A proton and an `alpha-`particle enter a uniform magnetic field moving with the same speed. If the proton takes `25 mu s` to make 5 revolutions, then the periodic time for the `alpha-` particle would be

Options

A. `50 mu s`
B. `25 mu s`
C. `10 mu s`
D. `5 mu s`

Explanation

To solve the problem, we need to find the periodic time for the alpha particle given that a proton takes 25 µs to make 5 revolutions in a uniform magnetic field.

### Step-by-Step Solution:

1. **Determine the Period of the Proton:**
- The proton takes 25 µs to make 5 revolutions.
- Therefore, the time period \( T_p \) for one revolution of the proton can be calculated as:
\[ T_p = \frac{25 \, \mu s}{5} = 5 \, \mu s \]

2. **Understand the Relationship Between Time Periods:**
- The time period \( T \) in a magnetic field is given by the formula:
\[ T = \frac{2\pi m}{qB} \]
- For the proton, we have:
\[ T_p = \frac{2\pi m}{qB} \]
- For the alpha particle, which has 4 times the mass of the proton and 2 times the charge, the time period \( T_\alpha \) can be expressed as:
\[ T_\alpha = \frac{2\pi (4m)}{(2q)B} \]

3. **Simplify the Expression for the Alpha Particle:**
- Simplifying \( T_\alpha \):
\[ T_\alpha = \frac{2\pi (4m)}{(2q)B} = \frac{4}{2} \cdot \frac{2\pi m}{qB} = 2 \cdot T_p \]

4. **Calculate the Time Period for the Alpha Particle:**
- Since we found that \( T_\alpha = 2 \cdot T_p \) and we know \( T_p = 5 \, \mu s \):
\[ T_\alpha = 2 \cdot 5 \, \mu s = 10 \, \mu s \]

### Final Answer:
The periodic time for the alpha particle is \( 10 \, \mu s \).

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