An electron moves with a constant speed `v` along a circle of radius `r`. Its magnetic moment will be (`e` is the electron's charge)

Options

A. `evr`
B. `(1)/(2)evr`
C. `pir^(2)ev`
D. `2pirev`

Explanation

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

### Step 1: Understand the definition of magnetic moment
The magnetic moment (m) is given by the formula:
\[ m = i \cdot A \]
where \( i \) is the current and \( A \) is the area of the loop.

### Step 2: Determine the current (i)
The current \( i \) can be defined as the charge (q) passing through a point in the circuit per unit time (t). For an electron moving in a circle:
\[ i = \frac{q}{t} \]

### Step 3: Calculate the time period (t)
The time \( t \) taken for one complete revolution around the circle can be calculated using the formula:
\[ t = \frac{\text{Circumference of the circle}}{\text{Speed}} = \frac{2\pi r}{v} \]

### Step 4: Substitute the time period into the current formula
Substituting the expression for \( t \) into the current formula, we get:
\[ i = \frac{e}{t} = \frac{e}{\frac{2\pi r}{v}} = \frac{e \cdot v}{2\pi r} \]

### Step 5: Calculate the area (A) of the circular path
The area \( A \) of the circular path is given by:
\[ A = \pi r^2 \]

### Step 6: Substitute current and area into the magnetic moment formula
Now, substituting the values of \( i \) and \( A \) into the magnetic moment formula:
\[ m = i \cdot A = \left(\frac{e \cdot v}{2\pi r}\right) \cdot (\pi r^2) \]

### Step 7: Simplify the expression
Simplifying the expression:
\[ m = \frac{e \cdot v}{2\pi r} \cdot \pi r^2 = \frac{e \cdot v \cdot r}{2} \]

### Final Answer
Thus, the magnetic moment of the electron moving in a circular path is:
\[ m = \frac{e \cdot v \cdot r}{2} \]

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