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Question #85744

Difficulty: 8/10 | 🟢 Free

The Question

A solid non-conducting sphere of radius $R$ carries a spherically symmetric volume charge density $\rho(r) = \rho_0 \left(1 - \frac{r}{R}\right)$ for $r \le R$ , where $\rho_0$ is a positive constant with units of $\text{C m}^{-3}$ . Let $r_{max}$ be the radial distance from the center where the electric field magnitude is maximum, $E_{max}$ be this maximum electric field magnitude, $E_{surf}$ be the electric field magnitude at the surface ( $r=R$ ), and $\Phi_{R/2}$ be the total electric flux through a concentric spherical Gaussian surface of radius $R/2$ . Which of the following calculated values is INCORRECT?

Options

A.
$r_{max} = R$
B.
$E_{max} = \frac{\rho_0 R}{9 \epsilon_0}$
C.
$E_{surf} = \frac{\rho_0 R}{12 \epsilon_0}$
D.
$\Phi_{R/2} = \frac{5\pi \rho_0 R^3}{48 \epsilon_0}$

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

Electric Field Line Geometry|Gauss Law|Surface Integration

Why This Question Is Tricky

Examiner Levers

Negative Thinking

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

Difficulty Boost : +2

Logical Assertion

Tests the causal relationship between two true statements.

Difficulty Boost : +4

Common Mistakes

This question tests the student's reliance on the 'Symmetry-First' heuristic derived from uniform charge distributions, where the electric field magnitude strictly increases linearly inside a solid sphere and reaches its maximum at the surface ( $r=R$ ).

By introducing a linearly decreasing volumetric charge density, the flux constraint forces the electric field to peak inside the volume ( $r=2R/3$ ) rather than at the boundary.

Students who misapply the uniform-sphere intuition will incorrectly assume $r_{max} = R$ without performing the full volume integral and subsequent derivative required by Gauss's Law for non-uniform geometries.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electric Charges and Fields - Class 12
Concept	Electric Field Line Geometry
Difficulty	8
Cognitive Load	8
Target Solve Time	180 sec
Access	Free