A uniform magnetic field acts right angles to the direction of motion of electrones. As a result, the electron moves in acircular path of radius 2 cm. If the speed of electrons is doubled, then the radius of the circular path will be

Options

A. 2.0 cm
B. 0.5 cm
C. 4.0 cm
D. 1.0 cm

Answer

To solve the problem, we need to understand the relationship between the radius of the circular path of an electron moving in a magnetic field and its speed. The formula that relates these quantities is given by:

\[ r = \frac{mv}{qB} \]

Where:
- \( r \) is the radius of the circular path,
- \( m \) is the mass of the electron,
- \( v \) is the speed of the electron,
- \( q \) is the charge of the electron,
- \( B \) is the magnetic field strength.

### Step-by-Step Solution:

1. **Identify the initial conditions**:
- The initial radius of the circular path \( r = 2 \, \text{cm} \).
- The speed of the electron is \( v \).

2. **Understand the effect of doubling the speed**:
- If the speed of the electron is doubled, the new speed \( v' = 2v \).

3. **Substitute the new speed into the radius formula**:
- The new radius \( r' \) can be calculated using the same formula:
\[ r' = \frac{m v'}{qB} = \frac{m (2v)}{qB} = 2 \left(\frac{mv}{qB}\right) = 2r \]

4. **Calculate the new radius**:
- Since the initial radius \( r = 2 \, \text{cm} \):
\[ r' = 2r = 2 \times 2 \, \text{cm} = 4 \, \text{cm} \]

5. **Conclusion**:
- The radius of the new circular path when the speed of the electron is doubled is \( 4 \, \text{cm} \).

### Final Answer:
The radius of the circular path when the speed of the electrons is doubled is **4 cm**.