358 



THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 



5. THE SPACE CHARGE EXPRESSION 



We have mentioned earlier that the space charge field is computed 

 from the disc-model suggested by Tien, Walker and Wolontis. In their 

 calculation, the force excited on one disc by the other is approximated 

 by an exponential function 



F. = 



— [a(z'— z)/ro] 



27rro-eo 



Here ro is the radius of the disc or the beam, q is the charge carried by 

 each disc, and eo is the dielectric constant of the medium. The discs are 

 supposed to be respectively at z and z . a is a constant and is taken 

 equal to 2. 



Consider two electrons which have their initial phases <pq and ^o and 

 which reach the position ij (or z) at times t and t' respectively. The time 

 difference, 



* - / = 1 



00 



wt — — Z — [bit — — Z) 



Vo \ Vq J 



CO 



multiplied by the velocity of the electron i<o[l + Cw(y, (po )] is obviously 

 the distance between the two electrons at the time t. Thus 



(z - z)t=t = - y(y, <Po) - <p(y, <Po)]uo[l + Cw(y,ipo)] (19a) 



In this equation, we are actually taking the first term of the Taylor's 

 expansion, 



(z — z)t=t = 



dzjij, cpo) 

 dt 



t=t 



(t _ /^ j_ ^ c?^2(y, <pq) 



it - ty 



t=t 



(19b) 



+ 



It is clear that the electrons at y may have widely different velocities 

 after having traveled a long distance from the input end, but changes in 

 their velocities, in the vicinity of y and in a time-period of around 2 tt, 

 are relatively small. This is why we must keep the first term of (19b) 

 and may neglect the higher order terms. From (19a) the space charge 

 field Es in (17) is 



2e 



Es = 





/+00 



-k]ip(.y ,<po+<t>)—<p(.U ,<Po) 1 li+Cw(y,(po+<t>)] 



d(f> sgn (<p(<po -\- <p) - <t>iy, <po)) 

 Here, e/m is the ratio of electron charge to mass, cop is the electron 



