334 Steady Currents in continuous Media [CH. x 



The same equation can be obtained at once on considering the current- 

 flow across the different faces of a small rectangular parallelepiped of edges 

 dx t dy, dz (cf. 49). 



Equation (310) of course expresses that the vector C of which the com- 

 ponents are u, v, w, must be solenoidal. The equation of continuity can 

 accordingly be expressed in the form 



div C = 0. 



Equation satisfied by the Potential. 



376. On substituting in equation (311) the values for u, v, w given by 

 equations (310), we obtain 



d -I\ 1 (1 3T\ _ ft 

 dy) + dz('T dz)~ 



The potential must accordingly be a solution of this differential equation. 

 The equation is the same as would be satisfied by the potential in an 

 uncharged dielectric in an electrostatic field, provided the inductive capacity 



at every point is proportional to - . If the specific resistance of the conduc- 

 tor is the same throughout, the differential equation to be satisfied by the 



potential reduces to 



V 2 V = 0. 



377. We may for convenience suppose that the current enters and leaves 

 by perfectly conducting electrodes, and that the conductor through which the 

 current flows is bounded, except at the electrodes, by perfect insulators. 

 Then, over the surface of contact between the conductor and the electrodes, 

 the potential will be constant. Over the remaining boundaries of the con- 

 ductor, the condition to be satisfied is that there shall be no flow of current, 



3F 

 and this is expressed mathematically by the condition that = shall vanish. 



Thus the problem of determining the current-flow in a conductor amounts 

 mathematically to determining a function V such that equation (312) is 



satisfied throughout the volume of the conductor, while either = 0, or else 



V has a specified value, at each point on the boundary. By the method used 

 in 188, it is easily shewn that the solution of this problem is unique. 



It is only in a very few simple cases that an exact solution of the problem 

 can be obtained. There are, however, various artifices by which approxima- 

 tions can be reached, and various ways of regarding the problem from which 

 it may be possible to form some ideas of the physical processes which deter- 

 mine the nature of the flow in a conductor. Some of these will be discussed 

 later ( 386 394). At present we consider general characteristics of the 

 flow of currents through conductors. 



