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Question #43095

Difficulty: 8/10 | 🟢 Free

The Question

A monochromatic light beam propagating in the $+z$ direction has an electric field given by $\vec{E}(z,t) = \sqrt{5} E_0 \cos(kz - \omega t) \hat{i} + \sqrt{3} E_0 \cos\left(kz - \omega t + \frac{\pi}{6}\right) \hat{j}$ where $E_0$ is a positive constant in $\text{V/m}$ . The total time-averaged intensity of this incident beam is $I_0$ . The beam passes through an ideal linear polarizer whose transmission axis lies in the $x$ - $y$ plane and can be rotated to any angle $\theta$ with respect to the positive $x$ -axis. As the polarizer is rotated, let $I_{max}$ and $I_{min}$ be the maximum and minimum possible values of the transmitted intensity, respectively. Which of the following gives the correct values of $I_{max}$ and $I_{min}$ ?

Options

A.
$I_{max} = \frac{15}{16}I_0, \quad I_{min} = \frac{1}{16}I_0$
B.
$I_{max} = I_0, \quad I_{min} = 0$
C.
$I_{max} = \frac{5}{8}I_0, \quad I_{min} = \frac{3}{8}I_0$
D.
$I_{max} = \frac{7}{8}I_0, \quad I_{min} = \frac{1}{8}I_0$

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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

Phase Relationship|Malus's Law|Vector Cross-Products

Why This Question Is Tricky

Common Mistakes

This question tests the application of Malus's Law to elliptically polarized light, specifically targeting the misconception that orthogonal field components can be treated as incoherent.

Students often incorrectly calculate the transmitted intensity by simply adding the individual intensities of the $x$ and $y$ components ( $I = I_x \cos^2\theta + I_y \sin^2\theta$ ), completely ignoring the coherent phase difference.

By forcing the projection of the complex electric field vector onto the polarizer's transmission axis, the problem requires the evaluation of the interference cross-term $\langle E_x E_y \rangle$ .

This cross-term dictates the true orientation and magnitude of the polarization ellipse's principal axes, demonstrating that phase rotation fundamentally alters the intensity extrema.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electromagnetic Waves - Class 12
Concept	Phase Relationship
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free