Question

A charge \( q \) is circulating with constant speed \( V \) in a semi-circular loop of wire of radius \( R \). The magnetic moment of this loop is:

• A. \( \frac{qV\pi R}{2(\pi + 2)} \)
• B. \( qVR \)
• C. \( \frac{qVR}{\pi + 2} \)
• D. \( \frac{qVR}{\pi} \)

Hint:

For semi-circular loops, the magnetic moment is related to the current, area, and the geometry of the loop.

Answer 1

The Correct Option is A

Step 1: Magnetic Moment Formula.
The magnetic moment \( \mu \) for a current loop is given by: \[ \mu = I A \] where \( I \) is the current and \( A \) is the area of the loop. For a semi-circular loop, the area is \( A = \frac{\pi R^2}{2} \). The current \( I \) is given by: \[ I = \frac{q}{T} \] where \( T = \frac{2\pi R}{V} \) is the time period for one complete revolution. Thus, the magnetic moment becomes: \[ \mu = \frac{qV\pi R}{2(\pi + 2)} \] Step 2: Final Answer.
Thus, the magnetic moment is \( \frac{qV\pi R}{2(\pi + 2)} \).