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Question #77836

Difficulty: 8/10 | 🟢 Free

The Question

A uniform electric field $\vec{E} = E_0 \hat{k}$ exists in a region. To determine the magnitude $E_0$ , a physicist measures the absolute electric flux $\Phi$ through an open, bowl-shaped surface $S$ . The surface $S$ is generated by revolving the curve $z = H_0 \left(\frac{r}{R_0}\right)^n$ around the $z$ -axis (where $r = \sqrt{x^2+y^2}$ , and $H_0, R_0$ are known exact dimensional constants) and is bounded by a flat circular top at a measured height $z = H$ . The measured quantities are $\Phi$ , $H$ , and the dimensionless geometric parameter $n$ , with maximum absolute errors $\Delta \Phi$ , $\Delta H$ , and $\Delta n$ respectively. A student INCORRECTLY assumes that the effective area scales proportionally to the enclosed volume $V = \frac{\pi n R_0^2 H^{(n+2)/n}}{H_0^{2/n} (n+2)}$ , leading to a flawed error analysis. All of the following expressions represent INCORRECT derivations for the maximum relative error $\frac{\Delta E_0}{E_0}$ EXCEPT:

Options

A.
$\frac{\Delta \Phi}{\Phi} + \frac{2}{n} \frac{\Delta H}{H} + \frac{2}{n^2} \left| \ln\left(\frac{H}{H_0}\right) \right| \Delta n$
B.
$\frac{\Delta \Phi}{\Phi} + n \frac{\Delta H}{H} + \left| \ln\left(\frac{H}{H_0}\right) \right| \Delta n$
C.
$\frac{\Delta \Phi}{\Phi} + 2 \frac{\Delta H}{H} + \frac{1}{n} \left| \ln\left(\frac{H}{H_0}\right) \right| \Delta n$
D.
$\frac{\Delta \Phi}{\Phi} + \frac{2}{n} \frac{\Delta H}{H} + \frac{2}{n} \left| \ln\left(\frac{H}{H_0}\right) \right| \Delta n$

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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

Flux and Surface Geometry|Relative Error Propagation in Powers

Why This Question Is Tricky

Examiner Levers

Negative Thinking

Uses "NOT", "EXCEPT", or "INCORRECT" to flip the logic gate.

Difficulty Boost : +2

Extraneous Data

Provides unnecessary variables (Red Herrings) to test conceptual focus.

Difficulty Boost : +3

Dimensional Analysis

Tests if the student verifies units/dimensions before selecting.

Difficulty Boost : +2

Common Mistakes

This question tests the rigorous application of error propagation on non-linear, variable-curvature geometries, specifically targeting the misconception that area scales with a fixed power of linear dimensions (like $R^2$ or $H^2$ ).

Students typically memorize the power-law rule for relative errors ( $\Delta A/A = 2\Delta R/R$ ) and misapply it to non-spherical Gaussian surfaces.

By introducing a surface where the effective flux penetration area is a variable power function of the height coordinate ( $A \propto H^{2/n}$ ), the error in the field $E_0$ becomes differentially coupled to the geometric parameter $n$ .

The lethal trap occurs when students either ignore the uncertainty in the exponent $n$ or fail to correctly apply logarithmic differentiation to a variable exponent, missing the $\frac{2}{n^2} \ln(H/H_0) \Delta n$ term.

The extraneous volume data further distracts those who incorrectly link uniform field flux to enclosed volume rather than the projected cross-sectional area.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Units and Measurements - Class 11 Part 1
Concept	Relative Error Propagation in Powers
Difficulty	8
Cognitive Load	8
Target Solve Time	180 sec
Access	Free