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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

lExplain what happens to the energy carried by light that it is dimmed by passing it through two crossed polarizing filters.
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hamidat Reply
Specific heat capacity is the amount of heat required to raise the temperature of one (Kg) of a substance through one Kelvin
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formula for measuring Joules
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I don't understand, do you mean the S.I unit of work and energy?
hamidat
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What limits the Magnification of an optical instrument?
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Lithography is 2 micron
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An hydraulic press is a type of machine that is operated by different pressure of water on pistons.
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it can simple defined as constant temperature
Muhammad
Boyles law states that the volume of a fixed amount of a gas is inversely proportional to the pressure acting on in provided that the temperature is constant.that is V=k(1/p) or V=k/p
FADILAT
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what is a atom
ADAMS
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what are the laws of motion
Mua
what is force
Ugwu Reply
Force is any quantity or a change that produces motion on an object body.
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A force is a push or a pull that has the tendency of changing a body's uniform state of rest or uniform state of motion in a straight line.
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plsoo give me the gravitational motion formulas
Okoye
f=Gm/d²
FADILAT
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electro magnetic force
shafiu
Electromotive force (emf) is a measurement of the energy that causes current to flow through a circuit.
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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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