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Question #42142

Difficulty: 8/10 | 🟢 Free

The Question

A cylindrical heating wire of initial resistance $R_0 = 1 \text{ }\Omega$ at $T_0 = 300 \text{ K}$ is submerged in an insulating vessel containing $m = 1 \text{ kg}$ of a liquid with specific heat capacity $s = 4000 \text{ J kg}^{-1} \text{K}^{-1}$ . A constant voltage $V = 20 \text{ V}$ is applied across the wire. The temperature coefficient of resistivity of the wire is $\alpha_\rho = 5 \times 10^{-3} \text{ K}^{-1}$ and its coefficient of linear thermal expansion is $\alpha_L = 1 \times 10^{-3} \text{ K}^{-1}$ . Assuming the heat capacity of the wire and the vessel are negligible, and no heat is lost to the surroundings, find the time $t$ required to heat the liquid to $T = 350 \text{ K}$ . (Use the approximation $(1+x)^n \approx 1+nx$ for small $x$ ).

Options

A.
$550 \text{ s}$
B.
$562.5 \text{ s}$
C.
$500 \text{ s}$
D.
$575 \text{ s}$

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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

Specific Heat Capacity of Phase Mixtures|Temperature Coefficient (Alpha)|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to couple Joule heating with thermodynamics while accounting for non-linear feedback and geometric thermal expansion.

A common trap is assuming constant power $P = V^2/R_0$ , which ignores the temperature dependence of resistance and leads to a simple linear time relation (yielding 500 s).

The secondary, more advanced trap is forgetting that resistance $R = \rho L / A$ depends on the physical dimensions of the wire, which also expand.

Students often use only the resistivity coefficient $\alpha_\rho$ (yielding 562.5 s) or incorrectly add the linear expansion coefficient instead of deriving the effective coefficient $\alpha_{eff} = \alpha_\rho - \alpha_L$ .

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Thermal Properties of Matter - Class 11 Part 2
Concept	Specific Heat Capacity of Phase Mixtures
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free