238 ENERGY OF CURRENTS. 



245. CONNECTION BETWEEN OHM'S AND JOULE'S LAWS. This 

 result can be arrived at in another way : 



Let us consider a conductor of capacity C, a battery for instance, 

 electrified to potential V ; the value of the potential energy is (89) 



-CV 2 . 



2 



Let us now suppose this battery connected to earth by a wire 

 whose resistance R is so great that its discharge has an appreciable 

 duration. During the time dt, a mass of electricity dlA flows out, 

 and the potential diminishes by^V; we have 



and the loss of energy in the same time is 



= C WV = WM = I Vdt. 



Ohm's law applies if the current remains sensibly constant during 

 the time dt ; from this it follows that 



-i- 



and 



V 2 



that is to say that the energy expended in unit time is expressed by 

 Joule's law. 



We have here deduced Joule's law from the principle of the 

 conservation of energy together with Ohm's law. Ohm's law might 

 conversely be deduced from the same principle combined with 

 Joule's law. For Joule's law gives 



W = I 2 R. 



We have further 



W-Elj 



from which follows 



E = IR, 



that is to say Ohm's law. 



246. We may here observe that in a multiple circuit, which does 

 not contain localised electromotive forces, the quantity of heat 

 developed is a minimum, when the currents come under Ohm's law. 



