Interference, Energy Conservation
In the interference pattern, if we take Imax = k (a + b)2, then average intensity of light in the interference of light in the interference pattern
Iav = Imax + Imin/2 = k (a + b)2 + k (a – b)2/2
Iav = 2k (a2 + b2)/2 = k (a2 + b2)
If there were no interference, intensity of light from two sources at every point on the screen would be I = I1 + I2 = k (a2 + b2), which is the same as Iav in the interference pattern.
This establishes that in the interference pattern, intensity of light is simply being redistributed i.e. energy is being transferred to the regions of destructive interference to the regions of constructive interference. No energy is being created or destroyed in the process. Thus the principle of energy conservation is being obeyed in the process of interference of light.
Conditions for sustained interference of light
Following are some of the important conditions for obtaining sustained interference of light:
(i) The two sources of plight must be coherent i.e. they should emit continuous light waves of same wavelength or frequency, which have either the same phase or a constant phase difference.
(ii) The two sources should be strong with least background.
(iii) The amplitudes of waves from two sources should preferably be equal.
(iv) The two sources should preferably be monochromatic.
(v) The coherent sources must be very close to each other.
(vi) The two sources should be point sources or very narrow sources.
Combining more than two waves
When more than two sinusiodally varying waves meet at a point, then two find their resultant,
(i) We construct a series of phasors representing the waves. Draw them end to end, maintaining the proper phase relations between adjacent phasors.
(ii) Draw the vector sum of this array. The length of the vector sum gives us the amplitude of the resultant phasor.
The angle between the vector sum and the first phasor is the phase of the resultant with respect to this first phasor.
If a transparent sheet of refractive index µ and thickness t is introduced in one of the paths of interfering waves, the optical path length of this path will become µ t instead of t, increasing by (µ - 1)t.
If present position of a particular fringe is y = D/d (?x), the new position of the same fringe will be given by
y’ = D/d[Δx + (µ - 1)t]
Therefore, lateral shift of the fringe
y0 = y’ – y = D/d (µ - 1)t
As β = λ D/d
Therefore, D/d = β/ λ
Therefore, y0 = β/ λ (µ - 1)t
As this expression is independent of m, therefore, each fringe or the entire fringe pattern is displaced byy0. The shifting is towards the side in which the transparent plate is introduced without any change in fringe width.
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