An $\alpha$-particle (mass 4 amu) and a singly charged sulfur ion (mass 32 amu) are initially at rest. They are accelerated through a potential V and then allowed to pass into a region of uniform magnetic field which is normal to the velocities of the particles. Within this region, the $\alpha$-particle and the sulfur ion move in circular orbits of radii $r_{\alpha}$ and $r_S$, respectively. The ratio ($r_S/r_\alpha$) is _____.

*Solution*

$r = \sqrt {\frac{{2mV}}{{q{B^2}}}} $

So, $r \propto \sqrt {\frac{m}{q}} $

$\frac{{{r_S}}}{{{r_\alpha }}} = \sqrt {\frac{{{m_S}}}{{{q_S}}} \times \frac{{{q_\alpha }}}{{{m_\alpha }}}} = \sqrt {\frac{{{m_S}}}{{{m_\alpha }}} \times \frac{{{q_\alpha }}}{{{q_S}}}} = \sqrt {\frac{{32}}{4} \times \frac{2}{1}} = 4$