A proton and an a-particle enter in a uniform magnetic field perpendicularly with same speed. The will be 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 perpendicularly, it experiences a magnetic force that causes it to move in a circular path. The time period \( T \) of this circular motion 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 2: Define the Parameters for Proton and Alpha Particle
- For the proton:
- Mass \( m_p = m \) (mass of proton)
- Charge \( q_p = q \) (charge of proton)

- For the alpha particle:
- Mass \( m_{\alpha} = 4m \) (since an alpha particle consists of 2 protons and 2 neutrons, its mass is approximately four times that of a proton)
- Charge \( q_{\alpha} = 2q \) (since it has 2 protons)

### Step 3: Write the Time Periods for Both Particles
Using the formula for the time period:

- For the proton:
\[ T_p = \frac{2\pi m_p}{q_p B} = \frac{2\pi m}{q B} \]

- For the alpha particle:
\[ T_{\alpha} = \frac{2\pi m_{\alpha}}{q_{\alpha} B} = \frac{2\pi (4m)}{2q B} = \frac{4\pi m}{q B} \]

### Step 4: Calculate the Ratio of Time Periods
Now, we can find the ratio of the time periods:

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

### Step 5: Conclusion
Thus, the ratio of the time periods of the proton and the alpha particle is:

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

This means that the time period of the proton is half that of the alpha particle.

### Final Answer
The ratio of the time periods \( \frac{T_p}{T_{\alpha}} \) is \( \frac{1}{2} \) or \( 1:2 \).

---