Resistance of Thin Metallic Films. 477 



When the electric field is absent the number of electrons 

 crossing one of the gaps in any given direction will of course 

 be balanced by the number crossing in the opposite direction. 

 When the field which we shall call X is on, however, the 

 symmetry of the flow will be disturbed, and we shall obtain a 

 resultant current. The current for a given field will be less 

 than when the gaps are absent, and since the conductivity of 

 the metal in the slabs separating the gaps has to remain the 

 same, the result will be that a displacement of electricity 

 will take place in the slabs, so that there will be a finite drop 

 of potential V across each gap, of such an amount as to 



V 



insure that the field X in the slabs is such as will drive 



a current through them equal to that which crosses the 

 gaps. ^ V 



If we imagine the field X— y , which we shall call X 3 , 



to urge the electrons from left to right, it must be noted that 

 owing to the potential drop across the gaps, the energy cor- 

 responding to the a velocity which an electron travelling 

 from right to left must have by the time it reaches the gap, 

 in order to cross, is less than ^mu 2 by a quantity of the order 

 of half the energy produced in the electron in a drop through 

 a potential V. The actual value of h necessary is such that 



2 , 2 v e Ye 

 m in 



(see Appendix, Problem 1), where r) is a function of V, and 

 is a quantity such that rjejuui 2 is of the order (Ye/mu 2 ) 2 

 compared with unity. On the other hand, for an electron 

 travelling in the opposite direction, the necessary value of x 

 is (see Appendix) such that 



.„ 2 , 2rje Ye 



x 2 >• u 2 H S- j . 



m m 



The reason for the existence of the term Ye/m in the above 

 is obvious, and it will not be advisable here to break in upon 

 the general train of reasoning in order to discuss the sio-ni- 

 ficance of the quantity 77, which is of the nature of a correction, 

 and is fully explained in the Appendix. 



The current density across the gap is to be obtained by 

 calculating the rate of flow of electrons per square centimetre 

 across the gap from left to right, subtracting the total rate 

 of flow from right to left, and multiplying the result by e. 



