Advertisements
Advertisements
Question
Derive an expression for path difference in Young’s double slit experiment and obtain the conditions for constructive and destructive interference at a point on the screen.
Advertisements
Solution
Diagrammatical arrangement for Young’s double slit experiment:
In the above figure, S1 and S2 are two narrow closely spaced slits illuminated by monochromatic light of wavelength λ. The screen on which the interference pattern is observed is represented as XY.
If S1 and S2 emit light in the same phase, then for point O, the path difference receives light in the same phase. The superposition at O is constructive producing a bright point called the central maxima. The intensity at any point P at a distance x from O depends on the path difference between light reaching P from S1 and S2. The path difference is S2P − S1P.
From the geometry of the figure, we have
`(S_2P)^2=D^2+(x+d/2)^2`
Similarly, we have
`(S_1P)^2=D^2+(x+d/2)^2`
`:.(S_2P)^2-(S_1P)^2=D^2+(x+d/2)^2-D^2-(x-d/2)^2`
`:.(S_2P)^2-(S_1P)^2=(x+d/2)^2-(x-d/2)^2`
`:.(S_2P)^2-(S_1P)^2=x^2+xd+d^2/4-x^2+xd-d^2/4`
∴ (S2P)2-(S1P)2=2xd
∴(S2P-S1P)(S2P+S1P)=2xd
`:.S_2P-S_1P=(2xd)/(S_2P+S_1P)`
Now, from the figure we can see that
S2P≈S1P = D
`:.S_2P-S_1P=(2xd)/(2D)=(xd)/D`
This is the expression for path difference.
Condition for constructive interference: Constructive interference will occur when the phase difference between the two superposing waves is an even multiple of π or the path difference is an integral multiple of wavelength λ.
Condition for destructive interference: Destructive interference will occur when the phase difference between the two superposing waves is an odd multiple of π or the path difference is an odd multiple of wavelength λ/2.
RELATED QUESTIONS
Using monochromatic light of wavelength λ in Young’s double slit experiment, the eleventh dark fringe is obtained on the screen for a phase difference of ______.
In a double-slit experiment the angular width of a fringe is found to be 0.2° on a screen placed 1 m away. The wavelength of light used is 600 nm. What will be the angular width of the fringe if the entire experimental apparatus is immersed in water? Take refractive index of water to be 4/3.
A monochromatic light of wavelength 500 nm is incident normally on a single slit of width 0.2 mm to produce a diffraction pattern. Find the angular width of the central maximum obtained on the screen.
Estimate the number of fringes obtained in Young's double slit experiment with fringe width 0.5 mm, which can be accommodated within the region of total angular spread of the central maximum due to single slit.
A beam of light consisting of two wavelengths, 800 nm and 600 nm is used to obtain the interference fringes in a Young's double slit experiment on a screen placed 1 · 4 m away. If the two slits are separated by 0·28 mm, calculate the least distance from the central bright maximum where the bright fringes of the two wavelengths coincide.
The intensity at the central maxima in Young’s double slit experimental set-up is I0. Show that the intensity at a point where the path difference is λ/3 is I0/4.
How does the fringe width get affected, if the entire experimental apparatus of Young is immersed in water?
Two coherent sources of light having intensity ratio 81 : 1 produce interference fringes. Calculate the ratio of intensities at the maxima and minima in the interference pattern.
The slits in a Young's double slit experiment have equal width and the source is placed symmetrically with respect to the slits. The intensity at the central fringe is I0. If one of the slits is closed, the intensity at this point will be ____________ .
A thin transparent sheet is placed in front of a Young's double slit. The fringe-width will _____________ .
In a Young's double slit experiment, two narrow vertical slits placed 0.800 mm apart are illuminated by the same source of yellow light of wavelength 589 nm. How far are the adjacent bright bands in the interference pattern observed on a screen 2.00 m away?
Find the angular separation between the consecutive bright fringes in a Young's double slit experiment with blue-green light of wavelength 500 nm. The separation between the slits is \[2 \cdot 0 \times {10}^{- 3}m.\]
White light is used in a Young's double slit experiment. Find the minimum order of the violet fringe \[\left( \lambda = 400\text{ nm} \right)\] which overlaps with a red fringe \[\left( \lambda = 700\text{ nm} \right).\]
In a Young's double slit experiment, using monochromatic light, the fringe pattern shifts by a certain distance on the screen when a mica sheet of refractive index 1.6 and thickness 1.964 micron (1 micron = 10−6 m) is introduced in the path of one of the interfering waves. The mica sheet is then removed and the distance between the screen and the slits is doubled. It is found that the distance between the successive maxima now is the same as the observed fringe-shift upon the introduction of the mica sheet. Calculate the wavelength of the monochromatic light used in the experiment.
