1. For a charged particle moving perpendicular to a uniform magnetic field, the magnetic force provides the centripetal force:

$qvB = \frac{mv^2}{r}$

So,

$r = \frac{mv}{qB}$

2. The particles have the same kinetic energy $K$.

Using

$K = \frac{1}{2}mv^2$

we get

$v = \sqrt{\frac{2K}{m}}$

Substitute into the radius formula:

$r = \frac{m}{qB}\sqrt{\frac{2K}{m}}$

$r = \frac{\sqrt{2Km}}{qB}$

Thus,

$r \propto \frac{\sqrt{m}}{q}$

for fixed $K$ and $B$.

3. Now compare for each particle.

• Electron: mass $m_e$, charge magnitude $e$

$r_e \propto \frac{\sqrt{m_e}}{e}$

• Proton: mass $m_p$, charge magnitude $e$

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

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

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

Hence,

$r_\alpha = r_p$

4. Compare electron with proton:

Since $m_e \ll m_p$,

$\frac{\sqrt{m_e}}{e} < \frac{\sqrt{m_p}}{e}$

Therefore,

$r_e < r_p$

and since $r_p = r_\alpha$,

$r_e < r_p = r_\alpha$

5. Checking options:

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

Therefore, the correct option is C.