ELECTRIC CHARGE AND THE CONDENSER. 131 



We may substitute CE for Q in equation (i), according to 

 equation (i) of Art. 77, and we get: 



W=%CE? (2) 



or we may substitute Q/C for E in equation (i), according to 

 equation (i) of Art. 77, and we get: 



IF-*? (3) 



Following is a rigorous derivation of equation (3) as applied 

 to a stretched spring. Let g be the elongation of the spring 

 when the stretching force is e. Then q and e are proportional, 



so that: 



q = Ce (4) 



where C is a constant for the given spring. Let Ag be the added 

 elongation due to an increment Ae of the stretching force, and 

 let &W be the work done on the spring to produce the added 

 elongation. Then: 



APF is greater than e-Ag 

 and 



AW is less than (e + Ae) Ag 

 or 



AW 



is greater than e and less than (e + Ae) 



Therefore APF/Ag approaches e as a limit when Ae and Ag 

 both approach zero; or, using differential notation, we have 

 dW/dq = e\ or, using the value of e from equation (4), we have: 



dW q 

 Hj ~C 



Now the potential energy W of the spring when its elongation is 

 Q, is the amount of work done in stretching the spring from 

 q = o to q = Q, and this is found by integrating equation (5) 

 from g = o to g = Q, which gives: 



