666 BELL SYSTEM TECHNICAL JOURNAL 



= f°° r V'{x,)V'*(xOe'''''--''' dx.dx, 



J— 00 •f— 00 



d\A 



-00 •'—00 



i2 • ,» 00 ^00 



= i ( ( V'(x,)V'*{x^)y(x, - x,)e^''^'-^^' dx.dx, . (hl8) 



P J— 00 •'— 00 



^7 

 Hence, we see 



B. Transverse Field 



Suppose we consider the additional power transfer because of deflections. 

 There will be two sources of energy transfer. First, imagine a fluctuating 

 y component of velocity, y. Let i(o be the x component of velocity and — 7o 

 the convection current to the right. In a distance <fjc this will flow against 

 the potential gradient in the y direction a distance 



dy = (y/uo)dx (h20) 



and the power flowing to the held from the beam will be 



dP = -^-^^(r/uo)dx. (h21) 



2 dy 



This is not the total power transfer, however. The beam will also suffer 

 a displacement y in the y direction. Now the x component of field varies 

 with displacement; hence the beam will encounter a varying field. We 

 can write the instantaneous power transferred from the beam to the field. 

 Let {V)i be the instantaneous value of V and (y)i be the instantaneous 

 value of y. The instantaneous power will be 



dp= - 



e-^' +§&■»■)- 



Let us compare this with the instantaneous power transferred from the 

 beam to the field by a fluctuating convection current (i)i 



dp = ^' {i)i . \ (h23) 



We see that according to our convention that ■ 



F = VI* (h24) 



