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Question #42644

Difficulty: 8/10 | 🟢 Free

The Question

An electromagnetic wave is incident from Medium 1 ( $x < 0$ , refractive index $n_1 = 5$ ) onto a planar interface at $x = 0$ with Medium 2 ( $x > 0$ , refractive index $n_2 = 4$ ). The incident electric field vector is given by $\vec{E}_i = E_0 (3\hat{i} - 4\hat{j}) \cos(\vec{k}_i \cdot \vec{r} - \omega t)$ , where $E_0 > 0 \text{ V/m}$ . Given that the incident wave vector $\vec{k}_i$ lies entirely in the $x-y$ plane, which of the following represents the unit vector in the direction of the transmitted electric field amplitude $\vec{E}_t$ ?

Options

A.
$\frac{3}{4}\hat{i} - \frac{\sqrt{7}}{4}\hat{j}$
B.
-\hat{j}
C.
$\frac{\sqrt{7}}{4}\hat{i} - \frac{3}{4}\hat{j}$
D.
$\frac{3}{5}\hat{i} - \frac{4}{5}\hat{j}$

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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

Particle Velocity vs. Wave Velocity|Snell’s Law Vector Form|Vector Cross-Products

Why This Question Is Tricky

Common Mistakes

This question tests the fundamental distinction between the phase velocity vector (wave vector) and the polarization vector (electric field) during refraction.

A common trap is applying Snell's Law directly to the polarization vector's angle, which leads to physically absurd results (like Option B).

Students must realize that Snell's Law, derived from boundary phase matching ( $\hat{n} \times (\vec{k}_i - \vec{k}_t) = 0$ ), strictly governs the wave vector $\vec{k}$ .

The electric field direction is then determined by the transversality condition ( $\vec{k} \cdot \vec{E} = 0$ ) and boundary continuity.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Waves - Class 11 Part 2
Concept	Particle Velocity vs. Wave Velocity
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free