A proton and an `alpha-` particle enter a uniform magnetic field at right angles to the field with same speed. The ratio of the periods of `alpha-` particle and proton is

Options

A. `1:1`
B. `1:4`
C. `1:2`
D. `2:1`

Explanation

To solve the problem, we need to find the ratio of the periods of an alpha particle and a proton when they enter a uniform magnetic field at right angles with the same speed.

### Step-by-Step Solution:

1. **Understanding the Motion in a Magnetic Field**:
When a charged particle moves in a magnetic field, it experiences a centripetal force that keeps it moving in a circular path. The radius \( r \) of the circular path is given by the formula:
\[ r = \frac{mv}{qB} \]
where \( m \) is the mass of the particle, \( v \) is its velocity, \( q \) is the charge, and \( B \) is the magnetic field strength.

2. **Finding the Time Period**:
The time period \( T \) of the circular motion is the time taken to complete one full circle. The circumference of the circle is \( 2\pi r \), and the time period can be expressed as:
\[ T = \frac{\text{Circumference}}{\text{Velocity}} = \frac{2\pi r}{v} \]
Substituting the expression for \( r \):
\[ T = \frac{2\pi \left( \frac{mv}{qB} \right)}{v} = \frac{2\pi m}{qB} \]

3. **Calculating the Time Periods for Proton and Alpha Particle**:
- For the proton:
\[ T_p = \frac{2\pi m_p}{q_p B} \]
- For the alpha particle (which has twice the charge and four times the mass of a proton):
\[ T_{\alpha} = \frac{2\pi m_{\alpha}}{q_{\alpha} B} = \frac{2\pi (4m_p)}{(2q_p) B} = \frac{4\pi m_p}{2q_p B} = \frac{2\pi m_p}{q_p B} \]

4. **Finding the Ratio of Time Periods**:
Now, we can find the ratio of the time periods of the alpha particle to the proton:
\[ \frac{T_{\alpha}}{T_p} = \frac{\frac{2\pi m_{\alpha}}{q_{\alpha} B}}{\frac{2\pi m_p}{q_p B}} = \frac{m_{\alpha}/q_{\alpha}}{m_p/q_p} \]
Substituting the values:
\[ = \frac{4m_p / 2q_p}{m_p / q_p} = \frac{4}{2} = 2 \]

### Final Answer:
The ratio of the periods of the alpha particle to the proton is:
\[ \frac{T_{\alpha}}{T_p} = 2 \]