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sin θ V = 3 . 80 × 10 7 m 1 . 00 × 10 6 m = 0 . 380. size 12{"sin"θ rSub { size 8{V} } = { {3 "." "80" times "10" rSup { size 8{ - 7} } `m} over {1 "." "00" times "10" rSup { size 8{ - 6} } `m} } =0 "." "380"} {}

Thus the angle θ V size 12{θ rSub { size 8{V} } } {} is

θ V = sin 1 0 . 380 = 22 . 33º. size 12{θ rSub { size 8{V} } ="sin" rSup { size 8{ - 1} } 0 "." "380"="22" "." 3°} {}

Similarly,

sin θ R = 7 . 60 × 10 7 m 1.00 × 10 6 m . size 12{"sin"θ rSub { size 8{R} } = { {7 "." "60" times "10" rSup { size 8{ - 7} } `m} over {1 "." "00" times "10" rSup { size 8{ - 6} } `m} } } {}

Thus the angle θ R size 12{θ rSub { size 8{R} } } {} is

θ R = sin 1 0 . 760 = 49.46º. size 12{θ rSub { size 8{R} } ="sin" rSup { size 8{ - 1} } 0 "." "760"="49" "." 5°} {}

Notice that in both equations, we reported the results of these intermediate calculations to four significant figures to use with the calculation in part (b).

Solution for (b)

The distances on the screen are labeled y V size 12{y rSub { size 8{V} } } {} and y R size 12{y rSub { size 8{R} } } {} in [link] . Noting that tan θ = y / x size 12{"tan"θ=y/x} {} , we can solve for y V size 12{y rSub { size 8{V} } } {} and y R size 12{y rSub { size 8{R} } } {} . That is,

y V = x tan θ V = ( 2.00 m ) ( tan 22.33º ) = 0.815 m size 12{y rSub { size 8{V} } =x"tan"θ rSub { size 8{V} } = \( 2 "." "00"`m \) \( "tan""22" "." 3° \) =0 "." "822"`m} {}

and

y R = x tan θ R = ( 2.00 m ) ( tan 49.46º ) = 2.338 m. size 12{y rSub { size 8{R} } =x"tan"θ rSub { size 8{R} } = \( 2 "." "00"`m \) \( "tan""49" "." 5° \) =2 "." "339"`m} {}

The distance between them is therefore

y R y V = 1.52 m. size 12{y rSub { size 8{R} } - y rSub { size 8{V} } =1 "." 52`m} {}

Discussion

The large distance between the red and violet ends of the rainbow produced from the white light indicates the potential this diffraction grating has as a spectroscopic tool. The more it can spread out the wavelengths (greater dispersion), the more detail can be seen in a spectrum. This depends on the quality of the diffraction grating—it must be very precisely made in addition to having closely spaced lines.

Test prep for ap courses

Which of the following cannot be a possible outcome of passing white light through several evenly spaced parallel slits?

  1. The central maximum will be white but the higher-order maxima will disperse into a rainbow of colors.
  2. The central maximum and higher-order maxima will be of equal widths.
  3. The lower wavelength components of light will have less diffraction compared to higher wavelength components for all maxima except the central one.
  4. None of the above.

(b)

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White light is passed through a diffraction grating to a screen some distance away. The n th-order diffraction angle for the longest wavelength (760 nm) is 53.13º. Find the n th-order diffraction angle for the shortest wavelength (380 nm). What will be the change in the two angles if the distance between the screen and the grating is doubled?

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Section summary

  • A diffraction grating is a large collection of evenly spaced parallel slits that produces an interference pattern similar to but sharper than that of a double slit.
  • There is constructive interference for a diffraction grating when d sin θ = (for m = 0, 1, –1, 2, –2, …) size 12{d"sin"θ=mλ,`m="0,"`"1,"`"2,"` dotslow } {} , where d size 12{d} {} is the distance between slits in the grating, λ is the wavelength of light, and m is the order of the maximum.

Conceptual questions

What is the advantage of a diffraction grating over a double slit in dispersing light into a spectrum?

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What are the advantages of a diffraction grating over a prism in dispersing light for spectral analysis?

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Can the lines in a diffraction grating be too close together to be useful as a spectroscopic tool for visible light? If so, what type of EM radiation would the grating be suitable for? Explain.

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If a beam of white light passes through a diffraction grating with vertical lines, the light is dispersed into rainbow colors on the right and left. If a glass prism disperses white light to the right into a rainbow, how does the sequence of colors compare with that produced on the right by a diffraction grating?

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Suppose pure-wavelength light falls on a diffraction grating. What happens to the interference pattern if the same light falls on a grating that has more lines per centimeter? What happens to the interference pattern if a longer-wavelength light falls on the same grating? Explain how these two effects are consistent in terms of the relationship of wavelength to the distance between slits.

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Source:  OpenStax, College physics for ap® courses. OpenStax CNX. Nov 04, 2016 Download for free at https://legacy.cnx.org/content/col11844/1.14
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