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Question #42895

Difficulty: 8/10 | 🟢 Free

The Question

A plane electromagnetic wave with a time-averaged total energy density $u_0$ propagates in a vacuum ( $x < 0$ ) along the $+x$ -direction. At $x = 0$ , it enters a non-magnetic, lossless dielectric medium extending through $x \ge 0$ . The relative permittivity of the medium, $K(x)$ , increases continuously and very slowly from $K(0) = 1$ to $K(L) = 4$ at a depth $x = L$ . The spatial gradient of $K(x)$ is sufficiently small such that any reflection of the wave can be neglected. Which of the following represents the time-averaged total energy density of the wave at $x = L$ ?

Options

A.
$2 u_0$
B.
$\frac{16}{9} u_0$
C.
$4 u_0$
D.
$u_0$

Check Your Answer

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

Dielectric Constant (K) Shift|Energy Density in Transverse Waves|Approximations

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to distinguish between wave propagation across a discontinuous boundary versus a continuously varying medium (WKB regime).

A common trap is to incorrectly apply the Fresnel equations for a sharp impedance mismatch, which would yield a transmitted intensity $I_t = I_0 \frac{4\sqrt{K}}{(1+\sqrt{K})^2}$ and an energy density of $\frac{16}{9}u_0$ .

By specifying an adiabatic gradient where reflection is suppressed, students must instead apply the conservation of energy flux (Poynting vector).

Because the wave slows down as $v = c/\sqrt{K}$ , the spatial energy density must increase proportionally to $\sqrt{K}$ to maintain a constant energy flux, highlighting the differential relationship between intensity, wave speed, and local energy density.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electric Charges and Fields - Class 12
Concept	Dielectric Constant (K) Shift
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free