40 Electrostatics Field of Force [en. n 



Hence /(r) = A + Br + Cr\ 



and by definition / (r) = J ( |< ( r ) rfr ) rdr > 



so that on equating the two values of/' (r), 



B+Cr = rf<j>(r)dr. 



Therefore 



or <M r J=- 2 > 



so that the law of force is that of the inverse square. 



48. Maxwell has examined what charge would be produced on the inner 

 sphere if, instead of the law of force being accurately B/r*, it were of the 

 form B/r^+Q, where q is some small quantity. In this way he found that if q 

 were even so great as ^y^, the charge on the inner sphere would have been 

 too great to escape observation. As we have seen, the limit which Cavendish 

 was able to assign to q was $. 



It may be urged that the form B/r 2+ 9 is not a sufficiently general 

 law of force to assume. To this Maxwell has replied that it is the most 

 general law under which conductors which are of different sizes but geometri- 

 cally similar can be electrified similarly, while experiment shews that in point 

 of fact geometrically similar conductors are electrified similarly. We may 

 say then with confidence that the error in the law of the inverse square, if 

 any, is extremely small. It should, however, be clearly understood that 

 experiment has only proved the law B/r* for values of r which are great 

 enough to admit of observation. The law of force between two electric 

 charges which are at very small distances from one another still remains 

 entirely unknown to us. 



III. The Equations of Poisson and Laplace. 



49. There is still a third way of expressing the law of the inverse 

 square, and this can be deduced most readily from 

 Gauss' Theorem. 



Let us examine the small rectangular paralleli- 

 piped, of volume dxdydz, which is bounded by the 

 six plane faces 



FIG. 11. 



We shall suppose that this element does not con- 

 tain any point charges of electricity, or part of 

 any charged surface, but for the sake of generality 

 we shall suppose that the whole space is charged 



