Proton, deuteron and alpha particles of the same kinetic energy are moving in circular trajectories in a constant magnetic field. The radii of proton, deuteron and alpha particles are respectively rp, rd, and rα. Which one of the following relations is correct?

The radius of the circular path of a charged particle in a magnetic field is given by:
r = (mv)/(qB) = (√(2mK))/(qB)
where:
m is the mass of the particle
K is the kinetic energy
q is the charge of the particle
B is the magnetic field strength
Since the kinetic energy K and the magnetic field B are the same for all three particles, we can write:
r ∝ √(m/q)
For a proton:
m = mp, q = e
rp ∝ √(mp/e)
For a deuteron:
m = 2mp, q = e
rd ∝ √(2mp/e) = √2 √(mp/e) = √2 rp
For an alpha particle:
m = 4mp, q = 2e
rα ∝ √(4mp/2e) = √2 √(mp/e) = √2 rp
Therefore, rd = rα = √2 rp. This means rα = rd > rp.
The correct relation is rα = rd > rp.