An electron, a proton and an alpha particle having the same kinetic energy are moving in circular orbits of radii $$r_e$$, $$r_p$$, $$r_\alpha$$ respectively in a uniform magnetic field B. The relation between $$r_e$$, $$r_p$$, $$r_\alpha$$ is:

We begin with the fact that a charged particle of mass $$m$$ and charge magnitude $$|q|$$, moving with speed $$v$$ perpendicular to a uniform magnetic field of induction $$B$$, describes a circular path whose radius is given by the cyclotron‐motion formula

$$r=\frac{m v}{|q|\,B}.$$

The electron, the proton and the alpha particle are all said to possess the same kinetic energy. Let us call this common value $$K$$. The usual expression for kinetic energy is

$$K=\frac{1}{2}\,m v^{2}.$$

Solving this relation for the speed $$v$$ of any one of the particles, we obtain

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

Now we substitute this value of $$v$$ back into the radius formula. Doing so gives

$$r=\frac{m}{|q|\,B}\;\sqrt{\frac{2K}{m}}.$$

Inside the square root the factor $$m$$ in the denominator cancels partly with the $$m$$ in the numerator, leaving us with

$$r=\frac{\sqrt{2K}}{B}\;\frac{\sqrt{m}}{|q|}.$$

Because $$\sqrt{2K}/B$$ is a constant common to all three particles (the kinetic energy and the magnetic field are the same for each of them), the radius for any particle is directly proportional to the factor $$\dfrac{\sqrt{m}}{|q|}$$. So we can write symbolically

$$r\;\propto\;\frac{\sqrt{m}}{|q|}.$$

We therefore compare the three particles only through their masses and charges:

• Electron: mass $$m_e$$, charge magnitude $$e$$.
• Proton:    mass $$m_p$$, charge magnitude $$e$$.
• Alpha particle: mass $$m_\alpha = 4m_p$$, charge magnitude $$2e$$ (because it carries two positive elementary charges).

Computing the relevant proportional factors one by one:

For the electron

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

For the proton

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

For the alpha particle we first take its mass $$m_\alpha = 4m_p$$, so $$\sqrt{m_\alpha}=\sqrt{4m_p}=2\sqrt{m_p}$$. Hence

$$r_\alpha \;\propto\;\frac{\sqrt{m_\alpha}}{2e} \;=\;\frac{2\sqrt{m_p}}{2e} \;=\;\frac{\sqrt{m_p}}{e}.$$

This shows immediately that

$$r_p = r_\alpha$$

because both are proportional to the identical factor $$\dfrac{\sqrt{m_p}}{e}$$, whereas the electron radius is proportional to $$\dfrac{\sqrt{m_e}}{e}$$ with $$\sqrt{m_e} \ll \sqrt{m_p}$$ (the electron’s mass is about 1836 times smaller than the proton’s mass).

Therefore we have the inequality

$$r_e \;<\; r_p = r_\alpha.$$

Among the given options this order corresponds exactly to option 3.

Hence, the correct answer is Option 3.