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Question #42969

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The Question

A very long dielectric slab of thickness $d$ (in meters) along the $y$ -axis and relative permittivity $\epsilon_r$ (dimensionless) is moving with a constant velocity $\vec{v} = v_0 \hat{i}$ (in m/s) along its length (the $x$ -axis). The slab is infinite in the $z$ -direction. It passes through a region of uniform external magnetic field $\vec{B}_{ext} = B_0 \hat{k}$ (in Tesla). Assume $v_0 \ll c$ , where $c$ (in m/s) is the speed of light in vacuum. Due to the motion of the polarized dielectric, an induced magnetic field $\vec{B}_{ind}$ is generated inside the slab. What is the magnitude and direction of $\vec{B}_{ind}$ far from the edges of the slab?

Options

A.
$\frac{v_0^2}{c^2} (\epsilon_r - 1) B_0 \hat{k}$
B.
$-\frac{v_0^2}{c^2} (\epsilon_r - 1) B_0 \hat{k}$
C.
$\frac{v_0^2}{c^2} \epsilon_r B_0 \hat{k}$
D.
$-\frac{v_0^2}{c^2} \epsilon_r B_0 \hat{k}$

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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

Magnetic Flux Scalar Sign|Non-Uniform Dielectric Filling|Electric Field Symmetry

Why This Question Is Tricky

Common Mistakes

Students often misapply Lenz's Law, assuming that any induced magnetic field must oppose the external magnetic field, leading them to incorrectly choose the negative sign (Option B).

Furthermore, they frequently confuse the total permittivity with the electric susceptibility, mistakenly using $\epsilon_r$ instead of $(\epsilon_r - 1)$ to calculate the polarization (Option C or D).

The core physical trap lies in failing to recognize that the moving bound charges constitute a polarization current that generates an effective magnetization $\vec{M} = \vec{P} \times \vec{v}$ .

Because the polarization $\vec{P}$ is induced by the motional EMF ( $\vec{v} \times \vec{B}$ ), the resulting cross-product yields an induced magnetic field that actually enhances the external field, a counter-intuitive result that rigorously tests the integral form of the Maxwell-Ampere law and relativistic field transformations.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electromagnetic Induction - Class 12
Concept	Magnetic Flux Scalar Sign
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free