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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 74
Chapter 33, Problem 74

Suppose the mirrors in a Michelson interferometer are perfectly aligned and the path lengths to mirrors M₁ and M₂ are identical. With these initial conditions, an observer sees a bright maximum at the center of the viewing area. Now one of the mirrors is moved a distance x. Determine a formula for the intensity at the center of the viewing area as a function of x, the distance the movable mirror is moved from the initial position.

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1
Understand the basic operation of a Michelson interferometer: It splits a beam of light into two paths, reflects them back with mirrors, and then recombines them. The interference pattern observed depends on the path difference between the two beams.
Recognize that the initial condition of a bright maximum at the center indicates constructive interference, meaning the path difference between the two beams is an integer multiple of the wavelength, starting with zero (i.e., the path lengths are initially equal).
Identify that moving one mirror a distance x changes the path length for that beam by 2x (since the light travels to the mirror and back). This changes the path difference between the two beams.
Recall the condition for constructive interference (bright fringes) is when the path difference is an integer multiple of the wavelength, \( n\lambda \), where \( n \) is an integer. The new path difference is \( 2x \).
Use the formula for intensity in terms of path difference: \( I(x) = I_0 \cos^2(\frac{\pi \cdot 2x}{\lambda}) \), where \( I_0 \) is the maximum intensity, and \( \lambda \) is the wavelength of the light used. This formula accounts for the variation in intensity due to the change in path difference caused by moving the mirror.

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Key Concepts

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

Interference of Light

Interference occurs when two or more light waves overlap, resulting in a new wave pattern. In the context of the Michelson interferometer, constructive interference leads to bright fringes when the path difference between the two beams is an integer multiple of the wavelength, while destructive interference results in dark fringes. Understanding this principle is crucial for analyzing how moving one mirror affects the observed intensity.
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Path Length Difference

The path length difference is the difference in distance traveled by two light beams before they recombine. In a Michelson interferometer, if one mirror is moved, the path length for that beam changes, affecting the interference pattern. The intensity at the center of the viewing area can be expressed as a function of this path length difference, which is directly related to the distance x that the mirror is moved.
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Intensity Formula

The intensity of light resulting from interference can be described by the formula I = I₀(1 + cos(Δφ)), where I₀ is the maximum intensity and Δφ is the phase difference between the two beams. The phase difference is related to the path length difference and the wavelength of the light used. By determining how the phase changes as the mirror is moved, one can derive the intensity at the center of the viewing area as a function of the distance x.
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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

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

Light of wavelength 5.0 x 10⁻⁷ passes through two parallel slits and falls on a screen 5.0 m away. Adjacent bright bands of the interference pattern are 2.0 cm apart.

(a) Find the distance between the slits.

(b) The same two slits are next illuminated by light of a different wavelength, and the fifth minimum for this light occurs at the same point on the screen as the fourth minimum for the previous light. What is the wavelength of the second source of light?

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Textbook Question

What would Brewster’s angle be for reflections off the surface of water for light coming from beneath the surface? Compare to the angle for total internal reflection, and to Brewster’s angle from above the surface.

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Textbook Question

Light of wavelength 690 nm passes through two narrow slits 0.66 mm apart. The screen is 1.75 m away. A second source of unknown wavelength produces its second-order fringe 1.23 mm closer to the central maximum than the 690-nm light. What is the wavelength of the unknown light?

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