(25),
in other words, the force at P is inversely as the cube of the distance from A. Suppose then we remove the negative point-charge, and let the sphere be supposed to become conductive and be connected to earth. If we make a distribution of negative electricity over it, which has a density σ varying according to the law
σ = −(d² − r²) q / 4πrAP³
(26),
that distribution, together with the point-charge +q at A, will make a distribution of electric force at all points outside the sphere exactly similar to that which would exist if the sphere were removed and a negative point charge −qr/d were placed at B. Hence this charge is the electrical image of the charge +q at A in the spherical surface.
We may generalize these statements in the following theorem, which is an important deduction from a wider theorem due to G. Green. Suppose that we have any distribution of electricity at rest over conductors, and that we know the potential at all points and consequently the level or equipotential surfaces. Take any equipotential surface enclosing the whole of the electricity, and suppose this to become an actual sheet of metal connected to the earth. It is then a zero potential surface, and every point outside is at zero potential as far as concerns the electric charge on the conductors inside. Then if U is the potential outside the surface due to this electric charge inside alone, and V that due to the opposite charge it induces on the inside of the metal surface, we must have U + V = 0 or U = −V at all points outside the earthed metal surface. Therefore, whatever may be the distribution of electric force produced by the charges inside taken alone, it can be exactly imitated for all space outside the metal surface if we suppose the inside charge removed and a distribution of electricity of the same sign made over the metal surface such that its density follows the law
σ = −(¼π) dU / dn
(27),
where dU/dn is the electric force at that point on the closed equipotential surface considered, due to the original charge alone.
Bibliography.—For further developments of the subject we must refer the reader to the numerous excellent treatises on electrostatics now available. The student will find it to be a great advantage to read through Faraday’s three volumes entitled Experimental Researches on Electricity, as soon as he has mastered some modern elementary book giving in compact form a general account of electrical phenomena. For this purpose he may select from the following books: J. Clerk Maxwell, Elementary Treatise on Electricity (Oxford, 1881); J.J. Thomson, Elements of the Mathematical Theory of Electricity and Magnetism (Cambridge, 1895); J.D. Everett, Electricity, founded on part iii. of Deschanel’s Natural Philosophy (London, 1901); G.C. Foster and A.W. Porter, Elementary Treatise on Electricity and Magnetism (London, 1903); S.P. Thompson, Elementary Lessons on Electricity and Magnetism (London, 1903)·