42-46] Gauss 1 Theorem 37 



Corollaries to Gauss Theorem. 



43. THEOREM. If a closed surface be drawn, such that every point on it 

 is occupied by conducting material, the total charge inside it is nil. 



We have seen that at any point occupied by conducting material, the 

 electric intensity must vanish. Hence at every point of the closed surface, 



N = 0, so that 1 1 NdS = Q, and therefore, by Gauss' Theorem, the total charge 



inside the closed surface must vanish. 



The two following special cases of this theorem are of the greatest 

 importance. 



44. THEOREM. There is no charge at any point which is occupied by con- 

 ducting material, unless this point is on the surface of a conductor. 



For if the point is not on the surface, it will be possible to surround the 

 point by a small sphere, such that every point of this sphere is inside the 

 conductor. By the preceding theorem the charge inside this sphere is nil, 

 hence there is no charge at the point in question. 



This theorem is often stated by saying : 



The charge of a conductor resides on its surface. 



45. THEOREM. If we have a hollow closed conductor, and place any 

 number of charged bodies inside it, the charge on its inner surface will be 

 equal in magnitude but opposite in sign, to the total charge on the bodies inside. 



For we can draw a closed surface entirely inside the material of the 

 conductor, and by the theorem of 43, the whole charge inside this surface 

 must be nil. This whole charge is, however, the sum of (i) the charge on the 

 inner surface of the conductor, and (ii) the charges on the bodies inside the 

 conductor. Hence these two must be equal and opposite. 



This result explains the property of the electroscope which led us to the 

 conception of a definite quantity of electricity. The vessel placed on the 

 plate of the electroscope formed a hollow closed conductor. The charge on 

 the inner surface of this conductor, we now see, must be equal and opposite 

 to the total charge inside, and since the total charge on this conductor is nil, 

 the charge on its outer surface must be equal and opposite to that on the 

 inner surface, and therefore exactly equal to the sum of the charges placed 

 inside, independently of the position of these charges. 



The Cavendish Proof of the Law of the Inverse Square. 



46. We have deduced from the law of the inverse square, that the 

 charge inside a closed conductor is zero. The converse theorem, as we shall 

 now shew, is also true, so that from the observed fact that the charge inside 



