Ch 36: Diffraction

Young & Freedman Calc - University Physics 14th Edition

Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook

Young & Freedman Calc 14th Edition

Ch 36: Diffraction

Problem 28a

Chapter 36, Problem 28a

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.

Verified step by step guidance

Step 1: Start by identifying the diffraction grating equation:

d sin(θ) = mλ

, where

d

is the distance between adjacent slits (grating spacing),

θ

is the diffraction angle,

m

is the diffraction order, and

λ

is the wavelength of light.

Step 2: Calculate the grating spacing

d

from the given number of slits per millimeter. Since there are 350 slits/mm, the spacing is

d = \(\frac{1}{350 \times 10^3}\)

meters.

Step 3: For the first-order diffraction (

m = 1

), use the grating equation to calculate the diffraction angle for the shortest wavelength (

λ = 380 \(\times\) 10^{-9}

m). Rearrange the equation to solve for

θ

θ = \(\arcsin\[\left\)(\(\frac{mλ}{d}\]\right\))

Step 4: Repeat the calculation for the longest wavelength (

λ = 750 \(\times\) 10^{-9}

m) to find the corresponding diffraction angle.

Step 5: The angular width of the visible spectrum in the first order is the difference between the two angles calculated in Steps 3 and 4:

Δθ = θ_{750} - θ_{380}

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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 splits and diffracts light into several beams. The angle at which light is diffracted depends on the wavelength and the spacing of the slits. The number of slits per millimeter determines the grating's resolving power, allowing for the separation of different wavelengths of light.

Order of Diffraction

The order of diffraction refers to the integer multiples of the wavelength that correspond to the angles at which constructive interference occurs. The first order (m=1) is the first angle where light is maximally intensified after passing through the grating. Higher orders (m=2, 3, etc.) correspond to angles where additional maxima appear, but they are less intense and can overlap.

Angular Width of the Spectrum

The angular width of the visible spectrum is the range of angles over which the visible wavelengths are diffracted by the grating. It can be calculated by determining the angles for the longest and shortest wavelengths in the visible spectrum. The difference between these angles gives the total angular width, which is crucial for understanding how much of the spectrum can be observed at a given order.

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The Electromagnetic Spectrum

Related Practice

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

If the planes of a crystal are 3.50 Å (1 Å = 10^-10 m = 1 Ångstrom unit) apart, what wavelength of electromagnetic waves is needed so that the first strong interference maximum in the Bragg reflection occurs when the waves strike the planes at an angle of 22.0°, and in what part of the electromagnetic spectrum do these waves lie?

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Textbook Question

If a diffraction grating produces its third-order bright band at an angle of 78.4° for light of wavelength 681 nm, find (a) the number of slits per centimeter for the grating and (b) the angular location of the first-order and second-order bright bands. (c) Will there be a fourth-order bright band? Explain.

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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

(a) What is the wavelength of light that is deviated in the first order through an angle of 13.5° by a transmission grating having 5000 slits/cm? (b) What is the second-order deviation of this wavelength? Assume normal incidence.

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Textbook Question

A laser beam of wavelength λ = 632.8 nm shines at normal incidence on the reflective side of a compact disc. (a) The tracks of tiny pits in which information is coded onto the CD are 1.60 μm apart. For what angles of reflection (measured from the normal) will the intensity of light be maximum? (b) On a DVD, the tracks are only 0.740 μm apart. Repeat the calculation of part (a) for the DVD.

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