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

Options

A. `(qvR)/(2)`
B. `qvR^(2)`
C. `(qvR_(2))/(2)`
D. `qvR`

Answer

To find the magnetic moment (\( \mu \)) associated with a charged particle moving in a circle, we can follow these steps:

### Step 1: Understand the definition of magnetic moment
The magnetic moment (\( \mu \)) for a current loop is defined as:
\[ \mu = I \cdot A \]
where \( I \) is the current and \( A \) is the area of the loop.

### Step 2: Calculate the area of the circular path
The area \( A \) of a circle with radius \( R \) is given by:
\[ A = \pi R^2 \]

### Step 3: Determine the current \( I \)
Current \( I \) can be defined as the charge \( q \) passing through a point in the circuit per unit time. For a particle moving in a circle, the time \( T \) taken for one complete revolution is:
\[ T = \frac{\text{Circumference}}{\text{Speed}} = \frac{2 \pi R}{v} \]
Thus, the current \( I \) can be expressed as:
\[ I = \frac{q}{T} = \frac{q}{\frac{2 \pi R}{v}} = \frac{qv}{2 \pi R} \]

### Step 4: Substitute \( I \) and \( A \) into the magnetic moment formula
Now, substituting the values of \( I \) and \( A \) into the magnetic moment formula:
\[ \mu = I \cdot A = \left(\frac{qv}{2 \pi R}\right) \cdot (\pi R^2) \]

### Step 5: Simplify the expression
Now simplifying the expression:
\[ \mu = \frac{qv}{2 \pi R} \cdot \pi R^2 = \frac{qv R}{2} \]

### Final Result
Thus, the magnetic moment \( \mu \) associated with the charged particle moving in a circle is:
\[ \mu = \frac{qv R}{2} \]