Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/263

177.] $\begin{array}{l} \sigma_{a}=\frac{E_{a}}{4\pi a^{2}}=\frac{\pi}{24b}\\ \\\sigma_{b}=\frac{E_{b}}{4\pi b^{2}}=\frac{1}{4\pi b},\\ \\\sigma_{a}=\frac{\pi^{2}}{6}\sigma_{b}.\end{array}$|undefined

Hence, if a very small sphere is made to touch a very large one, the mean density on the small sphere is equal to that on the large sphere multiplied by $$\tfrac{\pi^{2}}{6}$$, or 1.644936.

Application of Electrical Inversion to the case of a Spherical Bowl.

176.] One of the most remarkable illustrations of the power of Sir W. Thomson's method of Electrical Images is furnished by his investigation of the distribution of electricity on a portion of a spherical surface bounded by a small circle. The results of this investigation, without proof, were communicated to M. Liouville and published in his Journal in 1847. The complete investigation is given in the reprint of Thomson's Electrical Papers, Article XV. I am not aware that a solution of the problem of the distribution of electricity on a finite portion of any curved surface has been given by any other mathematician.

As I wish to explain the method rather than to verify the calculation, I shall not enter at length into either the geometry or the integration, but refer my readers to Thomson's work.

Distribution of Electricity on an Ellipsoid.

177.] It is shewn by a well-known method that the attraction of a shell bounded by two similar and similarly situated and concentric ellipsoids is such that there is no resultant attraction on any point within the shell. If we suppose the thickness of the shell to diminish indefinitely while its density increases, we ultimately arrive at the conception of a surface-density varying as the perpendicular from the centre on the tangent plane, and since the resultant attraction of this superficial distribution on any point within the ellipsoid is zero, electricity, if so distributed on the surface, will be in equilibrium.

Hence, the surface-density at any point of an ellipsoid undisturbed by external influence varies as the distance of the tangent plane from the centre.