From Engineering Physics: Fundamentals and Modern Applications
4.10 DIVISION OF WAVEFRONT
Theory of Interference Fringes
Let S be a narrow slit illuminated by monochromatic light of wavelength ?, and S 1 and S 2 be the two narrow slits separated by a small distance 2 d (as in Figure 5). Let a screen XY be placed at a distance D from the coherent sources S 1 and S 2. Let O be the foot of the perpendicular drawn from C, the mid point of S 1 and S 2 on the screen. Now consider a point P on the screen at a distance x from O at which the condition for bright and dark fringes are to be determined. Draw S 1M and S 2N perpendiculars from S 1 and S 2 on the screen and join S 1P and S 2P. The path difference between the waves reaching at P from S 1 and S 2 is given by
In a right angled triangle S 2NP, we have
Using the binomial theorem and neglecting higher terms, since D >> x or d, we get
Similarly in a right angled triangle S 1MP, we have
? Path difference,
If the sources S 1 and S 2 are in the same phase as the waves reaching at P, the path differences must only be due to path difference, ?
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