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Ch 36: Diffraction
Young & Freedman Calc - University Physics 15th Edition
Young & Freedman Calc15th EditionUniversity PhysicsISBN: 9780135159552Not the one you use?Change textbook
All textbooksYoung & Freedman Calc 15th EditionCh 36: DiffractionProblem 22
Chapter 35, Problem 22

Laser light of wavelength 500.0 nm illuminates two identical slits, producing an interference pattern on a screen 90.0 cm from the slits. The bright bands are 1.00 cm apart, and the third bright bands on either side of the central maximum are missing in the pattern. Find the width and the separation of the two slits.

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Step 1: Understand the problem. This is a double-slit interference problem where the wavelength of the laser light (λ = 500.0 nm), the distance to the screen (L = 90.0 cm), and the spacing between bright bands (Δy = 1.00 cm) are given. Additionally, the third bright bands are missing, indicating destructive interference due to the slit width.
Step 2: Use the formula for the position of bright fringes in a double-slit interference pattern: y=md⋅λL, where m is the fringe order, d is the slit separation, λ is the wavelength, and L is the distance to the screen. From the given data, the fringe spacing Δy corresponds to the distance between adjacent bright fringes, so Δy=λd⋅L. Rearrange this equation to solve for d: d=λΔy⋅L.
Step 3: Recognize that the missing third bright bands indicate destructive interference due to the finite width of the slits. This occurs when the condition for the first minimum in the single-slit diffraction pattern overlaps with the third-order bright fringe of the double-slit interference pattern. The condition for the first minimum in single-slit diffraction is a⋅sin⁡θ=mλ, where a is the slit width, θ is the angle of diffraction, and m = 1 for the first minimum.
Step 4: Use the small angle approximation (sin⁡θ≈tan⁡θ=yL) to relate the angle θ to the position of the third bright fringe. For the third-order bright fringe, y=m⋅Δy, where m = 3. Substitute this into the single-slit diffraction condition to find a: a=mL⋅λy.
Step 5: Substitute the given values (λ = 500.0 nm, L = 90.0 cm, Δy = 1.00 cm, m = 3) into the equations derived in Steps 2 and 4 to calculate the slit separation (d) and the slit width (a). Ensure all units are consistent (e.g., convert nm to cm) before performing the calculations.

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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 a double-slit experiment, constructive interference produces bright bands (maxima) where the waves are in phase, while destructive interference leads to dark bands (minima) where the waves are out of phase. This phenomenon is crucial for understanding how light behaves when passing through slits.
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Wavelength and Slit Separation

The wavelength of light is the distance between successive peaks of the wave. In a double-slit experiment, the separation of the slits and the wavelength of the light determine the spacing of the interference pattern on the screen. The relationship between these variables is described by the formula for fringe spacing, which helps in calculating the slit separation based on the observed pattern.
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Fringe Spacing

Fringe spacing refers to the distance between adjacent bright or dark bands in an interference pattern. It can be calculated using the formula: fringe spacing = (wavelength × distance to screen) / slit separation. Understanding this concept is essential for solving problems related to interference patterns, as it directly relates to the geometry of the setup and the properties of the light used.
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Related Practice
Textbook Question

The wavelength range of the visible spectrum is approximately 380–750 nm. White light falls at normal incidence on a diffraction grating that has 350 slits/mm. Find the angular width of the visible spectrum in the first order.

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Textbook Question

When laser light of wavelength 632.8 nm passes through a diffraction grating, the first bright spots occur at ±17.8° from the central maximum. (a) What is the line density (in lines/cm) of this grating? (b) How many additional bright spots are there beyond the first bright spots, and at what angles do they occur?

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Textbook Question

Parallel rays of monochromatic light with wavelength 568 nm illuminate two identical slits and produce an interference pattern on a screen that is 75.0 cm from the slits. The centers of the slits are 0.640 mm apart and the width of each slit is 0.434 mm. If the intensity at the center of the central maximum is 5.00 x 10-4 W/m2, what is the intensity at a point on the screen that is 0.900 mm from the center of the central maximum?

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Textbook Question

If a diffraction grating produces a third-order bright spot for red light (of wavelength 700 nm) at 65.0° from the central maximum, at what angle will the second-order bright spot be for violet light (of wavelength 400 nm)?

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Textbook Question

Monochromatic light of wavelength 592 nm from a distant source passes through a slit that is 0.0290 mm wide. In the resulting diffraction pattern, the intensity at the center of the central maximum (θ = 0°) is 4.00x10-5 W/m2. What is the intensity at a point on the screen that corresponds to θ = 1.20°?

Textbook Question

Monochromatic light of wavelength 580 nm passes through a single slit and the diffraction pattern is observed on a screen. Both the source and screen are far enough from the slit for Fraunhofer diffraction to apply. (a) If the first diffraction minima are at ±90.0°, so the central maximum completely fills the screen, what is the width of the slit? (b) For the width of the slit as calculated in part (a), what is the ratio of the intensity at θ = 45.0° to the intensity at θ = 0?

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