An alpha particle and a proton are moving with the same kinetic energy in a transverse uniform magne

Question

An alpha particle and a proton are moving with the same kinetic energy in a transverse uniform magnetic field. What is the ratio of the radii of their circular paths ($r_\alpha : r_p$)?

Professor PC Thomas Classes · Answer

Radius $r = \frac{mv}{qB} = \frac{\sqrt{2mK}}{qB}$. Since $m_\alpha = 4m_p$ and $q_\alpha = 2q_p$, $r_\alpha = \frac{\sqrt{2(4m_p)K}}{(2q_p)B} = \frac{2\sqrt{2m_p K}}{2q_p B} = r_p$. The ratio is 1:1.