434 Prof. Richardson and Dr. Bazzoni : Experiments 



V' = jiVat any point and i ! =n s i at the same point, where 

 n is constant and s is to be determined. From (1) 



»(?)'-©'€)' • w 



From (3) 

 and 



J= 2 



^^©v^. 



Thus 



p' = n*-*p (11) 



But from (2) 



4ttp' = V'V = n V 2 V = iirnp, 

 and 



p'=np (12) 



For (11) and (12) to be compatible 



5 -3/2 (13) 



Corresponding to any solution V=/(#, y, z) of the equations 

 there will be a solution for which Y' = nY, p' = ?ip, and 

 i' = n d / 2 i. If the total potential difference is. changed then 

 the current will vary as the potential difference raised to the 

 power 1*5. 



When the maximum potential difference V z exceeds the 

 critical value V l5 the problem becomes more complicated. 

 Since the electrons lose all their kinetic energy by collisions 

 at the critical value eV] the equation (1), for values of V 

 lying between Vi and 2V l5 will have to be replaced by 



Jmi; 2 =e(V-V 1 ) = «U (14) 



In equations (2) and (3) the variable V can be replaced by 

 the new variable U. Thus the differential equation for the 

 space between the critical equipotential surface and the anode 

 is the same as before, the only change being that the potential 

 differences are now measured from the critical value. From 

 (2), (3), and (14), 



2*1^ + 0= (g) 3 (15) 



The further treatment of the problem depends very much 

 on what one supposes to take place at the critical impacts. 

 The work of Franck and Hertz has shown that at this stage 

 the electrons lose all their energy, and that some positive 



