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`

Explanation

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

### Step-by-Step Solution:

1. **Determine the Time Period (T)**:
The time taken for the charged particle to complete one full revolution in a circular path is given by the formula:
\[ T = \frac{2\pi R}{v} \]
where `R` is the radius of the circle and `v` is the uniform speed of the particle.

2. **Calculate the Current (I)**:
The current `I` associated with the moving charge can be defined as the charge passing through a point in one second. It is given by:
\[ I = \frac{q}{T} \]
Substituting the expression for `T` from step 1, we have:
\[ I = \frac{q}{\frac{2\pi R}{v}} = \frac{qv}{2\pi R} \]

3. **Determine the Area (A)**:
The area `A` enclosed by the circular path of the charged particle is given by:
\[ A = \pi R^2 \]

4. **Calculate the Magnetic Moment (μ)**:
The magnetic moment `μ` is defined as the product of the current and the area through which it flows:
\[ \mu = I \cdot A \]
Substituting the expressions for `I` and `A`, we get:
\[ \mu = \left(\frac{qv}{2\pi R}\right) \cdot (\pi R^2) \]
Simplifying this expression:
\[ \mu = \frac{qvR}{2} \]

Thus, the associated magnetic moment `μ` for the charged particle moving in a circle of radius `R` with uniform speed `v` is:
\[ \mu = \frac{qvR}{2} \]