Question

A thin stiff insulated metal wire is bent into a circular loop with its two ends extending tangentially from the same point of the loop. The wire loop has mass \( m \) and radius \( r \) and is in a uniform vertical magnetic field \( B_0 \). When a current \( I \) is passed through the loop, the loop turns about the line PQ by an angle \( \theta \). The angle \( \theta \) is given by: 

• A. \( \tan\theta = \frac{\pi r I B_0}{mg} \)
• B. \( \tan\theta = \frac{2\pi r I B_0}{mg} \)
• C. \( \tan\theta = \frac{\pi r I B_0}{2mg} \)
• D. \( \tan\theta = \frac{mg}{\pi r I B_0} \)

Hint:

Analyze torques for equilibrium conditions in magnetic and gravitational fields.

Answer 1

The Correct Option is A

The torque due to magnetic force is: \[ \tau = I (\pi r^2) B_0 \cos\theta. \] For equilibrium: \[ \text{Gravitational torque } = mgr \sin\theta. \] Equating: \[ I (\pi r^2) B_0 \cos\theta = mgr \sin\theta \quad \Rightarrow \quad \tan\theta = \frac{\pi r I B_0}{mg}. \]