Young's double slit experiment derivation and Explanation
Young's double slit experiment derivation is performed by Thomas Young a scientist who established the wave nature of light as a interference pattern.
Yong's double slit experiment tells us that wave nature of light interfere their waves during travels to each other.(or light waves can interfere with each other during propagation). you can see the picture of Young's double slit experiment.
The separation between any two consecutive bright or dark fringes is called fringe width.
Position of bright fringe: If the path difference ( BP - AP) is an integral multiple of wave length λ, then position P is bright.
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Explanation of Young's double slit experiment
see the picture, in this picture a monochromatic source of light is taken to perform the experiment.
'S' is a very small hole or slit about 1mm of wide, is taken and a monochromatic source of light enter through S slit or hole.
A diffraction occur in slit S and spread the light wave in two fine slit called ( slit A and B) situated about 10 cm apart from the first slit. each slit A and B are ( 0.5 mm apart) and also equidistance from first slit 'S'. let take about ( λ/2) from first slit.
Again diffraction occur at slit A and B. the wave emerging from the slits A and B interfere with each other. the light waves emerging from these slits falls on a screen placed at a considerable distance apart 2m from the slits a and B.
Now the waves which is emerging from slits A and B originate from the same source S , so they are in phase and also having the same frequency ( or wavelength) and Amplitude due to the monochromatic source of light so the sources A and B are coherent and produce stable interference pattern on the screen.
on the screen these are two interference fringes a bright band and a dark band occurs.
alternate bright and dark bands appear on the screen these are called interference fringes.
Bright band occur when the light waves interference constructively.
alternate bright and dark bands appear on the screen these are called interference fringes.
Bright band occur when the light waves interference constructively.
constructive interference
constructive interference means crest of one wave falls on the crest of other waves and trough of one wave falls on the trough of other waves.
Note:- in the fig. dark wave called crest and dotted wave called trough.
you can see in the picture at the point O the two waves interfere is dark-dark waves or dotted-dotted waves namely called crest falls on crest or trough falls on trough.
constructive interference also occurs at position 2.
Dark bands occur where the waves interfere destructively.
Destructive interference
means a crest from one slit and trough from other slit arrives at the same time, interfere each other a dark bands ( or fringe) is produced.
therefore on the screen, we obtain a number of alternate bright and dark fringes which is parallel to the slits.
Note:- the bright and dark fringes are equally thick and equally spaced ( or equal in shape).
this picture shows the interference of two waves as single wave and alternate bright and dark fringe display on the screen.
Note:- the bright and dark fringes are equally thick and equally spaced ( or equal in shape).
this picture shows the interference of two waves as single wave and alternate bright and dark fringe display on the screen.
Derivation of Young's double slit experiment
consider a monochromatic source of light S emitting waves of wavelength λ and two narrow slits A and B placed at equal distance from S. the waves emerging from slits A and B are in phase and have the same frequency ( or wavelength) and also Amplitude is same at all times. so light sources A and B are coherent sources.
let (a>o read as a note) be the amplitude of each waves. suppose the phase difference between the two waves arriving at p on the screen is Φ.
then the waves can be represented as:
y1= a>o sin ωt
y2 = ( a>o sin ωt+ Φ)
According to superposition principle , the displacement ( y ) of the resultant given by,
y= y1+y2
= a>o sin ωt + ( a>o sin ωt+ Φ)
y = R sin( ωt+ Φ)
it is the equation of simple harmonic vibration of amplitude R.
Calculation of Fringe width
In Young's double slit experiment, alternate bright and dark fringes appear on the screen.The separation between any two consecutive bright or dark fringes is called fringe width.
Position of bright fringe: If the path difference ( BP - AP) is an integral multiple of wave length λ, then position P is bright.
xd/D = n λ D/d ( where n= 0, 1, 2, 3 ....)
Position of dark Fringe: If the path difference ( BP - AP) is an odd integral multiple of half wavelength, then point P is dark.
xd/D = (2n - 1) λ/2
x = (2n - 1) λ D/2 d
(where n = 1, 2, 3.....)
Fringe width formula:
The distance between two consecutive bright or dark fringe is called fringe width.
width of dark fringe β = λ D / d
width of bright fringe β' = λ D / d
Hence all bright and dark fringes are equal width. Therefore, expression for the fringe width in Young's double slit experiment is given by:
Fringe width, β = λ D / d
As all the bright and dark fringes are of the same width, therefore angular width of a fringe is given by:
θ = β /D
= λ D / d / D
θ = λ /d
Thus angular width of a fringe is independent of the position of the screen.
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