A proton and an alpha particle move in circular paths in a plane perpendicular to a uniform magnetic field. If the kinetic energy of the two particles is equal, the ratio of the radii of their paths is _________.

Explanation

To solve the problem, we need to find the ratio of the radii of the circular paths of a proton and an alpha particle when they are moving in a magnetic field with equal kinetic energy.

### Step-by-Step Solution:

1. **Understanding the Motion in a Magnetic Field:**
When a charged particle moves in a magnetic field, it experiences a magnetic force that acts as a centripetal force, causing it to move in a circular path. The expression for the radius \( R \) of the circular path is given by:
\[ R = \frac{mv}{qB} \]
where:
- \( m \) = mass of the particle,
- \( v \) = velocity of the particle,
- \( q \) = charge of the particle,
- \( B \) = magnetic field strength.

2. **Kinetic Energy of the Particles:**
The kinetic energy \( KE \) of a particle is given by:
\[ KE = \frac{1}{2} mv^2 \]
Since we are given that the kinetic energies of the proton and the alpha particle are equal, we can denote them as:
\[ KE_p = KE_{\alpha} \]

3. **Expressing Velocity in Terms of Kinetic Energy:**
From the kinetic energy formula, we can express the velocity \( v \) in terms of kinetic energy:
\[ v = \sqrt{\frac{2 \cdot KE}{m}} \]

4. **Substituting Velocity in the Radius Formula:**
Substitute \( v \) into the radius formula:
\[ R = \frac{m \sqrt{\frac{2 \cdot KE}{m}}}{qB} = \frac{\sqrt{2 \cdot KE \cdot m}}{qB} \]

5. **Finding the Ratio of the Radii:**
Let \( R_p \) be the radius for the proton and \( R_{\alpha} \) be the radius for the alpha particle. Thus:
\[ R_p = \frac{\sqrt{2 \cdot KE \cdot m_p}}{q_p B} \]
\[ R_{\alpha} = \frac{\sqrt{2 \cdot KE \cdot m_{\alpha}}}{q_{\alpha} B} \]
Now, taking the ratio \( \frac{R_p}{R_{\alpha}} \):
\[ \frac{R_p}{R_{\alpha}} = \frac{\sqrt{m_p}}{q_p} \cdot \frac{q_{\alpha}}{\sqrt{m_{\alpha}}} \]

6. **Substituting Masses and Charges:**
- The mass of the proton \( m_p \) is approximately \( 1 \, u \) (atomic mass unit).
- The mass of the alpha particle \( m_{\alpha} \) is approximately \( 4 \, u \).
- The charge of the proton \( q_p \) is \( +1e \).
- The charge of the alpha particle \( q_{\alpha} \) is \( +2e \).

Substituting these values into the ratio:
\[ \frac{R_p}{R_{\alpha}} = \frac{\sqrt{1}}{1} \cdot \frac{2}{\sqrt{4}} = \frac{2}{2} = 1 \]

### Final Answer:
The ratio of the radii of their paths is:
\[ \frac{R_p}{R_{\alpha}} = 1 \]