605-608] Metallic Media 527 



remain approximately unaltered. Finally, the values of a u , 6 n , ... become 

 so great that the equation assumes the limiting form (586). Thus the 

 supposition that the current may be treated as uniformly distributed through 

 the conductor the supposition on which our analysis has so far proceeded 

 has led us to undervalue the electrokinetic energy of the current by an 

 amount 



2wi O 2 + y 2 + w~) ........................... (587), 



where m is the mass of an electron, and the summation extends over all the 

 electrons which carry the current. 



607. We have to consider next how to modify our equations so as to 

 take account of this additional term in the kinetic energy. 



Let us suppose that in a unit volume of the conductor the current is 

 carried by N electrons, each of mass m and carrying a charge e. For 

 simplicity, suppose that the electrons all move with an equal velocity u 

 parallel to Osc. The current i is then parallel to Ox and is given by 



i = Neu ................................. (588). 



Let X be the aggregate electromotive force in the conductor given by 

 our former equations. Then the electrons in unit volume are acted on by a 

 force XNe. On our former theory, this force is just balanced by a force 

 arising from the resistance of the conductor. This force must accordingly be 

 of amount riNe, for on balancing these two forces we obtain 



XNe- riNe = Q ........ . ..................... (589), 



which is .equivalent to Ohm's Law. By D'Alembert's principle, the extra 

 term 



which we are now supposing to exist in the electrokinetic energy of the 

 electrons, can be allowed for by replacing equation (589) by 



XNe - riNe - ~ (Nmu) = 0, 



and this again, by equation (588), is equivalent to 



. m di 



/-rm\ 

 (090) ' 



Thus the required correction consists in replacing r by 



m d 



608. Returning to the analysis of 599, in which we were considering 

 the propagation of a plane wave of frequency p, we see that if K' is the true 

 value of the inductive capacity, we may suppose the equations of wave- 



