Advertisements
Advertisements
प्रश्न
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
उत्तर
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
संबंधित प्रश्न
State any one difference between interference of light and diffraction of light
A long narrow horizontal slit is paced 1 mm above a horizontal plane mirror. The interference between the light coming directly from the slit and that after reflection is seen on a screen 1.0 m away from the slit. Find the fringe-width if the light used has a wavelength of 700 nm.
The intensity at the central maximum (O) in a Young’s double slit experimental set-up shown in the figure is IO. If the distance OP equals one-third of the fringe width of the pattern, show that the intensity at point P, would equal `(I_0)/4`.
Answer in brief:
Explain what is the optical path length. How is it different from actual path length?
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.
Why two light sources must be of equal intensity to obtain a well-defined interference pattern?
Draw a neat labelled ray diagram of the Fresnel Biprism experiment showing the region of interference.
What are coherent sources of light?
What is phase of a wave?
How does wavefront division provide coherent sources?
What is a bandwidth of interference pattern?
In Young’s double slit experiment, the slits are 2 mm apart and are illuminated with a mixture of two wavelength λ0 = 750 nm and λ = 900 nm. What is the minimum distance from the common central bright fringe on a screen 2 m from the slits where a bright fringe from one interference pattern coincides with a bright fringe from the other?
The interference pattern is obtained with two coherent light sources of intensity ratio n. In the interference pattern, the ratio `("I"_"max" - "I"_"min")/("I"_"max" + "I"_"min")` will be ______
A metal rod has length, cross-sectional area and Young's modulus as L, A and Y, respectively. If the elongation in the rod produced is l, then work done is proportional to ______.
The light waves from two independent monochromatic light sources are given by, y1 = 2 sin ωt and y2 = 3 cos ωt. Then the correct statement is ____________.
In a biprism experiment, D = 1 m, `lambda` = 6000 Å. When a convex lens is interposed between the biprism ru1d the eyepiece, then the distance between the images of the slits given by the Jens at two positions are 1.5 mm and 6.0 mm. The fringe width will 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, the two slits act as coherent sources of equal amplitude A and wavelength `lambda`. In another experiment with the same set up the two slits are of equal amplitude A and wavelength `lambda`. but are incoherent. The ratio of the intensity of light at the mid-point of the screen in the first case to that in the second case is ____________.
In a biprism experiment, red light of wavelength 6500 Å was used. It was then replaced by green light of wavelength 5200 Å. The value of n for which (n + 1)th green bright band would coincide with nth red bright band for the same setting is ______.
Two waves with same amplitude and frequency superpose at a point. The ratio of resultant intensities when they arrive in phase to that when they arrive 90° out of phase is ______.
`[cos pi/2=0]`
In Young's double-slit experiment, the distance between the slits is 3 mm and the slits are 2 m away from the screen. Two interference patterns can be obtained on the screen due to light of wavelength 480 nm and 600 run respectively. The separation on the screen between the 5th order bright fringes on the two interference patterns is ______
In an interference experiment, the intensity at a point is `(1/4)^"th"` of the maximum intensity. The angular position of this point is at ____________.
(cos 60° = 0.5, `lambda` = wavelength of light, d = slit width)
Show graphically the intensity distribution in a single slit diffraction pattern.
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 ______.
Interference fringes are produced on a screen by using two light sources of intensities I and 9I. The phase difference between the beams is `pi/2` at point P and π at point Q on the screen. The difference between the resultant intensities at point P and Q is ______.
Describe Young's double-slit interference experiment.
