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Question #43160

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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 completely filled with a non-uniform dielectric whose permittivity varies radially as $\epsilon(r) = \epsilon_0 \left(1 + 2\frac{r^2}{R^2}\right)$ , where $r$ is the distance from the central axis of the plates. A constant conduction current $I_0$ is supplied to the capacitor, charging it. Neglecting edge effects, what is the magnitude of the induced magnetic field $B$ at a radial distance $r = \frac{R}{2}$ from the central axis between the plates? (Assume $I_0$ is in Amperes, $R$ is in meters, and $\mu_0$ is the vacuum permeability in $\text{T}\cdot\text{m/A}$ )

Options

A.
$\frac{5 \mu_0 I_0}{32 \pi R}$
B.
$\frac{\mu_0 I_0}{4 \pi R}$
C.
$\frac{\mu_0 I_0}{8 \pi R}$
D.
$\frac{3 \mu_0 I_0}{16 \pi R}$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Variable_Constraint

Develop pure symbolic logic skills by mastering algebraic constraints without relying on numerical data.

Concepts Behind This Question

Displacement Current Source|Non-Uniform Dielectric Filling|Vector Calculus

Why This Question Is Tricky

Common Mistakes

This question tests the common misconception that displacement current density $J_d$ is always uniform across a capacitor's cross-section.

Students often blindly apply $J_d = I_0 / A$ , failing to recognize that $J_d = \partial D / \partial t$ .

In a capacitor with a radially varying dielectric, the highly conducting plates remain equipotential, forcing the electric field $E$ to be spatially uniform.

Consequently, the displacement field $D = \epsilon(r)E$ and the displacement current density $J_d$ become radially dependent.

Distractor options specifically target the assumptions of uniform $J_d$ (Option B), ignoring the dielectric's polarization contribution by using $J_d = \epsilon_0 \partial E / \partial t$ (Option C), or incorrectly evaluating the non-uniform current integral by assuming $J_d$ is constant at its boundary value (Option D).

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electromagnetic Waves - Class 12
Concept	Displacement Current Source
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free