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Figure a shows horizontal rays passing from left to right through a vertical slit of length D. This is labeled theta equal to zero, bright. Figure b shows rays passing through the slit at an angle theta to the horizontal. This is labeled sine theta equal to lambda by D, dark. A dotted line perpendicular to the rays touches the top of the slit. Its perpendicular distance from the bottom of the slit is lambda and that from the center of the slit is lambda by 2. A separate view shows the dotted line being at an angle theta to the vertical. It intersects the ray starting from the bottom of the slit at a particular point. The horizontal distance of this point from the slit is delta l equal to D sine theta. Figure c shows rays passing through the slit at an angle theta to the horizontal. This is labeled sine theta equal to 3 lambda by 2 D, bright. A dotted line perpendicular to the rays touches the top of the slit. Its perpendicular distance from the bottom of the slit is 3 lambda by 2. Figure d shows rays passing through the slit at an angle theta to the horizontal. This is labeled sine theta equal to 2 lambda by D, dark. A dotted line perpendicular to the rays touches the top of the slit. Its perpendicular distance from the bottom of the slit is 2 lambda.
Light passing through a single slit is diffracted in all directions and may interfere constructively or destructively, depending on the angle. The difference in path length for rays from either side of the slit is seen to be D sin θ .

At the larger angle shown in part (c), the path lengths differ by 3 λ / 2 for rays from the top and bottom of the slit. One ray travels a distance λ different from the ray from the bottom and arrives in phase, interfering constructively. Two rays, each from slightly above those two, also add constructively. Most rays from the slit have another ray to interfere with constructively, and a maximum in intensity occurs at this angle. However, not all rays interfere constructively for this situation, so the maximum is not as intense as the central maximum. Finally, in part (d), the angle shown is large enough to produce a second minimum. As seen in the figure, the difference in path length for rays from either side of the slit is D sin θ , and we see that a destructive minimum is obtained when this distance is an integral multiple of the wavelength.

Thus, to obtain destructive interference for a single slit    ,

D sin θ = m λ , for m = ± 1 , ± 2 , ± 3 , ... ( destructive ) ,

where D is the slit width, λ is the light’s wavelength, θ 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 effect is explored in Double-Slit Diffraction .

Figure shows a graph of intensity versus sine theta. The intensity is maximum at sine theta equal to 0. There are smaller wave crests to either side of this, at sine theta equal to minus 2 lambda D, minus lambda D, lambda D, 2 lambda D and so on.
A graph of single-slit diffraction intensity showing the central maximum to be wider and much more intense than those to the sides. In fact, the central maximum is six times higher than shown here.

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 45.0 ° relative to the incident direction of the light, as in [link] . (a) What is the width of the slit? (b) At what angle is the first minimum produced?

Figure shows a vertical line on the left. This has a slit in the middle, of length D. A ray labeled lambda passes horizontally through the slit. This splits into 5 dotted lines that fall on a screen. The screen is shown as a vertical line. Of the five dotted lines, two make angles theta 1 and theta 2 with the horizontal. Theta 2 is 45 degrees. Theta 1 is smaller than theta 2 and is unknown. Intensity on the screen is shown as a vertical wave. The crest at the center, where a horizontal dotted line from the center of the slit falls on the screen, is the largest. The wave attenuates at the top and bottom. The remaining four dotted lines correspond to troughs in the wave.
In this example, we analyze a graph of the single-slit diffraction pattern.

Strategy

From the given information, and assuming the screen is far away from the slit, we can use the equation D sin θ = m λ first to find D , and again to find the angle for the first minimum θ 1 .

Solution

  1. We are given that λ = 550 nm , m = 2 , and θ 2 = 45.0 ° . Solving the equation D sin θ = m λ for D and substituting known values gives
    D = m λ sin θ 2 = 2 ( 550 nm ) sin 45.0 ° = 1100 × 10 −9 m 0.707 = 1.56 × 10 −6 m .
  2. Solving the equation D sin θ = m λ for sin θ 1 and substituting the known values gives
    sin θ 1 = m λ D = 1 ( 550 × 10 −9 m ) 1.56 × 10 −6 m .

