36 Electrostatics Field of Force [OH. n 



On the other hand if P l is outside the closed surface, as in fig. 9, the 

 cone through any element of area da) on the unit sphere may either not cut 

 the closed surface at all, or may cut twice, or four, six or any even number 

 of times. If the cone through dco intersects the surface at all, the first pair 

 of elements of surface which are cut off by the cone contribute e^dco and 



+ eidco respectively to I l^dS. The second pair, if they occur, make a similar 



contribution and so on. In every case the total contribution from any small 

 cone through P 1 is nil. By summing over all such cones we shall include 

 the contributions from all parts of the closed surface, so that if P 1 is outside 



the surface llN^S is equal to zero. 



We have now seen that 1 1 N^S is equal to 4>7re l when the charge e l is 



inside the closed surface, and is equal to zero when the charge e l is outside 

 the closed surface. Hence 



4<7r x (the sum of all the charges inside the surface) 



which proves the theorem. 



Obviously the theorem is true also when there is a continuous distribution 

 of electricity in addition to a number of point charges. For clearly we can 

 divide up the continuous distribution into a number of small elements and 

 treat each as a point charge. 



Since N, the normal component of intensity, is equal by 36 to -= , 



where ^ denotes differentiation along the outward normal, it appears that 



we can also express Gauss' Theorem in the form 



"dV, 



fo dS " 



Gauss' theorem forms the most convenient method at our disposal, of 

 expressing the law of the inverse square. 



We can best obtain an idea of the physical meaning underlying the 

 theorem by noticing that if the surface contains no charge at all, the theorem 

 3xpresses that the average normal intensity is nil. If there is a negative 

 charge inside the surface, the theorem shews that the average normal 

 intensity is negative, so that a positively charged particle placed at a point 

 on the imaginary surface will be likely to experience an attraction to the 

 interior of the surface rather than a repulsion away from it, and vice versa if 

 the surface contains a positive charge. 



