131-135] Gauss' Theorem 121 



of equation (61) to a sphere of radius r, having the point charge as centre, it 

 is found that the intensity at a distance r from the charge is 



e 



The force between two point charges e, e f , at distance r apart in a homo- 

 geneous unbounded dielectric is therefore 



ee 



and the potential of any number of charges, obtained by integration of this 

 expression, is 



Coulomb's Equation. 



134. The strength of a tube being measured by the charge at its end, it 

 follows that at a point just outside a conductor, P, the aggregate strength 

 of the tubes per unit of cross-section, becomes numerically equal to cr, the 

 surface density. We have also the general relation 



and on replacing P by a-, we arrive at the generalised form of Coulomb's 

 equation, 



R = ^ (67), 



in which K is the inductive capacity at the point under consideration. 



CONDITIONS TO BE SATISFIED AT THE BOUNDARY OF A DIELECTRIC. 



135. Let us examine the conditions which will obtain at a boundary at 

 which the inductive capacity changes abruptly from K l to K 2 . 



The potential must be continuous in crossing the boundary, for if P, Q, 

 are two infinitely near points on opposite sides of the boundary, the work done 

 in bringing a small charge to P must be the same as that done in bringing 

 it to Q. As a consequence of the potential being continuous, it follows that 

 the tangential components of the intensity must also be continuous. For if 

 P, Q are two very near points on different sides of the boundary, and P', Q' 

 a similar pair of points at a small distance away, we have V P V Q , and 

 Vp = VQ, so that 



PP' QQ' 



The expressions on the two sides of this equation are, however, the two 

 intensities in the direction PP', on the two sides of the boundary, which 

 establishes the result. 



