A proton and an alpha particle are moving along a circular path in a uniform magnetic field with the same kinetic energy. The ratio of radii of the circular paths described by them is, respectively.
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 1:1
A charged particle moving perpendicular to a uniform magnetic field moves in a circle, with the magnetic force providing the centripetal force. Writing the momentum in terms of the kinetic energy, \( p = \sqrt{2mK} \):
\( \displaystyle qvB = \frac{mv^{2}}{r} \;\Rightarrow\; r = \frac{mv}{qB} = \frac{\sqrt{2mK}}{qB} \)
For the proton (mass m, charge e) and the alpha particle (mass 4m, charge 2e) with the same K and B:
\( \displaystyle r_p = \frac{\sqrt{2mK}}{eB} \)
\( \displaystyle r_\alpha = \frac{\sqrt{2(4m)K}}{2eB} = \frac{2\sqrt{2mK}}{2eB} = \frac{\sqrt{2mK}}{eB} \)
\( r_p : r_\alpha = 1 : 1 \)
Shortcut: For the same K and B, \( r \propto \sqrt{m}/q \): the proton gives \( \sqrt{1}/1 = 1 \) and the alpha particle \( \sqrt{4}/2 = 1 \), so the ratio is 1 : 1.
Note: A similar question was also asked in CUET (UG) 2025 on 29-05-2025 (Slot 2), in CUET (UG) 2023 on 02-06-2023 (Slot 2), and in CUET (UG) 2022 on 18-08-2022 (Slot 2).