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Question #43429

Difficulty: 8/10 | 🟢 Free

The Question

A circular parallel plate capacitor of radius $R$ and plate separation $d$ is being charged by a constant current $I_0$ . The space between the plates is completely filled with a non-uniform dielectric material whose relative permittivity varies radially as $\epsilon_r(r) = 1 + \frac{r}{R}$ , where $r$ is the distance from the central axis of the capacitor. Assuming the electric field between the plates is spatially uniform at any instant and edge effects are negligible, what is the magnitude of the induced magnetic field $B$ at a radial distance $r = \frac{R}{2}$ from the central axis?

Options

A.
$\frac{\mu_0 I_0}{5 \pi R}$
B.
$\frac{\mu_0 I_0}{4 \pi R}$
C.
$\frac{3 \mu_0 I_0}{10 \pi R}$
D.
$\frac{7 \mu_0 I_0}{20 \pi R}$

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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

Ampere’s Law Displacement Current|Non-Uniform Dielectric Filling|Magnetic Flux Scalar Sign

Why This Question Is Tricky

Common Mistakes

This question tests the subtle distinction between the electric field ( $E$ ) and the electric displacement field ( $D$ ) in the context of the Ampere-Maxwell Law.

Students frequently fall into the trap of assuming that a uniform charging current implies a uniform displacement current density ( $J_d$ ), leading them to incorrectly scale the enclosed current by the simple area ratio $(r/R)^2$ (which yields Option B).

However, because the plates are equipotential, $E$ is spatially uniform, but the radially varying dielectric causes $D$ , and consequently $J_d = \partial D/\partial t$ , to be non-uniform.

Solving this requires recognizing that $J_d$ is proportional to the local permittivity $\epsilon(r)$ and correctly integrating this non-homogeneous density over the circular cross-section to find the enclosed displacement current.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Moving Charges and Magnetism - Class 12
Concept	Ampere’s Law Displacement Current
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free