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Ch. 34 - The Wave Nature of Light: Interference and Polarization
Giancoli Douglas - Physics for Scientists and Engineers 5th edition
Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook
All textbooksGiancoli Douglas 5th editionCh. 34 - The Wave Nature of Light: Interference and PolarizationProblem 85
Chapter 33, Problem 85

Unpolarized light falls on two polarizer sheets whose axes are at right angles. (a) What fraction of the incident light intensity is transmitted? (b) What fraction is transmitted if a third polarizer is placed between the first two so that its axis makes a 58° angle with the axis of the first polarizer? (c) What if the third polarizer is in front of the other two?

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1
Understand the concept of polarization: Unpolarized light consists of waves vibrating in all directions perpendicular to the direction of propagation. When it passes through a polarizer, only the component of light vibrating parallel to the axis of the polarizer is transmitted.
Apply Malus's Law for the first scenario (a): When unpolarized light passes through a polarizer, the intensity of the transmitted light is half of the incident light. If a second polarizer is placed at right angles (90°) to the first, no light will pass through, as the axes are orthogonal.
Consider the introduction of a third polarizer for scenario (b): Place the third polarizer between the first two with its axis at a 58° angle to the first polarizer. Use Malus's Law to calculate the intensity after the first polarizer, then apply the law again for the transition from the first to the third polarizer, and finally from the third to the second polarizer.
Calculate the intensity for scenario (c): If the third polarizer is placed in front of the first two, first calculate the transmission through the third polarizer (now the first in the sequence), then through the original first (now second), and finally through the original second (now third).
Summarize the results: For each scenario, the final transmitted intensity can be calculated by multiplying the successive transmission fractions obtained from applying Malus's Law at each stage.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Malus's Law

Malus's Law states that when polarized light passes through a polarizer, the intensity of the transmitted light is proportional to the square of the cosine of the angle between the light's polarization direction and the polarizer's axis. This principle is crucial for calculating the intensity of light after passing through polarizers, especially when their axes are oriented at specific angles.
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Unpolarized Light

Unpolarized light consists of waves vibrating in multiple planes perpendicular to the direction of propagation. When unpolarized light encounters a polarizer, it becomes polarized, with only the component of light aligned with the polarizer's axis being transmitted. This concept is essential for understanding how light intensity changes when it interacts with polarizing materials.
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Multiple Polarizers

When multiple polarizers are used, the transmission of light through each polarizer must be calculated sequentially. The intensity of light after passing through each polarizer depends on the angle between the light's polarization direction and the polarizer's axis, as described by Malus's Law. This concept is key to solving problems involving more than two polarizers, especially when angles are not at right angles.
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Related Practice
Textbook Question

Consider two antennas radiating 6.8-MHz radio waves in phase with each other. They are located at points S₁ and S₂, separated by a distance d = 175 m, Fig. 34–50. Determine the points on the positive y-axis where the signals from the two sources will be out of phase (crests of one meet troughs of the other).

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Textbook Question

At what angle should the axes of two Polaroids be placed so as to reduce the intensity of the incident unpolarized light by an additional factor (after the first Polaroid cuts it in half) of (a) 4, (b) 10, (c) 100?

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Textbook Question

Two narrow slits 0.070 mm apart are illuminated by a very bright 488-nm light source forming an interference pattern on a screen 4.0 m away. Calculate (a) the distance between the m = 0 and m = 1 lines in the pattern and (b) the distance between the m = 100 and m = 101 lines.

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Textbook Question

A highly reflective mirror can be made for a particular wavelength at normal incidence by using two thin layers of transparent materials of indices of refraction n₁ and n₂ ( 1 < n₁ < n₂ ) on the surface of the glass (n > n₂). What should be the minimum thicknesses d₁ and d₂ in Fig. 34–49 in terms of the incident wavelength λ, to maximize reflection?

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Textbook Question

Two polarizers are oriented at 55° to each other and plane-polarized light is incident on them. If only 25% of the light gets through both of them, what was the initial polarization direction of the incident light?

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Textbook Question

"Two identical sources S₁ and S₂, separated by distance d, coherently emit light of wavelength λ uniformly in all directions. Defining the x axis with its origin at S₁ as shown in Fig. 34–52, find the locations (expressed as multiples of λ ) where the signals from the two sources are out of phase along this axis for x > 0 , if d = 3λ.

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