73-75] Spheres and Cylinders 67 



and its direction as we have seen is normal to the surface. Applying Gauss' 

 Theorem to this sphere, we find that the surface integral of normal intensity 



NdS becomes simply R multiplied by the area of the surface 4?rr 2 , so that 



or R = ~ . 



7 .2 



This becomes e/a? at the surface, agreeing with the value previously 

 obtained. 



Thus the electric force at any point is the same as if the charged sphere 

 were replaced by a point charge e, at the centre of the sphere. And, just 

 as in the case of a single point charge e, the potential at a point outside the 

 sphere, distant r from its centre, is 



77 f r e j e 

 V = - dr = - , 



Joo r 9 r' 



p 



so that at the surface of the sphere the potential is - . 



Inside the sphere, as has been proved in 37, the potential is constant, 

 and therefore equal to e/a, its value at the surface, while the electric intensity 

 vanishes. 



As we gradually charge up the conductor, it appears that the potential 

 at the surface is always proportional to the charge of the conductor. 



It is customary to speak of the potential at the surface of a conductor as 

 " the potential of the conductor," and the ratio of the charge to this potential 

 is defined to be the " capacity " of the conductor. From a general theorem, 

 which we shall soon arrive at, it will be seen that the ratio of charge to 

 potential remains the same throughout the process of charging any conductor 

 or condenser, so that in every case the capacity depends only on the shape 

 and size of the conductor or condenser in question. For a sphere, as we 

 have seen, 



charge e 



capacitv = - -^r-7 = - = a, 

 potential e 



a 

 so that the capacity of a sphere is equal to its radius. 



A Cylindrical Conductor. 



75. Let us next consider the distribution of electricity on a circular 

 cylinder, the cylinder either extending to infinity, or else having its ends so 

 far away from the parts under consideration that their influence may be 

 neglected. 



As in the case of the sphere, the charge distributes itself symmetrically, 



52 



