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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 90
Chapter 33, Problem 90

"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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Define the condition for destructive interference (out of phase): For two sources to be out of phase, the path difference between the waves from the two sources must be an odd multiple of half the wavelength, i.e., ΔL = (m + 1/2)λ, where m is an integer (m = 0, 1, 2, ...).
Express the path difference ΔL in terms of the geometry of the problem: The path difference is the difference in distances from the point on the x-axis to the two sources. For a point at position x on the x-axis, the distance to S₁ is x, and the distance to S₂ is √(x² + d²). Thus, ΔL = √(x² + d²) - x.
Substitute the given value of d = 3λ into the path difference equation: ΔL = √(x² + (3λ)²) - x.
Set the path difference equal to the condition for destructive interference: √(x² + (3λ)²) - x = (m + 1/2)λ. Solve this equation for x in terms of m and λ. This will involve isolating x and squaring both sides to eliminate the square root.
Determine the locations of x for different integer values of m (m = 0, 1, 2, ...), ensuring that x > 0. Express these locations as multiples of λ, as required by the problem.

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Key Concepts

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

Coherent Sources

Coherent sources are light sources that emit waves with a constant phase difference and the same frequency. In this scenario, the two sources S₁ and S₂ are coherent, meaning they produce light waves that maintain a fixed relationship in their oscillations. This property is crucial for understanding interference patterns, as it allows for predictable constructive and destructive interference based on the path length differences between the waves from the two sources.
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Path Length Difference

The path length difference refers to the difference in distance traveled by waves from two sources to a given point. For two coherent sources, this difference determines whether the waves will interfere constructively (in phase) or destructively (out of phase). In this problem, the condition for destructive interference, where the waves are out of phase, is met when the path length difference is an odd multiple of half the wavelength (λ/2).
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Interference Patterns

Interference patterns arise when two or more waves overlap, resulting in regions of increased (constructive interference) or decreased (destructive interference) intensity. In the context of this question, the locations where the signals from S₁ and S₂ are out of phase correspond to points along the x-axis where destructive interference occurs. Understanding these patterns is essential for predicting the behavior of light waves emitted from the two sources.
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Related Practice
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

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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