Two electron beams having velocities in the ratio 1: 2 are subjected to the same transverse magnetic field. The ration of the radii is

Options

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

Answer

To solve the problem, we need to find the ratio of the radii of the circular paths of two electron beams moving in a magnetic field, given that their velocities are in the ratio of 1:2.

### Step-by-Step Solution:

1. **Understand the Formula for Radius in a Magnetic Field:**
The radius \( R \) of the circular path of a charged particle moving in a magnetic field is given by the formula:
\[ R = \frac{mv}{qB} \]
where:
- \( m \) = mass of the particle,
- \( v \) = velocity of the particle,
- \( q \) = charge of the particle,
- \( B \) = magnetic field strength.

2. **Identify the Parameters for Both Beams:**
- Let the velocity of the first beam be \( v_1 = v \).
- Let the velocity of the second beam be \( v_2 = 2v \).
- Since both beams consist of electrons, the mass \( m \) and charge \( q \) are the same for both beams.

3. **Calculate the Radius for Each Beam:**
- For the first beam:
\[ R_1 = \frac{mv_1}{qB} = \frac{mv}{qB} \]
- For the second beam:
\[ R_2 = \frac{mv_2}{qB} = \frac{m(2v)}{qB} = \frac{2mv}{qB} \]

4. **Find the Ratio of the Radii:**
Now, we need to find the ratio \( \frac{R_1}{R_2} \):
\[ \frac{R_1}{R_2} = \frac{\frac{mv}{qB}}{\frac{2mv}{qB}} = \frac{mv}{qB} \cdot \frac{qB}{2mv} = \frac{1}{2} \]

5. **Express the Ratio in Simplified Form:**
Thus, the ratio of the radii \( R_1 : R_2 \) is:
\[ R_1 : R_2 = 1 : 2 \]

### Final Answer:
The ratio of the radii of the two electron beams is \( 1 : 2 \).