Before You Begin

Question #42296

Difficulty: 8/10 | 🟢 Free

The Question

A cylindrical metallic wire of length $L_0$ , radius $r$ , and emissivity $\epsilon$ hangs vertically from a rigid support, carrying a mass $M$ at its lower end. At ambient temperature $T_0$ , the free electron density is $n$ and the relaxation time is $\tau_0$ . When a constant current $I$ flows through the wire, its temperature increases by $\Delta T$ . The relaxation time decreases as $\tau = \tau_0 / (1 + \beta \Delta T)$ , and the Young's modulus decreases as $Y = Y_0 - \alpha \Delta T$ . The wire loses heat only via radiation to the surroundings at $T_0$ (Stefan-Boltzmann constant $\sigma_s$ ). Assuming $\Delta T \ll T_0$ and ignoring linear thermal expansion, what is the steady-state increase in the length of the wire due to the current? (Let $m_e$ and $e$ be the mass and charge of an electron).

Options

A.
$\frac{M g L_0 \alpha I^2 m_e}{\pi r^2 Y_0^2 \left( 8 \pi^2 r^3 n e^2 \tau_0 \epsilon \sigma_s T_0^3 - I^2 \beta m_e \right)}$
B.
$\frac{M g L_0 \alpha I^2 m_e}{8 \pi^3 r^5 Y_0^2 n e^2 \tau_0 \epsilon \sigma_s T_0^3}$
C.
$\frac{M g L_0 \alpha I^2 m_e}{\pi r^2 Y_0 \left( 8 \pi^2 r^3 n e^2 \tau_0 \epsilon \sigma_s T_0^3 - I^2 \beta m_e \right)}$
D.
$\frac{M g L_0 \alpha I^2 m_e}{\pi r^2 Y_0^2 \left( 2 \pi^2 r^3 n e^2 \tau_0 \epsilon \sigma_s T_0^3 - I^2 \beta m_e \right)}$

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

Young's Modulus Sensitivity|Relaxation Time Dependency|Binomial Approximation

Why This Question Is Tricky

Common Mistakes

This question tests the coupled understanding of Drude's model, thermal radiation, and elasticity.

The primary trap lies in assuming the electrical heating power is independent of temperature ( $P = I^2 R_0$ ), ignoring the temperature dependence of the relaxation time which creates a feedback loop in the steady-state temperature.

Additionally, students often fail to correctly linearize the Stefan-Boltzmann law using the binomial approximation ( $(T_0+\Delta T)^4 - T_0^4 \approx 4T_0^3\Delta T$ ) or incorrectly apply the Taylor expansion for the change in elongation ( $\Delta L \propto 1/Y$ ), leading to missing the squared term $Y_0^2$ in the denominator.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Mechanical Properties of Solids - Class 11 Part 2
Concept	Young's Modulus Sensitivity
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free