Answer :

Because of electric charge the initial velocities are along –z for positron and +z for electron. Then it experiences Lorentz force and start rotation. The rotation is limited to the y-z plane as no involvement of any x directing components.

The momentum P is always perpendicular to the radius in circular motion.

Let us assume the momentum of the electron is Pelectron and the positron is Ppositron.

So, the coordinates of the center of their paths are going to be,

Ce = (0, Rsinθ, Rcosθ)

Cp = (0, -Rsinθ, 1.5R-Rcosθ)

The condition for the paths being non intersecting circles is that the distance between the centers is greater than 2R.

Distance between the center,

D2 = (Rsinθ - (-Rsinθ)2 + (Rcosθ - (1.5R – cosθ))2

= 4R2sin2θ + (2Rcosθ – 1.5R)2

= 4R2sin2θ + 4R2cos2θ – 6R2cosθ +2.25R � � �2

= 6.25R2 – 6R2cos2θ

From the condition; D2>4R2

6.25R2 – 6R2cos2θ > 4R2

2.25/6 > cosθ

cosθ < 0.375

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