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Question #42723

Difficulty: 8/10 | 🟢 Free

The Question

A spherical capacitor consists of two concentric conducting shells of inner radius $a$ and outer radius $b$ . The space between the shells ( $a < r < b$ ) is filled with a leaky, non-uniform dielectric material. The permittivity of the dielectric varies radially as $\epsilon(r) = \epsilon_1 \left(\frac{r}{a}\right)^2$ , and its electrical conductivity varies as $\sigma(r) = \sigma_1 \left(\frac{a}{r}\right)^2$ , where $\epsilon_1$ and $\sigma_1$ are positive constants. A steady DC voltage $V_0$ is applied across the capacitor, maintaining the inner shell at a higher potential than the outer shell. In the steady state, what is the total free charge accumulated strictly within the volume of the dielectric material (excluding the surfaces of the conducting shells)?

Options

A.
$\frac{4\pi \epsilon_1 V_0}{a^2 (b-a)} (b^4 - a^4)$
B.
0
C.
$4\pi \epsilon_1 V_0 (b+a)$
D.
$\frac{4\pi \epsilon_1 V_0 a^2}{b-a}$

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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

Current Density Vector|Non-Uniform Dielectric Filling|Electric Field Symmetry|Integration

Why This Question Is Tricky

Common Mistakes

This question tests a critical misconception regarding steady-state currents in non-uniform media.

Students often incorrectly assume that a steady DC state implies zero free charge accumulation ( $\rho_f = 0$ ) everywhere inside a conducting medium, leading them to choose 0.

However, in a non-uniform leaky dielectric, the spatial variation of conductivity $\sigma(r)$ dictates the electric field via $\nabla \cdot (\sigma \vec{E}) = 0$ .

Consequently, the displacement field $\vec{D} = \epsilon \vec{E}$ has a non-zero divergence ( $\nabla \cdot \vec{D} \neq 0$ ), meaning free charge must continuously accumulate and be replenished to maintain the steady state.

Another trap is calculating the surface charge on the inner shell instead of the volume charge, or ignoring the radial dependence of $\epsilon(r)$ when applying Gauss's Law.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Current Electricity - Class 12
Concept	Current Density Vector
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free