A proton and an a-particle enter in a uniform magnetic field perpendicularly with same speed. The will b ratio of time periods of both particle `(T_(p)/(T_(alpha))` will be

Options

A. `1:2`
B. `1:3`
C. `2:1`
D. `3:1`

Explanation

To find the ratio of the time periods of a proton and an alpha particle entering a uniform magnetic field perpendicularly with the same speed, we can follow these steps:

### Step 1: Understand the motion of charged particles in a magnetic field
When a charged particle moves through a magnetic field, it experiences a magnetic force that acts as a centripetal force, causing the particle to move in a circular path. The time period \( T \) for a charged particle moving in a magnetic field can be expressed as:

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

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 2: Determine the parameters for the proton
For a proton:
- Mass \( m_p = m \)
- Charge \( q_p = e \) (where \( e \) is the elementary charge)

Thus, the time period for the proton \( T_p \) is:

\[ T_p = \frac{2\pi m}{eB} \]

### Step 3: Determine the parameters for the alpha particle
For an alpha particle:
- Mass \( m_{\alpha} = 4m \) (since an alpha particle consists of 2 protons and 2 neutrons)
- Charge \( q_{\alpha} = 2e \) (since it has 2 protons)

Thus, the time period for the alpha particle \( T_{\alpha} \) is:

\[ T_{\alpha} = \frac{2\pi (4m)}{2eB} = \frac{4\pi m}{eB} \]

### Step 4: Find the ratio of the time periods
Now, we can find the ratio of the time periods \( \frac{T_p}{T_{\alpha}} \):

\[ \frac{T_p}{T_{\alpha}} = \frac{\frac{2\pi m}{eB}}{\frac{4\pi m}{eB}} = \frac{2\pi m}{eB} \cdot \frac{eB}{4\pi m} = \frac{2}{4} = \frac{1}{2} \]

### Final Answer
The ratio of the time periods of the proton and the alpha particle is:

\[ \frac{T_p}{T_{\alpha}} = \frac{1}{2} \]

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