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Question #43730

Difficulty: 8/10 | 🟢 Free

The Question

Two plane electromagnetic waves are propagating in a vacuum along the $+x$ direction. Their electric field components are given by: $E_1(x,t) = E_0 \sin(k_1 x - \omega_1 t)$ $E_2(x,t) = E_0 \sin(\omega_2 t - k_2 x)$ where $\omega_1$ and $\omega_2$ are slightly different angular frequencies such that $\Delta\omega = \omega_1 - \omega_2 > 0$ and $\Delta\omega \ll \omega_1, \omega_2$ . A detector placed at $x = 0$ measures the time-averaged intensity $I(t)$ (averaged over the fast optical cycles). If $I_0$ is the intensity of a single wave, what is the total energy per unit area $U$ recorded by the detector during the time interval from $t = 0$ to $t = \frac{\pi}{2\Delta\omega}$ ?

Options

A.
$\frac{I_0}{\Delta\omega} (\pi - 2)$
B.
$\frac{I_0}{\Delta\omega} (\pi + 2)$
C.
$\frac{\pi I_0}{\Delta\omega}$
D.
$\frac{2\pi I_0}{\Delta\omega}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Intensity Distribution in YDSE|Wave Equation Sign Convention|Superposition of Non-identical Waves

Why This Question Is Tricky

Common Mistakes

This question tests the limits of the Superposition Principle and the definition of wave intensity, specifically targeting the 'Wave Equation Sign Convention' trap.

Students often reflexively use the static interference formula $I = 4I_0\cos^2(\delta/2)$ or assume identical frequencies, leading to incorrect constant intensities.

Furthermore, the problem presents two waves traveling in the same direction but written with opposite phase conventions ( $kx - \omega t$ vs $\omega t - kx$ ).

A common mistake is ignoring this sign difference, which flips the sign of the interference term from $1 - \cos(\Delta\omega t)$ to $1 + \cos(\Delta\omega t)$ .

By requiring integration over a fraction of the beat period, the question ensures that only a rigorous application of superposition, time-averaging over optical cycles, and exact integration yields the correct energy.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Wave Optics - Class 12
Concept	Intensity Distribution in YDSE
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free