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Question #42703

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The Question

Two concentric conducting spherical shells of radii $a$ and $b$ ( $a < b$ ) form a spherical capacitor. The inner shell carries a free charge $+Q$ and the outer shell carries a free charge $-Q$ . The space between the shells ( $a < r < b$ ) is filled with a non-uniform isotropic dielectric material whose relative permittivity varies with the radial distance $r$ as $\epsilon_r(r) = \frac{\beta}{r}$ , where $\beta$ is a positive constant with dimensions of length. Which of the following expressions correctly represents the bound volume charge density $\rho_b(r)$ within the dielectric region?

Options

A.
$\frac{Q}{4\pi \beta r^2}$
B.
$-\frac{Q}{4\pi \beta r^2}$
C.
$\frac{Q \beta}{4\pi r^4}$
D.
0

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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

Potential of Concentric Shells|Dielectric Polarization Logic|Non-Uniform Dielectric Filling

Why This Question Is Tricky

Common Mistakes

This question tests the deep understanding of polarization in non-uniform dielectrics.

A common trap is assuming that if the free volume charge density is zero, the bound volume charge density $\rho_b$ must also be zero (Option D), which is only true for uniform dielectrics.

Another lethal trap is calculating the polarization vector $\vec{P}$ using the vacuum electric field $\vec{E} = \frac{Q}{4\pi \epsilon_0 r^2} \hat{r}$ instead of the actual modified electric field inside the dielectric, leading to an incorrect radial dependence (Option C).

Finally, students often miss the negative sign in $\rho_b = -\nabla \cdot \vec{P}$ or incorrectly apply the spherical divergence operator by forgetting the $r^2$ weighting factor, leading to Option B.

The problem rigorously enforces the hierarchy of Gauss's Law for $\vec{D}$ , the constitutive relations, and the definition of bound charges.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electrostatic Potential and Capacitance - Class 12
Concept	Potential of Concentric Shells
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free