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A particle of mass $m$ and charge $q$ enters a region of magnetic field (as shown) with speed $v$. There is a region in which the magnetic field is absent, as shown. The particle after entering the region collides elas tically with a rigid wall. Time after which the velocity of particle becomes anti parallel to its initial velocity is
$\frac{m}{{2qB}}\left( {\pi + 4} \right)$
$\frac{m}{{qB}}\left( {\pi + 2} \right)$
$\frac{m}{{4qB}}\left( {\pi + 2} \right)$
$\frac{m}{{4qB}}\left( {2\pi + 3} \right)$
Solution
As shown, the time after which velocity of particle becomes antiparallel is :
$\mathrm{T}=2 \mathrm{t}_{1}+2 \mathrm{t}_{2}$
$=2\left(\frac{\theta}{\omega}\right)+2\left(\frac{\mathrm{R}}{\mathrm{v}}\right)=2\left[\frac{\pi / 4}{\mathrm{qB} / \mathrm{m}}\right]+2\left[\frac{\mathrm{m}}{\mathrm{qB}}\right]$
$=\frac{\mathrm{m} \pi}{2 \mathrm{qB}}+\frac{2 \mathrm{m}}{\mathrm{qB}}=\frac{\mathrm{m}}{2 \mathrm{qB}} [\pi+4]$
