Before You Begin

Question #43425

Difficulty: 8/10 | 🟢 Free

The Question

A rigid circular wire loop of mass $m$ and radius $R$ carries a constant current $I$ . The loop is suspended from a frictionless pivot $P$ located on its circumference. The pivot restricts the motion such that the loop can only swing as a pendulum about the $y$ -axis. The pivot $P$ is fixed at the origin $(0,0,0)$ . In its stable equilibrium position, the loop lies in the $xy$ -plane with its center at $(R, 0, 0)$ . A static, non-uniform magnetic field is present in the region, given by $\vec{B} = \beta z \hat{i} + (B_0 + \beta x) \hat{k}$ , where $B_0$ and $\beta$ are positive constants. Assuming gravity is negligible, what is the angular frequency $\omega$ of small oscillations of the loop about the $y$ -axis?

Options

A.
$\sqrt{\frac{2 \pi I (B_0 + 4 \beta R)}{3 m}}$
B.
$\sqrt{\frac{2 \pi I (B_0 + 2 \beta R)}{3 m}}$
C.
$\sqrt{\frac{\pi I (B_0 + 4 \beta R)}{m}}$
D.
$\sqrt{\frac{2 \pi I B_0}{3 m}}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Multi-Concept_Bridge

Master problems that require linking physics and math concepts to solve complex competitive questions.

Concepts Behind This Question

Magnetic Moment of Orbiting Electron|Compound Pendulum Period|Torque and Potential Energy in B-Field

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to synthesize rigid body dynamics with the behavior of a magnetic dipole in a non-uniform magnetic field.

A common trap is to calculate the torque solely using $\vec{\tau} = \vec{\mu} \times \vec{B}$ evaluated at the center of mass, which yields a restoring torque coefficient proportional to $(B_0 + 2\beta R)$ and leads to Option B.

However, in a non-uniform field, the gradient exerts a net force $\vec{F} = \nabla(\vec{\mu} \cdot \vec{B})$ on the loop.

Because the loop is pivoted at its edge, this force generates an additional torque about the pivot.

Using the potential energy method $U = -\vec{\mu} \cdot \vec{B}$ elegantly captures both the pure couple and the force-induced torque, revealing the true $(B_0 + 4\beta R)$ dependence.

Another trap (Option C) occurs if students incorrectly treat the loop as a point mass at distance $R$ ( $I = mR^2$ ) rather than applying the parallel axis theorem for a compound pendulum ( $I = 1.5mR^2$ ).

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Moving Charges and Magnetism - Class 12
Concept	Magnetic Moment of Orbiting Electron
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free