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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.

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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Suppose a feather appears green but has no green pigment. Explain in terms of diffraction.

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It is possible that there is no minimum in the interference pattern of a single slit. Explain why. Is the same true of double slits and diffraction gratings?

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Problems&Exercises

A diffraction grating has 2000 lines per centimeter. At what angle will the first-order maximum be for 520-nm-wavelength green light?

5 . 97º size 12{5 "." "97"°} {}

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Find the angle for the third-order maximum for 580-nm-wavelength yellow light falling on a diffraction grating having 1500 lines per centimeter.

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Questions & Answers

what is physics
Rhema Reply
a15kg powerexerted by the foresafter 3second
Firdos Reply
what is displacement
Xolani Reply
movement in a direction
Jason
hello
Hosea
Explain why magnetic damping might not be effective on an object made of several thin conducting layers separated by insulation? can someone please explain this i need it for my final exam
anas Reply
Hi
saeid
hi
Yimam
What is thê principle behind movement of thê taps control
Oluwakayode Reply
while
Hosea
what is atomic mass
thomas Reply
this is the mass of an atom of an element in ratio with the mass of carbon-atom
Chukwuka
show me how to get the accuracies of the values of the resistors for the two circuits i.e for series and parallel sides
Jesuovie Reply
Explain why it is difficult to have an ideal machine in real life situations.
Isaac Reply
tell me
Promise
what's the s . i unit for couple?
Promise
its s.i unit is Nm
Covenant
Force×perpendicular distance N×m=Nm
Oluwakayode
İt iş diffucult to have idêal machine because of FRİCTİON definitely reduce thê efficiency
Oluwakayode
if the classica theory of specific heat is valid,what would be the thermal energy of one kmol of copper at the debye temperature (for copper is 340k)
Zaharadeen Reply
can i get all formulas of physics
BPH Reply
yes
haider
what affects fluid
Doreen Reply
pressure
Oluwakayode
Dimension for force MLT-2
Promise Reply
what is the dimensions of Force?
Osueke Reply
how do you calculate the 5% uncertainty of 4cm?
melia Reply
4cm/100×5= 0.2cm
haider
how do you calculate the 5% absolute uncertainty of a 200g mass?
melia Reply
= 200g±(5%)10g
haider
use the 10g as the uncertainty?
melia
which topic u discussing about?
haider
topic of question?
haider
the relationship between the applied force and the deflection
melia
sorry wrong question i meant the 5% uncertainty of 4cm?
melia
its 0.2 cm or 2mm
haider
thank you
melia
Hello group...
Chioma
hi
haider
well hello there
sean
hi
Noks
hii
Chibueze
10g
Olokuntoye
0.2m
Olokuntoye
hi guys
thomas
Practice Key Terms 2

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Source:  OpenStax, College physics. OpenStax CNX. Jul 27, 2015 Download for free at http://legacy.cnx.org/content/col11406/1.9
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