An electron revolves in a circular orbit of radius r with angular speed `omega` . The magnetic field at the centre of electron orbit is

Options

A. `(mu_(0) e omega)/(pi r)`
B. `(mu_(0)e omega)/(4pir)`
C. `(mu_(0) e^(2) omega)/(4 pi r)`
D. `(mu_(0) e omega)/(2pi r )`

Explanation

To find the magnetic field at the center of an electron orbit, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Parameters**:
- The electron revolves in a circular orbit of radius \( r \) with angular speed \( \omega \).
- The charge of the electron is \( e = 1.6 \times 10^{-19} \, \text{C} \).

2. **Determine the Current**:
- The current \( I \) due to the revolving electron can be calculated using the formula:
\[ I = \frac{q}{T} \]
- Here, \( q \) is the charge of the electron, and \( T \) is the time period of one complete revolution.

3. **Calculate the Time Period \( T \)**:
- The time period \( T \) for one complete revolution is given by:
\[ T = \frac{2\pi}{\omega} \]

4. **Substituting the Time Period into the Current Formula**:
- Substitute \( T \) into the current formula:
\[ I = \frac{e}{T} = \frac{e}{\frac{2\pi}{\omega}} = \frac{e \omega}{2\pi} \]

5. **Use the Formula for Magnetic Field**:
- The magnetic field \( B \) at the center of a circular loop carrying current \( I \) is given by:
\[ B = \frac{\mu_0 I}{2r} \]
- Here, \( \mu_0 \) is the permeability of free space.

6. **Substituting the Current into the Magnetic Field Formula**:
- Substitute \( I \) into the magnetic field formula:
\[ B = \frac{\mu_0 \left(\frac{e \omega}{2\pi}\right)}{2r} = \frac{\mu_0 e \omega}{4\pi r} \]

7. **Final Result**:
- Therefore, the magnetic field at the center of the electron orbit is:
\[ B = \frac{\mu_0 e \omega}{4\pi r} \]