﻿Electric Waves obtained by Valves. 175 



work done falls off much more rapidly on the short-wave 

 side of the maximum than on the long-wave side. 



9. To calculate the time T an electron from the filament 

 takes to go from the grid to the plate when the grid potential 

 is V volts above both filament and plate it is not necessary 

 to assume the grid and plate to be parallel, but they may be 

 taken, as they actually are, to be concentric cylinders of 



radii a, b. 



The retarding force on the electron when it is at distance v 



k b 



from the axis is -, where & = V/log e -. 

 t a 



The equation of motion is therefore 



l 2 r - eh 



m 



which oives when integrated twice, remembering that 

 dr 



= 0, when r = b, 

 at 



\/jv 1o 4J 



V 



log 



T = b \ / -w log, - ] e~ x \lx. 



In the actual valve used a = '5 cm., 6 = 1*25 cm., and, taking 

 -'= 5*3 x 10 17 E.S. units and measuring V in volts, 



711 



T = - r= second, 



w 



the accuracy of this being limited by the accuracy to 

 which a and b are known and probably from 5 to 10 per cent. 



The wave-length for any relation between T and 1/p can 

 now be at once calculated. If p = n7r/T, the time of one 

 oscillation is 2ir/p or 2T/n and the wave-length in cm. is 

 6xl0 10 T/>?,. 



The simple theory shows that for the oscillations of 

 maximum amplitude pT has a certain value about 37r/4. But 



Tec , andXx — , and hence the connexion between the 



v 7 V P 



grid voltage V and the wave-length X of maximum 

 oscillation is X 2 V = constant. 



10. The theory is thus in good general agreement with 

 the experimental results, but there is one fact unaccounted 

 for — that being the variation in the wave-length of the 



