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 P_{electron} and the positron is P_{positron.}

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

C_{e} = (0, Rsinθ, Rcosθ)

C_{p} = (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,

D^{2} = (Rsinθ - (-Rsinθ)^{2} + (Rcosθ - (1.5R – cosθ))^{2}

= 4R^{2}sin^{2}θ + (2Rcosθ – 1.5R)^{2}

= 4R^{2}sin^{2}θ + 4R^{2}cos^{2}θ – 6R^{2}cosθ +2.25R ^{� � �} �^{2}

= 6.25R^{2} – 6R^{2}cos^{2}θ

From the condition; D^{2}>4R^{2}

6.25R^{2} – 6R^{2}cos^{2}θ > 4R^{2}

2.25/6 > cosθ

cosθ < 0.375

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