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Question #42005

Difficulty: 8/10 | 🟢 Free

The Question

A point charge $Q_1$ is fixed at the origin. The surrounding space is filled with a linear, isotropic dielectric medium whose relative permittivity varies radially as $\epsilon_r(r) = \alpha r$ , where $\alpha$ is a known exact constant with units of $\text{m}^{-1}$ . A second point charge $Q_2$ is placed at a distance $r = R$ from the origin. The maximum percentage errors in the measurement of $Q_1$ , $Q_2$ , and $R$ are $1\%$ , $1\%$ , and $2\%$ respectively. Assuming $\alpha$ and the vacuum permittivity $\epsilon_0$ are exact, what is the maximum percentage error in the calculated magnitude of the electrostatic force on $Q_2$ ?

Options

A.
$8\%$
B.
$6\%$
C.
$4\%$
D.
$7\%$

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

Coulomb's Law Vector Form|Relative Error Propagation in Powers

Why This Question Is Tricky

Common Mistakes

This question tests the integration of Gauss's Law in dielectrics with error propagation.

A common trap is blindly applying the standard Coulomb's Law ( $F \propto r^{-2}$ ) and assuming the relative error contribution from distance is $2(\Delta R/R)$ , which would incorrectly yield $6\%$ .

However, in a medium with spatially varying permittivity $\epsilon_r(r) = \alpha r$ , the effective force law becomes $F \propto r^{-3}$ .

Students must first derive the correct functional dependence of the force using $\vec{D} = \epsilon \vec{E}$ before applying relative error formulas, shifting the power dependency and invalidating standard intuition.

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	7
Target Solve Time	160 sec
Access	Free