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Question #42919

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The Question

A parallel plate capacitor consists of two circular plates of radius $R$ separated by a small distance $d$ . The space between the plates is filled with a leaky dielectric material with constant permittivity $\epsilon$ . The conductivity of the dielectric varies radially as $\sigma(r) = \sigma_0 \frac{r}{R}$ , where $r$ is the distance from the central axis. At $t=0$ , a constant direct current $I_0$ begins to flow into the positive plate and out of the negative plate from an external circuit. Assuming the electric field magnitude $E$ between the plates is spatially uniform at any given instant and edge effects are negligible, what is the magnitude of the magnetic field $B$ at a distance $r$ ( $r < R$ ) from the central axis as a function of time $t$ ? (Assume quasi-static conditions where the induced electric field due to the time-varying magnetic field is negligible, and $\mu_0$ is the vacuum permeability).

Options

A.
$\frac{\mu_0 I_0}{2\pi r} \left[ \frac{r^3}{R^3} + \left( \frac{r^2}{R^2} - \frac{r^3}{R^3} \right) e^{-\frac{2 \sigma_0}{3 \epsilon} t} \right]$
B.
$\frac{\mu_0 I_0}{2\pi r} \left[ \frac{r^3}{R^3} + \left( \frac{r^2}{R^2} - \frac{r^3}{R^3} \right) e^{-\frac{\sigma_0}{\epsilon} t} \right]$
C.
$\frac{\mu_0 I_0 r^2}{2\pi R^3} \left( 1 - e^{-\frac{2\sigma_0}{3\epsilon} t} \right)$
D.
$\frac{\mu_0 I_0 r}{2\pi R^2}$

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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

Flux and Surface Geometry|Maxwell's Correction to Ampere's Law

Why This Question Is Tricky

Common Mistakes

This question tests the rigorous application of the Ampere-Maxwell Law in a system where both conduction and displacement currents coexist and vary spatially.

A common trap is assuming the conduction current enclosed by an Amperean loop is simply $J_c \times \pi r^2$ , ignoring the required area integral $\int J_c 2\pi r dr$ for a radially varying conductivity.

This leads to an incorrect time constant (Option B).

Another trap is ignoring the displacement current entirely (Option C) or assuming the magnetic field remains constant in time as it would in an ideal, non-leaky capacitor (Option D).

The student must correctly bridge Gauss's Law (to relate $E$ to total current) and Ampere's Law (to find $B$ ) while handling non-trivial surface geometry integrations.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electric Charges and Fields - Class 12
Concept	Flux and Surface Geometry
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free