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Question #43408

Difficulty: 8/10 | 🟢 Free

The Question

A rigid square loop of mass $M = 1.2 \text{ kg}$ and side length $L = 0.5 \text{ m}$ lies initially in the horizontal $xy$ -plane. One of its edges is hinged along the $y$ -axis (from $y=0$ to $y=L$ ) so that the loop can rotate freely. Gravity acts in the $-z$ direction ( $g = 10 \text{ m/s}^2$ ). A constant current $I = 1 \text{ A}$ is maintained in the loop by an external source, flowing in the $+y$ direction along the hinged edge. The system is immersed in a non-uniform, time-independent magnetic field $\vec{B} = B_0 \left(1 + \frac{x}{L}\right) \hat{i}$ , where $B_0 = 2 \text{ T}$ . An ideal string is attached to the center of the outer edge of the loop (at $x=L, y=L/2$ ). The string passes vertically upwards (in the $+z$ direction) over a small ideal pulley and is connected to a hanging block of mass $m = 0.2 \text{ kg}$ . If the system is released from rest, calculate the initial angular acceleration of the loop. (Neglect any self-induction effects).

Options

A.
$\frac{40}{7} \text{ rad/s}^2$
B.
$\frac{50}{7} \text{ rad/s}^2$
C.
$\frac{40}{3} \text{ rad/s}^2$
D.
8 \text{ rad/s}^2

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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

Torque on Current Loop|Constraint Relations via String Length|Vector Cross-Products

Why This Question Is Tricky

Common Mistakes

This question tests the application of magnetic torque in a non-uniform field coupled with mechanical constraints.

A common lethal trap is blindly applying the magnetic dipole formula $\vec{\tau} = \vec{\mu} \times \vec{B}_{center}$ (which yields an incorrect torque of $1.5 I L^2 B_0$ and leads to Option B).

This formula is strictly valid only for uniform magnetic fields.

In a non-uniform field, students must integrate the Lorentz force $d\vec{F} = I d\vec{l} \times \vec{B}$ along the specific geometry of the loop to find the true torque ( $2 I L^2 B_0$ ).

Furthermore, it tests the correct sign convention in the string constraint linking rotational and translational acceleration, ensuring students recognize that the downward rotation of the loop forces the hanging mass to accelerate upwards.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Moving Charges and Magnetism - Class 12
Concept	Torque on Current Loop
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free