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Question #42770

Difficulty: 8/10 | 🟢 Free

The Question

Consider an ideal parallel-plate capacitor with circular plates of radius $R$ and separation $d$ . The space between the plates is completely filled with a linear dielectric of relative permittivity $\epsilon_r$ . The geometric capacitance is independent of the applied potential. A time-varying voltage $V(t) = \alpha t$ (where $\alpha$ is a positive constant) is applied across the plates. The dielectric is modeled as a collection of atoms, each containing revolving electrons. The changing electric field shifts these electron orbits, inducing a time-varying microscopic electric dipole moment per unit volume, which constitutes a macroscopic polarization current. Let $c$ be the speed of light in vacuum. By applying the superposition principle to the local magnetic fields generated by these shifting microscopic dipoles, determine the magnitude of the magnetic field $B_p$ at the radial edge of the capacitor ( $r = R$ ) contributed solely by the bound polarization currents in the dielectric.

Options

A.
$\frac{(\epsilon_r - 1) \alpha R}{2 d c^2}$
B.
$\frac{\epsilon_r \alpha R}{2 d c^2}$
C.
$\frac{\alpha R}{2 d c^2}$
D.
$\frac{(\epsilon_r - 1) \alpha R}{d c^2}$

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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

Capacitance Independence|Superposition Principle|Magnetic Dipole Moment of Revolving Electron|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to differentiate between the total effective displacement current and the physical bound current (polarization current) in a dielectric.

A common trap is to blindly apply the standard formula for the total magnetic field inside a charging capacitor ( $B = \frac{\mu_0 \epsilon_0 \epsilon_r R}{2} \frac{\partial E}{\partial t}$ ), which leads to option B.

By explicitly asking for the magnetic field generated solely by the shifting microscopic revolving electrons (the atomic dipoles), the student is forced to isolate the polarization current density $J_p = \frac{\partial P}{\partial t} = \epsilon_0(\epsilon_r - 1)\frac{\partial E}{\partial t}$ .

This prevents formula-plugging and requires a fundamental synthesis of electrostatics (dielectric polarization) and magnetostatics (Ampere's law applied to bound currents).

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electrostatic Potential and Capacitance - Class 12
Concept	Capacitance Independence
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free