1. Radius of circular motion in a magnetic field

For a charged particle moving perpendicular to a uniform magnetic field,

so,

where $p$ is the momentum.

2. Use the fact that kinetic energy is same

Given same kinetic energy $K$ for all particles,

Therefore,

Hence,

Since $K$ and $B$ are same for all, we get

3. Now compare proton, deuteron, and alpha particle

• Proton: mass $m_p$, charge $e$

  $$r_p \propto \frac{\sqrt{m_p}}{e}$$

• Deuteron: mass $\approx 2m_p$, charge $e$

  $$r_d \propto \frac{\sqrt{2m_p}}{e} = \sqrt{2}\,\frac{\sqrt{m_p}}{e}$$

  So,

  $$r_d = \sqrt{2}\,r_p > r_p$$

• Alpha particle: mass $\approx 4m_p$, charge $2e$

  $$r_\alpha \propto \frac{\sqrt{4m_p}}{2e} = \frac{2\sqrt{m_p}}{2e} = \frac{\sqrt{m_p}}{e}$$

  So,

  $$r_\alpha = r_p$$

4. Final comparison

Thus,

5. Check options

• A: $r_\alpha = r_p = r_d$ ❌
• B: $r_\alpha = r_p < r_d$ ✅
• C: $r_\alpha > r_d > r_p$ ❌
• D: $r_\alpha = r_d > r_p$ ❌

Therefore, the correct option is B.