338 Steady Currents in continuous Media [CH. x 



In order that there may be no accumulation at any point under this new 

 distribution, the components of current must satisfy the equation of con- 

 tinuity, so that we must have 



du dv dw 



,-+ + = ........................... (315). 



dx dy dz 



By the same reasoning as in 383, we find for the rate at which heat is 

 generated under the new system of currents, 



fff (/ 1 3V y / i W Y / i dV v) j 



I II Hi ~ 3- + *) + (-- ^T + v ) +(-- a"+ w ) \dccdydz, 



J JJ (\ r dx ) V T 9y / V T dz ) } 



which, on expanding, is equal to 



f/Yi f/ary /ary fi^Y) j , 7 



I - 1 -5- )+(-*- 1 + -5- ) r dxdydz 



JJ] r \\dxj \dyj \d2j J 



3V dV dV 



dxdydz .............................. (316). 



Ofi transforming by Green's Theorem, the second term 



xyz - V(lu 



The volume integral vanishes by equation (315), the integrand of the 

 surface integral vanishes over each electrode from the condition that the total 

 flow of current across the electrode is to remain unaltered, and at every point 

 of the insulating boundary from the condition that there is to be no flow 

 across this boundary. Thus the new rate of generation of heat is represented 

 by the first and third terms of expression (316). The first term represents 

 the old rate of generation of heat, the third term is an essentially positive 

 quantity. Thus the rate of heat generation is increased by any deviation 

 from the natural distribution of currents, proving the result. 



385. An immediate result of this is that any increase or decrease in the 

 specific resistance of any part of a conductor is accompanied by an increase 

 or decrease of the resistance of the conductor as a whole. For on decreasing 

 the value of r at any point and keeping the distribution of currents 

 unaltered, the rate of heat production will obviously decrease. On allow- 

 ing the currents to assume their natural distribution, the rate of heat 

 production will further decrease. Thus the rate of heat production with a 

 natural distribution of currents is lessened by any decrease of specific 

 resistance. But if / is the total current transmitted by the conductor, and 

 R the resistance of the conductor, this rate of heat production is RP. 

 Thus R decreases when r is decreased at any point, and obviously the 

 converse must be true (cf. 359). 



