Superposition Principle: The superposition principle states that, when two or more than two wave travelled through a medium simultaneously super impose than a new wave is formed which has resultant displacement () at any instant is equal to the vector sum of the displacements due to individual waves. If , be the displacements due to individual waves than according to the principle of superposition of waves, the resultant displacement () of the resultant wave is given by = +
Coherent Sources: Two sources of light which continuously emit light waves of same wave length, same frequency and are always in phase or have a constant phase differences are called coherent sources. e.g Laser light is a coherent source of light. Coherent sources can be obtained from a single source S. Light waves which are in phase at S reach in phase on slit S 1 and S 2 as both S 1 and S 2 lies on the same wave front and are at equal distance from S. Since they are derived from a single wave front, the waves from S 1 and S 2 have same frequency or wave length so they are coherent sources of light.
Note : The two independent sources of light can not be coherent sources. It is because the light emitted by one source would have random phase with respect to other sources so the light from these two sources are not in the same phase. Thus two independent sources of light are incoherent sources.
Mono chromatic Light: The light having single wave length and frequency is called mono chromatic light.
Interference of Light: The phenomenon of non- uniform distribution of light energy in a medium due to superposition of light waves from two coherent sources is called interference of light.
Explanation:
Let us consider two point sources of light S 1 and S 2 have exactly same frequency and amplitude having always in the same phase with each other. The light waves from the two sources are always in phase when they reach point B. It is because the distance S 1 B is equal to the distance S 2 B. Thus the constructive interference occurs at point B and the resultant wave has amplitude double to that of either wave. Since the energy of the wave is directly proportional to the square of the amplitude so the light energy at point B is 4 times that from S 1 or S 2 alone. As waves from S 1 and S 2 arriving at B are in same phase so the point B is a permanent seat of brightness. Let us now consider a point D, where distance from S 1 is half a wavelength longer than from S 2 i.e. S 1 D - S 2 D = Therefore, at point D the waves from the two sources differ in phase by 180 0 or (Ï€) and the destructive interference occurs at point D. and the intensity at that point is zero. So the point D is a permanent seat of darkness. Thus the light energy is redistributed due to the superposition of waves from two coherent sources. At a point of maximum intensity, the energy is 4 times that from a single source where as at a point of minimum intensity the energy is zero. However the average energy is the same as if two sources acted separately without interferences. Hence the law of conservation of energy holds good. Therefore, no violation of the law of conservation of energy is involved in the interference.
1. Constructive Interferences : When two waves having same frequency or wave length and are in the same phase, superimpose on each other than the amplitude of the resultant wave is equal to the sum of the amplitudes of the two waves such type of interference is called constructive interference. In constructive interference, crest of one wave fall on the crest of other wave and the trough of one wave fall on the trough of other wave.
2. Destructive Interference :
When two waves having same frequency or wave length and the phase of the waves is differ by π, superimpose on each other than the resultant amplitude is equal to the differences between the amplitudes of two waves, such type of interference is called destructive interference. In destructive interference, crest of one wave fall on the trough of another waves.
Phase differences and Path differences
A common cause of phase diff. between two waves is the path length travelled by the waves. Let two sources of light S 1 and S 2 that have same frequency and amplitude and are in the same phase. The waves from S 1 and S 2 arriving at point P on the screen have a path difference x is given by
Path difference (x) = S 2 P - S 1 P
If the path diff. between the two waves is wave length () of the wave then the phase diff = 2
If the path diff. between two waves is x then phase diff =
For a path diff. , phase diff.= 2
For a path diff. x, phase diff.= x
Phase diff. () = x path diff.
The point P will be a seat of brightness or darkness depending upon whether the waves from S 1 and S 2 arriving at P are in phase or out of phase. If the waves arriving at P are in phase, constructive interference will take place at P on the other hand, if the waves arriving at P are out of phase, destructive interference will take place at P.
Optical Path: It is defined as the distance through which the light travels in vacuum during the same time for which it travels in the medium.
Let us consider an optically denser medium at which light travels with a velocity v. If d
is the distance travelled in the medium than time taken by it is given by t= …………..(I)
Where d= geometrical path in the medium
If C and V are the velocities of light in vaccum and medium respectively than the
refractive index of the medium is given by = ……………………….(II)
If light travel for same time't' in vacuum, than the distance covered will be
L= c× t
Where L is called optical path P
L= c x
L = x d
Hence the optical path can also be defined as the product of refractive index of the medium and geometrical distance.
Conditions for sustained interference: In order to produce a stable interference pattern, the following conditions should be satisfied.
1. The two sources must be coherent i.e they should emit continuous light wave of same frequency having zero phase difference.
2. The two sources should be monochromatic light. Otherwise different colours will produce different interference and fringes of different colours will overlap.
3. The amplitudes of the waves from the two sources should be equal. Otherwise the dark fringe will not be completely dark.
