CHAPTER VIII. 



ELECTROKINETICS. STEADY FLOW IN CONDUCTORS. 



162. Ohm's Law. The condition of equilibrium of electri- 

 city in homogeneous conductors has been found to be that in each 

 conductor the potential has a constant value. If this condition is 

 not fulfilled in any conductor, the electrification changes with the 

 time, if the conductor be left to itself, or in ordinary terms elec- 

 tricity moves from one place to another in the conductor. The 

 laws of this flow of electricity were enunciated in 1827 by Georg 

 Simon Ohm*, although the notion of the potential was unknown 

 to him. If at any point in the conductor we construct an element 

 of surface dS, the quantity of electricity q crossing the surface in 

 the unit of time per unit of area will vary according to the direc- 

 tion of the normal to dS at the point. That direction of normal 

 for which the quantity per unit of time is greatest is called the 

 direction of the current at the point, and the quantity q is called 

 the current density. The current density is a vector quantity, 

 and its components according to the axes will be denoted by 

 u, v, w. If the quantities u, v, w are independent of the time, we 

 call the state of the conductor a state of steady flow. We shall 

 now consider the properties of the steady state. If we consider 

 any portion of a conductor in which there is no electricity created 

 nor destroyed, as much electricity must enter the space during any 

 interval as leaves it, or the whole flow resolved along the inward 

 normal must be zero. Accordingly 



( I ) = 1 1 q cos (qn) dS = II {u cos (nx) + v cos (ny) + w cos (nz)} dS 



Mdu dv dw\ 

 ~- -f ~- + ~- }dr ) 

 dx dy fa) 



* G. S. Ohm, Die galvanische Kette mathematisch bearbeitet. Berlin, 1827. 



