168 BRIDGMAN. 



Theoretical Discussion. • 



I have already mentioned that J. J. Thomson has suggested that 

 at extreme current densities the current will increase as the square 

 root of the E.M.F., instead of linearly. As far as I know, no theoreti- 

 cal discussion has ever been given of the magnitude of the effect to 

 be expected at lower current densities, where departures are jugt 

 beginning to be perceptible. Swann ^ has discussed the resistance of 

 thin films from the standpoint of the classical electron theory of con- 

 ductivity, and has retained the second order terms, subject to the 

 assumption of the classical theory that after every collision of elec- 

 tron with atom all vestige of the effect of the electric force acting on 

 the electron before the collision is wiped out. Under these assump- 

 tions he obtains a departure from Ohm's law in the opposite direction 

 from that found experimentally above, or given by Thomson's theory. 

 Subject to the same assumption I have also carried through an exami- 

 nation from the classical point of view, retaining the second and third 

 order terms in the expressions for the velocity imparted to the free 

 electron by the applied force during its free path, and arrive at a result 

 similar to Swann's, that is, a departure from Ohm's law in the drection 

 of a decrease of resistance with high E.M.F.'s. The first term in the 

 departure involves the square of the 'E.M.F., as considerations of 

 symmetry show that it must. This simple analysis must be incorrect, 

 however. It is not likely that the trend at very high E.M.F.'s found 

 by Thomson is reversed at lower E.M.F.'s, and it is also exceedingly 

 probable that the assumptions at the basis of the classical theory are 

 not exact. One would certainly expect an effect of higher order on 

 the velocity distribution by the applied force, even if there is no effect 

 of zero order. 



I have not been able to make the modifications in the classical analy- 

 sis which would be necessary to take account of terms of higher order 

 in the velocity distribution. I believe that it must involve the 

 details of atomic and crystal structure. 



Not being able to give the exact analysis that we could desire, we 

 may fall back on a dimensional analysis. If the mechanism of con- 

 duction is a free path mechanism, as is supposed in the classical 

 theory, and as I think most likely, we see that the departure from 

 Ohm's law is going to depend only on the kinematics of the motion of 

 an electron moving with a certain normal velocity in a free path when 

 acceleration is impressed by an additional force from without. This 

 assumes that the number of free electrons is not changed by the exter- 



