<< Chapter < Page Chapter >> Page >
The graph shows the variation of intensity as a function of sine theta. The curve has a strong peak at sine theta equals zero, then has small oscillations spreading symmetrically to the left and right of this central peak. The oscillations all appear to be of the same height. Between each oscillation, the curve appears to go to zero, and each zero is labeled. The first zero to the left of the main peak is labeled minus lambda over d and the first zero to the right is labeled lambda over d. The second zero to the left is labeled minus two lambda over d and the second zero to the right is labeled two lambda over d. The third zero to the left is labeled minus three lambda over d and the third zero to the right is labeled three lambda over d.
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.

Thus, to obtain destructive interference for a single slit    ,

D sin θ = , for m = 1, –1, 2, –2, 3, (destructive), size 12{D`"sin"θ= ital "mλ",~m="1,"`"2,"`"3,"` dotslow } {}

where D size 12{D} {} is the slit width, λ size 12{λ} {} is the light’s wavelength, θ size 12{θ} {} is the angle relative to the original direction of the light, and m size 12{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 45.0º size 12{"45" "." 0°} {} 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?

The schematic shows a single slit to the left and the resulting intensity pattern on a screen is graphed on the right. The single slit is represented by a gap of size d in a vertical line. A ray of wavelength lambda enters the gap from the left, then five rays leave from the gap center and head to the right. One ray continues on the horizontal centerline of the schematic. Two rays angle upward: the first at an unknown angle theta one above the horizontal and the second at an angle theta two equals forty five degrees above the horizontal. The final two rays angle downward at the same angles, so that they are symmetric about the horizontal with respect to the two rays that angle upward. The intensity on the screen is a maximum where the central ray hits the screen, whereas it is a minimum where the angled rays hit the screen.
A graph of the single slit diffraction pattern is analyzed in this example.

Strategy

From the given information, and assuming the screen is far away from the slit, we can use the equation D sin θ = size 12{D`"sin"θ= ital "mλ"} {} first to find D size 12{D} {} , and again to find the angle for the first minimum θ 1 size 12{θ rSub { size 8{1} } } {} .

Solution for (a)

We are given that λ = 550 nm size 12{λ="500"`"nm"} {} , m = 2 size 12{m=2} {} , and θ 2 = 45.0º . Solving the equation D sin θ = size 12{D`"sin"θ= ital "mλ"} {} for D size 12{D} {} and substituting known values gives

D = sin θ 2 = 2 ( 550 nm ) sin 45.0º = 1100 × 10 9 0.707 = 1.56 × 10 6 .

Solution for (b)

Solving the equation D sin θ = size 12{D`"sin"θ= ital "mλ"} {} for sin θ 1 size 12{"sin"θ rSub { size 8{1} } } {} and substituting the known values gives

sin θ 1 = D = 1 550 × 10 9 m 1 . 56 × 10 6 m . size 12{"sin"θ rSub { size 8{1} } = { {mλ} over {D} } = { {1 left ("550" times "10" rSup { size 8{ - 9} } m right )} over {1 "." "56" times "10" rSup { size 8{ - 6} } m} } } {}

Thus the angle θ 1 size 12{θ rSub { size 8{1} } } {} is

θ 1 = sin 1 0.354 = 20.7º. size 12{θ rSub { size 8{1} } ="sin" rSup { size 8{ - 1} } 0 "." "354"="20" "." 7°} {}

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 20.7º on either side of the original beam, for a width of about 41º . The angle between the first and second minima is only about 24º ( 45.0º 20.7º ) size 12{"24"°` \( "45" "." 0° - "20" "." 7° \) } {} . Thus the second maximum is only about half as wide as the central maximum.

Got questions? Get instant answers now!

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 sin θ = , (for m = 1, –1, 2, –2, 3, …) size 12{D`"sin"θ= ital "mλ",~m="1,"`"2,"`"3,"` dotslow } {} , 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. Note that there is no m = 0 size 12{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?

Got questions? Get instant answers now!

Problems&Exercises

(a) At what angle is the first minimum for 550-nm light falling on a single slit of width 1 . 00 μm size 12{1 "." "00"`"μm"} {} ? (b) Will there be a second minimum?

(a) 33 . size 12{"33" "." 4°} {}

(b) No

Got questions? Get instant answers now!

