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Question #77730

Difficulty: 8/10 | 🟢 Free

The Question

A thin, rigid, non-conducting ring of mass $m$ (in $\text{kg}$ ) and radius $r$ (in $\text{m}$ ) has a total positive charge $q$ (in $\text{C}$ ) uniformly distributed over it. The ring is constrained to rotate frictionlessly about its central axis, which is fixed along the $z$ -axis. At $t=0$ , the ring is rotating with an initial angular velocity $\vec{\omega}_0 = \omega_0 \hat{k}$ (in $\text{rad/s}$ ). A uniform, time-varying magnetic field $\vec{B}(t) = B_0 \cos(\Omega t) \hat{k} + B_0 \sin(\Omega t) \hat{i}$ (in $\text{T}$ ) is applied, where $B_0$ and $\Omega$ are positive constants. Ignoring self-inductance, what is the angular velocity vector $\vec{\omega}$ of the ring at time $t = \frac{\pi}{\Omega}$ (in $\text{s}$ )?

Options

A.
$\left( \omega_0 + \frac{q B_0}{m} \right) \hat{k}$
B.
$\left( \omega_0 - \frac{q B_0}{m} \right) \hat{k}$
C.
$\sqrt{\omega_0^2 + \frac{2 q B_0 \omega_0}{m}} \hat{k}$
D.
$\left( \omega_0 + \frac{q B_0}{2m} \right) \hat{k}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Variable_Constraint

Develop pure symbolic logic skills by mastering algebraic constraints without relying on numerical data.

Concepts Behind This Question

Dimensional Independence vs. Scalar/Vector Nature|Magnetic Moment of Orbiting Electron|Vector Cross-Products

Why This Question Is Tricky

Examiner Levers

Visualization Stress

Requires mental rotation or 3D mapping (Optics/Geometry/Vectors).

Difficulty Boost : +4

Common Mistakes

This question tests a critical misconception regarding the work-energy theorem in time-varying magnetic fields.

Students often incorrectly assume that the orientational potential energy $U = -\vec{\mu} \cdot \vec{B}$ is conserved or can be used in a standard work-energy balance ( $K_i + U_i = K_f + U_f$ ).

This is a trap because the magnetic moment $\vec{\mu}$ is state-dependent (it changes magnitude as the ring accelerates) and the external magnetic field is explicitly time-dependent, meaning the system is not isolated and the external agent rotating the field does work.

By forcing the student to rely on the fundamental scalar flux and Faraday's law of induction to find the non-conservative induced electric field, the problem decouples the vector cross-product torque (which is neutralized by the mechanical constraint) from the scalar-driven EMF, exposing the flaw in applying conservative potential energy formulas to non-conservative, time-varying systems.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Units and Measurements - Class 11 Part 1
Concept	Dimensional Independence vs. Scalar/Vector Nature
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free