A proton and an`alpha`-particles enters in a uniform magnetic field with same velocity, then ratio of the radii of path describe by them

Options

A. 0.042361111111111
B. 0.043055555555556
C. 0.084027777777778
D. None of these

Explanation

To solve the problem of finding the ratio of the radii of the paths described by a proton and an alpha particle when they enter a uniform magnetic field with the same velocity, we can follow these steps:

### Step 1: Understand the motion of charged particles 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 magnetic force \( F \) on a charged particle is given by:
\[ F = qvB \]
where \( q \) is the charge of the particle, \( v \) is its velocity, and \( B \) is the magnetic field strength.

### Step 2: Set the magnetic force equal to the centripetal force
The centripetal force required to keep an object moving in a circular path is given by:
\[ F = \frac{mv^2}{r} \]
where \( m \) is the mass of the particle, \( v \) is its velocity, and \( r \) is the radius of the circular path.

### Step 3: Equate the magnetic force and centripetal force
Setting the magnetic force equal to the centripetal force, we have:
\[ qvB = \frac{mv^2}{r} \]

### Step 4: Solve for the radius \( r \)
Rearranging the equation to solve for the radius \( r \):
\[ r = \frac{mv}{qB} \]

### Step 5: Calculate the radius for the proton and alpha particle
1. **For the proton**:
- Mass \( m_p = 1 \, \text{amu} \)
- Charge \( q_p = 1e \)

The radius \( r_p \) for the proton is:
\[ r_p = \frac{m_p v}{q_p B} = \frac{1 \, \text{amu} \cdot v}{1e \cdot B} \]

2. **For the alpha particle**:
- Mass \( m_{\alpha} = 4 \, \text{amu} \) (since an alpha particle is made of 2 protons and 2 neutrons)
- Charge \( q_{\alpha} = 2e \)

The radius \( r_{\alpha} \) for the alpha particle is:
\[ r_{\alpha} = \frac{m_{\alpha} v}{q_{\alpha} B} = \frac{4 \, \text{amu} \cdot v}{2e \cdot B} \]

### Step 6: Find the ratio of the radii
Now, we can find the ratio of the radii \( \frac{r_p}{r_{\alpha}} \):
\[ \frac{r_p}{r_{\alpha}} = \frac{\frac{1 \, \text{amu} \cdot v}{1e \cdot B}}{\frac{4 \, \text{amu} \cdot v}{2e \cdot B}} = \frac{1 \cdot 2e}{4 \cdot 1e} = \frac{2}{4} = \frac{1}{2} \]

### Conclusion
Thus, the ratio of the radii of the paths described by the proton and the alpha particle is:
\[ \frac{r_p}{r_{\alpha}} = \frac{1}{2} \]