An $\alpha$ particle and a proton are moving along the same direction with the same kinetic energy. They enter a uniform magnetic field acting perpendicular to their velocities. The ratio of radii of their paths is:

$1: 1$

The correct answer is Option (4) → $1: 1$

The radii (r) of charged particle is,

$r=\frac{mv}{qB}$

since K.E. is same for both $α$-particle and proton,

$\frac{1}{2}m_α{v_α}^2=\frac{1}{2}m_p{v_p}^2$

$⇒\frac{v_α}{v_p}=\sqrt{\frac{m_p}{m_α}}$

$∴\frac{r_α}{r_p}=\frac{\frac{m_αv_α}{q_α}}{\frac{m_pv_p}{q_p}}=\frac{\sqrt{m_αm_p}}{q_α}.\frac{q_p}{m_p}$

$=\frac{\sqrt{m_p.4m_p}}{2q_p}×\frac{q_p}{m_p}=\frac{1}{1}$