Advertisements
Advertisements
Question
Describe geometry of the Young’s double slit experiment with the help of a ray diagram. What is fringe width? Obtain an expression of it. Write the conditions for constructive as well as destructive interference.
Advertisements
Solution
i. Let S1 and S2 be the two coherent monochromatic sources that are separated by short distance d. They emit light waves of wavelength λ.
ii. Let D = horizontal distance between screen and source.
iii. Draw S1M and S2N ⊥ AB
OP = perpendicular bisector of the slit.
Since S1P = S2P, the path difference between waves reaching P from S1 and S2 is zero, therefore there is a bright point at P.
iv. Consider a point Q on the screen which is at a distance y from the central point P on the screen. Light waves from S1 and S2 reach at Q simultaneously by covering paths S1Q and S2Q, where they superimpose. 
Derivation:
In Δ S1MQ, (S1Q)2 = (S1M)2 + (MQ)2
`("S"_1"Q")^2 = "D"^2 + ["y" - "d"/2]^2` ....(1)
In `Δ "S"_2"NQ", ("S"_2"Q")^2 = ("S"_2"N")^2 + ("NQ")^2`
∴ `("S"_2"Q")^2 = "D"^2 + ["y" + "d"/2]^2` ....(2)
Subtract equation (1) from (2),
`("S"_2"Q")^2 - ("S"_1"Q")^2 = ["D"^2 + ("y" + "d"/2)^2] - ["D"^2 + ("y" - "d"/2)^2]`
= `"D"^2 + ("y" + "d"/2)^2 - "D"^2 - ("y" - "d"/2)^2`
= `("y" + "d"/2)^2 - ("y" - "d"/2)^2`
= `("y"^2 + "d"^2/4 + "yd") - ("y"^2 + "d"^2/4 - "yd")`
= `"y"^2 + "d"^2/4 + "yd" - "y"^2 - "d"^2/4 + "yd"("S"_2"Q")^2 - ("S"_1"Q")^2`
= 2yd
∴ `("S"_2"Q" + "S"_1"Q")("S"_2"Q" - "S"_1"Q")` = 2yd
∴ `"S"_2"Q" - "S"_1"Q" = (2"yd")/("S"_2"Q" + "S"_1"Q")` ....(3)
If y << D and d << D then, S1Q ≈ S2Q ≈ D
S2Q + S1Q = 2D
∴ Equation (3) becomes,
`"S"_2"Q" - "S"_1"Q" = (2"yd")/(2"D")`
∴ `"S"_2"Q" - "S"_1"Q" = "yd"/"D"`
∴ Δl = `"yd"/"D"` ....(4)
Equation (4) gives the path difference of two interfering light waves. Point Q will be bright if,
Δl = nλ = 2n`lambda/2`
where n = 0, 1, 2,….
∴ `("y"_"n""d")/"D" = "n"λ = 2"n"lambda/2` ........….[From equation (4)]
∴ `"y"_"n" = "n"(lambda"D")/"d"` ….(5)
Equation (5) represents the distance of the nth bright fringe from the central bright fringe.
Point Q will be a dark point if,
`Deltal = (2"n" - 1)lambda/2`
where n = 1, 2, 3,……
∴ `("y"_"n"^'"d")/"D" = (2"n" - 1)lambda/2`
∴ `"y"_"n"^' = (2"n" - 1)(lambda"D")/(2"d") = ("n" - 1/2) (lambda"D")/"d"` .........(6)
Equation (6) represents the distance of nth dark fringe from the central maximum.
Fringe width:
The distance between any two successive dark or any two successive bright fringes is equal. This is called the fringe width and is given by, Fringe width = W = Δy = `"y"_("n" + 1) - "y"_"n" = "y"_"n + 1"^' - "y"_"n"^'`
W = `λ"D"/"d"`
Thus, both dark and bright fringes are equidistant and have equal widths.
Conditions for constructive and destructive interference:
The phase difference between the two waves reaching P, from `"S"_1` and `"S"_2` is given by,
`Deltaphi = "y""d"/"D"((2pi)/lambda)` ..........`(∵ Deltal = "yd"/"D")`
The condition for constructive interference in terms of phase difference is given by,
ΔΦ = n2π, where, n = 0, ±1 ±2
∴ The condition for destructive interference in terms of phase difference is given by
`Deltaphi = ("n" - 1/2)2pi,` where, n = ±1, ±2
APPEARS IN
RELATED QUESTIONS
Write the important characteristic features by which the interference can be distinguished from the observed diffraction pattern.
