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 causes it to move in a circular path. The magnetic force acting on the particle is given by:
\[ F_{\text{magnetic}} = qvB \]
where \( q \) is the charge of the particle, \( v \) is its velocity, and \( B \) is the magnetic field strength.

### Step 2: Relate magnetic force to centripetal force
For a particle moving in a circular path, the magnetic force provides the necessary centripetal force. The centripetal force is given by:
\[ F_{\text{centripetal}} = \frac{mv^2}{r} \]
where \( m \) is the mass of the particle, \( v \) is its velocity, and \( r \) is the radius of the circular path.

### Step 3: Set the magnetic force equal to the centripetal force
Setting the two forces equal gives us:
\[ qvB = \frac{mv^2}{r} \]

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

### Step 5: Calculate the radius for proton and alpha particle
Let’s denote:
- \( m_p \) and \( q_p \) as the mass and charge of the proton.
- \( m_{\alpha} \) and \( q_{\alpha} \) as the mass and charge of the alpha particle.

For the proton:
\[ r_p = \frac{m_p v}{q_p B} \]

For the alpha particle:
\[ r_{\alpha} = \frac{m_{\alpha} v}{q_{\alpha} B} \]

### Step 6: Find the ratio of the radii
Now, we can find the ratio of the radii:
\[ \frac{r_{\alpha}}{r_p} = \frac{m_{\alpha}}{m_p} \cdot \frac{q_p}{q_{\alpha}} \]

### Step 7: Substitute the known values
- The mass of an alpha particle is approximately 4 times the mass of a proton: \( m_{\alpha} = 4m_p \).
- The charge of an alpha particle is 2 times the charge of a proton: \( q_{\alpha} = 2q_p \).

Substituting these values into the ratio:
\[ \frac{r_{\alpha}}{r_p} = \frac{4m_p}{m_p} \cdot \frac{q_p}{2q_p} = \frac{4}{2} = 2 \]

### Step 8: Final ratio
Thus, the ratio of the radii of the paths described by the alpha particle to the proton is:
\[ \frac{r_{\alpha}}{r_p} = 2 \]

### Step 9: Inverse ratio for the question
Since the question asks for the ratio of the radius of the proton to the radius of the alpha particle:
\[ \frac{r_p}{r_{\alpha}} = \frac{1}{2} \]

### Conclusion
The ratio of the radii of the paths described by the proton and the alpha particle is \( \frac{1}{2} \).

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