Why are the fringes of Newton's rings (Fig. 24-31) closer together as you look farther from the center?
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The light ray number 2 is reflected from the curved surface of the lens. The light ray number 1 reflect from the lower glass as passing through the thin layer of air remaining between lens and lower glass. These two reflected rays has 1800 phase difference. Because of 1800 of phase difference the observer sees inference pattern as in the figure.
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Since the thickness of the thin air film between the lens and glass is not increasing uniformly. The increase of the thickness of the film is depends on the curvature of the surface of the spherical lens.
The radius of the dark bright fringes on the interference pattern increases according to the relations below from the geometry.
The radius of the lens is R
The radius of the dark bright fringe is r
The wavelength of the light used is λ
The order of maxima or minima is m
radius of for the dark fringes
radius of bright fringes
The radius of the fringes are proportional to the square of the order number. The radius increase of the fringes decreases with the increase of the order number. That means the fringes get thinner and thiner as we go farther from the center.