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 find the ratio of the radii of the circular paths of a proton and an alpha particle moving in a magnetic field with equal kinetic energy, we can follow these steps:

### Step 1: Understand the relationship between radius, mass, charge, and velocity in a magnetic field
The radius \( r \) of the circular path of a charged particle moving in a magnetic field is given by the equation:
\[ r = \frac{mv}{qB} \]
where:
- \( m \) is the mass of the particle,
- \( v \) is the velocity of the particle,
- \( q \) is the charge of the particle,
- \( B \) is the magnetic field strength.

### Step 2: Relate kinetic energy to velocity
The kinetic energy \( KE \) of a particle is given by:
\[ KE = \frac{1}{2} mv^2 \]
From this, we can express the velocity \( v \) in terms of kinetic energy and mass:
\[ v = \sqrt{\frac{2KE}{m}} \]

### Step 3: Substitute velocity into the radius formula
Substituting the expression for \( v \) into the radius formula gives:
\[ r = \frac{m \sqrt{\frac{2KE}{m}}}{qB} = \frac{\sqrt{2m \cdot KE}}{qB} \]

### Step 4: Set up the ratio of the radii for the proton and alpha particle
Let \( r_p \) be the radius for the proton and \( r_{\alpha} \) be the radius for the alpha particle. Since the kinetic energies are equal, we can write:
\[ \frac{r_p}{r_{\alpha}} = \frac{\sqrt{2m_p \cdot KE}}{q_p B} \div \frac{\sqrt{2m_{\alpha} \cdot KE}}{q_{\alpha} B} \]
This simplifies to:
\[ \frac{r_p}{r_{\alpha}} = \frac{\sqrt{m_p}}{q_p} \div \frac{\sqrt{m_{\alpha}}}{q_{\alpha}} = \frac{\sqrt{m_p} \cdot q_{\alpha}}{\sqrt{m_{\alpha}} \cdot q_p} \]

### Step 5: Substitute known values for mass and charge
For a proton:
- Mass \( m_p = m \) (mass of proton)
- Charge \( q_p = e \) (charge of proton)

For an alpha particle:
- Mass \( m_{\alpha} = 4m \) (mass of alpha particle, which is 4 times that of a proton)
- Charge \( q_{\alpha} = 2e \) (charge of alpha particle, which is 2 times that of a proton)

### Step 6: Calculate the ratio
Substituting these values into the ratio gives:
\[ \frac{r_p}{r_{\alpha}} = \frac{\sqrt{m} \cdot 2e}{\sqrt{4m} \cdot e} = \frac{2\sqrt{m}}{2\sqrt{m}} = \frac{1}{2} \]

### Final Answer
Thus, the ratio of the radii of their paths is:
\[ \frac{r_p}{r_{\alpha}} = \frac{1}{2} \]

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