Again take the case of two parallel infinite planes under the influence of a point-charge between them. This system inverted from P as centre gives the equilibrium distribution on two charged insulated spheres in contact (Fig. [8]); for this system is the inverse of the planes and the charges upon them. Another interesting case is that of the "electric kaleidoscope" referred to above. Here the two infinite conducting planes are inclined at an angle 360° ⁄ n, where n is a whole number, and are therefore bounded in one direction by the straight line which is their intersection. The image points I1, J1, ..., of P placed in the angle between the planes are situated as shown in Fig. [3], and are n − 1 in number. This system inverted from P as centre gives two spherical surfaces which cut one another at the same angle as do the planes. This system is one of electrical equilibrium in free space, and therefore the problem of the distribution on two intersecting spheres is solved, for the case at least in which the angle of intersection is an aliquot part of 360°. When the planes are at right angles the result is that for two perpendicularly intersecting planes, for which Fig. [9] gives a diagram.
Fig. 9.
But the greatest achievement of the method was the determination of the distribution on a segment of a thin spherical shell with edge in one plane. The solution of this problem was communicated to M. Liouville in the letter of date September 16, 1846, referred to above, but without proof, which Thomson stated he had not time to write out owing to preparation for the commencement of his duties as Professor of Natural Philosophy at Glasgow on November 1, 1846. It was not supplied until December 1868 and January 1869; and in the meantime the problem had not been solved by any other mathematician.
As a starting point for this investigation the distribution on a thin plane circular disk of radius a is required. This can be obtained by considering the disk as a limiting case of an oblate ellipsoid of revolution, charged to potential V, say. If Fig. [10] represent the disk and P the point at which the density is sought, so that CP = r, and CA = a, the density is V ⁄ {2π2√(a2 − r2)}.
The ratio q ⁄ V, of charge to potential, which is called the electrostatic capacity of the conductor, is thus 2a ⁄ π, that is a ⁄ 1.571. It is, as Thomson notes in his paper, very remarkable that the Hon. Henry Cavendish should have found long ago by experiment with the rudest apparatus the electrostatic capacity of a disk to be 1 ⁄ 1.57 of that of a sphere of the same radius.
| Fig. 10. | Fig. 11. |
Now invert this disk distribution with any point Q as centre of inversion, and with radius of inversion a. The geometrical inverse is a segment of a spherical surface which passes through Q. The inverse distribution is the induced distribution on a conducting shell uninsulated and coincident with the segment, and under the influence of a charge − aV situated at Q (Fig. [11]). Call this conducting shell the "bowl." If the surface-densities at corresponding points on the disk and on the inverse, say points P and P', be s and s', then, as on page [51], s' = sa3 ⁄ QP'3. If we put in the value of s given above, that of s' can be put in a form given by Thomson, which it is important to remark is independent of the radius of the spherical surface. This expression is applicable to the other side of the bowl, inasmuch as the densities at near points on opposite sides of the plane disk are equal.
If v, v' be the potentials at any point R of space, due to the disk and to its image respectively, − v' = av ⁄ QR. If then R be coincident with a point P' on the spherical segment we have (since then v = V) V' = aV ⁄ QP', which is the potential due to the induced distribution caused by the charge − aV at Q as already stated.