This is exactly analogous to the replacement by a distribution on an equipotential surface of the electrical charge or charges within the surface, by a distribution over the surface, with fulfilment of Coulomb's theorem (p. [43] below) at the surface. Thomson's paper on the "Uniform Motion of Heat" gave an intuitive proof of this great theorem of electrostatics, which the statements above may help to make clear to those who have, or are willing to acquire, some elementary knowledge of electricity.
Returning to the distribution on any isothermal surface surrounding the sink (or sinks) we see that it represents a surface-sink in equilibrium with the flow in the field. The distribution on a metal shell, coinciding with the surface, which keeps the surface at a potential which is the analogue of the temperature at the isothermal surface, while the shell is under the influence of a point-charge of electricity—the analogue of the thermal source—is the distribution as affected by the induction of the point-charge. If the shell coincide with the spherical equipotential surface referred to above, and the distribution given by the theorem of replacement be made upon it, the shell will be at zero potential, and the charge will be that which would exist if the shell were uninsulated, that is, the "induced charge."
The consideration of the following simple problem will serve to make clear the meaning of an electric image, and form a suitable introduction to a description of the application of the method to the electrification of spherical surfaces. Imagine a very large plane sheet of tinfoil connected by a conducting wire with the earth. If there are no electrified bodies near, the sheet will be unelectrified. But let a very small metallic ball with a charge of positive electricity upon it be brought moderately close to one face of the tinfoil. The tinfoil will be electrified negatively by induction, and the distribution of the negative charge will depend on the position of the ball. Now, it can be shown that the field of electric force, on the same side of the tinfoil as the ball, is precisely the same as would be produced if the foil (and everything behind it) were removed, and an equal negative charge of electricity placed behind the tinfoil on the prolonged perpendicular from the ball to the foil, and as far from the foil behind as the ball is from it in front. Such a negative charge behind the tinfoil sheet is called an electric image of the positive charge in front. It is situated, as will be seen at what would be, if the tinfoil were a mirror, the optical image of the ball in the mirror.
Fig. 2.
Fig. 3.
Now, suppose a second very large sheet of tinfoil to be placed parallel to the first sheet, so that the small electrified sphere is between the two sheets, and that this second sheet is also connected to the earth. The charge on the ball induces negative electricity on both sheets, but besides this each sheet by its charge influences the other. The problem of distribution is much more complicated than in the case of a single sheet, but its solution is capable of very simple statement. Let us call the two sheets A and B (Fig. [2]), and regard them for the moment as mirrors. A first image of an object P between the two mirrors is produced directly by each, but the image I1 in A is virtually an object in front of B, and the image J1 in B an object in front of A, so that a second image more remote from the mirror than the first is produced in each case. These second images I2 and J2 in the same way produce third images still more remote, and so on. The positions are determined just as for an object and a single mirror. There is thus an infinite trail of images behind each mirror, the places of which any one can assign.