Ch. 35 - Diffraction

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook

Giancoli Douglas 5th edition

Ch. 35 - Diffraction

Problem 45

Chapter 34, Problem 45

Red laser light from a He–Ne laser (λ = 632.8 nm) creates a second-order fringe at 53.2° after passing through a grating. What is the wavelength λ of light that creates a first-order fringe at 21.2°?

Verified step by step guidance

Step 1: Recall the diffraction grating equation:

d \, \(\sin\) \(\theta\) = m \, \(\lambda\)

, where

d

is the spacing between adjacent slits in the grating,

\(\theta\)

is the diffraction angle,

m

is the order of the fringe, and

\(\lambda\)

is the wavelength of the light.

Step 2: Use the given information for the red laser light to calculate the grating spacing

d

. For the second-order fringe (

m = 2

) at an angle of

\(\theta\) = 53.2^\(\circ\)

and wavelength

\(\lambda\) = 632.8 \, \(\text{nm}\)

, rearrange the equation to solve for

d

d = \(\frac{m \, \lambda}{\sin \theta}\)

Step 3: Substitute the known values for the red laser light into the equation to calculate

d

. This value of

d

will be the same for the second part of the problem, as the grating is unchanged.

Step 4: Use the diffraction grating equation again to find the wavelength

\(\lambda\)

of the light that creates a first-order fringe (

m = 1

) at an angle of

\(\theta\) = 21.2^\(\circ\)

. Rearrange the equation to solve for

\(\lambda\)

\(\lambda\) = \(\frac{d \, \sin \theta}{m}\)

Step 5: Substitute the value of

d

(calculated in Step 3) and the given values for

m

and

\(\theta\)

into the equation to find the wavelength

\(\lambda\)

of the light.

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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 component wavelengths. When light passes through or reflects off a grating, it creates interference patterns due to the superposition of light waves. The angles at which these patterns occur depend on the wavelength of the light and the spacing of the grating lines.

Interference Patterns

Interference patterns arise when two or more coherent light waves overlap, resulting in regions of constructive and destructive interference. In the context of diffraction gratings, these patterns are characterized by bright and dark fringes, where the position of the fringes is determined by the wavelength of the light and the angle of observation.

Order of Fringes

The order of fringes refers to the integer multiples of the wavelength that correspond to specific angles of constructive interference in a diffraction pattern. The first-order fringe (m=1) occurs at a certain angle for a given wavelength, while higher-order fringes (m=2, m=3, etc.) occur at angles that are multiples of the first-order angle, allowing for the calculation of unknown wavelengths based on known angles.

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Related Practice

Textbook Question

A diffraction grating has 15,000 rulings in its 1.9 cm width. Determine (a) its resolving power in first and second orders, and (b) the minimum wavelength resolution (∆λ) it can yield for λ = 410 nm.

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Textbook Question

(II) White light passes through a 640-slit/ mm diffraction grating. First-order and second-order visible spectra (“rainbows”) appear on the wall 32 cm away as shown in Fig. 35–40. Determine the widths ℓ₁ and ℓ₂ of the two “rainbows” (400 nm to 700 nm). In which order is the “rainbow” dispersed over a larger distance?

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Textbook Question

(II) X-rays of wavelength 0.138 nm fall on a crystal whose atoms, lying in planes, are spaced 0.315 nm apart. At what angle Φ (relative to the surface, Fig. 35–28) must the X-rays be directed if the first diffraction maximum is to be observed?

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Textbook Question

A diffraction grating has 6.5 x 10⁵ slits/m. Find the angular spread in the second-order spectrum between red light of wavelength 7.0 x 10⁻⁷ m and blue light of wavelength 4.5 x 10⁻⁷ m.

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Textbook Question

Suppose the angles measured in Problem 42 were produced when the spectrometer (but not the source) was submerged in water. What then would be the wavelengths (in air)?

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Textbook Question

Show that the second- and third-order spectra of white light produced by a diffraction grating always overlap. What wavelengths overlap?