# 27.5 Single slit diffraction  (Page 2/4)

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Thus, to obtain destructive interference for a single slit    ,

$D\text{sin}\phantom{\rule{0.25em}{0ex}}\theta =\mathrm{m\lambda },\phantom{\rule{0.25em}{0ex}}\text{for}\phantom{\rule{0.25em}{0ex}}m=\text{1,}\phantom{\rule{0.25em}{0ex}}\text{–1,}\phantom{\rule{0.25em}{0ex}}\text{2,}\phantom{\rule{0.25em}{0ex}}\text{–2,}\phantom{\rule{0.25em}{0ex}}\text{3,}\phantom{\rule{0.25em}{0ex}}\dots \phantom{\rule{0.25em}{0ex}}\text{(destructive),}$

where $D$ is the slit width, $\lambda$ is the light’s wavelength, $\theta$ is the angle relative to the original direction of the light, and $m$ is the order of the minimum. [link] shows a graph of intensity for single slit interference, and it is apparent that the maxima on either side of the central maximum are much less intense and not as wide. This is consistent with the illustration in [link] (b).

## Calculating single slit diffraction

Visible light of wavelength 550 nm falls on a single slit and produces its second diffraction minimum at an angle of $\text{45.0º}$ relative to the incident direction of the light. (a) What is the width of the slit? (b) At what angle is the first minimum produced?

Strategy

From the given information, and assuming the screen is far away from the slit, we can use the equation $D\text{sin}\phantom{\rule{0.25em}{0ex}}\theta =\text{mλ}$ first to find $D$ , and again to find the angle for the first minimum ${\theta }_{1}$ .

Solution for (a)

We are given that $\lambda =\text{550 nm}$ , $m=2$ , and ${\theta }_{2}=\text{45.0º}$ . Solving the equation $D\phantom{\rule{0.25em}{0ex}}\text{sin}\phantom{\rule{0.25em}{0ex}}\theta =\text{mλ}$ for $D$ and substituting known values gives

$\begin{array}{lll}D& =& \frac{\mathrm{m\lambda }}{\phantom{\rule{0.25em}{0ex}}\text{sin}\phantom{\rule{0.25em}{0ex}}{\theta }_{2}}=\frac{2\left(\text{550 nm}\right)}{\text{sin 45.0º}}\\ & =& \frac{\text{1100}×{\text{10}}^{-9}}{0.707}\\ & =& 1.56×{\text{10}}^{-6}.\end{array}$

Solution for (b)

Solving the equation $D\phantom{\rule{0.25em}{0ex}}\text{sin}\phantom{\rule{0.25em}{0ex}}\theta =\text{mλ}$ for $\text{sin}\phantom{\rule{0.25em}{0ex}}{\theta }_{1}$ and substituting the known values gives

$\text{sin}\phantom{\rule{0.25em}{0ex}}{\theta }_{1}=\frac{\mathrm{m\lambda }}{D}=\frac{1\left(\text{550}×{\text{10}}^{-9}\phantom{\rule{0.25em}{0ex}}\text{m}\right)}{1\text{.}\text{56}×{\text{10}}^{-6}\phantom{\rule{0.25em}{0ex}}\text{m}}\text{.}$

Thus the angle ${\theta }_{1}$ is

${\theta }_{1}={\text{sin}}^{-1}\phantom{\rule{0.25em}{0ex}}\text{0.354}=\text{20.7º.}$

Discussion

We see that the slit is narrow (it is only a few times greater than the wavelength of light). This is consistent with the fact that light must interact with an object comparable in size to its wavelength in order to exhibit significant wave effects such as this single slit diffraction pattern. We also see that the central maximum extends $\text{20.7º}$ on either side of the original beam, for a width of about $\text{41º}\phantom{\rule{0.25em}{0ex}}$ . The angle between the first and second minima is only about $\text{24º}\phantom{\rule{0.25em}{0ex}}\left(\text{45.0º}-20.7º\right)$ . Thus the second maximum is only about half as wide as the central maximum.

## Section summary

• A single slit produces an interference pattern characterized by a broad central maximum with narrower and dimmer maxima to the sides.
• There is destructive interference for a single slit when $D\phantom{\rule{0.25em}{0ex}}\text{sin}\phantom{\rule{0.25em}{0ex}}\theta =\text{mλ},\phantom{\rule{0.25em}{0ex}}\text{(for}\phantom{\rule{0.25em}{0ex}}m=\text{1,}\phantom{\rule{0.25em}{0ex}}\text{–1,}\phantom{\rule{0.25em}{0ex}}\text{2,}\phantom{\rule{0.25em}{0ex}}\text{–2,}\phantom{\rule{0.25em}{0ex}}\text{3,}\phantom{\rule{0.25em}{0ex}}\dots \right)$ , where $D$ is the slit width, $\lambda$ is the light’s wavelength, $\theta$ is the angle relative to the original direction of the light, and $m$ is the order of the minimum. Note that there is no $m=0$ minimum.