Two transparent slabs having equal thickness but different refractive indices µ1 and µ2are pasted side by side to form a composite slab. This slab is placed just after the double slit in a Young's experiment so that the light from one slit goes through one material and the light from the other slit goes through the other material. What should be the minimum thickness of the slab so that there is a minimum at the point P0 which is equidistant from the slits?
A double slit S1 − S2 is illuminated by a coherent light of wavelength \[\lambda.\] The slits are separated by a distance d. A plane mirror is placed in front of the double slit at a distance D1 from it and a screen ∑ is placed behind the double slit at a distance D2 from it (see the following figure). The screen ∑ receives only the light reflected by the mirror. Find the fringe-width of the interference pattern on the screen.
In a Young's double slit interference experiment, the fringe pattern is observed on a screen placed at a distance D from the slits. The slits are separated by a distance d and are illuminated by monochromatic light of wavelength \[\lambda.\] Find the distance from the central point where the intensity falls to (a) half the maximum, (b) one-fourth the maximum.
In a Young's double slit experiment, \[\lambda = 500\text{ nm, d = 1.0 mm and D = 1.0 m.}\] Find the minimum distance from the central maximum for which the intensity is half of the maximum intensity.
In Young’s double-slit experiment, show that:
`beta = (lambda "D")/"d"` where the terms have their usual meaning.
When a beam of light is used to determine the position of an object, the maximum accuracy is achieved, if the light is ______.
A projectile can have the same range R for two angles of projection. If t1 and t2 be the times of flight in two cases, then what is the product of two times of flight?
Young's double slit experiment is made in a liquid. The 10th bright fringe lies in liquid where 6th dark fringe lies in vacuum. The refractive index of the liquid is approximately
The Young's double slit experiment is performed with blue and with green light of wavelengths 4360Å and 5460Å respectively. If x is the distance of 4th maxima from the central one, then:
In Young's double slit experiment, the minimum amplitude is obtained when the phase difference of super-imposing waves is: (where n = 1, 2, 3, ...)
Two slits, 4mm apart, are illuminated by light of wavelength 6000 A° what will be the fringe width on a screen placed 2 m from the slits?
In a Young’s double slit experiment, the path difference at a certain point on the screen between two interfering waves is `1/8`th of the wavelength. The ratio of intensity at this point to that at the centre of a bright fringe is close to ______.
How will the interference pattern in Young's double-slit experiment be affected if the source slit is moved away from the plane of the slits?
In a Young's double slit experiment, the width of the one of the slit is three times the other slit. The amplitude of the light coming from a slit is proportional to the slit- width. Find the ratio of the maximum to the minimum intensity in the interference pattern.
Interference fringes are observed on a screen by illuminating two thin slits 1 mm apart with a light source (λ = 632.8 nm). The distance between the screen and the slits is 100 cm. If a bright fringe is observed on a screen at distance of 1.27 mm from the central bright fringe, then the path difference between the waves, which are reaching this point from the slits is close to :
In Young's double slit experiment the two slits are 0.6 mm distance apart. Interference pattern is observed on a screen at a distance 80 cm from the slits. The first dark fringe is observed on the screen directly opposite to one of the slits. The wavelength of light will be ______ nm.
- Assertion (A): In Young's double slit experiment all fringes are of equal width.
- Reason (R): The fringe width depends upon the wavelength of light (λ) used, the distance of the screen from the plane of slits (D) and slits separation (d).
In Young's double-slit experiment, the separation between the two slits is d and the distance of the screen from the slits is 1000 d. If the first minima fall at a distance d from the central maximum, obtain the relation between d and λ.
In Young's double-slit experiment, the screen is moved away from the plane of the slits. What will be its effect on the following?
- The angular separation of the fringes.
- Fringe-width.
In an interference experiment, a third bright fringe is obtained at a point on the screen with a light of 700 nm. What should be the wavelength of the light source in order to obtain the fifth bright fringe at the same point?
In Young's double slit experiment, show that:
`β = (λ"D")/"d"`
Where the terms have their usual meaning.
If the monochromatic source in Young’s double slit experiment is replaced by white light, then ______.
