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Question #42976

Difficulty: 8/10 | 🟢 Free

The Question

A parallel plate capacitor with circular plates of radius $R$ is filled completely with a linear dielectric material of relative permittivity $\epsilon_r$ . The electric field $\vec{E}$ between the plates is uniform, directed along the central axis, and its magnitude is increasing at a constant rate $\frac{dE}{dt} = \alpha$ . What is the magnitude of the magnetic field $B$ at a radial distance $r$ ( $r < R$ ) from the central axis inside the dielectric? (Assume edge effects are negligible, and $\mu_0, \epsilon_0$ are the permeability and permittivity of free space, respectively).

Options

A.
$\frac{1}{2} \mu_0 \epsilon_0 \alpha r$
B.
$\frac{1}{2} \mu_0 \epsilon_0 (\epsilon_r - 1) \alpha r$
C.
$\frac{1}{2} \mu_0 \epsilon_0 \epsilon_r \alpha r$
D.
$\frac{1}{2} \mu_0 \epsilon_0 \epsilon_r \alpha \frac{R^2}{r}$

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

Faraday’s Law Vector Form|Displacement Current Source|Dielectric Polarization Logic|Vector Calculus

Why This Question Is Tricky

Common Mistakes

This question tests a deep understanding of the Ampere-Maxwell law in the presence of a dielectric medium.

A common trap for students is to memorize the displacement current strictly as $I_d = \epsilon_0 \frac{d\Phi_E}{dt}$ , which is only valid in a vacuum.

When a dielectric is present, the changing electric field induces a time-varying polarization $\vec{P}$ , creating a polarization current density $\vec{J}_p = \frac{\partial \vec{P}}{\partial t}$ .

The true source of the magnetic field is the total displacement current density $\vec{J}_d = \frac{\partial \vec{D}}{\partial t} = \epsilon_0 \frac{\partial \vec{E}}{\partial t} + \frac{\partial \vec{P}}{\partial t} = \epsilon_0 \epsilon_r \frac{\partial \vec{E}}{\partial t}$ .

Ignoring the polarization term leads to the incorrect vacuum-only answer (Option A), while using only the polarization term leads to Option B.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electromagnetic Induction - Class 12
Concept	Faraday’s Law Vector Form
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free