(a) Calculate the angle at which a 2 . 00 -μm size 12{2 "." "00""-μm"} {} -wide slit produces its first minimum for 410-nm violet light. (b) Where is the first minimum for 700-nm red light?

Got questions? Get instant answers now!

Questions & Answers

Calculate the work done by an 85.0-kg man who pushes a crate 4.00 m up along a ramp that makes an angle of 20.0º20.0º with the horizontal. (See [link] .) He exerts a force of 500 N on the crate parallel to the ramp and moves at a constant speed. Be certain to include the work he does on the crate an
Collins Reply
What is thermal heat all about
Abel Reply
why uniform circular motion is called a periodic motion?.
Boniface Reply
when a train start from A & it returns at same station A . what is its acceleration?
Mwdan Reply
what is distance of A to B of the stations and what is the time taken to reach B from A
BELLO
the information provided is not enough
aliyu
Hmmmm maybe the question is logical
yusuf
where are the parameters for calculation
HENRY
there is enough information to calculate an AVERAGE acceleration
Kwok
mistake, there is enough information to calculate an average velocity
Kwok
~\
Abel
what is the unit of momentum
Abel
wha are the types of radioactivity ?
Worku Reply
what are the types of radioactivity
Worku
what is static friction
Golu Reply
It is the opposite of kinetic friction
Mark
static fiction is friction between two surfaces in contact an none of sliding over on another, while Kinetic friction is friction between sliding surfaces in contact.
MINDERIUM
I don't get it,if it's static then there will be no friction.
author
It means that static friction is that friction that most be overcome before a body can move
kingsley
static friction is a force that keeps an object from moving, and it's the opposite of kinetic friction.
author
It is a force a body must overcome in order for the body to move.
Eboh
If a particle accelerator explodes what happens
Eboh
why we see the edge effect in case of the field lines of capacitor?
Arnab
what is wave
Muhammed Reply
what is force
Muhammed
force is something which is responsible for the object to change its position
MINDERIUM
more technically it is the product of mass of an object and Acceleration produced in it
MINDERIUM
wave is disturbance in any medium
iqra
energy is distributed in any medium through particles of medium.
iqra
If a particle accelerator explodes what happens
Eboh Reply
we have to first figure out .... wats a particle accelerator first
Teh
What is surface tension
Subi Reply
The resistive force of surface.
iqra
Who can tutor me on simple harmonic motion
yusuf Reply
on both a string and peldulum?
Anya
spring*
Anya
Yea
yusuf
Do you have a chit-chat contact
yusuf
I dont have social media but i do have an email?
Anya
Which is
yusuf
Where are you chatting from
yusuf
I don't understand the basics of this group
Jimmy
teach him SHM init
Anya
Simple harmonic motion
yusuf
how.an.equipotential.line is two dimension and equipotential surface is three dimension ?
syed Reply
definition of mass of conversion
umezurike Reply
Force equals mass time acceleration. Weight is a force and it can replace force in the equation. The acceleration would be gravity, which is an acceleration. To change from weight to mass divide by gravity (9.8 m/s^2).
Marisa
how many subject is in physics
Adeshina Reply
the write question should be " How many Topics are in O- Level Physics, or other branches of physics.
effiom
how many topic are in physics
Praise
Praise what level are you
yusuf
If u are doing a levels in your first year you do AS topics therefore you do 5 big topic i.e particles radiation, waves and optics, mechanics,materials, electricity. After that you do A level topics like Specific Harmonic motion circular motion astrophysics depends really
Anya
Yeah basics of physics prin8
yusuf
Heat nd Co for a level
yusuf
yh I need someone to explain something im tryna solve . I'll send the question if u down for it
Tamdy Reply
a ripple tank experiment a vibrating plane is used to generate wrinkles in the water .if the distance between two successive point is 3.5cm and the wave travel a distance of 31.5cm find the frequency of the vibration
Tamdy
hallow
Boniface
please send the answer
Boniface
the range of objects and phenomena studied in physics is
Bethel Reply
I don't know please give the answer
Boniface
Practice Key Terms 1

Get the best College physics course in your pocket!





Source:  OpenStax, College physics. OpenStax CNX. Jul 27, 2015 Download for free at http://legacy.cnx.org/content/col11406/1.9
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'College physics' conversation and receive update notifications?

Ask