An electron of charge e is moving in a circular orbit of radius r around a nucleus, at a frequency v. The magnetic moment associated with the orbital motion of the electron is

Options

A. `mu_(0)/4pi ev/R^(2)`
B. `mu_(0)/2 ev/R`
C. `mu_(0)e vR`
D. `mu_(0)ev/R^(2)`

Answer

To find the magnetic moment associated with the orbital motion of an electron moving in a circular orbit around a nucleus, we can follow these steps:

### Step-by-Step Solution

1. **Understanding the Magnetic Moment**:
The magnetic moment (\( \mu \)) associated with a current-carrying loop is given by the formula:
\[ \mu = I \cdot A \]
where \( I \) is the current and \( A \) is the area of the loop.

2. **Calculating the Area**:
Since the electron is moving in a circular orbit of radius \( r \), the area \( A \) of the circular orbit can be calculated using the formula for the area of a circle:
\[ A = \pi r^2 \]

3. **Finding the Current**:
The current \( I \) can be defined as the charge passing through a point in the circuit per unit time. For an electron moving in a circular path, the current can be expressed as:
\[ I = \frac{Q}{T} \]
where \( Q \) is the charge of the electron (denoted as \( e \)) and \( T \) is the time period of one complete revolution. The time period \( T \) can be related to the frequency \( v \) by:
\[ T = \frac{1}{v} \]
Thus, the current can be rewritten as:
\[ I = e \cdot v \]

4. **Substituting Values**:
Now we can substitute the expressions for \( I \) and \( A \) into the magnetic moment formula:
\[ \mu = I \cdot A = (e \cdot v) \cdot (\pi r^2) \]

5. **Final Expression**:
Therefore, the magnetic moment associated with the orbital motion of the electron is:
\[ \mu = e \cdot v \cdot \pi r^2 \]

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

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