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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 is the fumula for calculating specific heat capacity, fusion,fission and vaporization?
Dohn Reply
Q=cm(∆t)
Emmanuel
Q=cm∆T
Muhammad
what is difference b/w vaporization and evaporation
Muhammad
evaporation is the process of extracting moisture while vaporization is process of becoming a vapor or gas
Emmanuel
From a molecular standpoint they are both cooling processes. Also, you may want to explore states of matter😊 #myTwoCents ~Shi~
Shii
cooling is a similarlity in both process I am confused in difference
Muhammad
1- Evaporation is a process where a liquid change to gas without reaching its boiling point. 2- Vaporization is a process where a liquid change to gas after reaching its boiling point. 3- Sublimation is a process where a solid changes into vapour without passing through a liquid state
Victor
I see. Evaporation is a type of vaporization, that occurs on the surface of a liquid as it changes into the gaseous phase before reaching its boiling point. hope that aids
Shii
vaporisation is cooling process while vaporization is heating process
Emmanuel
I mean to write evaporation is an heating process while vaporization is cooling process
Emmanuel
Yea here are two applications. 1- your wet washed clothes dry under the sun, the water EVAPORATES 2- when u are cooking, it reaches a point where u need to add more water because the water you added previously is getting dried. this is VAPORIZATION. Am not sure which is a cooling or heating process
Victor
vaporization occur only when the evaporation get to level where the above cloud is been (saturated) so cooling take place and started to change to liquid (eg rain fall)
Emmanuel
They are both properties of the same process so they're both cooling
Shii
what about sublimation? cooling or heating process?
Victor
exact
Muhammad
evaporation is the increase in kinetic energy of the liquid which can be gone by adding heat
Emmanuel
so its an heating process
Emmanuel
sublimation is when a solid change to gas
Emmanuel
evaporation is very definitely a cooling process. respectfully@Emmanuel when liquid turns to gas it requires more energy from its surroundings, this energy is in the form of heat, and when heat energy leaves the evaporating liquid it leaves it cooler. Thus, cooling process.
Shii
.
Shii
evaporation is very definitely a cooling process. respectfully@Emmanuel
Shii
kk
Emmanuel
You're right @Shi. I get your point
Victor
eascape velocity on the surface of Earth is 11.2 kms-1 the escape velocity on the surface of another planet of same mass as that of Earth but of 1/4 times of radius of Earth is a5.6kms-1 b11.2 kms-1 c22.4kms-1 d5.6ms-1
Muhammad
Emm.. is that a question? or..
Victor
it is McQ
Muhammad
a)5.6km/s
Alvis
c= Q/cm◇T
A.d
it's answer is 22.4
Muhammad
units...
Shii
vital
Shii
the time period of the artificial satellite is given by ?
raza
Why is there no 2nd harmonic in the classical electron orbit?
Shree Reply
how to reform magnet after been demagneted
Inuwa Reply
A petrol engine has a output of 20 kilowatts and uses 4.5 kg of fuel for each hour of running. The energy given out when 1 kg of petrol is burnt is 4.8 × 10 to the power of 7 Joules. a) What is the energy output of the engine every hour? b) What is the energy input of the engine every hour?
Morris Reply
what is the error during taking work done of a body..
Aliyu Reply
what kind of error do you think? and work is held by which force?
Daniela
I am now in this group
smart
theory,laws,principles and what-a-view are not defined. why? you
Douglas Reply
A simple pendulum is used in a physics laboratory experiment to obtain an experimental value for the gravitational acceleration, g . A student measures the length of the pendulum to be 0.510 meters, displaces it 10 o from the equilibrium position, and releases it. Using a s
Emmanuel Reply
so what question are you passing across... sir?
Olalekan
Two masses of 2 kg and 4 kg are held with a compressed spring between them. If the masses are released, the spring will push them away from each other. If the smaller mass moves off with a velocity of 6m/s, what is the stored energy in the spring when it is compressed?
Emmanuel Reply
54 joule
babar
how?
rakesh
Reduce that two body problem into one body problem. Apply potential and k. E formula to get total energy of the system
rakesh
i dont think dere is any potential energy... by d virtue of no height present
Olalekan
there is compressed energy,dats only potential energy na?
rakesh
yes.. but... how will u approach that question without The Height in the question?
