A proton and an alpha particle moving with equal kinetic energies enter perpendicularly into a magnetic field. If \( r_p \) and \( r_\alpha \) are the respective radii of their circular paths, what will be the ratio \( r_p/r_\alpha \)?
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \( 1 : 1 \)
For a charge moving perpendicular to a magnetic field, the magnetic force provides the centripetal force, and the momentum can be written in terms of the kinetic energy:
\( \displaystyle qvB = \frac{mv^{2}}{r} \;\Rightarrow\; r = \frac{mv}{qB} = \frac{\sqrt{2mK}}{qB} \)
With the same \( K \) and \( B \), \( r \propto \sqrt{m}/q \).
\( m_\alpha = 4m_p \)
\( q_\alpha = 2q_p \)
\( \displaystyle \frac{r_p}{r_\alpha} = \sqrt{\frac{m_p}{m_\alpha}} \times \frac{q_\alpha}{q_p} = \sqrt{\frac{1}{4}} \times 2 = 1 \)
So \( r_p : r_\alpha = 1 : 1 \).
Note: A similar question was also asked in CUET (UG) 2026 on 26-05-2026 (Slot 2), in CUET (UG) 2023 on 02-06-2023 (Slot 2), and in CUET (UG) 2022 on 18-08-2022 (Slot 2).