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Question #42674

Difficulty: 8/10 | 🟢 Free

The Question

A spherical capacitor consists of two concentric perfectly conducting shells of inner radius $a$ and outer radius $b$ . The space between the shells ( $a < r < b$ ) is completely filled with a leaky dielectric material of uniform resistivity $\rho$ and permittivity $\epsilon$ . A battery of constant electromotive force $\mathcal{E}$ and finite internal resistance $R_0$ is connected across the two shells via thin insulated wires. After a long time, steady-state conditions are reached. Which of the following expressions gives the magnitude of the electric field $E(r)$ at a radial distance $r$ ( $a < r < b$ ) from the center? (Assume all quantities are in standard SI units)

Options

A.
$\frac{\rho a b \mathcal{E}}{r^2 [4\pi a b R_0 + \rho(b-a)]}$
B.
$\frac{a b \mathcal{E}}{r^2 (b-a)}$
C.
0
D.
$\frac{\rho \mathcal{E}}{4\pi r^2 R_0}$

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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

Conductor Interior Potential|Kirchhoff’s Loop Rule|Internal Resistance Variation

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to distinguish between electrostatic equilibrium and steady-state electrodynamics in a circuit containing a non-ideal (leaky) capacitor.

The primary trap is the reflexive assumption that a capacitor in a DC circuit acts as an open circuit in steady state, leading students to incorrectly apply the electrostatic boundary condition ( $I=0$ , $V_c = \mathcal{E}$ ) and choose the standard ideal capacitor field.

Alternatively, students might invoke the 'Faraday Cage' theorem, assuming $E=0$ inside a conducting medium.

By introducing a finite resistivity, the problem forces the transition from Gauss's Law to the microscopic form of Ohm's Law ( $J = \sigma E$ ), requiring the calculation of a steady-state radial current via Kirchhoff's Loop Rule to find the correct non-zero potential gradient.

Quick Facts About This Question

Question Information

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