Question

The magnetic moment $\mu$ associated with a charged particle carrying charge $q$ moving in a circle of radius $a$ with uniform speed $v$ is

• A. $qva$
• B. $\frac{qva}{4}$
• C. $\frac{qva}{2}$
• D. $\frac{qva}{16}$
• E. $\frac{qva}{8}$

Hint:

The magnetic moment of a current-carrying loop depends on the charge, velocity, and radius of the path.

Answer 1

The Correct Option is C

Step 1: The magnetic moment of a charged particle moving in a circular path is given by: \[ \mu = I A \] where $I$ is the current and $A$ is the area of the circular path. 
Step 2: The current $I$ is given by: \[ I = \frac{q}{T} \] where $T$ is the time period of the circular motion. The time period is: \[ T = \frac{2\pi a}{v} \] Step 3: Substituting $T$: \[ I = \frac{q}{2\pi a / v} = \frac{q v}{2\pi a} \] 
Step 4: The area of the circular path is: \[ A = \pi a^2 \] 
Step 5: Compute the magnetic moment: \[ \mu = \left(\frac{q v}{2\pi a}\right) (\pi a^2) \] \[ = \frac{q v a}{2} \] 
Step 6: Therefore, the correct answer is (C).