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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 temperature
Adeleye Reply
temperature is the measure of degree of hotness or coldness of a body. measured in kelvin
Ahmad
a characteristic which tells hotness or coldness of a body
babar
Average kinetic energy of an object
Kym
average kinetic energy of the particles in an object
Kym
A measure of the quantity of heat contained in an object which arises from the average kinetic energy of the constituent particles of that object. It can be measured using thermometers. It has a unit of kelvin in the thermodynamic scale and degree Celsius in the Celsius scale.
ibrahim
Mass of air bubble in material medium is negative. why?
Hrithik Reply
a car move 6m. what is the acceleration?
Umaru Reply
depends how long
Peter
What is the simplest explanation on the difference of principle, law and a theory
Kym Reply
how did the value of gravitational constant came give me the explanation
Varun Reply
how did the value of gravitational constant 6.67×10°-11Nm2kg-2
Varun
A steel ball is dropped onto a hard floor from a height of 1.50 m and rebounds to a height of 1.45 m. (a) Calculate its velocity just before it strikes the floor.
Kris Reply
9.8m/s?
Muhammad
Sqrt(2*1.5m*9.81m/s^2)
Richard
0.5m* mate.
Muhammad
0.05 I meant.
Muhammad
Guess your solution is correct considering the ball fall from 1.5m height initially.
Muhammad
Sqrt(2*1.5m*9.81m/s^2)
Deepak
How can we compare different combinations of capacitors?
Prakash Reply
find the dimension of acceleration if it's unit is ms-2
Happiness Reply
lt^-2
Ahmad
b=-2 ,a =1
Ahmad
M^0 L^1T^-2
Sneha
what is botany
Masha
it is a branch of science which deal with the study of plants animals and environment
Varun
what is work
Sunday Reply
a boy moving with an initial velocity of 2m\s and finally canes to rest with a velocity of 3m\s square at times 10se calculate it acceleration
Sunday
.
Abdul
6.6 lol 😁😁
Abdul
show ur work
Sunday
sorry..the answer is -10
Abdul
your question is wrong
Abdul
If the boy is coming to rest then how the hell will his final velocity be 3 it'll be zero
Abdul
re-write the question
Nicolas
men i -10 isn't correct.
Stephen
using v=u + at
Stephen
1/10
Happy
ya..1/10 is very correct..
Stephen
hnn
Happy
how did the value 6.67×10°-11Nm2kg2 came tell me please
Varun
Work is the product of force and distance
Kym
physicist
Michael
what is longitudinal wave
Badmus Reply
A longitudinal wave is wave which moves parallel or along the direction of propagation.
sahil
longitudinal wave in liquid is square root of bulk of modulus by density of liquid
harishree
Is British mathematical units the same as the United States units?(like inches, cm, ext.)
Nina Reply
We use SI units: kg, m etc but the US sometimes refer to inches etc as British units even though we no longer use them.
Richard
Thanks, just what I needed to know.
Nina
What is the advantage of a diffraction grating over a double slit in dispersing light into a spectrum?
Uditha Reply
can I ask questions?
Boniface Reply
yes.
Abdul
Yes
Albert
sure
Ajali
yeap
Sani
yesssss
bilal
hello guys
Ibitayo
when you will ask the question
Ana
anybody can ask here
bichu
is free energy possible with magnets?
joel
no
Mr.
you could construct an aparatus that might have a slightly higher 'energy profit' than energy used, but you would havw to maintain the machine, and most likely keep it in a vacuum, for no air resistance, and cool it, so chances are quite slim.
Mr.
calculate the force, p, required to just make a 6kg object move along the horizontal surface where the coefficient of friction is 0.25
Gbolahan
Yes ask
Albert
if a man travel 7km 30degree east of North then 10km east find the resultant displacement
Ajali Reply
11km
Dohn
disagree. Displacement is the hypotenuse length of the final position to the starting position. Find x,y components of each leg of journey to determine final position, then use final components to calculate the displacement.
Daniel
1.The giant star Betelgeuse emits radiant energy at a rate of 10exponent4 times greater than our sun, where as it surface temperature is only half (2900k) that of our sun. Estimate the radius of Betelgeuse assuming e=1, the sun's radius is s=7*10exponent8metres
James Reply
2. A ceramic teapot (e=0.20) and a shiny one (e=0.10), each hold 0.25 l of at 95degrees. A. Estimate the temperature rate of heat loss from each B. Estimate the temperature drop after 30mins for each. Consider only radiation and assume the surrounding at 20degrees
James
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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