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d sin θ = , for m = 0, 1, –1, 2, –2, (constructive), 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 size 12{λ} {} is the wavelength of light, and m size 12{m} {} is the order of the maximum. Note that this is exactly the same equation as for double slits separated by d size 12{d} {} . However, the slits are usually closer in diffraction gratings than in double slits, producing fewer maxima at larger angles.

The figure shows a schematic of a diffraction grating, which is represented by a vertical black line into which are cut five small gaps. The gaps are evenly spaced a distance d apart. From the left five rays arrive, with one ray arriving at each gap. To the right of the line with the gaps the rays all point down and to the right at an angle theta below the horizontal. At each gap a triangle is formed where the hypotenuse is length d, one angle is theta, and the side opposite theta is labeled delta l. At the top is written delta l equals d sine theta.
Diffraction grating showing light rays from each slit traveling in the same direction. Each ray travels a different distance to reach a common point on a screen (not shown). Each ray travels a distance d sin θ size 12{d`"sin"θ} {} different from that of its neighbor.

Where are diffraction gratings used? Diffraction gratings are key components of monochromators used, for example, in optical imaging of particular wavelengths from biological or medical samples. A diffraction grating can be chosen to specifically analyze a wavelength emitted by molecules in diseased cells in a biopsy sample or to help excite strategic molecules in the sample with a selected frequency of light. Another vital use is in optical fiber technologies where fibers are designed to provide optimum performance at specific wavelengths. A range of diffraction gratings are available for selecting specific wavelengths for such use.

Take-home experiment: rainbows on a cd

The spacing d size 12{d} {} of the grooves in a CD or DVD can be well determined by using a laser and the equation d sin θ = , for m = 0, 1, –1, 2, –2, size 12{d"sin"θ=mλ,`m="0,"`"1,"`"2,"` dotslow } {} . However, we can still make a good estimate of this spacing by using white light and the rainbow of colors that comes from the interference. Reflect sunlight from a CD onto a wall and use your best judgment of the location of a strongly diffracted color to find the separation d size 12{d} {} .

Calculating typical diffraction grating effects

Diffraction gratings with 10,000 lines per centimeter are readily available. Suppose you have one, and you send a beam of white light through it to a screen 2.00 m away. (a) Find the angles for the first-order diffraction of the shortest and longest wavelengths of visible light (380 and 760 nm). (b) What is the distance between the ends of the rainbow of visible light produced on the screen for first-order interference? (See [link] .)

The image shows a vertical black bar at the left labeled grating. From the midpoint of this bar four lines fan out to the right, with two lines angled above the horizontal centerline and two lines angled symmetrically below the horizontal centerline. These four lines hit a vertical black line to the right that is labeled screen. On the screen between the two upper lines is a rainbow region, with violet nearer the centerline and red farther from the centerline. The same is true for the two lower lines, except that they are below the centerline instead of above. The distance from the centerline to the upper violet zone is labeled y sub v equals question mark and the distance from the centerline to the upper red zone is labeled y sub r equals question mark. The angle between the centerline and the line leading to the upper violet zone is labeled theta V equals question mark and the angle between the line leading to the upper red zone is labeled theta R equals question mark. The distance between the grating and the screen is labeled x equals two point zero zero meters.
The diffraction grating considered in this example produces a rainbow of colors on a screen a distance x = 2 . 00 m size 12{x=2 "." "00"`m} {} from the grating. The distances along the screen are measured perpendicular to the x size 12{x} {} -direction. In other words, the rainbow pattern extends out of the page.

Strategy

The angles can be found using the equation

d sin θ = (for m = 0, 1, –1, 2, –2, …) size 12{d"sin"θ=mλ,`m="0,"`"1,"`"2,"` dotslow } {}

once a value for the slit spacing d size 12{d} {} has been determined. Since there are 10,000 lines per centimeter, each line is separated by 1/10,000 of a centimeter. Once the angles are found, the distances along the screen can be found using simple trigonometry.

