380-384] Generation of Heat 337 



382. As an example of the distribution of a surface charge, we may 

 notice that the surface-density of the charge on the surface of the sphere 



dV 

 considered in 380 will be proportional to either value of -= , and therefore 



to cos 0, where 6 is the angle between the radius through the point and the 

 direction of flow of the undisturbed current. 



GENERATION OF HEAT. 



383. Consider any small element of a tube of flow, length ds, cross- 

 section ft). The current per unit area is, by equations (310), -- -= , so 



1 dV 

 that the current flowing through the tube is -- -=- co. The resistance of 



T 08 



the element of the tube under consideration is -- . Hence, as in 355, the 

 amount of heat generated per unit time in this element is 



/I dV y rds 1 flV\* , 



- -5- I - or - ( -5- a) ds. 



\T OS ) ft) T V OS J 



Thus the heat generated per unit time per unit volume is - f^r] , and 

 the total generation of heat per unit time will be 



+ 



Thus the heat generated per unit time is STT times the energy of the 

 whole field in the analogous electrostatic problem ( 168). 



Rate of generation of heat a minimum. 



384. It can be shewn that for a given current flowing through a con- 

 ductor, the rate of heat generation is a minimum when the current distributes 

 itself as directed by Ohm's Law. To do this we have to compare the rate of 

 heat generation just obtained with the rate of heat generation when the 

 current distributes itself in some other way. 



Let us suppose that the components of current at any point have no 

 longer the values 



_idv __idv _idv 



r dx ' r dy ' T dz 

 assigned to them by Ohm's Law, but that they have different values 



IdV IdV IdV 



-~- + u, -^-+^> -*- + w > 

 T ox T dy T oz 



j. 22 



