278 ELECTROSTATICS. [PT. II. CH. VI. 



or the energy of a condenser is proportional to the square of the 

 charge of either plate. Since the forces tend to cause W v to in- 

 crease, if the potentials remain constant, and W e to decrease, if 

 the charges remain constant, the capacity K tends in any case to 

 increase. 



In the case of the sphere, if R 2 RI be denoted by r, and if S 

 be the area of the sphere whose radius R is the geometrical mean 

 of R lf R 2 , 



s 



j - - . 



T T 47TT 



If T is small, S is approximately the surface of either plate. 



144. Concentric Cylinders. If the internal conductor is 

 a circular cylinder of radius R^ t the external a hollow cylinder of 

 radii R z and ^ 3 , both of very great length, at a sufficient distance 

 from the ends we have V dependent only on r, and in the space 

 between the conductors ( 88 (9)), 



; 

 dr 2 r dr 



dV^A 



;F~7' 



F = A log r + B. 



The potentials of the conductors being V l and F 2 , 

 V!=A log R l 4- B, 



At the inner surface 



_!_ 8F_ 1 9F 

 " ] ~ 47r8/? 47r dr 



The charge on a length I of the cylinder is 



