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प्रश्न
(i) In Young's double-slit experiment, deduce the condition for (a) constructive and (b) destructive interferences at a point on the screen. Draw a graph showing variation of intensity in the interference pattern against position 'x' on the screen.
(b) Compare the interference pattern observed in Young's double-slit experiment with single-slit diffraction pattern, pointing out three distinguishing features.
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उत्तर
Expression for Fringe Width in Young’s Double-Slit Experiment
Let S1 and S2 be two slits separated by a distance d. `GG'` is the screen at a distance D from the slits S1and S2. Point C is equidistant from both the slits. The intensity of light will be maximum at this point because the path difference of the waves reaching this point will be zero.
At point P, the path difference between the rays coming from the slits is given by
S1= S2P − S1P
Now, S1 S2 = d, EF = d, and S2F = D
In ΔS2PF,
`S_2P=[S_2F^2+PF^2]^(1/2)`
`S_2P=[D2+(x+d/2)^2]^(1/2)`
`=D[1+(x+d/2)^2/D^2]^(1/2)`
Similarly, in ΔS1PE
`S_1P=D[1+(x-d/2)^2/D^2]^(1/2)`
`:.S_2P-S_1P=D[1+1/2(x+d/2)^2/D^2)]-D[1+1/2(x-d/2)^2/D^2]`
On expanding it binomially, we get
`S_2P-S_1P=1/(2D)[4xd/2]=(xd)/D`
For constructive interference, the path difference is an integral multiple of wavelengths, that is, path difference is nλ.
`:.nlambda=(xd)/D`
`x=(nlambdaD)/d`
where n = 0, 1, 2, 3, 4, …
Similarly, for destructive interference,
`x_n=(2n-1)lambda/2D/d`
Graph of Intensity Distribution in Young’s Double-Slit Experiment
(ii) On comparing the interference pattern observed in Young's double slit experiment (interference) with single-slit diffraction pattern (diffraction), we can have three distinguishing features:
In the interference pattern, all the bright fringes have the same intensity. In a diffraction pattern, all the bright fringes are not of the same intensity
In the interference pattern, the dark fringe has zero or very small intensity so that the bright and dark fringes can easily be distinguished. In diffraction pattern, all the dark fringes are not of zero intensity
In the interference pattern, the widths of all the fringes are almost the same, whereas in diffraction pattern, the fringes are of different widths
संबंधित प्रश्न
Show that the angular width of the first diffraction fringe is half that of the central fringe.
The fringes produced in diffraction pattern are of _______.
(A) equal width with same intensity
(B) unequal width with varying intensity
(C) equal intensity\
(D) equal width with varying intensity
How does an unpolarized light incident on a polaroid get polarized? Describe briefly, with the help of a necessary diagram, the polarization of light by reflection from a transparent medium.
Two polaroids ‘A’ and ‘B’ are kept in crossed position. How should a third polaroid ‘C’ be placed between them so that the intensity of polarized light transmitted by polaroid B reduces to 1/8th of the intensity of unpolarized light incident on A?
In Young’s experiment interference bands were produced on a screen placed at 150 cm from two slits, 0.15 mm apart and illuminated by the light of wavelength 6500 Å. Calculate the fringe width.
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 \[2 \cdot 0 \times {10}^{- 3}m.\]
Write the conditions on path difference under which constructive interference occurs in Young’s double-slit experiment.
In Young’s double slit experiment, what should be the phase difference between the two overlapping waves to obtain 5th dark band/fringe on the screen?
Two slits in Young's interference experiment have width in the ratio 1 : 2. The ratio of intensity at the maxima and minima in their interference is ______.
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:
