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Question #42982

Difficulty: 8/10 | 🟢 Free

The Question

A large rectangular dielectric slab of thickness $d$ (spanning from $y=0$ to $y=d$ ) moves with a constant velocity $\vec{v} = v_0 \hat{i}$ through a uniform, time-independent magnetic field $\vec{B} = B_0 \hat{k}$ . The relative permittivity of the dielectric material varies non-uniformly along the $y$ -axis as $\epsilon_r(y) = K_0 e^{y/d}$ , where $K_0 > 1$ is a constant. Assuming there are no free charges present anywhere and the magnetic Reynolds number is negligibly small, what is the magnitude of the steady-state potential difference $\Delta V$ between the surfaces $y=0$ and $y=d$ in the laboratory frame?

Options

A.
$v_0 B_0 d \left( 1 - \frac{e-1}{K_0 e} \right)$
B.
$v_0 B_0 d \left( \frac{e-1}{K_0 e} \right)$
C.
$v_0 B_0 d$
D.
$v_0 B_0 d \left( 1 - \frac{1}{K_0 e} \right)$

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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

Motional EMF for Arbitrary Shapes|Electric Field Symmetry|Non-Uniform Dielectric Filling

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of motional induction and electrostatics in non-uniform dielectrics.

A common trap is for students to incorrectly assume that the dielectric simply attenuates the motional EMF by a factor of $\epsilon_r$ , leading to $E = vB/\epsilon_r$ (Option B).

This misapplies the concept of dielectric shielding, which is valid for external fields but not for internal motional EMFs.

The correct physical framework requires applying Gauss's Law for the displacement field ( $\nabla \cdot \vec{D} = 0$ ).

Because the polarization is driven by the Lorentz force in the moving frame ( $\vec{E}' = \vec{E} + \vec{v} \times \vec{B}$ ), the boundary condition $D_y = 0$ dictates that the internal electric field is $E_y = vB(1 - 1/\epsilon_r)$ .

Ignoring the depolarizing effect of the bound charges induced by the motional EMF leads to fundamental errors in calculating the potential difference.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electromagnetic Induction - Class 12
Concept	Motional EMF for Arbitrary Shapes
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free