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 15

Chapter 33, Problem 15

The two most prominent wavelengths in the light emitted by a hydrogen discharge lamp are 656 nm (red) and 486 nm (blue). Light from a hydrogen lamp illuminates a diffraction grating with 500 lines/mm, and the light is observed on a screen 1.50 m behind the grating. What is the distance between the first-order red and blue fringes?

Verified step by step guidance

Step 1: Begin by understanding the diffraction grating formula: \( d \sin \theta = m \lambda \), where \( d \) is the distance between adjacent lines on the grating, \( \theta \) is the diffraction angle, \( m \) is the order of the fringe, and \( \lambda \) is the wavelength of the light. Here, \( m = 1 \) for the first-order fringe.

Step 2: Calculate \( d \), the distance between adjacent lines on the grating. Since the grating has 500 lines per millimeter, \( d = \frac{1}{500 \times 10^3} \) meters.

Step 3: Use the formula \( \sin \theta = \frac{m \lambda}{d} \) to calculate the diffraction angle \( \theta \) for both the red light (\( \lambda = 656 \, \text{nm} \)) and the blue light (\( \lambda = 486 \, \text{nm} \)). Convert the wavelengths from nanometers to meters before substituting into the formula.

Step 4: Once \( \theta \) is calculated for both wavelengths, use the geometry of the setup to find the position of the fringes on the screen. The position \( y \) of a fringe on the screen is given by \( y = L \tan \theta \), where \( L = 1.50 \, \text{m} \) is the distance from the grating to the screen.

Step 5: Calculate the distance between the first-order red and blue fringes by finding the difference between their respective \( y \) positions: \( \Delta y = y_{\text{red}} - y_{\text{blue}} \).

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Key Concepts

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

Diffraction Grating

A diffraction grating is an optical component with a periodic structure that disperses light into its constituent wavelengths. It works by exploiting the wave nature of light, causing interference patterns when light waves overlap. The number of lines per millimeter on the grating determines the angle at which different wavelengths are diffracted, which is crucial for analyzing spectral lines.

Wavelength and Color

Wavelength is the distance between successive peaks of a wave, and it determines the color of light in the visible spectrum. For instance, red light has a longer wavelength (around 656 nm) compared to blue light (around 486 nm). The difference in wavelengths affects how light is diffracted by the grating, leading to distinct positions for the red and blue fringes on the observation screen.

Interference Pattern

An interference pattern is created when two or more overlapping waves combine, resulting in regions of constructive and destructive interference. In the context of a diffraction grating, the first-order fringes correspond to specific angles where the path difference between light waves leads to constructive interference. The distance between these fringes on a screen can be calculated using the grating equation and the geometry of the setup.

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Related Practice

Textbook Question

A diffraction grating produces a first-order maximum at an angle of 20.0°. What is the angle of the second-order maximum?

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Textbook Question

In a single-slit experiment, the slit width is 200 times the wavelength of the light. What is the width (in mm) of the central maximum on a screen 2.0 m behind the slit?

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Textbook Question

Light of 630 nm wavelength illuminates a single slit of width 0.15 mm. FIGURE EX33.22 shows the intensity pattern seen on a screen behind the slit. What is the distance to the screen?

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Textbook Question

In a double-slit experiment, the slit separation is 200 times the wavelength of the light. What is the angular separation (in degrees) between two adjacent bright fringes?

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Textbook Question

A double-slit interference pattern is created by two narrow slits spaced 0.25 mm apart. The distance between the first and the fifth minimum on a screen 60 cm behind the slits is 5.5 mm. What is the wavelength (in nm) of the light used in this experiment?

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Textbook Question

FIGURE EX33.17 shows the interference pattern on a screen 1.0 m behind an 800 lines/mm diffraction grating. What is the wavelength (in nm) of the light?

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