Ch 33: Wave Optics

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 33: Wave Optics

Problem 34

Chapter 33, Problem 34

Light from a helium-neon laser (λ = 633 nm) passes through a circular aperture and is observed on a screen 4.0 m behind the aperture. The width of the central maximum is 2.5 cm. What is the diameter (in mm) of the hole?

Verified step by step guidance

Understand the problem: The central maximum in the diffraction pattern is caused by light passing through a circular aperture. The width of the central maximum is related to the aperture diameter by the diffraction formula for a circular aperture.

Use the formula for the angular width of the central maximum in a circular aperture: \( \theta = 1.22 \frac{\lambda}{D} \), where \( \lambda \) is the wavelength of light, \( D \) is the diameter of the aperture, and \( \theta \) is the angular width of the central maximum.

Relate the angular width \( \theta \) to the physical width \( w \) of the central maximum on the screen using the small-angle approximation: \( \theta \approx \frac{w}{2L} \), where \( w \) is the width of the central maximum (2.5 cm) and \( L \) is the distance to the screen (4.0 m).

Combine the two equations: \( \frac{w}{2L} = 1.22 \frac{\lambda}{D} \). Rearrange to solve for \( D \): \( D = 1.22 \frac{\lambda \cdot 2L}{w} \).

Substitute the known values into the equation: \( \lambda = 633 \text{ nm} = 633 \times 10^{-9} \text{ m} \), \( L = 4.0 \text{ m} \), and \( w = 2.5 \text{ cm} = 0.025 \text{ m} \). Simplify the expression to find \( D \) in meters, then convert to millimeters.

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Key Concepts

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

Diffraction

Diffraction is the bending of waves around obstacles and the spreading of waves when they pass through small openings. In the context of light, diffraction patterns are created when coherent light, such as that from a laser, encounters an aperture. The resulting pattern on a screen can be analyzed to determine properties of the aperture, such as its size.

Central Maximum

The central maximum is the brightest and widest part of the diffraction pattern produced when light passes through an aperture. It occurs directly in line with the light source and is flanked by successive dark and bright fringes. The width of the central maximum is influenced by the wavelength of the light and the size of the aperture, making it a key factor in determining the aperture's diameter.

Wavelength and Aperture Size Relationship

The relationship between the wavelength of light and the size of the aperture is crucial in diffraction phenomena. According to the principles of wave optics, smaller apertures relative to the wavelength produce more pronounced diffraction effects. This relationship can be quantitatively described using formulas that relate the width of the central maximum to the wavelength and the diameter of the aperture, allowing for calculations of the aperture size.

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Textbook Question

Two 50-μm-wide slits spaced 0.25 mm apart are illuminated by blue laser light with a wavelength of 450 nm. The interference pattern is observed on a screen 2.0 m behind the slits. How many bright fringes are seen in the central maximum that spans the distance between the first missing order on one side and the first missing order on the other side?

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Textbook Question

Infrared light of wavelength 2.5 μm illuminates a 0.20-mm-diameter hole. What is the angle of the first dark fringe in radians? In degrees?

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Textbook Question

Your artist friend is designing an exhibit inspired by circular-aperture diffraction. A pinhole in a red zone is going to be illuminated with a red laser beam of wavelength 670 nm, while a pinhole in a violet zone is going to be illuminated with a violet laser beam of wavelength 410 nm. She wants all the diffraction patterns seen on a distant screen to have the same size. For this to work, what must be the ratio of the red pinhole’s diameter to that of the violet pinhole?

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Textbook Question

FIGURE P33.39 shows the light intensity on a screen 2.5 m behind an aperture. The aperture is illuminated with light of wavelength 620 nm. Is the aperture a single slit or a double slit? Explain.

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Textbook Question

You want to photograph a circular diffraction pattern whose central maximum has a diameter of 1.0 cm. You have a helium-neon laser (λ=633 nm) and a 0.12-mm-diameter pinhole. How far behind the pinhole should you place the screen that's to be photographed?

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Textbook Question

A Michelson interferometer uses red light with a wavelength of 656.45 nm from a hydrogen discharge lamp. How many bright-dark-bright fringe shifts are observed if mirror M₂ is moved exactly 1 cm?

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