72 Conductors and Condensers [CH. in 



which upon integration gives 



V=C + e - (27). 



We can determine the constant of integration as soon as we know the 

 potential of either of the spheres. Suppose for instance that the outer 

 sphere is put to earth so that F= over the sphere r = b, then we obtain at 

 once from equation (27) 



= C 4 



so that G= e/b, and equation (27) becomes 



On taking r a, we find that the potential of the inner sphere is e ( r), 



\CL 1 



and its charge is e, so that the capacity of the condenser is 



1 ab 



: r or 



b-a' 



80. In the more general case in which the outer sphere is not put to 

 earth, let us suppose that V a , V b are the potentials of the two spheres 

 of radii a and b, so that, from equation (27) 



Then we have on subtraction 



so that the capacity is 



The lines of force which starb from the inner sphere must all end on the 

 inner surface of the outer sphere, and each line of force has equal and 

 opposite charges at its two ends. Thus if the charge on the inner sphere is 

 e, that on the inner surface of the outer sphere must be e. We can there- 

 fore regard the capacity of the condenser as being the charge on either of 

 the two spheres divided by the difference of potential, the fraction being 

 taken always positive. On this view, however, we leave out of account any 

 charge which there may be on the outer surface of the outer sphere : this 

 is not regarded as part of the charge of the condenser. 



