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Question #42819

Difficulty: 8/10 | 🟢 Free

The Question

A point charge $Q$ (in $\text{C}$ ) is placed at the origin. It is surrounded by a thick spherical dielectric shell of inner radius $R$ and outer radius $2R$ (both in $\text{m}$ ). The relative permittivity (dielectric constant) of the shell varies with the radial distance $r$ from the origin as $K(r) = \alpha r$ , where $\alpha$ is a positive constant with units of $\text{m}^{-1}$ . Which of the following expressions correctly represents the volume bound charge density $\rho_b$ inside the dielectric shell at a distance $r$ ( $R < r < 2R$ ) from the origin?

Options

A.
$-\frac{Q}{4\pi \alpha r^4}$
B.
$\frac{Q(\alpha r - 1)}{4\pi \alpha r^4}$
C.
$-\frac{Q\alpha}{4\pi r^2}$
D.
$\frac{Q}{4\pi \alpha r^4}$

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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

Dielectric Constant (K) Shift|Electric Field Symmetry|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the fundamental distinction between the displacement field $\vec{D}$ and the electric field $\vec{E}$ in a non-uniform dielectric medium.

A common trap is assuming that the bound charge density can be found by treating the dielectric constant $K$ as a global scalar, or by incorrectly applying Gauss's Law directly to $\vec{E}$ without accounting for the spatial gradient of $K(r)$ .

By defining $K(r) = \alpha r$ , the displacement field $\vec{D}$ remains strictly inverse-square due to the enclosed free charge, but $\vec{E}$ becomes an inverse-cube field.

Students must correctly identify $\vec{P} = \vec{D} - \epsilon_0 \vec{E}$ and apply the divergence operator in spherical coordinates ( $\rho_b = -\nabla \cdot \vec{P}$ ) to find the local bound charge density.

Failing to differentiate the spatially varying polarization field correctly leads to dimensionally consistent but physically wrong radial dependencies.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electric Charges and Fields - Class 12
Concept	Dielectric Constant (K) Shift
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free