Theory of Interference Fringes

Consider a narrow monochromatic source S and two pinholes A and B, equidistant from S. A and B act as two coherent sources separated by a distance d. Let a screen be placed at a distance D from the coherent source. The point C on the screen is equidistant from A and B. Therefore, the path difference between the two waves is zero. Thus the point C has maximum intensity.

Consider a point P at a distance x from C. The waves reach at a the point P from A and B.

Here, PQ = x – d/2, PR = x + d/2



(BP)2 – (AP)2 = 2xd



But, BP = AP = D    (approximately)

(i) Bright fringes: If the path difference is a whole number multiple of wavelength λ, the point P is bright.

∴ xd/D = n λ

Where n = 0, 1, 2, 3, ….

Or, x = nλD/d                                                              (iii)

This equation gives the distances of the bright fringes from the point C. At C, the path difference is zero and a bright fringe is formed.



Therefore the distance between any two consecutive bright fringes


    
(ii) Dark Fringes: If the path difference is an odd number multiple of half wavelength, the point P is dark.



This equation gives the distances of the dark fringes from the point C.

The distance between any two consecutive dark fringes,



The distance between any two consecutive bright or dark fringes is known as fringe width. Therefore, alternately bright and dark parallel fringes are formed. The fringes are formed on both sides of C. Moreover, from equations (v) and (vi), it is clear that the width of the bright fringe is equal to the width of the dark fringe. All the fringes are equal in width and are independent of the order of the fringe. The breadth of a bright or a dark fringe is, however, equal to half the fringe width and is equal to λD/2d. The fringe width = λD/d.

Therefore, (i) the width of the fringe is directly proportional to the wavelength of light, ∝ λ. (ii) The width of the fringe is directly proportional to the distance between the two sources, ∝ 1/d. Thus, the width of the fringe increases (a) with increase in wavelength (b) with increase in the distance D and (c) by bringing the two sources A and  B close to each other.

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