What is the magnetic moment of an electron orbiting in a circular orbit of radius r with a speed v?

Options

A. `evr/2`
B. evr
C. `(er)/(2v)`
D. None of these

Explanation

To find the magnetic moment of an electron orbiting in a circular orbit of radius \( r \) with speed \( v \), we can follow these steps:

### Step 1: Understand the Formula for Magnetic Moment
The magnetic moment \( M \) of a current loop is given by the formula:
\[ M = n \cdot I \cdot A \]
where:
- \( n \) is the number of turns in the loop,
- \( I \) is the current,
- \( A \) is the area of the loop.

### Step 2: Determine the Current \( I \)
The current \( I \) can be defined as the charge passing through a point per unit time. For an electron, the charge \( Q \) is equal to the elementary charge \( e \). The time \( T \) taken for one complete revolution (the period) can be calculated as:
\[ T = \frac{\text{Distance}}{\text{Velocity}} = \frac{2\pi r}{v} \]
Thus, the current \( I \) can be expressed as:
\[ I = \frac{Q}{T} = \frac{e}{T} = \frac{e}{\frac{2\pi r}{v}} = \frac{e \cdot v}{2\pi r} \]

### Step 3: Calculate the Area \( A \)
The area \( A \) of the circular orbit is given by:
\[ A = \pi r^2 \]

### Step 4: Substitute Values into the Magnetic Moment Formula
Now we can substitute the values of \( n \), \( I \), and \( A \) into the magnetic moment formula:
- Since the electron is making one complete turn, \( n = 1 \).
- We already found \( I = \frac{e \cdot v}{2\pi r} \).
- The area \( A = \pi r^2 \).

Thus, we have:
\[ M = n \cdot I \cdot A = 1 \cdot \left(\frac{e \cdot v}{2\pi r}\right) \cdot \left(\pi r^2\right) \]

### Step 5: Simplify the Expression
Now, simplifying the expression:
\[ M = \frac{e \cdot v}{2\pi r} \cdot \pi r^2 = \frac{e \cdot v \cdot r}{2} \]

### Final Result
The magnetic moment \( M \) of the electron orbiting in a circular orbit of radius \( r \) with speed \( v \) is:
\[ M = \frac{e \cdot v \cdot r}{2} \]