Olalekan
Can you explain how you get 54J?
Emmanuel
Because mine is 36J
Emmanuel
got 36J too
Douglas
OK the answer is 54J Babar is correct
Emmanuel
Conservation of Momentum
Emmanuel
woow i see.. can you give the formula for this
joshua
Two masses of 2 kg and 4 kg are held with a compressed spring between them. If the masses are released, the spring will push them away from each other. If the smaller mass moves off with a velocity of 6m/s, what is the stored energy in the spring when it is compressed? Asume there is no external force.
Emmanuel Reply
Please help!
Emmanuel
please help find dy/dx 2x-y/x+y
Inuwa
By using the Quotient Rule dy/dx = 3y/(x +y)²
Emmanuel
3y/(x+y)²
Emmanuel
may be by using MC^2=MC^2 and Total energy=kinetic energy +potential energy so 1st find kinetic energy and den find potential energy which is stored energy
rakesh
i think i m correct
rakesh
But how?
Emmanuel
3y/(x+y)²
Douglas
what's the big bang?
kwame Reply
yes what is it?
LamaBbake
it is the explanation of how the universe began
Zainab
yes
Ana
explain
Chinagorom
in
Chinagorom
it is a theory on how the universe began. to understand more I would suggest researching the topic online.
david
thanks guys
kwame
if a force of 12N is applied to load of 200g what us the work done
Joshua Reply
We can seek accelation first
Nancy
we are given f=12 m=200g which is 0.2kg now from 2nd law of newton a= f/m=60m/s*2 work done=force applied x displacement cos (theta) w= 12x60 =720nm/s*2
Mudang
this very interesting question very complicated for me, í need urgent help. 1,two buses A and B travel along the same road in the same direction from Harper city (asume They both started from the same point) to Monrovia. if bus A maintains a Speedy of 60km/h and bus B a Speedy of 75km/h, how many
mohammed
hours Will it take bus B to overtake bus A assuming bus B starts One hour after bus A started. what is the distance travelled by the buses when They meet?.
mohammed
pls í need help
mohammed
4000 work is done
Ana
speed=distance /time distance=speed/time
Ana
now use this formula
Ana
what's the answer then
Julius
great Mudang
Kossi
please Ana explain 4000 ?
babar
hey mudang there is a product of force and acceleration not force and displacement
babar
@Mohammed answer is 0.8hours or 48mins
Douglas
nice
A.d
its not possible
Olalekan
í want the working procedure
mohammed
the answer is given but how Will One arrive at it. the answers are 4hours and 300m.
mohammed
physics is the science that studies the non living nature
isidor Reply
ancient greek language physis = nature
isidor
what is phyacs
technical Reply
if i am going to start studying physics where should i start?
BRIAN Reply
I think from kinematics
Nancy
You can find physics books at the library or online. That's how I started.
Chelsea
And yes, kinematics is usually where you can begin.
Chelsea
study basic algebra and calculus and can start from classical mechanics
Mudang
yes think so but dimension is the best starting point
Obed
3 formula's of equations of motion
benjamin Reply
vf=vi+at........1 s=vit+1/2(at)2 vf2=vi2+2as
Ana
solve the formula's please
benjamin
those are the three .. what you wanna solve ?
Nihrantz
For first equation simply integrate formula of acceleration in the limit v and u
Tripti
For second itegrate velocity formula by ising first equation
Tripti
similarly for 3 one integrate acceleration again by multiplying and dividing term ds
Tripti
any methods can take to solve this eqtions
a=vf-vi/t vf-vi=at vf=vi+at......1
Ana
suppose a body starts with an initial velocity vi and travels with uniform acceleration a for a period of time t.the distance covered by a body in this time is "s" and its final velocity becomes vf
Ana
what is the question dear
Zeeshan
average velocity=(vi+vf)/2 distance travelled=average velocity ×time therefore s=vi+vf/2×t from the first equation of motion ,we have vf =vi+at s=[vi+(vi+at)]/2×t s=(2vi+at)/2×t s=bit+1/2at2
Ana
find the distance
Ana
how
Zeeshan
v=u+at,v^2=(u^2) +2as,s=2u +(1/2)gt^2
Emmanuel
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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