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Question #42837

Difficulty: 8/10 | 🟢 Free

The Question

A solid non-conducting sphere of radius $R$ contains a uniform free volume charge density $\rho_0$ . The sphere is filled with a non-uniform linear dielectric material whose relative permittivity varies with the radial distance $r$ from the center as $\epsilon_r(r) = \frac{R}{r}$ for $0 < r \le R$ . Which of the following expressions correctly represents the volume bound charge density $\rho_b(r)$ inside the dielectric at a distance $r$ from the center?

Options

A.
$\rho_0 \left( \frac{4r}{3R} - 1 \right)$
B.
$\rho_0 \left( \frac{r}{R} - 1 \right)$
C.
$\rho_0 \left( \frac{2r}{3R} - \frac{1}{3} \right)$
D.
$\rho_0 \left( 1 - \frac{r}{R} \right)$

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

Gauss Law Applicability|Shell Theorem (Internal vs External)|Non-Uniform Dielectric Filling

Why This Question Is Tricky

Common Mistakes

This question tests the application of Gauss's Law in non-uniform dielectrics.

A common student trap is to blindly apply the formula $\rho_b = -\rho_f (1 - 1/\epsilon_r)$ , which is only valid for homogeneous linear dielectrics where $\nabla \epsilon_r = 0$ .

In an inhomogeneous medium, the spatial variation of permittivity acts as an additional source of bound charge.

To solve correctly, students must revert to fundamental principles: finding the displacement field $\vec{D}$ from free charges, determining $\vec{E}$ and $\vec{P}$ , and finally applying the differential operator $\rho_b = -\nabla \cdot \vec{P}$ in spherical coordinates.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electric Charges and Fields - Class 12
Concept	Gauss Law Applicability
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free