Solution for (a)

The distance between slits is d = ( 1 cm ) / 10 , 000 = 1 . 00 × 10 4 cm size 12{d= \( 1`"cm" \) /"10","000"=1 "." "00" times "10" rSup { size 8{ - 4} } `"cm"} {} or 1 . 00 × 10 6 m size 12{1 "." "00" times "10" rSup { size 8{ - 6} } `m} {} . Let us call the two angles θ V size 12{θ rSub { size 8{V} } } {} for violet (380 nm) and θ R size 12{θ rSub { size 8{R} } } {} for red (760 nm). Solving the equation d sin θ V = size 12{d"sin"θ rSub { size 8{V} } =mλ} {} for sin θ V size 12{"sin"θ rSub { size 8{V} } } {} ,

sin θ V = V d , size 12{"sin"θ rSub { size 8{V} } = { {mλ rSub { size 8{V} } } over {d} } ,} {}

where m = 1 size 12{m=1} {} for first order and λ V = 380 nm = 3 . 80 × 10 7 m size 12{λ rSub { size 8{V} } ="380"`"nm"=3 "." "80" times "10" rSup { size 8{ - 7} } `m} {} . Substituting these values gives

Questions & Answers

what the difference between laws and principles
Mary Reply
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Megryan
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clifford
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Megryan
explanations on harmonic motion
Tumwiine Reply
example ofchange of state of the body in the effectof heat
Abiodun Reply
solid ice = liquid water due to radical changes in temperature which are controlled by the capacity of heat an object can withstand.
Shii
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Mohamed
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Mohamed
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sununu
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Neyaz Reply
the force that pushes upward on us. the force that opposes gravity
clifford
upthrust of air
Abdikadir
Newton's 3rd law. the force of the ground (earth) that pushes back on gravity, keeping us on the ground instead of sinking into it.
clifford
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Ogboru Reply
Questions or answers?
Shii
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Shii
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Mohit
no friction is a force just like the gravitational force
clifford
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Mohit
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clifford
eg. when walking there are two forces acting on us gravitational and frictional force. friction helps us move forward and gravity keeps us on the ground
clifford
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clifford
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clifford
the friction force which oppose while it contact with surrounding. there are two kind of friction. slidding and rolling friction.
Neyaz
Two unequal masses M1 and M2 are connected by a string of tension T on a plane,find the acceleration and tension in the string
Ogboru
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Ogboru
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Olamide
What is physics?
Jeuloriz Reply
physics is a branch of science in which we are dealing with the knowledge of our physical things. macroscopic as well as microscopic. we are going look inside the univers with the help of physics. you can learn nature with the help of physics. so many branches of physics you have to learn physics.
vijay
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Breanna Reply
6 type of quarks
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Akani Reply
Candela is the unit for the measurement of light intensity.
Osei
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Neyaz Reply
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Impulse and momentum
Fauzia
force×time and mass× velocity
vijay
Good
Neyaz
What is the simple harmonic motion?
Fauzia Reply
oscillatory motion under a retarding force proportional to the amount of displacement from an equilibrium position
Yuri
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Yuri
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Yuri
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Fauzia
Oscillatory motion under a regarding force proportional to the amount of displacement from an equilibrium position
Neyaz
examples of work done by load of gravity
Maureen Reply
What is ehrenfest theorem?
Fauzia Reply
You can look it up, faster and more reliable answer.
Yuri
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Yuri
what is the work done by gravity on the load 87kj,11.684m,mass xkg[g=19m/s
Maureen
What is law of mass action?
Fauzia Reply
rate of chemical reactions is proportional to concentration of reactants ...
muhammad
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Fauzia
what the difference between laws and principles
Mary
what is lenses
Ndobe Reply
lenses are two types
Fauzia
concave and convex
muhammad
right
Fauzia
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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