38-42] Gauss' 1 Theorem 33 



41. If the electric distribution is not confined to points, we can imagine 

 it divided into small elements which may be treated as point charges. For 

 instance if the electricity is spread throughout a volume, let the charge on 

 any element of volume dx'dy'dz' be pdx'dy'dz so that p may be spoken of as 

 the " density " of electricity at x, y', z . Then in formula (11) we can replace 

 e l by pdx'dy'dz', and x lt y 1} z lt by x', y', z'. Instead of summing the charges 

 61, ... we of course integrate pdx'dy'dz' through all those parts of the space 

 which contain electrical charges. In this way we obtain 



' 



-fff- PMWM 



111 - X + - + *- t 



and 



X J + (y- yj 



These equations are one form of mathematical expression of the law of 

 the inverse square of the distance. An attempt to perform the integration, 

 in even a few simple cases, will speedily convince the student that the form 

 is not one which lends itself to rapid progress. A second form of mathe- 

 matical expression of the law of the inverse square is supplied by a Theorem 

 of Gauss which we shall now prove, and it is this expression of the law which 

 will form the basis of our development of electrostatical theory. 



II. Gauss Theorem. 



42. THEOREM. If any closed surface is taken in the electric field, and 

 if N denotes the component of the electric intensity at any point of this surface 

 in the direction of the outward normal, then 



where the integration extends over the whole of the surface, and E is the total 

 charge enclosed by the surface. 



Let us suppose the charges in the field, both inside and outside the closed 

 surface, to be e l at P lf e 2 at P 2 > and so on. The intensity at any point is 

 the resultant of the intensities due to the charges separately, so that at any 

 point of the surface, we may write 



N = N, + N,+ (12), 



where N lt J\ r 2 , ... are the normal components of intensity due to e lf e 2 , ... 

 separately. 



Instead of calculating i\NdS we shall calculate separately the values of 



jJN.dS, (JN 2 dS, . . .. The value of [JNdS will, by equation (12), be the 



of these integrals. 



J. i 



sum 



