A charge particle (charge q ) is moving in a circle of radius R with uniform speed. v The associated magnetic moment m is given by

Options

A. `qvR^2`
B. `qvR^2//2`
C. `qvR`
D. `qvR//2`

Explanation

To find the magnetic moment \( m \) associated with a charged particle of charge \( q \) moving in a circle of radius \( R \) with uniform speed \( v \), we can follow these steps:

### Step 1: Understand the Concept of Current
The first step is to recognize that a moving charge creates a current. The current \( I \) associated with the moving charge can be defined as:
\[ I = \frac{q}{T} \]
where \( T \) is the time period of one complete revolution.

### Step 2: Calculate the Time Period \( T \)
The time period \( T \) can be calculated using the formula for the circumference of the circle and the speed of the particle:
\[ T = \frac{\text{Circumference}}{\text{Speed}} = \frac{2\pi R}{v} \]

### Step 3: Substitute \( T \) into the Current Formula
Now, substituting \( T \) back into the current formula gives:
\[ I = \frac{q}{T} = \frac{q}{\frac{2\pi R}{v}} = \frac{qv}{2\pi R} \]

### Step 4: Calculate the Area \( A \) of the Circle
The area \( A \) of the circle in which the charge is moving can be calculated using the formula:
\[ A = \pi R^2 \]

### Step 5: Calculate the Magnetic Moment \( m \)
The magnetic moment \( m \) is given by the product of the current \( I \) and the area \( A \):
\[ m = I \cdot A \]
Substituting the expressions for \( I \) and \( A \):
\[ m = \left(\frac{qv}{2\pi R}\right) \cdot (\pi R^2) \]

### Step 6: Simplify the Expression
Now, simplifying the expression for \( m \):
\[ m = \frac{qv}{2\pi R} \cdot \pi R^2 = \frac{qvR}{2} \]

### Final Result
Thus, the magnetic moment \( m \) associated with the charged particle is:
\[ m = \frac{qvR}{2} \]