## Conceptual questions

As the width of the slit producing a single-slit diffraction pattern is reduced, how will the diffraction pattern produced change?

## Problems&Exercises

(a) At what angle is the first minimum for 550-nm light falling on a single slit of width $1\text{.}\text{00}\phantom{\rule{0.25em}{0ex}}\text{μm}$ ? (b) Will there be a second minimum?

(a) $\text{33}\text{.}4º$

(b) No

(a) Calculate the angle at which a $2\text{.}\text{00}\text{-μm}$ -wide slit produces its first minimum for 410-nm violet light. (b) Where is the first minimum for 700-nm red light?

#### Questions & Answers

and yea about that hypothesis thing....For electrons and photons and other microscopic particles ...The dual nature applicable but Macroscopic object follows De Broglie's Hypothesis or Theory
Duh !! And guys I can't see any conversation! All I can guess what is going on ,on board is through notifications ! otherwise this screen is blank ! Does anybody know what is happening ?
Bibekbir
I am facing problem ....I am not able to view the latest conversation in group discussion. ..its just showing a blank page even after reinstalling the app...although the Internet connection is also working
Bibekbir
its ok
Salima
Therefore it was summarised that electrons or any particle for instance (*De Broglie's hypothesis ) shows Dual nature
ok i get it but it is still a hypothesis
Salima
can i ask qusetions sometimes
Salima
yup it was as if electrons knew they were being watched ...
Which I personally find very creepy !
Bibekbir
Which I personally find very creepy !
Bibekbir
it is creepy
Salima
Which confused scientists so much ...so tried the same experiment using a microscope this time to watch the motion of the electrons...and surprisingly this time two slits were formed. !
really
Salima
Well what common sense says is ...There will be two slits on photographic plate...
But what was observed was totally unexpected. ..There were wave like imprint on the photographic plate....like water waves...
Bibekbir
ok then...
Salima
Well to explain the dual nature of light ...We need to study Young's double slit experiment .In which,.. okay I want you to imagine I have gun loaded with electrons. ..and before me I have this sheet with two slit in it and behind this sheet a photographic plate. .....what do you think my observation will be ?
can't say anything
Salima
Well light can exist as wave as well as a particle....There is something called Dual nature of light
yes you got it
i know about it but how is it possible
Salima
it depends upon given condition to photon
Bashir
Bashir
Mr. Bashir there is no question that have no answer
Salima
ok
Bashir
Why the Big Bang create more matter then antimatter?
there are a few theories but it is still mostly a mystery
James
Is it possible that the whole concept of antimatter is wrong after all its just theories
Salima
no its not theory
ghulam
but there is no strong evidence about it they are trying to make antimatter what if they are searching for, turns out something different other then antimatter
Salima
half of physics does not consists of the strong evidence
ghulam
salima are you pakistani ?
ghulam
yeah
Salima
I read in class 10th
Salima
And i am not talking about evidence
Salima
Actually there is an evidence .....its called cosmic inflation. ..
Bibekbir
can you explain
Salima
sure
Bibekbir
It's like this....okay imagine for a second that all the galaxies and stars ...and planet with all their energies vanishes
Bibekbir
Salima
even the black holes dark matter and dark energies also vanishes...
Bibekbir
Can anyone tell me what should be left ?
Bibekbir
ok
Salima
nothing but space
Salima
You will say an empty void ...but that's not the case after this called universe mapping we got this an active energetic background ! which was expanding at a very fast rate
Bibekbir
Now if something is expanding at a speed of nearly speed of light !! then it must have stated from a very small sized object and there you have big bang theory
Bibekbir
yeah i get it
Salima
You will say a void right ? But that was not what mathematical calculations said ....even there is something called universe mapping which mapped the energies in this background. ... and we found that this universe as whole itself was increasing with the speed faster than light !!!
Bibekbir
faster then light!!!
Salima
interesting question. would love a proper explanation.
Beatrax
questions are always intersting only if you know the answer
Salima
or understand the question
Salima
yup faster than light ...now here some people will ask that Einstein had claimed nothing can be faster than light right ? ....but the Universe doesn't count in 'nothing ' and everything is in universe including light ...
Bibekbir
Beatrax do you have a question?
Bibekbir
universe is expending right so there must be a limit for its expension and after that would it be possible that it start contracting again?
Salima
Well we simply don't know to what universe is expanding to .....so here comes string theory....which is story of another time though. ... yes there is a theory though which claims that the universe will again collapse in a small mass just as it started in big bang
