Orbits in the Field of a Doublet. 1011 



velocity, and the transverse velocity still satisfies the 



conditions 



r^Yi 2 - 2/4 cos 0! > 2/*, 



the particle approaches the origin to a distance r = r 2 



« iVenb ^ ,/,'/ = W + WZ-Vn 



in times given by 



t-t x = r 2 (x/Vi 2 - r 2 2 - v/'^^?) /\/V^ 

 and then recedes from the origin according to the law 



£ — t 1 = r 2 \/r 2 — r 2 2 /v 2yitm. 



While the particle is first approaching and then receding 

 from the origin, the variation of 6 with t is given by 



t-t x = i\ 2 J \ r 2/jbm tan \ / 2m((j> — (j) 1 ) ; 

 so that the orbit circumscribes the origin until that value 

 of 6 is reached for which (<fi — <p 1 )VZm = 7r/2. 

 A typical orbit is shown in fig. 5. 



Fig. 5. 



As in the previous cases, if the initial radial velocity is 

 towards the origin there is a finite excursion towards the 

 origin before the radial velocity becomes positive and increases 

 steadily as time goes on. 



Sum mary. 



The general characteristics of the orbits in the field of a 

 •doublet may be summed up as follows : — 



(a) The only periodic orbits are the semicircles 



r = r 



li 



-tt/2 < 6 < irj2, 



■lor various values of r x . The corresponding periodic time is 

 K(l/VWa. 



r,* 



3T2 



