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Question #43163

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The Question

A region of space ( $x > 0$ ) is filled with a non-uniform, time-varying dielectric medium. The permittivity of the medium is given by $\epsilon(x,t) = \epsilon_0 (1 + \beta x)(1 + \alpha t)$ , where $\epsilon_0$ is the vacuum permittivity, and $\alpha$ and $\beta$ are positive constants with units of $\text{s}^{-1}$ and $\text{m}^{-1}$ respectively. The electric field in this region is maintained as $\mathbf{E}(x,t) = E_0 t \hat{i}$ , where $E_0$ is a positive constant with units of $\text{V} \cdot \text{m}^{-1} \cdot \text{s}^{-1}$ and $t > 0$ is time. According to the macroscopic Maxwell's equations, which of the following represents the correct expressions for the free volume charge density $\rho_f(x,t)$ and the magnitude of the displacement current density $J_d(x,t)$ in this medium?

Options

A.
$\rho_f(x,t) = \epsilon_0 E_0 \beta t (1 + \alpha t)$ , \quad $J_d(x,t) = \epsilon_0 E_0 (1 + \beta x)(1 + 2\alpha t)$
B.
$\rho_f(x,t) = 0$ , \quad $J_d(x,t) = \epsilon_0 E_0 (1 + \beta x)(1 + \alpha t)$
C.
$\rho_f(x,t) = \epsilon_0 E_0 \beta t (1 + \alpha t)$ , \quad $J_d(x,t) = \epsilon_0 E_0 (1 + \beta x)(1 + \alpha t)$
D.
$\rho_f(x,t) = \epsilon_0 E_0 \beta (1 + \alpha t)$ , \quad $J_d(x,t) = \epsilon_0 E_0 (1 + 2\alpha t)$

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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

Maxwell's Correction to Ampere's Law|Gauss Law Applicability|Continuous Charge Distribution

Why This Question Is Tricky

Common Mistakes

This question tests the fundamental definitions of the electric displacement field $\mathbf{D}$ and its role in the macroscopic Maxwell's equations.

A common trap is conflating the vacuum displacement current $\epsilon_0 \frac{\partial \mathbf{E}}{\partial t}$ with the total displacement current $\frac{\partial \mathbf{D}}{\partial t}$ in a medium.

When the permittivity itself is time-varying, students often incorrectly apply the product rule, either calculating $J_d = \epsilon \frac{\partial \mathbf{E}}{\partial t}$ (ignoring the polarization current arising from changing material properties) or assuming $\rho_f = \epsilon_0 \nabla \cdot \mathbf{E} = 0$ because the electric field is uniform in space.

By requiring the calculation of both $\rho_f$ via $\nabla \cdot \mathbf{D}$ and $J_d$ via $\frac{\partial \mathbf{D}}{\partial t}$ , the problem forces a rigorous application of vector calculus to time-varying, non-uniform dielectrics.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electromagnetic Waves - Class 12
Concept	Maxwell's Correction to Ampere's Law
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free