452 Dr. W. F. G. Swann on the Expression for 



of electrical conduction to take full cognizance of the 

 essential phenomena by expressing the mean free path as 

 a function of the velocity with which the electron started its 

 journey, since we should expect that the factor which would 

 determine whether an electron suffered a collision in any 

 given element would be its velocity while in that element, 

 and this would depend upon the distance which the electron 

 had travelled resolved parallel to the field as well as upon 

 its initial velocity. In fact, it is probably more reasonable to 

 state the case in the following way. 



Let \ c be the mean free path for a velocity c in the absence 

 of the field, and let the fraction of N electrons which suffer 

 collisions in passing through an element of thickness d£ be 



f(l)d& so that 



"^-=N/(0 and N = N € ~V ( ^ 



• 

 When J is a constant over the path of an electron and equal 

 to the velocity c with which the electron started out, the 

 integral becomes f/(c), which shows that /(c) = 1/A C . In 

 order to obtain the number of electrons Bn' coming out from 

 the ring of volume 2irR 2 sin ty difr dR (p. 444) and passing 

 through E per square centimetre per second, and whose 

 velocities lie between c and c + dc, it is necessary to replace 



^r— n equation (10) by 



A€- hmc '~c 2 cdc» 



2X 



, v by c, and e E/A by 



, 2Xex\ 



*+—!-« gince i£ 



is the velocity with which an 



electron starts out, its velocity after travelling a distance 

 whose component parallel to the field is x amounts to 

 2Xex 



( , + 



m J i 



Now if we expand / ( 



c 2 + 



2X< 



?)> 



Taylor's theorem, 



then integrate with respect to R remembering that x = Rcos>>, 

 and finally replace /(c) by 1/X e , we readily find 



R 2 Xccosi/r BX C 



_ fB ( C 2 + «-! )*j R * R^COS^ ,&p __ c 



V v " = e -B/Xc € 2m*$ c be ~ \ l+ 2mX 2 c ' "be 



} 



.-B A 



* It will be noticed that the total number of electrons starting out from 

 an element after suffering collisions therein cannot be affected by the field 



Xe 

 to the first power of -—5" 



