GUN FOR STARTING ELECTRONS STRAIGHT IN MAGNETIC FIELD 827 



between Ai and A 2, the electron will move in a circular arc of some radius 

 fm, and at A 2 the radial velocity will be equal and opposite to that at ri; 

 that is, it will be — vn. 



The change in radial velocity of the electron in passing through the 

 aperture in A 2, Vr2, is 



Vr2= - ^ V2 (3) 



P2 



where F2 is the focal length of the lens at A 2 and V2 is the longitudinal 

 electron velocity at A2. F2 is made such that 



Vr2 = Vrl (4) 



Hence, the radial velocity —Vri of the approaching electrons is overcome 

 in passing through the aperture in A 2 and the electrons move parallel to the 

 axis to the right of A2. 



For temperature-limited emission and small space charge, we may assume 

 a uniform gradient between the cathode and Ai, and between Ai and A2. 

 Further, we may use the relation 



V^i = VK/Vi (5) 



From (l)-(5) we easily find that the required relation between Li, the spac- 

 ing from cathode to Ai, L2, the spacing between Ai and A2, and Vi and F2, 

 the potentials of Ai and A 2 with respect to the cathode, is 



L2/L1 = (VFV^2 + 1)(F2/Fi - 1) (6) 



In case of space-charge-Hmited emission, the space charge will cause the 

 gradient to the left of ^1 to be 3 times as great as in the absence of space 

 charge. If space charge is taken into account in this region only, L2/L1 as 

 obtained from (6) should be multiplied by f . 



We have still to determine the magnetic field required to return the elec- 

 trons leaving ^1 at a radius r to the radius r a,t A2. 



From Fig. 2 we see that the electrons turn through an angle $. Since the 

 angular velocity of electrons in a magnetic field is (e/m)B, 



$ = {e/m)B T (7) 



where r is the transit time between Ai and ^2. 

 As the electron moves between Ai and A 2 with a constant acceleration 



2L. 



2L2 



V2{e/m)V2 (1 + VTV^) 



(8) 



