Suppose you illuminate two thin slits by monochromatic...

Suppose you illuminate two thin slits by monochromatic coherent light in air and find that they produce their first interference minima at $\pm 35.09^{\circ}$ on either side of the central bright spot. You then immerse these slits in a transparent liquid and illuminate them with the same light. Now you find that the first minima occur at $\pm 19.36^{\circ}$ instead. What is the index of refraction of this liquid?

Key Concepts

Trigonometric Application in Interference

The analysis of interference patterns often involves trigonometric relationships, particularly sine functions, which relate the experimental geometry (angle of fringes relative to the central maximum) to the physical parameters such as slit separation and wavelength. By comparing the angles corresponding to minima under different conditions, one can derive quantitative information about changes in the wavelength, and hence determine the index of refraction of an intervening medium.

Index of Refraction

The index of refraction of a material is a measure of how much the speed of light is reduced inside that medium compared to the speed of light in vacuum. It also relates to the bending of light as it enters the medium. In interference experiments, knowing the index of refraction is important because it determines the effective wavelength of the interfering light waves and thus the spatial arrangement of the interference fringes.

Wavelength Change in a Medium

When light travels from one medium to another, its wavelength changes while its frequency remains constant. In a medium with an index of refraction n, the wavelength of light becomes the wavelength in vacuum divided by n. This adjustment impacts the positions of interference fringes, since the interference conditions depend directly on the wavelength, thereby causing the angular positions of minima and maxima to shift in a predictable manner.

Interference of Light

Interference is a phenomenon where light waves superimpose to form regions of constructive and destructive interference. In experiments such as the double?slit, the interference pattern arises due to the path difference between light from the two slits, leading to bright fringes (constructive interference) and dark fringes or minima (destructive interference). The conditions for these minima are typically given by equations that relate the slit separation, the wavelength of light, and the angle of the fringe from the central maximum.

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