﻿172 Messrs. E. W. B. Gill and J. H. Morrell on Short 



is the time the electron takes to pass from grid to plate, 

 v = fT. A further interval of time T brings the electron back 

 to the grid with velocity v. 



Assume now that superposed on the fixed potentials is an 

 alternating potential V sin pt between plate and grid; the 

 electric force due to this in the space between plate and grid 



y 



is ~ sin pt, and if — e is the charge on an electron the corre- 

 spending force on it is — -—sin pt towards the plate. 



Since V is taken to be very, small compared with V the 

 motion of the electron may to a first approximation be taken 

 as determined solely by V, i. e., its time across is T and 

 retardation /. 



The work done by V sinjt?£ depends upon the time t at 

 which the electron passes the grid, and for a particular value 



of t the work is equal to l •— -psin ptdx. The axis of x 



being perpendicular to plate and grid and x = being on the 

 grid. 



But the velocity at time t is 



JW-/(«-i„)=/T -/(*-<„), 



and the above work reduces to 



J 



fe — j (T -f t — t) sin pt dt, 



to 



(1) 



which finally gives ; 



Work on electron going from grid to plate = 



2eY / r Y oospto sinpt — sinpT-ht \ 



similarly, the work done on the same electron as it returns 

 from plate to grid comes out as 



-rpr{-cospt + 2L+ — C-s p2 —)■ • 12) 



Thus the velocities of the electrons on their arrival at the 

 plate or on their return to the grid depends on t , that is, on 

 the value of V sinp£ at the instant they pass through the 

 grid. Assuming a constant stream of electrons through the 

 grid, it is easily seen by integrating (1) for values of t 



between and — that the total work done per period is 

 p 