Bibekbir
so we can say that big bang will happen again
Salima
Let's come back to your question on antimatter right ? ....well we simply don't have a lot of information about anti matter...though there is a theory which claims that at the time of big bang antimatter was made with matter ....but not in the same quantity. ..
Bibekbir
Because if it were to happen .....The matter won't simply exist ...but we know it does exist....
Bibekbir
there must be some reason that why it creat more matter
Salima
and no the concept of antimatter is NOT wrong!!! Don't ever say that !!! Because some scientists spent their whole life studying on this .....and you will be surprised to know that there are antimatter found on the exosphere of Earth.
Bibekbir
and Ghulam you wrong there Bro. ..70% of physics is mathematics.....which language of universe....which is kinda right every time. ..Einstein gave all his theory solely based on thoughts and mathematics...
Bibekbir
really how is it possible was it natural i know that they are making antimatter but i don't believe it
Salima
And before something become a solid point ... ..It has to go through a long process of scientists in panels and all those stuff. ....and surprisingly it's all observable phenomenons....so yeah physics is actually defination of logic....mind that ..
Bibekbir
Do anyone have any question? feel free to ask ...
Bibekbir
yeah i also think that physics is the basic of all the universe and every thing we see in it
Salima
oh yeah i will
Salima
Yeah we kinda don't know....a lot on this ....moreover....I have not been keeping tap on this topic....so maybe they have got some stuff up their sleeves but I just don't know...
Bibekbir
what is actully light photons or electromagnetic waves
Salima
Well main topic which I focus on is dark matter dark energy and exoplanets. ..
Bibekbir
and maybe little bit of quantum mechanics
Bibekbir
ah there good question !! very good question! !
Bibekbir
are you a student of physics and where are you from
Salima
thanks
Salima
okay now ....We don't only have proton electron and neutron duh ! that's what students are taught!
Bibekbir
Actually from CERN till now 35 subatomic particles have been found !!!
Bibekbir
and now let's come back to your question about what is light photons....I guess you all from 12th grade am I right ?
Bibekbir
oh whats the 35 subatomic particals
Salima
no actully 10th grade
Salima
What? I am actually umm. ...In my first year of college ....Sem1 so yea I am 18 years old....and I am doing undergraduate course in Maths Chemistry and Physics ... I am from India ! Duh ! I thought my name would give away my nationality !
Bibekbir
There are a lot of those subatomic particles like higgs boson particle is one of them
Bibekbir
oh
Salima
Not a lot 35
Bibekbir
i am physics professor any problm dear friend
ghulam
oh what about my question
Salima
Salima you know a lot according to your age. ...don't worry you have a while chapter in 11th and 12th physics in electromagnetic waves and light photon ....
Bibekbir
*whole
Bibekbir
i am 16 years old
Salima
Mr. Ghulam No hard feelings I just said what I felt right . And I just love physics a lot can't just read any bad comment about it.Again I said what I felt right. If you are offended by anything I said Sir,then I am sorry . I am not here offend anyone,.just give my honest answers andwhatever I study
Bibekbir
Mr.Ghulam are you a Pakistani
Salima
why physics is more difficult for commerce student?
Bashir
i am a fsc pre medical student
Sameer
hello
James
am james
James
physics is not difficult but when you don't have any interest it seems difficult
Salima
James can you tell us about big bang theory?
Bashir
Bashir you can read it in above comments
Salima
Yes but is very necessary for us but how to understand that cause our direction is another side
Bashir
hi Bibekbir are you there
Salima
stay on your direction Mr. Bashir
Salima
Yes
Bashir
can anyone answer my question?
Salima
Yes can but ..
Bashir
but..
Salima
i just want answer not "but"
Salima
Ok you can ask
Bashir
?
Bashir
what light actually photon or electromagnetic waves
Salima
Both
how? can a partical be wave at the same time
Salima
yes photon depend upon condition
photon is the energy packet ..light consist of photons in the form of EM waves
ghulam
What is the difference between potential difference and electromotive force
basically potentiometer is series circuit or parallel circuit?
What is half-life
the life in which half of the radioactive element decay
Arif
what is fluid
anything that flows is Liquid.
prakash
a substance that has no specific shape
Saleemulhaq
How submarines floats one water the same time sink in water
A submarine has the ability to float and sink. The ability to control buoyancy comes from the submarine'strim or ballast tanks which can be filled with either water or air, depending on whether the submarine needs to floator sink. When the submarine floats it means its trim tanks are filled with air
Arif
Anthony
what is work
Force times distance
Karanja
product of force and distance...
Arif
Is physics a natural science?
Nature is all physics
Salima