4. The two coherent sources must be close to each other.
5. The two light sources should be narrow.
6. The two interfering waves must be propagated along the same line.
Youngs double slit Experiment :
In 1801, Thomas young established the wave nature of light by showing that light waves can interfere with each other. Young's double slit experimental arrangement is shown in the figure. S is a narrow slit (about 1mm wide) illuminated by a monochromatic source of light. At some distance from S (about 10 cm) there are two fine slits S 1 and S 2 very close to each other (about 0.5mm apart) equidistant from S. The light transmitted by these slits then falls on a screen placed at a considerable distance (about 2m) from the slits. Since the waves emerging from slits S 1 and S 2 originate from the same source S, are in the same phase having same wavelength and amplitude at all times. Therefore, the two light sources S 1 and S 2 are coherent and produce stable interference pattern on the screen. It is observed that the alternate bright and dark bands running parallel to the slits are appears on the screen. These are called interference fringes. At bright fringe crest of one wave (represented by solid line) coincide with the crest of other wave. At dark fringe crest of one wave coincide with the trough (represented by dotted line) of the other wave. The alternate bright and dark fringes are equally thick and equally spaced.
Analytical Theory of Interference: [expression for fringe width]
Let us consider a monochromatic source of light S, emitting light wave of wavelength and two narrow slits S 1 and S 2 are equidistant from S and act as coherent sources separated by a distance
d as shown in fig. Let the screen be placed at a distance D from the coherent sources. Let O be the mid points of slits S 1 and S 2 and line OC is the perpendicular drawn from mid point of slits to the screen. Thus the point C is equidistant from S 1 and S 2 . Therefore, the path difference between two waves at point C is zero and the constructive interference is produced and a bright fringe is observed at point C. This is called central bright fringe.
Let us now consider a point P at a distance ' ' from point C. Waves from S 1 and S 2 reach at the point P after travelling distance S 1 P and S 2 P respectively and will be in the same phase or out of phase depending upon the path difference. Path difference = S 2 P- S 1 P To calculate path difference perpendiculars S 1 Q and S 2 R are drawn from S 1 and S 2 on the screen.
Here, PQ = and PR =
(S 2 P) 2 - (S 1 P) 2 = -
(S 2 P- S 1 P) ( S 2 P + S 1 P)= 2
S 2 P- S 1 P =
Since, d D so S 1 P S 2 P=D
[Point P lies close to point C]
S 2 P+ S 1 P=2D
Path difference = S 2 P- S 1 P==
The waves from S 1 and S 2 arriving at point on the screen will interfere constructively or distructively depending upon the path difference. The phase diff. for this path diff. is:Phase diff. () =× .
1. Bright Fringe: If the path difference is an integral multiple of wavelength then the point P
is bright. Therefore for bright fringe
= n ; n= 0, 1, 2, 3…
= n
This given the n th bright fringe from C therefore writing n for we get
n =
For n= 0, 0 = 0 Central bright fringe
n=1 1= First bright fringe
n=2 2= 2 nd bright fringe
And So on
The distance between any two consecutive bright fringes is called fringe width and is denoted
by β.
Fringe width () = -
=
2. Dark Fringe: If the path diff. is an odd integral multiple of half wave length, then the point P is
dark. Therefore for dark fringes
= (2n-1)
= (2n-1) ; n = 1, 2, 3,…
This gives the n th dark fringe from C
= (2n-1)
for n=1 = First dark fringe
n=2 = 2 nd dark fringe
And so on.
The distance between any two consecutive dark fringes is called fringe width () is given by
Fringe width () = -
= -
=
Thus the width of bright fringe is equal to the width of dark fringe i.e the fringes are equally spaced.
From this equation it is clear that;
1. The fringe width increases as (i) the wave length increases (ii) the distance D of the screen from the source increases (iii) the distance d between the source decreases.
2. If one of the slits (S 1 or S 2 ) is covered up, the fringes disappear. This shows that interference pattern is due to superposition of waves from the two slits.
3. When the apparatus is immersed in a liquid of refractive index , the wavelength of light decreases therefore, the fringe width = decreases.
4. Since is directly proportional to , the fringe produced by shorter wave length will be narrower compared to those produced by light of longer wavelength.
5. If white light is used, the central fringe is white and the fringes on either side are coloured. Blue is the colour nearer to the central fringe and red is further away.
The path difference to point C on the perpendicular bisector of the slits S 1 and S 2 is zero for all colours and as a result each colour produces a bright fringe here. As they overlap, a white fringe is formed. But far away from point C, in a direction parallel to the slits the shortest visible wave, blue produces a bright fringe first.
Q. Two slits 0.45mm apart is placed 75cm from a screen. What is the diatance between the second and third dark lines of the interference pattern on the screen when the slits are illuminated with monochromatic light of wavelength 500nm?
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