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Question #76759

Difficulty: 8/10 | 🟢 Free

The Question

A perfectly black spherical body of radius $R \text{ m}$ is maintained at an unknown absolute temperature $T \text{ K}$ . The total radiated thermal power is measured to be $P \text{ W}$ using a detector with a least count of $\Delta P \text{ W}$ . The radius of the sphere is measured using an instrument with a least count of $\Delta R \text{ m}$ . The Stefan-Boltzmann constant $\sigma \text{ W} \cdot \text{m}^{-2} \cdot \text{K}^{-4}$ is assumed to be exact. Which of the following expressions correctly represents the maximum percentage error in the calculated value of the absolute temperature $T$ ?

Options

A.
$\frac{1}{4} \left( \frac{\Delta P}{P} + 2\frac{\Delta R}{R} \right) \times 100 \%$
B.
$4 \left( \frac{\Delta P}{P} + 2\frac{\Delta R}{R} \right) \times 100 \%$
C.
$\frac{1}{4} \left( \frac{\Delta P}{P} + \frac{\Delta R}{R} \right) \times 100 \%$
D.
$\left( \frac{\Delta P}{P} + 2\frac{\Delta R}{R} \right) \times 100 \%$

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

Least Count and Resolution Limits|Stefan-Boltzmann Power Law

Why This Question Is Tricky

Examiner Levers

Visualization Stress

Requires mental rotation or 3D mapping (Optics/Geometry/Vectors).

Difficulty Boost : +4

Dimensional Analysis

Tests if the student verifies units/dimensions before selecting.

Difficulty Boost : +2

Variable Manipulation

Algebraic solutions only (No numbers); tests pure symbolic logic.

Difficulty Boost : +4

Common Mistakes

This question tests the student's ability to correctly apply error propagation to non-linear physical laws, specifically the Stefan-Boltzmann law.

A common trap is the misapplication of the power rule in error analysis.

Because $P \propto T^4$ , students often erroneously multiply the relative error of the measured quantities by 4 instead of dividing by 4, confusing the dependent and independent variables in the error propagation formula.

Additionally, students may forget that the surface area of a sphere depends on $R^2$ , leading to a missing factor of 2 in the radius error term, or they may entirely ignore the power dependence, treating the errors linearly.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Units and Measurements - Class 11 Part 1
Concept	Least Count and Resolution Limits
Difficulty	8
Cognitive Load	8
Target Solve Time	180 sec
Access	Free