Question

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 this region collides elastically with a rigid wall. The time after which the velocity of the particle becomes anti-parallel to its initial velocity is $\frac{m}{\alpha qB}(\pi +\beta )$. Find $(\alpha +\beta )$.

Answer 1

Answer: 6

As shown, the time after which velocity of the particle becomes anti-parallel is
$T=2t_1+2t_2=2\left(\frac{\theta }{\omega }\right)+2\left(\frac{R}{v}\right)$
$=2\left[\frac{\pi /4}{qB/m}\right]+2\left[\frac{\dot{m}}{qB}\right]$
$=\frac{m\pi }{2qB}+\frac{2m}{qB}=\frac{m}{2qB}\cdot [\pi +4]$