﻿166 Mr, G. H. Livens on Lorentz's 



The metallic atoms being considered as practically immov- 

 able, the velocity of an electron will not be altered by a 

 collision. Let us, therefore, now fix our attention on a 

 certain group of electrons moving along their zigzag lines 

 with the definite Telocity u. Consider one of these electrons 

 and let us calculate the chance of its colliding with an atom 

 at rest in a unit of time. This chance is obviously equal to 

 the number of atoms in a cylinder of base 7rR 2 and height u, 

 R beino - as before the sum of the radii of an atom and an 

 electron ; it is therefore equal to 



^7rR 2 ^, 



n being the number of atoms per cubic centimetre in the 

 metal. 



But in unit time the electron under consideration travels 

 a distance iu hence the chance of a collision of the electron 

 with an atom per unit length of its path is 



n inrRht _ 



L= —nirii-, 



u 



and thus the mean free path of an electron is, as before, 



/ =i= 1 * 

 m c mrR? ' 



It is important to notice for future reference that l m is 

 independent of u. This is a consequence of the assumed 

 rigidity of the atoms. 



Now during the time 6 one of the electrons moving with 

 a velocity u describes a large number of paths, this number 

 being given by 



u6 



and we now want to know how many of these paths are of 

 oiven length /. 



For this, let f(l) be the probability that the electron shall 

 describe a path at least equal to I, then f(l + dl) is the prob- 

 ability that the electron has described a path I and shall 

 describe a further distance dl, and this will necessarily be 

 the product of f(l) and another factor, this second factor 

 expressing the probability of no collision occurring within 



* Lorentz does not make it clear that the l m introduced here is, in 

 fact, identical with that l m used iu the formula for the conductivity ; 

 the expanded argument here given, however, proves directly what was 

 probably already known to him. 



