Fig. 7.
Thus we have the conclusion that by the process of inversion we get from a distribution in equilibrium, on a conductor of any form, an induced distribution on the inverse surface supposed insulated and conducting; and conversely we obtain from a given induced distribution on an insulated conducting surface, a natural equilibrium distribution on the inverse surface. In each case the inducing charge is situated at the centre of inversion. The charges on the conductor (or conductors) after inversion are always obtainable at once from the fact that they are the inverses of the charges on the conductor (or conductors) in the direct case, and the surface-densities or volume-densities can be found from the relations stated above.
Now take the case of two concentric spheres insulated and influenced by a point-charge q placed at a point P between them as shown in Fig. [7]. We have seen at p. [49] how the induced distribution, and the amount of the charge, on each sphere is obtained from the two convergent series of images, one outside the outer sphere, the other inside the inner sphere. We do not here calculate the density of distribution at any point, as our object is only to explain the method; but the quantities on the spheres S1 and S2, are respectively − q.OA.PB ⁄ (OP.AB), − q.OB.AP ⁄ (OP.AB).
It may be noticed that the sum of the induced charges is − q, and that as the radii of the spheres are both made indefinitely great, while the distance AB is kept finite, the ratios OA ⁄ OP, OB ⁄ OP approximate to unity, and the charges to − q.PB ⁄ AB, − q.AP ⁄ AB, that is, the charges are inversely as the distances of P from the nearest points of the two surfaces. But when the radii are made indefinitely great we have the case of two infinite plane conducting surfaces with a point-charge between them, which we have described above.
Now let this induced distribution, on the two concentric spheres, be inverted from P as centre of inversion. We obtain two non-intersecting spheres, as in Fig. [5], for the inverse geometrical system, and for the inverse electrical system an equilibrium distribution on these two spheres in presence of one another, and charged with the charges which are the inverses of the induced charges. These maintain the system of two spheres at one potential. From this inversion it is possible to proceed as shown by Maxwell in his Electricity and Magnetism, vol. i, § 173, to the distribution on two spheres at two different potentials; but we have shown above how the problem may be dealt with directly by the method of images.
Fig. 8.