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प्रश्न
Describe Young's double-slit interference experiment and derive conditions for occurrence of dark and bright fringes on the screen. Define fringe width and derive a formula for it.
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उत्तर
Description of Young's double-slit interference experiment:
- a plane wavefront is made to fall on an opaque screen AB having two similar narrow slits S1 and S2.
- The plane wavefront can be either obtained by placing a linear source S far away from the screen or by placing it at the focus of a convex lens kept close to AB.
- The rays coming out of the lens will be parallel rays and the wavefront will be a plane wave front as shown in Figure.
- The figure shows a cross-section of the experimental set up and the slits have their lengths perpendicular to the plane of the paper. For better results, the slits should be about 2-4 mm apart from each other. An observing screen PQ is placed behind of AB.
- For simplicity, we assume that the slits S1 and S2 are equidistant from the S so that the wavefronts starting from S and reaching the S1 and S2 at every instant of time are in phase.
Young's double-slit experiment - S1 and S2 act as secondary sources. The crests/troughs of the secondary wavelets superpose and interfere constructively along straight lines joining the black dots shown in the above figure. The point where these lines meet the screen have high intensity and is bright.
- Similarly, there are points shown with red dots where the crest of one wave coincides with the trough of the other. The corresponding points on the screen are dark due to destructive interference. These dark and bright regions are called fringes or bands and the whole pattern is called an interference pattern.
Conditions for the occurrence of dark and bright lunges on the screen:
Consider Young's double-slit experimental set up. Wavefront splitting produces two narrow coherent light sources as monochromatic light of wavelength emerges from two narrow and closely spaced, parallel slits S1 and S2 of equal widths. The separation S1 S2 = d is very small. The interference pattern is observed on a screen placed parallel to the plane of S1S2 and at a considerable distance D (D >> d) from the slits. OO' is the perpendicular bisector of a segment S1S2.
Geometry of the double-slit experiment
Consider, a point P on the screen at a distance y from O' (y << 0). The two light waves from S1 and S2 reach P along paths S1P and S2P, respectively. If the path difference (Δl) between S1P and S2P is an integral multiple of λ, the two waves arriving there will interfere constructively producing a bright fringe at P. On the contrary, if the path difference between S1P and S2P is a half-integral multiple of λ, there will be destructive interference and a dark fringe will be produced at P.
From the above figure,
(S2P)2 = (S2S2')2 + (PS2')2
= (S2S2')2 + (PO' + O'S2')2
`= "D"^2 + ("y" + "d"/2)^2` ....(1)
and (S1P)2 = (S1S1')2 + (PS1')2
= (S1S1')2 + (PQ' - Q'S1)2
= `"D"^2 + ("y" - "d"/2)^2` .....(2)
(S2P)2 - (S1P)2 = `{"D"^2 + ("y" + "d"/2)^2} - {"D"^2 + ("y" - "d"/2)^2}`
∴ (S2P + S1P)(S2P - S1P)
`= ["D"^2 + "y"^2 + "d"^2/4 + "yd"] - ["D"^2 + "y"^2 + "d"^2/4 - "yd"] = 2"yd"`
∴ S2P + S1P = Δ l = 2yd/S2P + S1P
In practice, D >> y and D >> d,
∴ S2P + S1P ≅ 2D
∴ Path difference,
Δ l = S2P + S1P ≅ 2 `"yd"/"2D" = "y" "d"/"D"` ....(3)
The expression for the fringe width (or band width):
The distance between consecutive bright (or dark) fringes is called the fringe width (or bandwidth) W. Point P will be bright (maximum intensity), if the
path difference, Δ l = `"y"_"n" "d"/"D" = "n" lambda` where n = 0, 1, 2, 3, .....
Point P will be dark (minimum intensity equal to zero), if `"y"_"m" "d"/"D" = ("2m" - 1) lambda/2`, where, m = 1,2,3...,
Thus, for bright fringes (or bands),
`"y"_"n" = 0, lambda "D"/"d", (2lambda"D")/"d"` ...
and for dark fringes (or bands),
`"y"_"n" = lambda/2 "D"/"d", 3 lambda/2 "D"/"d", 5lambda/2 "D"/"d"` ....
