An electron and a positron are released from (0, 0, 0)

Question:

An electron and a positron are released from (0, 0, 0) and (0, 0, 1.5R) respectively, in a uniform magnetic field

$B=B_{0} \hat{i}$ each with an equal momentum of magnitude $\mathrm{p}=\mathrm{eBR}$. Under what conditions on the direction

of momentum will the orbits be non-intersecting circles?

Solution:

When the centres are greater than 2R, then the circular orbits of electron and positron shall not overlap.

Let d be the distance between Cp and Ce

Then d2 = 4R2 + 9/4R2 – 6R2 cosθ

As d is greater than 2R,

9/4 > 6 cos θ or cos θ < 3/8

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