A proton and an α-particle, having kinetic energies Kₚ and K α respectively, enter into a magnetic field at right angles.
The ratio of the radii of trajectory of proton to that of α-particle is 2 : 1. The ratio of Kₚ : K α is :

Solution

Physics sir · 2 steps

Write the radius in terms of kinetic energy, flip it round for K, then put in each particle's charge and mass.

1. Radius from K. The magnetic force supplies the centripetal force, so r = mv/qB = p/qB. Momentum and kinetic energy are linked by p = √2mK, so r = √2mK/qB.

   K = q²B²r²/2m ⇒ K ∝ q²r²/m

   Particle	Charge q	Mass m	Radius r	q²r²/m
   Proton	e	m	2	(1 × 4)/1 = 4
   α-particle	2e	4m	1	(4 × 1)/4 = 1

2. So. Kₚ : K_α = 4 : 1, which is option (b). The α-particle's doubled charge, once squared, exactly cancels its four times larger mass.

The idea to keep. In a magnetic field, r = √2mK/qB, so K ∝ q²r²/m. Always square both the charge and the radius, and keep the mass in the denominator.

Answer from the NTA final answer key. Solution by PrepWiser.