EI.ECTKOSTATICAL THEORY.] E L E C T K I I T Y E At + AP APA~~2AP" /B + AP 2/AP aAP 3 We might have any number of external points and find the image of each. We should thus get a system which might be called the image of the external system. The distribution induced in an uninsulated sphere by such an external system could easily be found by adding up the effect of each external element found by means of its image. Similar methods might also be applied to an in ternal system. The solution cm be generalized without difficulty to the case where either the charge or potential of the sphere is given. Suppose the charge Q given ; superpose on the distribution found above a uniform distribution of amount Q + T E. This will pro- O V duce a constant potential ~+7 all over the sphere, and therefore will not disturb the, equilibrium. We have thus got the required distribution of the given charge Q under the influence of A. The density of any point is given by 4ira/ (41). Fi So far the method of images is simply a synthetical method for obtaining distributions on a sphere. But Sir William Thomson has shown us how to convert it into an instrument for transforming any electrical problem into a variety of others. f P be any point (fig. 13), a fixed point, and P be taken in OP such that OP. OP = a 2 , then P is called the inverse of P with respect to O, which is called the origin of inver sion, or simply the origin ; a is the radius of inversion. We may 1 thus invert any locus of points into another locus of points, which we may call the inverse of the former. Let P, Q and P , Q be any two points and their inverses. Let us suppose that there is a charge E at Q, and a charge E at Q , which is the image of E in a sphere with radius a and centre ; so that K = /A/^E. Let V and V be the respective potentials of E and E at P and P . Then we have obviously where OP -- > , Op**-/. It is very easy to show that, if ds, rfS, dr, a; p, be elements of length, surface, and volume, and surface and volume densities, and the same symbols with dashes the inverses ot the.se, then we have ils ds r 3 a also ., -= = ; -77 p p V -77 V (42) By means of these equations it is easy to invert any electrical system. Take, for example, the case of any con ductor in electrical equilibrium ; then, since its potential is everywhere constant, it inverts into a surface distribu tion, the potential at any point of which distant r from the origin is by (42) , C, where C is the constant poten tial of the conductor. The surface density at any point of the system is found from that of the corresponding point on the conductor by the equation 1 For the general properties of curves and their inverses, the reader may consult Salmon s Solid Geometry. He will have no difficulty in provinc for himself such as we shall require here. , a ff ^* ""/> Again, if we consider the system thus found, it is obvioua that, if we place a quantity - aC of electricity at the origin, this will make the potential at every point of the system zero, and we have a solution of the case of an uninsulated conductor, whose surface is the inverse of that of the given conductor, under the influence of an electrified point. As an example of the use of this method, let us invert the uni form distribution on a sphere with respect to an origin on its cir cumference, the radius of inversion being the diameter of the sphere. The sphere inverts into an infinite plane, touching at the other end A of the diameter through the origin. Let C be the C potential on the sphere so that ff^^vd w ^ ere ^ *" e diameter. Hence the density at any point Pon an infinite plane influenced by a quantity - Cd of electricity placed at a point distant d from it is given by </ _ ^C Again, inverting points inside the sphere, for which the poten tial is constant, we get the potential due to the distribution on the infinite plane, at points on the other side from the inducing point, the result being which is the same aa that due to dC at 0. Hence the potential at a point on the same side as is that due to a quantity rfC placed at , where A = OA. is in fact the image of 0. If we write Q for - Cd, then we get -Qd (43). These results might of course have been deduced as particular cases of a sphere and point. Many beautiful applications of these methods will be found in the Reprint of Sir William Thomson s papers and in Maxwell s Electricity and Magnetism. Two of these are of especial importance. Adopting the method of succes sive influences given by Murphy (Electricity, 1833, p. 93), and conjoining with it the method of images, Sir William Thomson treated the problem of two spheres. For his results, see Reprint, pp. 86-97. At the end of that paper two valuable tables are given T. "Showing the quantities of electricity on two equal spherical conductors of radius r, and the mutual force between them, when charged to potentials?* and v respectively " II. " Giving the potentials and force when the charges D and E are given." The ratio of u to v in the first case and of D to E in the second is also given, for which at a given distance there is neither attraction nor repulsion. An interesting experiment on this curious phenomenon is described in Riess, Bd. i. 186. For an application of dipolar co-ordinates to the problem of two spheres, see Maxwell. Thomson also applied his methods to determine the dis- tribution on spherical bowls of different apertures. See Reprint, p. 178 sqq. His numerical results on p. 186 aro extremely interesting, as affording a picture of the effect of gradually closing a conductor, and are of great value in giving the experimenter an idea as to what aper ture he may allow himself in a vessel which he desires should be for practical purposes electrically closed. It would lead us too far to discuss here the analytical method of conjugate functions, and the allied geometrica method of inversion in two dimensions. A full account of these, with important applications, will be found in Max well, vol. i. 182 ffjq. We shall conclude our applications with a brief notice of a few of the ordinary electrostatical instruments, refer ring the reader for an account of some others to the article ELECTROMETER. If two plates be placed parallel to each other, and one VIII. - 5 Spherical bowl.
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