P' is in each case called the image of P in the sphere S, and the charge − qa ⁄ f there supposed situated is the electric image of the charge q at P. It will be seen that an electric image is a charge, or system of charges, on one side of an electrified surface which produces on the other side of that surface the same electrical field as is produced by the actual electrification of the surface.
While by the theorem of replacement there is only one distribution over a surface which produces at all points on one side of a surface the same field as does a distribution D on the other side of the surface, this surface distribution may be equivalent to several different arrangements of D. Thus the point-charge at P' is only one of various image-distributions equivalent to the surface-distribution in the sense explained. For example, a uniform distribution over any spherical surface with centre at P' (Fig. [4]) would do as well, provided this spherical surface were not large enough to extend beyond the surface S.
In order to find the potential of the sphere (Fig. [4]) when insulated with a charge Q upon it, in presence of the influencing charge q at the external point P, it is only necessary to imagine uniformly distributed over the sphere, already electrified in the manner just explained, the charge Q + aq ⁄ f. Then the whole charge will be Q, and the uniformity of distribution will be disturbed, as required by the action of the influencing point-charge. The potential will be Q ⁄ a + q ⁄ f. For a given potential V of the sphere, the total charge is aV − aq ⁄ f, that is the charge is aV over and above the induced charge.
If instead of a single influencing point-charge at P there be a system of influencing point-charges at different external points, each of these has an image-charge to be found in amount and situation by the method just described, and the induced distribution is that obtained by superimposing all the surface distributions found for the different influencing points.
The force of repulsion between the point-charge q and the sphere (with total charge Q) can be found at once by calculating the sum of the forces between q at P and the charges Q + aq ⁄ f at C and − aq ⁄ f at P'.
This can be found also by calculating the energy of the system, which will be found to consist of three terms, one representing the energy of the sphere with charge Q uninfluenced by an external charge, one representing the energy on a small conductor (not a point) at P existing alone, and a third representing the mutual energy of the electrification on the sphere and the charge q at P existing in presence of one another. By a known theorem the energy of a system of conductors is one half of the sum obtained by multiplying the potential of each conductor by its charge and adding the products together. It is only necessary then to find the variation of the last term caused by increasing f by a small amount df. This will be the product F . df of the force F required and the displacement.
Either method may be applied to find the forces of attraction and repulsion for the systems of electrified spheres described below.
The problem of two mutually influencing non-intersecting spheres, S1, S2 (Fig. [5]), insulated with given charges, q1, q2, may now be dealt with in the following manner. Let each be supposed at first charged uniformly. By the known theorem referred to above, the external field of each is the same as if its whole charge were situated at the centre. Now if the distribution on S2, say, be kept unaltered, while that on S1 is allowed to change, the action of S2 on S1 is the same as if the charge q2 were at the centre C2 of S2. Thus if f be the distance between the centres C1, C2, and a1 be the radius of S1, the distribution will be that corresponding to q1 + a1q2 ⁄ f uniformly distributed on S1 together with the induced charge − a1q2 ⁄ f, which corresponds to the image-charge at the point I1 (within S1), the inverse of C2 with respect to S1. Now let the charge on S1 be fixed in the state just supposed while that on S2 is freed. The charge on S2 will rearrange itself under the influence of q1 + a1q2 ⁄ f ( = q') and − a1q2 ⁄ f, considered as at C1 and I1 respectively. The former of these will give a distribution equivalent to q2 + a2q' ⁄ f uniformly distributed over S2, and an induced distribution of amount − a2q' ⁄ f at J1, the inverse point of C1 with regard to S2. The image-charge − a1q2 ⁄ f at I1 in S1 will react on S2 and give an induced distribution − a2 (− a1q2 ⁄ f ) f', (I1C2 = f' ) corresponding to an image-charge a2a1q2 ⁄ ff' at the inverse point J2 of P1 with respect to C2S2. Thus the distribution on S2 is equivalent to q2 + a2q' ⁄ f − a2a1q2 ⁄ ff' at the inverse point J2 of P1 distributed uniformly over it, together with the two induced distributions just described.