The bright and dark fringes (or bands) alternate and are evenly spaced in these situations. For Point O', the path difference (S2O' - S1O') = 0. Hence, point O' will be bright. It corresponds to the centre of the central bright fringe (or band). On both sides of O', the interference pattern consists of alternate dark and bright fringes (or band) parallel to the slit.
Let `"y"_"n"` and `"y"_"n + 1"`, be the distances of the nth and (m + 1)th bright fringes from the central bright fringe.
∴ `("y"_"n""d")/"D" = "n" lambda`
∴ `"y"_"n" = ("n" lambda "D")/"d"` .....(4)
and `("y"_("n + 1")"d")/"D" = ("n + 1")lambda`
∴ `("y"_("n + 1")) = (("n + 1") lambda "D")/"d"` .....(5)
The distance between consecutive bright fringes
`= "y"_("n + 1") - "y"_"n" = (lambda "D")/"d" [("n + 1") - "n"] = (lambda"D")/"d"` ....(6)
Hence, the fringe width,
∴ W = `triangle "y" = "y"_("n + 1") - "y"_"n" = (lambda"D")/"d"` (for bright fringes) ... (7)
Alternately, let `"y"_"m"` and `"y"_"m + 1"` be the distances of the m th and (m + 1)th dark fringes respectively from the central bright fringe.
∴ `("y"_"m""d")/"D" = (2"m" - 1) lambda/2` and
`("y"_("m+1")"d")/"D" = [2("m + 1") - 1] lambda/2 = (2"m" + 1) lambda/2` ....(8)
∴ `"y"_"m" = (2"m - 1") (lambda"D")/"2d"` and
`"y"_"m + 1" = (2"m" + 1) (lambda"D")/"2d"` .....(9)
∴ The distance between consecutive dark fringes,
`"y"_"m + 1" - "y"_"m" = (lambda"D")/"2d" [(2"m" + 1) - (2"m" - 1)] = (lambda"D")/"d"` ....(10)
∴ W = `"y"_"m + 1" - "y"_"m"`
`= (lambda"D")/"d"` (for dark fringes) .....(11)
Eqs. (7) and (11) show that the fringe width is the same for bright and dark fringes.
संबंधित प्रश्न
Write the important characteristic features by which the interference can be distinguished from the observed diffraction pattern.
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. If the mirror reflects only 64% of the light energy falling on it, what will be the ratio of the maximum to the minimum intensity in the interference pattern observed on the screen?
In Young’s double slit experiment, the slits are separated by 0.5 mm and screen is placed 1.0 m away from the slit. It is found that the 5th bright fringe is at a distance of 4.13 mm from the 2nd dark fringe. Find the wavelength of light used.
Answer in brief:
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What are the conditions for obtaining a good interference pattern? Give reasons.
Why two light sources must be of equal intensity to obtain a well-defined interference pattern?
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What is interference of light?
What is phase of a wave?
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Does diffraction take place at Young’s double-slit?
In Young’s double-slit experiment, 62 fringes are seen in the visible region for sodium light of wavelength 5893 Å. If violet light of wavelength 4359 Å is used in place of sodium light, then what is the number of fringes seen?
In Young's double slit experiment green light is incident on the two slits. The interference pattern is observed on a screen. Which one of the following changes would cause the observed fringes to be more closely spaced?
A thin transparent sheet is placed in front of a slit in Young's double slit experiment. The fringe width will ____________.
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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 interference experiment, intensity at a point is `(1/4)^"th"` of the maximum intensity. The angular position of this point is at (sin30° = cos60° = 0.5, `lambda` = wavelength of light, d = slit width) ____________.
If the two slits in Young's double slit experiment have width ratio 9 : 1, the ratio of maximum to minimum intensity in the interference pattern is ______.
In the biprism experiment, the fringe width is 0.4 mm. What is the distance between the 4th dark band and the 6th bright band on the same side?
Light waves from two coherent sources arrive at two points on a screen with a path difference of zero and λ/2. The ratio of the intensities at the points is ______
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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?
A ray of light AO in a vacuum is incident on a glass slab at an angle of 60° and refracted at an angle of 30° along OB as shown in the figure. The optical path length of the light ray from A to B is ______.
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`[sin 45^circ = cos 45^circ = 1/sqrt 2]`
