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 of finding the ratio of the radii of two electron beams subjected to the same transverse magnetic field, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Motion of Charged Particles in a Magnetic Field:**
When a charged particle (like an electron) moves through a magnetic field, it experiences a magnetic force that acts perpendicular to its velocity. This results in circular motion.

2. **Formula for the Radius of Circular Motion:**
The radius \( r \) of the circular path of a charged particle 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

3. **Identify Constants:**
In this problem, we have two electron beams:
- Both beams consist of electrons, so their mass \( m \) and charge \( q \) are the same.
- The magnetic field \( B \) is also the same for both beams.

4. **Set Up the Ratios:**
Let the velocities of the two electron beams be \( v_1 \) and \( v_2 \) such that:
\[ \frac{v_1}{v_2} = \frac{1}{2} \]
This means \( v_1 = v \) and \( v_2 = 2v \) for some velocity \( v \).

5. **Calculate the Radii:**
Using the formula for radius, we can write:
\[ r_1 = \frac{mv_1}{qB} \quad \text{and} \quad r_2 = \frac{mv_2}{qB} \]
Substituting \( v_1 \) and \( v_2 \):
\[ r_1 = \frac{m(v)}{qB} \quad \text{and} \quad r_2 = \frac{m(2v)}{qB} \]

6. **Find the Ratio of the Radii:**
Now, we can find the ratio of the radii:
\[ \frac{r_1}{r_2} = \frac{\frac{m(v)}{qB}}{\frac{m(2v)}{qB}} = \frac{v}{2v} = \frac{1}{2} \]

7. **Conclusion:**
Therefore, the ratio of the radii of the two electron beams is:
\[ \frac{r_1}{r_2} = \frac{1}{2} \]

### Final Answer:
The ratio of the radii \( r_1 : r_2 \) is \( 1 : 2 \).

---