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Question #43181

Difficulty: 8/10 | 🟢 Free

The Question

A plane electromagnetic wave of angular frequency $\omega$ propagates in the $+x$ direction through a non-magnetic dielectric medium ( $\mu = \mu_0$ ) occupying the region $0 \le x \le L$ . The relative permittivity of the medium varies spatially as $\epsilon_r(x) = \left(1 + \frac{x}{L}\right)^4$ . If the electric field at $x=0$ is given by $E(0,t) = E_0 \cos(\omega t)$ , what is the phase difference $\Delta \phi$ between the wave at $x=0$ and $x=L$ at any given instant? (Assume the WKB approximation is valid, i.e., reflections due to the gradient are negligible, and $c$ is the speed of light in vacuum).

Options

A.
$\frac{7\omega L}{3c}$
B.
$\frac{5\omega L}{2c}$
C.
$\sqrt{\frac{31}{5}} \frac{\omega L}{c}$
D.
$\frac{9\omega L}{4c}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Variable_Constraint

Develop pure symbolic logic skills by mastering algebraic constraints without relying on numerical data.

Concepts Behind This Question

Medium Dependency|Phase Constant (Epoch)|Dielectric Polarization Logic|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the application of the WKB (eikonal) approximation for wave propagation in an inhomogeneous medium.

A common trap is treating the medium as uniform by calculating an average relative permittivity $\epsilon_{avg}$ and using it to find an effective refractive index, or by simply averaging the refractive index at the boundaries.

The correct physical principle dictates that the phase must be evaluated as a spatial integral of the local wave vector $k(x) = \frac{\omega}{c}n(x)$ .

Failing to integrate the spatially varying refractive index $n(x) = \sqrt{\epsilon_r(x)}$ directly leads to incorrect phase constants and demonstrates a misunderstanding of phase accumulation in gradient dielectrics.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electromagnetic Waves - Class 12
Concept	Medium Dependency
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free