FIELD SOLUTIONS 647 



Let us expand in terms of the quantity A /{Be — OY, assuming this to be 

 small compared with unity. We obtain 



Pe = 



1 + 77~-7^. + 



20(0e - eyi 4(de - d) 



(14.77) 



The theory of Chapter VII is developed by assuming that all electrons 

 are acted on by the same a-c field. When this is not so, it is applied approxi- 

 mately by using an "effective current" or "effective field" as in Chapter 

 IV; either of these concepts leads to the same averaging over the electron 

 flow. An effective current can be obtained by averaging over the flow the 

 current density times the square of the field, evaluated in the absence of 

 electrons, and dividing by the square of the field at the reference position. 

 This is equivalent to the method used in evaluating the effective field in 

 Chapter III. 



In the device of Fig. 14.2, if we take as a reference position y = ±d, 

 the effective current /o per unit depth 



/o = 



Jo / cosh^ (yy) dy 



(14.78) 



cosh- yd 



{Jd/2) ("^^ + sech^ yd) (14.79) 



This is the effective current associated with the half of the flow from y — 

 to y = d. Here y is the value for no electrons. For « |8, 7 = ^. For 

 large values of 6, then 



/o = Jod/2d (14.80) 



Now, the corresponding a-c convection current per unit depth will be: 



Here E is the total field acting on the electrons in the 2-direction. From 

 (7.1) we see that we assumed this to be the field due to the circuit (the first 

 term in the brackets) plus a quantity which we can write 



£a = ^ i (14.82) 



Accordingly 



E = E,-\- £,1 (14.83) 



