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Question #42988

Difficulty: 8/10 | 🟢 Free

The Question

A highly flexible, tightly wound solenoid of natural length $L_0$ , total number of turns $N$ , cross-sectional area $A$ , and total mass $M$ is suspended vertically from its upper end in a uniform downward gravitational field $g$ . The solenoid behaves elastically like a helical spring with an overall equivalent spring constant $k$ . Assuming the cross-sectional area $A$ remains constant and the solenoid reaches mechanical equilibrium under its own weight, what is the total self-inductance $L$ of the suspended solenoid? (Ignore end effects and assume $\mu_0$ is the permeability of free space)

Options

A.
$\frac{\mu_0 N^2 A k}{M g} \ln\left(1 + \frac{M g}{k L_0}\right)$
B.
$\frac{\mu_0 N^2 A}{L_0 + \frac{M g}{2 k}}$
C.
$\frac{\mu_0 N^2 A}{L_0 + \frac{M g}{k}}$
D.
$\frac{\mu_0 N^2 A k}{2 M g} \ln\left(1 + \frac{2 M g}{k L_0}\right)$

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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

Self-Inductance of Solenoid|Wire Elongation under Self-Weight|Work Done in Stretching

Why This Question Is Tricky

Common Mistakes

This question tests the coupling of elastic mechanics and electromagnetism, specifically targeting the misconception that a non-uniformly stretched solenoid can be modeled using an average turn density or average length.

Students often fall into the trap of calculating the total extension of the center of mass ( $\Delta L = Mg/2k$ ) and substituting the new total length into the standard formula $L = \mu_0 N^2 A / L_{new}$ .

However, inductance depends on the integral of the square of the local turn density $n(x)^2$ .

Because the self-weight causes a linearly varying tension, the local strain and turn density vary continuously.

The correct approach requires defining the local inductance differential $dL = \mu_0 n(x)^2 A dx$ based on the local stretching $dx$ of an unextended element $dy$ , and integrating over the natural length.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electromagnetic Induction - Class 12
Concept	Self-Inductance of Solenoid
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free