384-386] Special Problems 339 



THE SOLUTION OF SPECIAL PROBLEMS. 

 Current-flow in an Infinite Conductor. 



386. A good approximation to the conditions of electric flow can 

 occasionally be obtained by neglecting the restrictive influence of the 

 boundaries of a conductor, and regarding the problem as one of flow between 

 two electrodes in an infinite conductor. For simplicity, we shall consider only 

 the case in which the conductor is homogeneous. 



The conditions to be satisfied by the potential V are as follows. We 

 must have V V^ over one electrode, and VV Z over the second electrode, 



dV 1 



while -^- must vanish at infinity to a higher order than and throughout 



the conductor we must have V 2 F= ( 376). We can easily see (cf. 186, 

 187) that these conditions determine V uniquely. 



Consider now an analogous electrostatic problem. Let the conducting 

 medium be replaced by air, while the electrodes remain conductors. Let 

 the electrodes receive equal and opposite charges of electricity until their 

 difference of potential is T^ V 2 . At this stage let ty denote the electro- 

 static potential at any point in the field. Let fa, fa be the values of ty over 

 the two electrodes, so that fa fa = Vi TJ. Then there will be a constant 

 C (namely V[ fa\ such that ty + C assumes the values T^, TJ respectively 

 over the two electrodes. Moreover VS/r = throughout the field, so that 

 V 2 (ty + C) = throughout the field, and i|r = at infinity except for terms 



-I O 



in (cf. 67) so that ~- (ty + C) vanishes at infinity to a higher order 



than . 

 r 2 



Hence \/r -f G satisfies the conditions which, as we have seen, must be 

 satisfied by the potential V in the current problem, and these are known to 

 suffice to determine V uniquely. It follows that the value of V must be 

 ^ + C. 



Thus the lines of flow in the current problem are identical with the lines 

 of force when the two electrodes are charged to different potentials in air. 



The normal current-flow at any point on the surface of an electrode is 



_1 d_V 

 r dn' 



so that the total flow of current outwards from this electrode 



222 



