R) respectively, in a
uniform magnetic field 
= 
, each with an equal
momentum of magnitude p = e BR. Under what conditions on the direction of
momentum will the orbits be non-intersecting circles ?
Answer:
Here, 
is
acting along the x-axis. For a circular orbit, the momentum of the electron and
positron are in y-z plane. Let 
and 
be the momentum of the electron
and positron respectively. Both of them due to same momentum (= e BR) move on
circular orbits, each of radius R, but in opposite sense. Let make an angle 
with the y-axis and must make the same angle with y-axis. Refer Fig.
3(EP).7. The centres of the respective circular orbits must be perpendicular to
the momenta at a distance R. Let the centre of the circular orbit of
the electron be at 
and of the
positron be at 
The
coordinates of is (0, R sin , R cos )
The coordinates
of is 
The
two circular orbits will not intersect if the distance between their two
centres is greater than 2 R.
Let
d be the distance between and Then
The
two circular orbits will not intersect each other if d > 2 R or 
> 4
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