Motion of Electrons in Solids. Ill 



In the steady state in which a steady current i x is main- 

 tained by an electric force X, the left-hand member must 

 vanish, so that we have 



Xe 

 u = — 



or by equation (2), 



Xe 2 



; * = V X (6) 



giving Ohm's law. The conductivity for steady currents, 

 which we shall call k 9 is given by 



K = ~J W 



If we replace u and 7 in equation (3) by their values as 

 given by equations (2) and (7), we obtain 



f = ^ 2 (X-L^ (8) 



at m \ k) 



7. Corresponding to a periodic impressed force 



X = X cos^>£, 



the solution of this equation is 



i x = tfX cos (pt — e) cos 6, .... (9) 



where 



m 

 tan€ = /cp^. 



We can readily calculate the rate at which energy is 

 dissipated by the resistance. In time dt and in volume 

 dxdydz, an amount of electricity i x dtdydz falls through a 

 potential-difference X^cos ptda\ its gain in momentum being- 

 nil. The work dissipated is accordingly 



i x X cos pt dx dy dz dt, 

 so that the loss of energy per unit volume per unit time is 

 fccX cos pt = kK 2 cos e cos (pt — e) cospt 



= *X 2 (cos 2 e cos 2 pt + cos e sin e cos pt sin pt). 



The average value of this, averaged over a number of 

 complete periods, is 



1 "V o 9 1 "V 2 ^ 



iKA ~ COS - 6 = -k A ^r~o 



J- ~\ XT^ 4 



