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 find the periodic time for the alpha particle, we can follow these steps:

### Step 1: Determine the time period of the proton
We know that the proton takes 25 microseconds to make 5 revolutions. Therefore, the time period (T_p) for one revolution of the proton can be calculated as follows:

\[ T_p = \frac{\text{Total time}}{\text{Number of revolutions}} = \frac{25 \, \mu s}{5} = 5 \, \mu s \]

### Step 2: Use the formula for the time period in a magnetic field
The time period for a charged particle moving in a magnetic field is given by the formula:

\[ T = \frac{2\pi m}{qB} \]

Where:
- \( T \) is the time period,
- \( m \) is the mass of the particle,
- \( q \) is the charge of the particle,
- \( B \) is the magnetic field strength.

### Step 3: Relate the time periods of the proton and alpha particle
Let \( T_{\alpha} \) be the time period of the alpha particle. Since both particles are in the same magnetic field and moving with the same speed, we can set up the ratio of their time periods:

\[ \frac{T_{\alpha}}{T_p} = \frac{m_{\alpha}/q_{\alpha}}{m_p/q_p} \]

### Step 4: Substitute known values
We know:
- The mass of the alpha particle \( m_{\alpha} = 4m_p \) (4 times the mass of the proton),
- The charge of the alpha particle \( q_{\alpha} = 2q_p \) (twice the charge of the proton).

Substituting these into the ratio gives:

\[ \frac{T_{\alpha}}{T_p} = \frac{4m_p / (2q_p)}{m_p / q_p} = \frac{4}{2} = 2 \]

### Step 5: Calculate the time period of the alpha particle
From the ratio, we can express the time period of the alpha particle in terms of the time period of the proton:

\[ T_{\alpha} = 2T_p \]

Now substituting \( T_p = 5 \, \mu s \):

\[ T_{\alpha} = 2 \times 5 \, \mu s = 10 \, \mu s \]

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

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