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Question

A charged particle +q of mass m is placed at a distanced from another charged particle -2q of mass 2m in a uniform magnetic field B as shown in fig. If the particles are projected towards each other with same speed v,
find the maximum value of projection speed $v_{m}$ so that the two particles do not collide

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Solution

The particle will move in circular paths, as velocity vector is perpendicular to the magnetic field. Time period of both the particles is same $(T = \frac{2 π m}{B q})$ .
So, for collision not to take place,
$r_{1} + r_{2} < d$
$\frac{m v}{B q} + \frac{2 m v}{2 B q} < d$ or $v < \frac{B q d}{2 m}$
Therefore, maximum speed should be $\frac{B q d}{2 m}$
i.e., $v_{m} < \frac{B q d}{2 m}$

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