    Thus the angle θ 1 is
    θ 1 = sin −1 0.354 = 20.7 ° .

Significance

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 20.7 ° on either side of the original beam, for a width of about 41 ° . The angle between the first and second minima is only about 24 ° ( 45.0 ° 20.7 ° ) . Thus, the second maximum is only about half as wide as the central maximum.

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Questions & Answers

The photoelectric effect is the emission of electrons when light shines on a material. 
Chris Reply
What is photoelectric effect
Amit Reply
it gives practical evidence of particke nature of light.
Omsai
particle nature
Omsai
photoelectric effect is the phenomenon of emission of electrons from a material(i.e Metal) when it is exposed to sunlight. Emitted electrons are called as photo electrons.
Anil
what are the applications of quantum mechanics to medicine?
Neptune
application of quantum mechanics in medicine: 1) improved disease screening and treatment ; using a relatively new method known as BIO- BARCODE ASSAY we can detect disease-specific clues in our blood using gold nanoparticles. 2) in Genomic medicine 3) in protein folding 4) in radio theraphy(MRI)
Anil
Quantam physics ki basic concepts?
Laxmikanta Reply
why does not electron exits in nucleaus
Kabbo Reply
electrons have negative
YASH
Proton and meltdown has greater mass than electron. So it naturally electron will move around nucleus such as gases surrounded earth
Amalesh
.......proton and neutron....
Amalesh
excuse me yash what negative
Rika
coz, electron contained minus ion
Manish
negative sign rika shrestha ji
YASH
electron is the smallest negetive charge...An anaion i.e., negetive ion contains extra electrons. How ever an atom is neutral so it must contains proton and electron
Amalesh
yes yash ji
Rika
yes friends
Prema
koantam theory
Laxmikanta
yes prema
Rika
quantum theory tells us that both light and matter consists of tiny particles which have wave like propertise associated with them.
Prema
proton and nutron nuclear power is best than proton and electron kulamb force
Laxmikanta
what is de-broglie wave length?
Ramsuphal
plot a graph of MP against tan ( Angle/2) and determine the slope of the graph and find the error in it.
Ime Reply
expression for photon as wave
BARISUA Reply
Are beta particle and eletron are same?
Amalesh Reply
yes
mari
how can you confirm?
Amalesh
sry
Saiaung
If they are same then why they named differently?
Amalesh
because beta particles give the information that the electron is ejected from the nucleus with very high energy
Absar
what is meant by Z in nuclear physic
Shubhu Reply
atomic n.o
Gyanendra
no of atoms present in nucleus
Sanjana
Note on spherical mirrors
Shamanth Reply
what is Draic equation? with explanation
M.D Reply
what is CHEMISTRY
trpathy Reply
it's a subject
Akhter
it's a branch in science which deals with the properties,uses and composition of matter
Eniabire
what is a Higgs Boson please?
FRANKLINE Reply
god particles is know as higgs boson, when two proton are reacted than a particles came out which is used to make a bond between than materials
M.D
bro little abit getting confuse if i am wrong than please clarify me
M.D
the law of refraction of direct current lines at the boundary between two conducting media of
BATTULA Reply
what is the black body of an ideal radiator
Areej Reply
uncertainty principles is applicable to
Areej
fermions
FRANKLINE
what is the cause of the expanding universe?
FRANKLINE
microscopic particles or gases
Areej
Astronomers theorize that the faster expansion rate is due to a mysterious, dark force that is pulling galaxies apart. One explanation for dark energy is that it is a property of space.
Areej
Thanks for your contribution Areej.
FRANKLINE
no problem
Areej
what is photoelectric equation
HIMANSHU Reply
Practice Key Terms 2

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Source:  OpenStax, University physics volume 3. OpenStax CNX. Nov 04, 2016 Download for free at http://cnx.org/content/col12067/1.4
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