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Question #42860

Difficulty: 8/10 | 🟢 Free

The Question

A semi-infinite string of linear mass density $\mu$ and tension $T$ lies along the negative x-axis ( $x \le 0$ ). Its end at $x=0$ is attached to a massless conducting rod of length $l$ oriented parallel to the z-axis. The rod can slide without friction along two parallel conducting rails that are parallel to the y-axis. A uniform magnetic field $B$ points in the positive x-direction. The rails are connected to a DC voltage source $V_0$ , a resistor $R$ , and an inductor $L_0$ to form a closed series circuit. At $t=0$ , the circuit is closed. As the rod moves, it generates transverse waves in the string. Assuming the string only supports transverse waves and remains taut, what is the steady-state energy stored in the inductor after a long time? (All given quantities are in standard SI units)

Options

A.
$\frac{L_0 V_0^2 \mu T}{2 (R \sqrt{\mu T} + B^2 l^2)^2}$
B.
$\frac{L_0 V_0^2}{2 R^2}$
C.
$\frac{L_0 V_0^2 \mu T}{2 (R \sqrt{\mu T} - B^2 l^2)^2}$
D.
$\frac{L_0 V_0^2 \mu T}{2 (R \sqrt{\mu T} + 2 B^2 l^2)^2}$

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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

Standing Wave Nodes/Antinodes|Transient Current in RL Circuit|Work-Energy Theorem Validity

Why This Question Is Tricky

Common Mistakes

This question tests the coupling of mechanical wave boundary conditions with electromagnetic transients.

A common trap is treating the mechanical boundary as a static or purely inertial load, ignoring the continuous energy carried away by the generated wave.

This leads to omitting the motional back-EMF in the circuit equation, effectively ignoring the 'radiation resistance' ( $R_{rad} = B^2 l^2 / \sqrt{\mu T}$ ) of the string.

Students falling for this will incorrectly assume the steady-state current is simply $V_0/R$ , failing to realize that the drifting node continuously extracts electrical energy to propagate mechanical waves.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Waves - Class 11 Part 2
Concept	Standing Wave Nodes/Antinodes
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free