Question

A charge \(q\) is circulating at a constant speed \(v\) with a semicircular loop of radius \(r\). The magnetic moment of this loop is

• A. \((qv\pi r)/(\pi +2)\)
• B. \(qvr/3\)
• C. \((\pi rqv)/2(\pi +2)\)
• D. \(qvr\)

Hint:

Find the time period using the full perimeter of the closed loop, then the current, and multiply by the area.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
The closed semicircular loop has perimeter \(\pi r+2r=(\pi+2)r\).

Step 2: Current
Time period \(T=\dfrac{(\pi+2)r}{v}\)
\(I=\dfrac{q}{T}=\dfrac{qv}{(\pi+2)r}\)

Step 3: Magnetic moment
Area \(A=\dfrac{\pi r^2}{2}\)
\(M=IA=\dfrac{qv}{(\pi+2)r}\cdot\dfrac{\pi r^2}{2}=\dfrac{\pi rqv}{2(\pi+2)}\)

Final Answer:
The magnetic moment is pi r q v over two times pi plus two. \[ \boxed{\text{(C) }\dfrac{\pi rqv}{2(\pi+2)}} \]