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Question #43300

Difficulty: 8/10 | 🟢 Free

The Question

A small conducting ring of mass $m$ , radius $a$ , resistance $R$ , and self-inductance $L$ is attached to a horizontal spring of force constant $K$ . The ring is constrained to move along the $x$ -axis such that its plane remains parallel to the $y$ - $z$ plane. A time-independent, non-uniform magnetic field $\vec{B} = (B_0 + \beta x) \hat{i}$ exists in the region, where $B_0$ and $\beta$ are positive constants. The ring is forced to oscillate along the $x$ -axis with a displacement $x(t) = x_0 \sin(\omega t)$ . Due to electromagnetic induction, a force $F_{em}$ acts on the ring. This force can be decomposed into a damping component proportional to velocity $v$ and a restoring component proportional to displacement $x$ , such that $F_{em} = - b v - \Delta K x$ . Determine the effective additional stiffness $\Delta K$ provided by the electromagnetic interaction.

Options

A.
$\frac{\pi^2 a^4 \beta^2 \omega^2 L}{R^2 + \omega^2 L^2}$
B.
$\frac{\pi^2 a^4 \beta^2 \omega R}{R^2 + \omega^2 L^2}$
C.
$\frac{\pi^2 a^4 \beta^2}{L}$
D.
$\frac{\pi^2 a^4 \beta^2 R^2}{(R^2 + \omega^2 L^2)L}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Lenz’s Law Energy Conservation|L-C Circuit Analogy|Physical Pendulum Inertia

Why This Question Is Tricky

Common Mistakes

Students often misapply Lenz's Law by assuming the magnetic force is purely a damping force ( $F \propto -v$ ), which is only true for a purely resistive loop ( $L=0$ ).

By ignoring the self-inductance $L$ , they fail to recognize the L-C circuit analogy where the inductive reactance causes the induced current to lag the induced EMF.

This phase shift means the current has a component proportional to the displacement $x$ (in-phase with $-x$ ), which acts as an additional restoring force (stiffness $\Delta K$ ).

The trap exposes the transition from a simple mechanical oscillator to a coupled electro-mechanical system where complex impedance alters the natural frequency.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electromagnetic Induction - Class 12
Concept	Lenz’s Law Energy Conservation
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free