A proton and an alpha particle are projected perpendicular to a uniform magnetic field with the same kinetic energy. The ratio of the radii of their…

The radius of a charged particle in a magnetic field is given by $r = \frac{\sqrt{2mK}}{qB}$. For the same kinetic energy $K$ and magnetic field $B$, $r \propto \frac{\sqrt{m}}{q}$. Mass of alpha particle $m_\alpha = 4m_p$ and charge $q_\alpha = 2q_p$. Thus, $r_p / r_\alpha = (\sqrt{m_p}/q_p) / (\sqrt{4m_p}/2q_p) = 1 / \sqrt{2}$.