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Question #42657

Difficulty: 8/10 | 🟢 Free

The Question

A semi-infinite solid rod of constant cross-sectional area $A$ extends along the x-axis from $x = 0$ to $x \to \infty$ . The thermal conductivity of the material varies spatially as $k(x) = k_0 \left(1 + \frac{x}{L}\right)^2$ , where $k_0$ and $L$ are positive constants. Heat is generated internally throughout the rod at a steady volumetric rate (power per unit volume) given by $q(x) = \frac{q_0}{\left(1 + \frac{x}{L}\right)^2}$ , where $q_0$ is a positive constant. The end of the rod at $x = 0$ is perfectly insulated. If the temperature as $x \to \infty$ approaches a constant reference temperature $T_\infty$ , what is the steady-state temperature $T_0$ at the insulated end $x = 0$ ?

Options

A.
$T_\infty + \frac{q_0 L^2}{2 k_0}$
B.
$T_\infty + \frac{q_0 L^2}{k_0}$
C.
$T_\infty + \frac{q_0 L^2}{3 k_0}$
D.
$T_\infty + \frac{2 q_0 L^2}{k_0}$

Check Your Answer

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

Reference Potential Convention|First Law Application|Steady State Temperature Gradient|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to apply the First Law of Thermodynamics (energy conservation) and Fourier's Law in a distributed parameter system, serving as an exact electro-thermal analogy to Gauss's Law for non-uniform charge distributions.

The primary trap (Option B) exploits the misconception of lumping distributed generation into a single point source.

Students often calculate the total heat generated ( $Q_{total} = q_0 A L$ ) and the total thermal resistance ( $R_{total} = L / (k_0 A)$ ), and incorrectly multiply them ( $\Delta T = Q_{total} R_{total}$ ) as if all heat must traverse the entire resistance.

This ignores the spatial distribution of the heat flux $H(x)$ , which is zero at the insulated boundary and grows asymptotically, requiring a rigorous integration of the local temperature gradient.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electrostatic Potential and Capacitance - Class 12
Concept	Reference Potential Convention
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free