An electron moving around the nucleus with an angular momenturm `l` has a magnetic moment

Options

A. `(e)/(m)l`
B. `(e)/(2m)l`
C. `(2e)/(m)l`
D. `(e)/(2pim)l`

Explanation

To find the magnetic moment of an electron moving around the nucleus with a given angular momentum \( l \), we can follow these steps:

### Step-by-step Solution:

1. **Understanding the Motion of the Electron**:
- The electron is moving in a circular path around the nucleus. Its motion can be characterized by its angular momentum \( L \) and its charge \( e \).

2. **Current Due to Electron Motion**:
- The current \( I \) due to the moving electron can be expressed as:
\[ I = \frac{Q}{T} \]
- Here, \( Q \) is the charge of the electron, and \( T \) is the time period of one complete revolution.

3. **Calculating the Time Period**:
- The time period \( T \) can be calculated using the circumference of the circular path:
\[ T = \frac{2\pi r}{v} \]
- Where \( r \) is the radius of the circular path and \( v \) is the speed of the electron.

4. **Substituting for Current**:
- Substituting the expression for \( T \) into the equation for current:
\[ I = \frac{e}{T} = \frac{e v}{2\pi r} \]

5. **Magnetic Moment Formula**:
- The magnetic moment \( M \) is given by:
\[ M = I \cdot A \]
- Where \( A \) is the area of the circular path, which is:
\[ A = \pi r^2 \]

6. **Substituting for Area**:
- Now substituting for \( I \) and \( A \):
\[ M = \left(\frac{e v}{2\pi r}\right) \cdot (\pi r^2) = \frac{e v r}{2} \]

7. **Relating Angular Momentum to Velocity**:
- The angular momentum \( L \) of the electron can be expressed as:
\[ L = m v r \]
- Rearranging gives:
\[ v = \frac{L}{m r} \]

8. **Substituting for Velocity in Magnetic Moment**:
- Now substitute \( v \) back into the equation for magnetic moment:
\[ M = \frac{e}{2} \cdot \frac{L}{m} \]

9. **Final Expression for Magnetic Moment**:
- Thus, the magnetic moment \( M \) can be expressed as:
\[ M = \frac{e L}{2m} \]

### Conclusion:
The magnetic moment of an electron moving around the nucleus with angular momentum \( L \) is given by:
\[ M = \frac{e L}{2m} \]