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14.12 Young's Double-slit Experiment
14.12.1 Analytical treatment of Young's Double slit experiment
To analyze the interference pattern and investigate the spacing of the interference fringes, consider the geometry in figure. Let S be a narrow slit illuminated by monochromatic light, and S1 and S2 two parallel narrow slits very close to each other and equidistant from S. The light from S arrives at S1 and S2 in the same phase. Beyond S1 and S2 the waves proceed as if they started from S1 and S2 with the same phase because the two slits are equidistant from S.
To find the intensity at a point P on the screen, we join S1P and S2P. The two waves arrive at P from S1 and S2 having traversed different paths S1P and S2P. Let us calculate this path difference S2P – S1P. Let,
y = distance of P from P0, the central point on the screen
d = separation of two slits S1 and S.
D= distance between slits and screen
S2A = d sin q¢ = d sin q = d tan q,
as for small q, sin q = tan q. As can be seen from the figure, tan q = y/D.
Now the intensity at the point P is a maximum or minimum according as the path difference
Hence, for bright fringes (maxima).
The number n is called the order of the fringe. Thus the fringes with n = 0, 1, 2, …. etc. are called zero, first, second…..etc. orders. The zeroth order fringe corresponds to the central maximum, the first order fringe corresponds to the first bright fringe on either side of the central maximum, and so on.
For dark fringes (minima),
where n= ± 1, ± 2, ± 3 ….
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