A thin slit illuminated by light of frequency f produces its...

A thin slit illuminated by light of frequency $f$ produces its first dark band at $\pm$38.2$^\circ$ in air. When the entire apparatus (slit, screen, and space in between) is immersed in an unknown transparent liquid, the slit's first dark bands occur instead at $\pm$21.6$^\circ$. Find the refractive index of the liquid.

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Key Concepts

Wavelength in Different Media

When light enters a medium with a refractive index n from a less refractive medium (like air), its wavelength decreases while its frequency remains constant. The new wavelength in the medium becomes ?/n. This change in wavelength affects all optical phenomena based on interference and diffraction, including the positions of dark and bright fringes.

Refractive Index Determination Using Diffraction

The condition for minima in a diffraction pattern can be used to determine the refractive index of a medium. By comparing the diffraction angles in air and in another medium, and knowing that the wavelength in the medium is ?/n, one can relate the sine of the diffraction angles in the two media to calculate the refractive index.

Single-Slit Diffraction

Single-slit diffraction is a wave phenomenon in which light passing through a narrow aperture spreads out and forms a characteristic pattern of bright and dark regions on a screen. The positions of the dark bands (minima) are determined by the equation a sin(?) = m?, where a is the slit width, ? is the diffraction angle, m is the order of the minimum (an integer), and ? is the wavelength of light.

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