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 $\frac{m}{\alpha q B}(\pi+\beta)$. Find $(\alpha+\beta)$.

Answer 1

As shown, the time after which velocity of the particle becomes anti-parallel is
$T =2 t_{1}+2 t_{2}=2\left(\frac{\theta}{\omega}\right)+2\left(\frac{R}{v}\right) $
$=2\left[\frac{\pi / 4}{q B / m}\right]+2\left[\frac{\dot{m}}{q B}\right]$
$=\frac{m \pi}{2 q B}+\frac{2 m}{q B}=\frac{m}{2 q B} \cdot[\pi+4]$