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प्रश्न
Answer in brief:
Explain what is the optical path length. How is it different from actual path length?
What is Optical Path length? How is it different from the actual Path length?
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उत्तर १
Consider, a light wave with an angular frequency of w and a wave vector of k travelling in the x-direction through a vacuum. The phase of this wave is (kx - ωt). In a vacuum, light speed at c, but in a medium, it speeds at v.
k = `(2pi)/lambda = (2pi v)/(v lambda) = omega/v` as ω = 2πv and v = vλ, where v is the frequency of light.
If the wave travels a distance Δ x, its phase changes by Δ Φ = kΔx = ω Δx/v.
Similarly, if the wave is travelling in vacuum,
k = ω/c and Δ Φ = ω Δ x/c
Now, consider a wave travelling a distance Δ x in the medium, the phase difference generated is,
Δ Φ' = k' Δ x = ωn Δ x/c = ω Δ x'/c ...(1)
where Δ x' = n Δ x .....(2)
The distance nΔ x is called the optical path length of the light in the medium; it is the distance the light would have travelled in the same time t in vacuum (with the speed c).
The optical path length in a medium is the corresponding path in a vacuum that light traverses at the same time as it does in the medium.
Now, speed = `"distance"/"time"`
∴ time = `"distance"/"speed"`
∴ t = `"d"_"medium"/"v"_"medium" = "d"_"vaccum"/"v"_"vaccum"`
Hence, the optical path = `"d"_"vacuum"`
`= "v"_"vaccum"/"v"_"medium" xx "d"_"medium"`
`= "n" xx "d"_"medium"`
Thus, a distance d travelled in a medium of refractive index n introduces a path difference = nd - d = d (n - 1) over a ray travelling equal distance through vacuum.
उत्तर २
i. When a wave travels a distance Δx through a medium having a refractive index of n, its phase changes by the same amount as it would if the wave had travelled a distance nΔx in a vacuum.
ii. Thus, a path length of Δx in a medium of refractive index n is equivalent to a path length of nΔx in a vacuum.
iii. nΔx is called the optical path travelled by a wave.
iv. This means, the optical path through a medium is the effective path travelled by light in a vacuum to generate the same phase difference.
v. Optical path in a medium can also be defined as the corresponding path in a vacuum that the light travels at the same time as it takes in the given medium.
i.e., time = `"d"_"medium"/"v"_"medium" = "d"_"vacuum"/"v"_"vacuum"`
∴ `"d"_"vacuum" = "v"_"vacuum"/"v"_"medium" xx "d"_"medium" = "n" xx "d"_"medium"`
But `"d"_"vacuum"` = Optical path
∴ Optical path = n × `"d"_"medium"`
Thus, a distance d travelled in a medium of refractive index n introduces a path difference = nd - d = d(n - 1) over a ray travelling an equal distance through the vacuum.
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?
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.
What is meant by coherent sources?
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.
What are the conditions for obtaining a good interference pattern? Give reasons.
Explain constructive and destructive interference with the help of a diagram?
In a Young’s double-slit experiment, the slit separation is doubled. To maintain the same fringe spacing on the screen, the screen-to-slit distance D must be changed to ______.
What is interference of light?
Does diffraction take place at Young’s double-slit?
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?
If the monochromatic source in Young's double slit experiment is white light, then ____________.
A thin transparent sheet is placed in front of a slit in Young's double slit experiment. The fringe width will ____________.
On a rainy day, a small oil film on water shows brilliant colours. This is due to ____________.
In a Young's experiment, two coherent sources are placed 0.60 mm apart and the fringes are observed one metre away. If it produces the second dark fringe at a distance of 1 mm from the central fringe, the wavelength of monochromatic light used would be ____________.
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 Young's double slit experiment fifth dark fringe is formed opposite to one of the slits. If D is the distance between the slits and the screen and d is the separation between the slits, then the wavelength of light used is ______.
In Young's double slit experiment the source is white light. One slit is covered with red filter and the other with blue filter. There shall be ____________.
Two identical light sources s1 and s2 emit light of same wavelength `lambda`. These light rays will exhibit interference if their ______.
`phi "and" phi_2 (phi_1 > phi_2)` are the work functions of metals A and B. When light of same wavelength is incident on A and B, the fastest emitted electrons from A are ____________ those emitted from B.
In a biprism experiment, the slit separation is 1 mm. Using monochromatic light of wavelength 5000 Å, an interference pattern is obtained on the screen. Where should the screen be moved? so that the change in fringe width is 12.5 x 105 m?
If two light waves reaching a point produce destructive interference, then the condition of phase difference is ______
In a biprism experiment, monochromatic light of wavelength (λ) is used. The distance between two coherent sources is kept constant. If the distance between slit and eyepiece (D) is varied as D1, D2, D3, and D4, the corresponding measured fringe widths are z1, z2, z3, and z4 then ______
If we have two coherent sources S1 and S2 vibrating in phase, then for an arbitrary point P constructive interference is observed whenever the path difference 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?
Describe Young's double-slit interference experiment.
With a neat labelled ray diagram explain the use of Fresnel's biprism to obtain two coherent sources.
In biprism experiment, the distance of 20th bright band from the central bright band is 1.2 cm. Without changing the experimental set-up, the distance of 30th bright band from the central bright band will be ______.
