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Question #43767

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The Question

A thin opaque cylindrical disk of initial density $\rho_0$ and initial face area $A_0$ at temperature $T_0$ is suspended in a transparent coolant gas maintained at $T_0$ . A laser beam of constant total power $P$ is completely absorbed by one flat face of the disk. The disk loses heat to the gas exclusively from this illuminated face via convection, with a constant convective heat transfer coefficient $h$ . The coefficient of linear expansion of the disk material is $\alpha$ . Assume the face area expands as $A = A_0(1 + 2\alpha\Delta T)$ and the density varies as $\rho = \rho_0(1 - 3\alpha\Delta T)$ , where $\Delta T = T - T_0$ . Which of the following expressions gives the steady-state density of the disk?

Options

A.
$\rho_0 \left[ 1 - \frac{3}{4} \left( \sqrt{1 + \frac{8\alpha P}{h A_0}} - 1 \right) \right]$
B.
$\rho_0 \left[ 1 - \frac{3\alpha P}{h A_0} \right]$
C.
$\rho_0 \left[ 1 - \frac{3}{2} \left( \sqrt{1 + \frac{4\alpha P}{h A_0}} - 1 \right) \right]$
D.
$\rho_0 \left[ 1 - \frac{1}{2} \left( \sqrt{1 + \frac{12\alpha P}{h A_0}} - 1 \right) \right]$

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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

Intensity and Photon Flux|Thermal Expansion of Area and Volume|Variation of Density with Temperature

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to handle non-linear feedback loops in thermal steady-state systems.

The primary trap lies in treating the target's area as a static parameter ( $A_0$ ) during the energy balance phase, which incorrectly yields $\Delta T = P / (h A_0)$ and leads to Option B.

Students must recognize that as the laser heats the target, thermal expansion increases the surface area available for convective heat loss, dynamically altering the cooling rate.

By forcing the solution of a quadratic equation coupling the temperature-dependent area $A(\Delta T)$ with the convective heat flux, the problem exposes the misconception of decoupling thermal expansion from thermodynamic energy balance.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Dual Nature of Radiation and Matter - Class 12
Concept	Intensity and Photon Flux
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free