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Question #42321

Difficulty: 8/10 | 🟢 Free

The Question

A perfectly conducting straight rod of mass $m$ and length $l$ slides without friction down two vertical, parallel conducting rails. A uniform horizontal magnetic field $B$ exists perpendicular to the plane of the rails. The top ends of the rails are connected to an ideal solenoid of inductance $L$ and internal volume $V_0$ . The system is released from rest at $t=0$ . Assuming the magnetic field inside the solenoid is uniform and confined to its volume $V_0$ , which of the following expressions represents the time-averaged magnetic energy density $\langle u_B \rangle$ inside the solenoid over one complete period of oscillation?

Options

A.
$\frac{3m^2g^2L}{4B^2 l^2 V_0}$
B.
$\frac{m^2g^2L}{4B^2 l^2 V_0}$
C.
$\frac{m^2g^2L}{2B^2 l^2 V_0}$
D.
$\frac{2m^2g^2L}{B^2 l^2 V_0}$

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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

Potential Energy Reference|L-C Circuit Analogy|Energy Density Partitioning

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to map gravitational potential energy to magnetic energy in an electromechanical oscillator and correctly identify the shifted reference frame.

The 'lethal collision' occurs when students assume the induced current oscillates symmetrically around zero (yielding a time-averaged factor of 1/2 instead of 3/2).

Because gravity provides a constant bias force, the 'ground' state of the system shifts, causing the current to oscillate between 0 and 2I_eq.

Integrating the time-varying magnetic field density exposes those who blindly apply standard AC power formulas without analyzing the transient DC offset.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Work, Energy, and Power - Class 11 Part 1
Concept	Potential Energy Reference
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free