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Question #43168

Difficulty: 8/10 | 🟢 Free

The Question

Two electromagnetic waves propagate in a vacuum along the $+z$ -axis. Their electric field vectors are given by: $\vec{E}_1(z,t) = E_0 \cos(kz - \omega t) \hat{i}$ $\vec{E}_2(z,t) = 2E_0 \cos(kz - \omega t + \Omega t) \left( \frac{1}{2}\hat{i} + \frac{\sqrt{3}}{2}\hat{j} \right)$ where $E_0$ is a positive constant, $\omega = 3 \times 10^{15} \text{ rad/s}$ , and $\Omega = 10^3 \text{ rad/s}$ . A detector is placed at $z = 0$ to measure the intensity of the resultant wave. The detector's response time $\tau$ satisfies $\frac{2\pi}{\omega} \ll \tau \ll \frac{2\pi}{\Omega}$ . What is the ratio of the maximum to the minimum intensity recorded by the detector?

Options

A.
$\frac{7}{3}$
B.
9
C.
3
D.
1

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

Transverse Nature of EM Waves|Phase Constant (Epoch)|Superposition of Non-identical Waves

Why This Question Is Tricky

Common Mistakes

Students often incorrectly apply the scalar interference formula $I = I_1 + I_2 + 2\sqrt{I_1 I_2}\cos(\Delta \phi)$ without accounting for the vector nature of electromagnetic waves.

This scalar approach assumes the electric fields are perfectly parallel, which would yield a maximum intensity of $9I_0$ and a minimum of $I_0$ (a ratio of 9).

By introducing non-orthogonal polarization vectors, the problem forces the use of vector superposition, where the interference term is strictly modulated by the dot product of the polarization states.

The time-varying phase constant $\Omega t$ ensures the interference is dynamic, requiring the student to correctly average over the optical frequency while preserving the slow beat frequency.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electromagnetic Waves - Class 12
Concept	Transverse Nature of EM Waves
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free