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Question #42737

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The Question

A spherical battery consists of an inner conducting electrode of radius $a$ and an outer conducting electrode of radius $b$ . The space between the electrodes ( $a < r < b$ ) is filled with a leaky dielectric material characterized by a constant relaxation time $\tau$ . The relative permittivity of the material varies radially as $\epsilon_r(r) = \frac{\lambda}{r^2}$ , where $\lambda$ is a positive constant with units of $\text{m}^2$ and $r$ is the radial distance from the center. The battery has an ideal electromotive force $\mathcal{E}$ generated at the inner electrode-dielectric interface. If the battery is short-circuited by connecting the inner and outer electrodes with an ideal zero-resistance wire, what is the magnitude of the steady-state electric field $E(r)$ at a distance $r$ inside the dielectric?

Options

A.
$\frac{\mathcal{E}}{b-a}$
B.
$\frac{\mathcal{E} a b}{(b-a) r^2}$
C.
$\frac{\mathcal{E} r}{b^2-a^2}$
D.
$\frac{\mathcal{E} (a+b)}{2 r^2 \ln(b/a)}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Internal Resistance Variation|Non-Uniform Dielectric Filling|Electrostatic Potential

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of macroscopic circuit laws and microscopic electromagnetic theory within a non-uniform medium.

The primary trap exploits the student's reflexive assumption that the electric field in any spherical geometry must follow an inverse-square law ( $E \propto 1/r^2$ ), leading them to incorrectly choose the standard spherical capacitor field (Option B).

By requiring the derivation of local resistivity from the relaxation time and spatially varying permittivity, the problem reveals a counter-intuitive physical reality: the $r^2$ dependence of the resistivity perfectly cancels the $1/r^2$ dependence of the current density.

Consequently, the electric field is uniform, challenging the misconception that geometry alone dictates the field profile without accounting for material inhomogeneities.

Quick Facts About This Question

Question Information

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