Laser light of wavelength 630 nm is incident on a pair of slits which are separated by 1.8 mm. If the screen is kept 80 cm away from the two slits, calculate:
1) fringe separation i.e. fringe width.
2) distance of 10th bright fringe from the centre of the interference pattern
A narrow slit S transmitting light of wavelength λ is placed a distance d above a large plane mirror, as shown in the following figure. The light coming directly from the slit and that coming after the reflection interfere at a screen ∑ placed at a distance D from the slit. (a) What will be the intensity at a point just above the mirror, i.e. just above O? (b) At what distance from O does the first maximum occur?
What is meant by coherent sources?
Answer in brief:
In Young's double-slit experiment what will we observe on the screen when white light is incident on the slits but one slit is covered with a red filter and the other with a violet filter? Give reasons for your answer.
The intensity of the light coming from one of the slits in Young's experiment is twice the intensity of the light coming from the other slit. What will be the approximate ratio of the intensities of the bright and dark fringes in the resulting interference pattern?
Two coherent sources whose intensity ratio is 25:1 produce interference fringes. Calculate the ratio of amplitudes of light waves coming from them.
One of Young’s double slits is covered with a glass plate as shown in figure. The position of central maximum will,
Discuss the interference in thin films and obtain the equations for constructive and destructive interference for transmitted and reflected light.
The ratio of maximum and minimum intensities in an interference pattern is 36 : 1. What is the ratio of the amplitudes of the two interfering waves?
A graph is plotted between the fringe-width Z and the distance D between the slit and eye-piece, keeping other adjustment same. The correct graph is
A. |
B. |
C. |
D. |
If the monochromatic source in Young's double slit experiment is white light, then ____________.
On a rainy day, a small oil film on water shows brilliant colours. This is due to ____________.
Two identical light waves having phase difference 'Φ' propagate in same direction. When they superpose, the intensity of the resultant wave is proportional to ______.
In Young's experiment, the distance between the slits is doubled and the distance between the slit and screen is reduced to half, then the fringe width ____________.
In a Young's experiment, two coherent sources are placed 0.60 mm apart and the fringes are observed one metre away. If it produces the second dark fringe at a distance of 1 mm from the central fringe, the wavelength of monochromatic light used would be ____________.
In Young's double-slit experiment, an interference pattern is obtained on a screen by a light of wavelength 4000 Å, coming from the coherent sources S1 and S2 At certain point P on the screen, third dark fringe is formed. Then the path difference S1P - S2P in microns is ______.
In Young's double slit experiment with a source of light of wavelength 5860 Å, the first maxima will occur when ____________.
If two waves represented by `"y"_1 = 3 "sin" omega "t"` and `"y"_2 = 5 "sin" (omega "t" + pi/3)` interfere at a point, then the amplitude of the resulting wave will be about ____________.
In a double slit experiment, the separation between the slits is d and distance of screen from slits is D. If the wavelength of light used is `lambda` and I is the intensity of central bright fringe, then intensity at distance x from central maximum is given by ____________.
`phi "and" phi_2 (phi_1 > phi_2)` are the work functions of metals A and B. When light of same wavelength is incident on A and B, the fastest emitted electrons from A are ____________ those emitted from B.
Waves from two coherent sources of light having an intensity ratio I1 : I2 equal to 'x' interfere. Then in the interference pattern obtained on the screen, the value of (Imax - Imin)/(Imax + Imin) is ______
In Young's double slit experiment, for wavelength λ1 the nth bright fringe is obtained at a point P on the screen. Keeping the same setting, source of light is replaced by wavelength λ2 and now (n + 1)th bright fringe is obtained at the same point P on the screen. The value of n is ______.
Young's double slit experiment is performed in water, instead of air, then fringe width ______.
A double slit experiment is immersed in water of refractive index 1.33. The slit separation is 1 mm, distance between slit and screen is 1.33 m. The slits are illuminated by a light of wavelength 6300 Å. The fringe width is ______.
How will the interference pattern of Young's double slit change if one of the two slits is covered by a paper which transmits only half of the light intensity?
The interference pattern is obtained with two coherent light sources of intensity ratio 4 : 1. And the ratio `("I"_"max" - "I"_"min")/("I"_"max" + "I"_"min")` is `5/x`. Then the value of x will be equal to ______.
Two coherent sources P and Q produce interference at point A on the screen where there is a dark band which is formed between 4th bright band and 5th bright band. Wavelength of light used is 6000 Å. The path difference between PA and QA is ______.
