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Question #42742

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The Question

A spherical capacitor consists of an inner conducting sphere of radius $a$ and an outer conducting shell of inner radius $c$ . The space between them is filled with two concentric layers of weakly conducting solid dielectrics. The inner layer ( $a < r < b$ ) has electrical conductivity $\sigma_1$ and permittivity $\epsilon_1$ . The outer layer ( $b < r < c$ ) has electrical conductivity $\sigma_2$ and permittivity $\epsilon_2$ . A constant potential difference is maintained across the electrodes such that a steady radial current $I$ flows outward from the inner sphere to the outer shell. What is the total free charge $Q$ accumulated at the dielectric interface $r = b$ in the steady state?

Options

A.
$I \left( \frac{\epsilon_2}{\sigma_2} - \frac{\epsilon_1}{\sigma_1} \right)$
B.
$0$
C.
$I \epsilon_0 \left( \frac{1}{\sigma_2} - \frac{1}{\sigma_1} \right)$
D.
$I \left( \frac{\epsilon_1}{\sigma_2} - \frac{\epsilon_2}{\sigma_1} \right)$

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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

Kirchhoff’s Junction Rule|Electric Field Line Geometry|Vector Calculus

Why This Question Is Tricky

Common Mistakes

This question tests a common misconception regarding steady-state currents in continuous media.

Students often incorrectly assume that a steady-state current ( $\nabla \cdot \mathbf{J} = 0$ ) implies zero net charge everywhere, misapplying Kirchhoff's junction rule by assuming no charge can accumulate at a 'junction' (the interface).

They fail to transition to the field theory perspective (Gauss's Law), where differing relaxation times ( $\epsilon/\sigma$ ) in the two media necessitate a discontinuity in the electric field.

This discontinuity requires a static accumulation of free charge at the interface to maintain the continuous, steady current flow.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Current Electricity - Class 12
Concept	Kirchhoff’s Junction Rule
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free