A proton, a deutron and an α-particle with same kinetic energy enter into a uniform magnetic field at right angle to magnetic field. The ratio of the radii of their respective circular paths is :

Solution

Physics sir · 2 steps

Write the radius in terms of kinetic energy, then the alpha's double charge cancels its extra mass.

1. Radius formula. The magnetic force supplies the centripetal force, so qvB = mv²/r, which gives r = mv/qB. Kinetic energy is fixed, so write the momentum as mv = √2mK.

   r = √2mK/qB ⇒ r ∝ √m/q

   Particle	Mass	Charge	√m/q
   Proton	m	e	1
   Deuteron	2m	e	√2
   α-particle	4m	2e	√4/2 = 1

2. So. The radii are in the ratio 1 : √2 : 1, which is option (d).

The idea to keep. Pick the form of r that matches what is held equal: same speed gives r ∝ m/q, same kinetic energy gives r ∝ √m/q, same accelerating voltage gives r ∝ √(m/q).

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