A proton, a deuteron and an alpha particle having the same kinetic energy are allowed to pass through a uniform magnetic field perpendicular to their direction of motion. The ratio of the radii of their respective circular paths is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \( 1 : \sqrt{2} : 1 \)
In a perpendicular magnetic field the magnetic force supplies the centripetal force, \( qvB = mv^{2}/r \), so \( r = mv/qB \). In terms of the kinetic energy \( K \), \( p = \sqrt{2mK} \):
\( \displaystyle r = \frac{\sqrt{2mK}}{qB} \;\Rightarrow\; r \propto \frac{\sqrt{m}}{q} \quad (K, B \text{ same}) \)
Proton: \( m \), \( q = e \). Deuteron: \( 2m \), \( q = e \). Alpha particle: \( 4m \), \( q = 2e \).
\( \displaystyle r_p : r_d : r_\alpha = \frac{\sqrt{1}}{1} : \frac{\sqrt{2}}{1} : \frac{\sqrt{4}}{2} = 1 : \sqrt{2} : 1 \)