Advertisements
Advertisements
प्रश्न
Using the geometry of the double slit experiment, derive the expression for fringe width of interference bands.
उत्तर
Let S1 and S2 be the two sources of light separated by distance d. A screen is placed at a distance of D from the two light sources. Let 'A be the wavelength of light.
Draw S1 S'1 and S2 S'2 perpendicular to the screen. Let O' be the midpoint of S'1 S'2.
∴ `O"'"S"'"_1 = O"'"S"'"_1 = d/2`
Let 'P' be the point at distance y from point 'O' on screen.
Finding whether point P is bright or dark depends upon the path difference. The path difference is S2P − S1P.
In ΔS1S'1P,
(S1P)2 = (S1S'1)2 + (S'1P)2
`(S_1P)^2 = D^2 + (y - d/2)^2`
`(S_1P)^2 = D^2 + y^2 - yd + d^2/4` ...(i)
In ΔS2S'2P,
(S2P)2 = (S2S'2)2 + (S'2P)2
`(S_2P)^2 = D^2 + (y + d/2)^2`
`(S_2P)^2 = D^2 + y^2 + yd + d^2/4` ...(ii)
Subtracting equation (i) from (ii),
(S2P)2 − (S1P)2
= `(D^2 + y^2 + yd + d^2/4) - (D^2 + y^2 - yd + d^2/4)`
∴ (S2P + S1P)(S2P − S1P) = 2yd
We know (S2P − S1P) = Path difference
∴ Path difference = `(2yd)/((S_P + S_1P))`
Now, D is very large as compared to y and
i.e. D >> y and D >> d.
∴ S2P ≈ S1P ≈ D
∴ Path difference = `(2yd)/(D + D) = (2yd)/(2D)`
∴ Path difference = `(yd)/D` ...(iii)
Case-I:
The point P will be bright if the path difference = nλ
`therefore (yd)/D = n lambda`
`y = (D n lambda)/d`
Equation for position of nth bright band is
`therefore y_n = (D n lambda)/d` ...(iv)
Case-II:
The point P will be dark if path difference = `(2n - 1)lambda/2`
`therefore (yd)/D = (2n - 1)lambda/2`
`therefore y = (D(2n - 1)lambda)/(2 d)`
The equation for position of the nth dark band is
`therefore y_n = (D(2n - 1)lambda)/(2d)` ...(v)
Fringe width (W): The distance between two successive bright or dark bands in an interference pattern is called fringe width or bandwidth.
∴ Fringe width (W) = yn + 1 − yn
From equation (iv),
Equation of nth bright band
`y_n = (D n lambda)/d`
The equation for (n + 1)th bright band
`y_(n + 1) = (D(n + 1)lambda)/d`
∴ Fringe width (W) = yn + 1 − yn
= `(D(n + 1)lambda)/d - (D n lambda)/d`
= `(D lambda)/d [(n + 1) - n]`
= `(D lambda)/d(1)`
∴ Fringe width (W) = `(lambda D)/d`
APPEARS IN
संबंधित प्रश्न
Derive the conditions for bright and dark fringes produced due to diffraction by a single slit.
In a biprism experiment, the fringes are observed in the focal plane of the eyepiece at a distance of 1.2 m from the slits. The distance between the central bright and the 20th bright band is 0.4 cm. When a convex lens is placed between the biprism and the eyepiece, 90 cm from the eyepiece, the distance between the two virtual magnified images is found to be 0.9 cm. Determine the wavelength of light used.
The bending of a beam of light around corners of obstacle is called ______
What is the difference between Fresnel and Fraunhofer diffraction?
In biprism experiment, the distance between source and eyepiece is 1.2 m, the distance between two virtual sources is 0.84 mm. Then the wavelength of light used if eyepiece is to be moved transversely through a distance of 2.799 cm to shift 30 fringes is ______.
A lens having focal length f gives Fraunhofer type diffraction pattern of a slit having width a. If wavelength of light is λ, the distance of first dark band and next bright band from axis is given by ____________.
A beam of light of wavelength 600 nm from a distant source falls on a single slit 1 mm wide and the resulting diffraction pattern is observed on a screen 4 m away. The distance between the first dark fringes on either side of the central bright fringe is ______.
In a diffraction pattern, width of a fringe ____________.
The angular spread of central maximum, in diffraction pattern, does not depend on ______.
In a single slit diffraction experiment, first minimum for a light of wavelength 480 nm coincides with the first maximum of another wavelength `lambda.` Then `lambda'` is ____________.
Select the CORRECT statement from the following.
A parallel beam of monochromatic light falls normally on a single narrow slit. The angular width of the central maximum in the resulting diffraction pattern ______
A plane wavefront of wavelength `lambda`. is incident on a slit of width a. The angular width of principal maximum is ______.
When two coherent sources in Young's experiment are far apart, then interference pattern ______
In Fraunhofer diffraction due to single slit, the angular width of central maximum does 'NOT' depend on ______
In young 's double slit experiment the two coherent sources have different amplitudes. If the ratio of maximum intensity to minimum intensity is 16 : 1, then the ratio of amplitudes of the two source will be _______.
The fringe width in a Young's double slit experiment can be increased if we decrease ______.
The increase in energy of a metal bar of length 'L' and cross-sectional area 'A' when compressed with a load 'M' along its length is ______.
(Y = Young's modulus of the material of metal bar)
In Fraunhofer diffraction pattern, slit width is 0.2 mm and screen is at 2 m away from the lens. If wavelength of light used is 5000 Å, then the distance between the first minimum on either side of the central maximum is ______. (θ is small and measured in radian)
In Young's double silt experiment, two slits are d distance apart. The interference pattern is observed on a screen at a distance D from the slits. The first dark fringe is observed on the screen directly opposite to one of the slits. The wavelength of light is ______.
In Fraunhoffer diffraction experiment, L is the distance between the screen and the obstacle, b is the size of the obstacle and λ is the wavelength of the incident light. The general condition for the applicability of Fraunhoffer diffraction is ______.
In a Young’s experiment, two coherent sources are placed 0.90 mm apart and the fringes are observed one metre away. If it produces the second dark fringe at a distance of 1mm from the central fringe, then the wavelength of monochromatic light used will be ______.
In a Young's double slit experiment carried out with light of wavelength λ = 5000 Å, the distance between the slits is 0.2 mm and the screen is at 200 cm from the slits. The central maximum is at x = 0. The third maximum (taking the central maximum as zeroth maximum) will be at x equal to ______.
The angular width of the central maximum of the diffraction pattern in a single slit (of width a) experiment, with λ as the wavelength of light, is ______.
In Young's double slit experiment, which of the following graph represents the correct variation of fringe width β versus distance D between sources and screen?
The first diffraction minimum due to single slit diffraction is θ, for a light wavelength 5000 Å. If the width of slit is 1 × 10-4 cm, then the value of θ is ______.
Two coherent monochromatic light beams of amplitudes E10 and E20 produce an interference pattern. The ratio of the intensities of the maxim a and minima in the interference pattern is ______.
