A particle of charge $q$ and mass $m$ enters normally (at point $P$ ) in a region of magnetic field with speed $v$ . It comes out normally from $Q$ after time $T$ as shown in figure. The magnetic field B is present only in the region of radius R and is uniform. Initial and final velocities are along radial direction and they are perpendicular to each other. For this to happen, which of the following expression(s) is/are correct?

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$B = \frac{m v}{q R}$

$T = \frac{π m}{2 q B}$

$T = \frac{π R}{2 v}$

None of these

answer is A, B, C.

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Detailed Solution

The particle will move along an arc which is part of a circle of radius:
$r = \frac{m v}{B q}$
From figure, we can see that $r = R$
$∴ R = \frac{m v}{B q}$
$T = \frac{π r / 2}{v} = \frac{π r}{2 v} = \frac{π R}{2 v} (∵ r = R = \frac{m v}{B q})$
$∴ T = \frac{π m}{2 B q}$
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