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Question #43315

Difficulty: 8/10 | 🟢 Free

The Question

A thin uniform non-conducting ring of mass $m$ , radius $R$ , and uniformly distributed positive charge $Q$ is initially at rest in the $xy$ -plane. It is free to rotate about its central axis (the $z$ -axis). A uniform magnetic field $\vec{B} = (B_0 + \alpha t) \hat{k}$ exists in the region, where $B_0$ and $\alpha$ are positive constants, and $t$ is time. Given $B_0 = 2 \alpha t_0$ , find the magnitude of the internal compressive force developed in the ring at time $t = t_0$ . Assume the ring remains perfectly circular.

Options

A.
$\frac{5 Q^2 \alpha^2 t_0^2 R}{8 \pi m}$
B.
$\frac{7 Q^2 \alpha^2 t_0^2 R}{8 \pi m}$
C.
$\frac{3 Q^2 \alpha^2 t_0^2 R}{8 \pi m}$
D.
$\frac{Q^2 \alpha^2 t_0^2 R}{8 \pi m}$

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

Induced Electric Fields|Angular Velocity vs Linear Velocity|Centripetal Acceleration Vector

Why This Question Is Tricky

Common Mistakes

This question tests the coupling of induced non-conservative electric fields with rotational dynamics and frame-dependent magnetic forces.

A common lethal mistake (Trap D) is calculating the induced velocity but completely ignoring the $q(\vec{v} \times \vec{B})$ Lorentz force that arises because the charge is now moving in a magnetic field.

Another trap (Trap B) is incorrectly assuming the magnetic force is outward and adding it to the centripetal requirement.

Trap C occurs if students use the initial magnetic field $B_0$ instead of the instantaneous field $B(t_0)$ for the Lorentz force.

The correct approach requires a rigorous radial force balance on a differential element, recognizing that the massive inward magnetic force forces the ring into compression to maintain its circular path.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electromagnetic Induction - Class 12
Concept	Induced Electric Fields
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free