

Conductors by Electric Currents. 2G9 



6. General Case of a Homogeneous Isotropic Conductor. 

 Since identically 



( dV 2 dV 2 r/V 2 ^ 



(B) reduces by means of (A) to 



Now the conductor being initially homogeneous, s, c, and 

 A; are functions of d only> therefore we may put 



*=?<*)=<%; w 



and thus (B) finally reduces to 



l^-v/i/w+i/v 1 } < G > 



In the general case of varying movement the simultaneous 

 equations (A) and (G) are utterly intractable except by 

 methods of successive approximation, say, beginning with the 

 assumption of s, c, and k, independent of S. Passing, then, 

 at once to the consideration of the steady state, we have, 

 instead of (G), 



v;u( d )+4i v n=° (H) 



The general treatment of (A) and (H) is still extremely 

 difficult, but these equations being of the same form, we can 

 extend the method of Section 4 to give results similar to and 

 equally general with those of that section, and, further, to 

 obtain the final in terms of the initial distribution of potential. 

 In fact 



/(S)+i/V 2 = A + BV 



is an obvious solution, and therefore suitably determining A 

 and B we find that : — 



a. If any two parts of the surface of an initially homogeneous 

 conductor for which k = cf'($) are maintained at potentials V, 

 and V 2 , and at temperatures Sj and 3 2 respectively, and the 

 remainder of the boundary is such that there is no flux of heat 

 or electricity across it, then, when the steady state supervenes, 

 the equipotential surfaces will be isothermal, and the tempera- 

 ture at any point will be given in terms of the potential at 

 that point by the formula 



/(*) =i>(V,-V)(V-V 1 ) +{/(5 2 )(V-V 1 ) 



+./W(V 2 -V)}/(V 2 -V 1 ). . (2) 



