TRAVELING-WAVE TUBES 743 



This shows that applying a small magnetic field tends to decrease the gain. 

 This does not mean, however, that the gain with a longitudinal and a trans- 

 verse field and a magnetic field is less than the gain with the longitudinal 

 field alone. To see this, we can assume that not ^o but a^ is small. Differ- 

 entiating (55) we obtain 



dd 

 If a = 



3 3 r -2dd 2a^d8 da^ "1 



-ijS'C' = 8' (62) 



1 d^da^ 

 ^^ = 3(7T^)- (^^) 



li^l is zero, a small transverse field (small increase ina^) increases the magni- 

 tude of 5 without changing the phase angle. If /So ^ | 5 | , then 



d& = ZJ^ da' (64) 



Po 



and the change in 5 is purely imaginary. For the increasing wave, the change 

 in 6 as a transverse field is added will range from an increase in the real part 

 for small magnetic fields to no change in the real part for large magnetic 

 fields. 



APPENDIX 



Study of The Algebraic Equation 



' = -^^'^'\}^ + WTJ^ <»-^) 



5^5^ + /3o) + j^'C'id' -{- ^1 + a'52) = 



