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प्रश्न
Answer the following question.
Describe any two characteristic features which distinguish between interference and diffraction phenomena. Derive the expression for the intensity at a point of the interference pattern in Young's double-slit experiment.
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उत्तर
Difference between interference and diffraction:
In the interference pattern, the intensity of the dark fringe is completely zero.
In the diffraction pattern, the intensity of secondary minima is minimum, but not completely zero.
In interference pattern the width of all the interference fringes is equal. In diffraction pattern the width of central maxima is large, and on increasing distance, the width of maxima decreases.
In interference pattern the intensity of all the bright bands is equal.
In the diffraction pattern, the intensity of all the secondary maxima is not equal.
The principle of superposition of light waves:
When two or more wave trains of light travelling in a medium superpose upon each other, the resultant displacement at any instant is equal to the vector sum of the displacements due to individual waves.
If `vecy_1,vecy_2,vecy_3,....` be the displacements due to different waves, then the resultant displacement is given by, `vecy = vecy_1 + vecy_2 + vecy_3 +` ... conditions for constructive and destructive interference:
Let the displacement of the waves from the source `S_1` and `S_2` at a point P on the screen at any time 't' be given by,
`"y"_1 = a_1 sin omegat`
and
`"y"_2 = a_2 sin(omegat + phi)`
where `phi` is the constant phase difference between the two waves
by the superposition principal , the resultant displacement at point P is given by,
`"y" = "y"_1 + "y"_2`
= `a_1 sin omegat + a_2 sin (omegat + phi)`
= `a_1 sin omegat + a_2 sin omegat cos phi + a_2 cos omegat sin phi`
`y = (a_1 + a_2 cos phi) sin omegat + a_2 sin phi cos omegat` ...(i)
Let `a_1 + a_2 cos phi = A cos theta` ...(ii)
`a_2 sin phi = A sin theta` ...(iii)
Then, equation (i) becomes
`"y" = Acostheta sin omegat + A sin theta cos omegat`
`"y" = Asin(omegat + theta)`
Squaring and adding both sides of the equations (ii) and (iii), we obtain
`A^2 cos^2theta + A^2 sin^2 theta = (a_1 + a_2 cos phi)^2 + a_2^2 sin^2 phi`
`A^2 = a_1^2 + a_2^2 (cos^2phi + sin^2phi) + 2a_1a_2 cos phi`
`A^2 = a_1^2 + a_2^2 + 2a_1a_2 cos phi`
The intensity of light is directly proportional to the square of the amplitude of the wave. The intensity of light at point P on the screen is given by,
`I = a_1^2 + a_2^2 + 2a_1a_2 cos phi` ...(iv)
संबंधित प्रश्न
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?
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`.
How does wavefront division provide coherent sources?
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. |
Band width for red light of wavelength 6600 Å is 0.42 mm. If red Light is replaced by blue light of wavelength 4400 Å, then the change m bandwidth will be ____________.
A thin mica sheet of thickness 4 x 10-6 m and refractive index 1.5 is introduced in the path of the first wave. The wavelength of the wave used is 5000 A. The central bright maximum will shift ______.
In biprism experiment, the 4th dark band is formed opposite to one of the slits. The wavelength of light used is ______.
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 a double-slit experiment, the optical path difference between the waves coming from two coherent sources at a point P on one side of the central bright is 7.5 µm and that at a point Q on the other side of the central bright fringe and 1.8 µm. How many bright and dark fringes are observed between points P and Q if the wavelength of light used is 600 nm?
With a neat labelled ray diagram explain the use of Fresnel's biprism to obtain two coherent sources.
