142, 143] CONDENSERS. 277 



143. Condensers. The capacity q n decreases as R z in- 

 creases, becoming equal to ^ when R. 2 = oo . Accordingly by the 

 presence of the envelope the capacity of the sphere is increased 



T> 



in the ratio 2 which may be made very large. Such an 



Jl> 2 KT, 



arrangement of two conductors, by which the presence of the 

 second largely increases the capacity of the first, is called a con- 

 denser, for by it a larger quantity of electricity is condensed on 

 the first by raising it to a given potential, the second being to 

 earth. The coefficient q u , or <? 12 which is here equal to it, is 

 called the capacity of the condenser, and will be denoted by K. 

 It is not necessary that one conductor shall surround the other. 

 If it does not, we shall not have q 12 = q u , but in any condenser 

 we shall suppose the coefficients q u , q 22 , q l2 to be nearly equal. 

 In that case we need not distinguish between the two conductors, 

 or plates, of the condenser. 



The energy is 



= I ?n ( F, - F 2 ) 2 + *(- ?") Vf + (fc, + q u ) V, F,. 



In virtue of the supposition made regarding the <^s, the last 

 two terms are small compared to the first, and we may write 



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

 of the difference of potentials of the plates. If one of the plates 

 is to earth this is accurately true, and this is generally the condi- 

 tion in practice. 



Now e l = q n V, + q u V a = q n ( V l - F 2 ) + (q u + q u ) F 2 , 



e 2 = ?i2 Pi + q* F 2 = q u ( F 2 - FO + (q a - q u ) F 2 + (q l2 + q u ) V l , 

 or e = -e 



to the same order of approximation, and the two plates of a con- 

 denser receive equal and opposite charges, proportional to the 

 difference of their potentials. Using 



we get 



w __ _ 

 "~2 K~2 K' 



