100 Systems of Conductors [OH. iv 



We notice that q^ = q^i, that the value of each is negative, and that 

 #11 = guy in accordance with 113. The value of q u is the capacity of 

 sphere 1 when 2 is to earth, and is in agreement with the result of 79. 



The capacity of 2 when 1 is to earth, q w , is seen to be j- . This can 



~~ a 



also be seen by regarding the system as composed of two condensers, the 

 inner sphere and the inner surface of the outer sphere form a single spherical 



condenser of capacity j- , while the outer surface of the outer sphere has 



o a 



capacity b. The total capacity accordingly 



ab 6 2 



b a b a' 



Two spheres at a great distance apart. 



116. Suppose we have two spheres, radii a, b, placed with their centres 

 at a great distance c apart. Let us first place unit charge on the former, the 



FIG. 39. 



charge being placed so that the surface density is constant. This will not 

 produce uniform potential over 2 ; at a point distant r from the centre of 1 

 it will produce potential 1/r. We can, however, adjust this potential to the 

 uniform value 1/c by placing on the surface of 2 a distribution of electricity 



such that it produces a potential over this surface. 



Take B, the centre of the second sphere, as origin, and AB as axis of x. 



Then we may write 



I I r ex f 1 



= - - = , as far as - . 



c r cr c 2 c- 



Let a be the surface density required to produce this potential, then 

 clearly o- is an odd function of a?, and therefore the total charge, the value of 

 a integrated over the sphere, vanishes. Thus the potential of 2 can be 

 adjusted to the uniform value 1/c without altering the total charge on 2 

 from zero, neglecting 1/c 3 . The new surface density being of the order of 

 1/c 2 , the additional potential produced on 1 by it will be at most of order 1/c 3 , 

 so that if we neglect 1/c 3 we have found an equilibrium arrangement which 

 makes 



^ = 1, E 2 =0, , = -, F 2 = i. 



a c 



