An electron moving in a circular orbit of radius r makes n rotations per second per second. The magnetic moment of the orbital electron is

Options

A. zero
B. `pi r^(2)n e`
C. `pi r^(2) n^(2)e`
D. `(r^(2) n e)/(2pi)`

Answer

To find the magnetic moment of an electron moving in a circular orbit, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Motion of the Electron**:
An electron moving in a circular orbit can be considered as a current loop. The current (I) due to the electron's motion can be defined based on the number of rotations it makes per second.

2. **Calculating the Current (I)**:
The current generated by the electron can be calculated using the formula:
\[ I = \frac{e}{T} \]
where \( e \) is the charge of the electron and \( T \) is the time period of one complete rotation. Since the electron makes \( n \) rotations per second, the time period \( T \) can be expressed as:
\[ T = \frac{1}{n} \]
Therefore, the current becomes:
\[ I = e \cdot n \]

3. **Calculating the Area of the Circular Orbit**:
The area \( A \) of the circular orbit can be calculated using the formula:
\[ A = \pi r^2 \]
where \( r \) is the radius of the circular orbit.

4. **Calculating the Magnetic Moment (μ)**:
The magnetic moment \( \mu \) of a current loop is given by the formula:
\[ \mu = I \cdot A \]
Substituting the expressions for \( I \) and \( A \):
\[ \mu = (e \cdot n) \cdot (\pi r^2) \]
Thus, the magnetic moment of the orbital electron is:
\[ \mu = e \cdot n \cdot \pi r^2 \]

### Final Answer:
The magnetic moment of the orbital electron is:
\[ \mu = e \cdot n \cdot \pi r^2 \]

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