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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 89
Chapter 33, Problem 89

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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Understand the concept of Malus's Law, which states that the intensity of light passing through a polarizing filter is given by I = I_0 \(\cos\)^2(\(\theta\)), where I_0 is the initial intensity and \(\theta\) is the angle between the light's initial polarization direction and the axis of the polarizer.
Recognize that the first Polaroid polarizes the light and reduces its intensity by half, so after the first Polaroid, the intensity is I_1 = \(\frac{1}{2}\)I_0.
Determine the additional reduction factor required after the second Polaroid. For (a) a factor of 4, (b) a factor of 10, and (c) a factor of 100, calculate the final intensity I_f as I_f = \(\frac{I_1}{factor}\).
Use Malus's Law for the second Polaroid to find the angle \(\theta\). Set up the equation \(\frac{1}{2}\)I_0 \(\cos\)^2(\(\theta\)) = I_f, and solve for \(\theta\).
Calculate the angle \(\theta\) for each case: (a) \(\theta\) when the factor is 4, (b) \(\theta\) when the factor is 10, and (c) \(\theta\) when the factor is 100.

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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 polarizing filter, 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 axis of the filter. Mathematically, it is expressed as I = I0 * cos²(θ), where I0 is the initial intensity and θ is the angle between the light's polarization direction and the filter's axis.
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Unpolarized Light

Unpolarized light consists of waves vibrating in multiple planes perpendicular to the direction of propagation. Common sources of unpolarized light include sunlight and incandescent bulbs. When unpolarized light passes through a polarizer, it becomes polarized, with its intensity reduced to half of the original intensity, as only the component of light aligned with the polarizer's axis is transmitted.
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Intensity Reduction Factor

The intensity reduction factor refers to the ratio of the transmitted intensity to the incident intensity of light. In this context, after passing through the first polarizer, the intensity is halved. To achieve further reductions, the angle between the axes of the two polarizers must be adjusted according to Malus's Law, allowing for specific factors of reduction such as 4, 10, or 100, which correspond to different angles.
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Related Practice
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.

Textbook Question

A radio telescope, whose two antennas are separated by 55 m, is designed to receive 3.0-MHz radio waves produced by astronomical objects. The received radio waves create 3.0-MHz electronic signals in the telescope’s left and right antennas. These signals then travel by equal-length cables to a centrally located amplifier, where they are added together. The telescope can be “pointed” to a certain region of the sky by adding the instantaneous signal from the right antenna to a “time-delayed” signal received by the left antenna a time ∆t ago. (This time delay of the left signal can be easily accomplished with the proper electronic circuit.) If a radio astronomer wishes to “view” radio signals arriving from an object oriented at a 12° angle to the vertical as in Fig. 34–54, what time delay ∆t is necessary?

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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

Describe how to rotate the plane of polarization of a plane-polarized beam of light by 90° and produce only a 10% loss in intensity, using polarizers. Let N be the number of polarizers and θ be the (same) angle between successive polarizers.

Textbook Question

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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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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