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Question #43190

Difficulty: 8/10 | 🟢 Free

The Question

A plane electromagnetic wave of angular frequency $\omega$ is normally incident from vacuum onto a semi-infinite, non-magnetic dielectric medium of refractive index $n$ . Let $J_{d,i}$ and $J_{d,t}$ be the amplitudes of the displacement current densities of the incident and transmitted waves, respectively. Let $\langle u_i \rangle$ and $\langle u_t \rangle$ be the time-averaged electromagnetic energy densities of the incident and transmitted waves. Which of the following expressions correctly relates the ratio $\frac{J_{d,t}}{J_{d,i}}$ to the ratio $\frac{\langle u_t \rangle}{\langle u_i \rangle}$ ?

Options

A.
$\frac{J_{d,t}}{J_{d,i}} = n \sqrt{\frac{\langle u_t \rangle}{\langle u_i \rangle}}$
B.
$\frac{J_{d,t}}{J_{d,i}} = \frac{1}{n} \sqrt{\frac{\langle u_t \rangle}{\langle u_i \rangle}}$
C.
$\frac{J_{d,t}}{J_{d,i}} = \sqrt{\frac{\langle u_t \rangle}{\langle u_i \rangle}}$
D.
$\frac{J_{d,t}}{J_{d,i}} = n^2 \sqrt{\frac{\langle u_t \rangle}{\langle u_i \rangle}}$

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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

Refractive Index Relativity|Maxwell's Correction to Ampere's Law|Energy Density Locality

Why This Question Is Tricky

Common Mistakes

This question tests the student's fundamental understanding of Maxwell's correction to Ampere's law and electromagnetic energy density within a polarizable medium.

A common trap for students is to memorize the vacuum definition of displacement current density ( $J_d = \epsilon_0 \frac{\partial E}{\partial t}$ ) and incorrectly apply it inside the dielectric, forgetting the material permittivity factor $\epsilon_r = n^2$ .

If a student makes this error, they will find $J_{d,t} \propto E_t$ instead of $n^2 E_t$ , leading them to incorrectly select Option B.

Alternatively, if they forget the permittivity factor in both the displacement current and the energy density, they will select Option C.

The problem forces students to abandon rote formulas and derive the relationship using the strict local definitions of $\vec{D}$ and energy density in matter.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Ray Optics and Optical Instruments - Class 12
Concept	Refractive Index Relativity
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free