can only be allowed for approximately.

The law as respects distance is proved in a somewhat similar way. The repulsions at two different distances are measured in terms of the torsion of the wire, the charges on the two balls remaining the same. The same corrections must be introduced as in the former case.

254. Distribution.—The law of electrical force has been stated in terms of the charges of two bodies. We may, however, consider electricity as a quantity which has an existence independent of matter and which is distributed in space. The fact cited in § 252 (4) shows that this distribution must be looked on as being on the surfaces of conductors, and not m their interiors. If we define surface density of electrification at any point on the surface of a charged conductor as the limit of the ratio of the quantity of electricity on an element of the surface at that point to the area of the element as that area approaches zero, we may measure quantities of electricity in terms of surface density. The surface density of electricity is usually designated by $$\sigma.$$

If the law of electrical force hold true not only for charges on bodies, but also for quantities of electricity on the surface elements, of a conductor, it is evident, from the fact that within an electrified conductor there is no electrical force, that its surface density of electrification must be proportional at every point on its surface to the thickness at that point of a shell of matter which is so distributed on that surface that there is no force at any point enclosed by the surface. The distribution on a charged sphere may, from symmetry, be assumed uniform. The fact that there is no