White light (400-700 nm) incident on a 600 line/mm...

White light $(400-700 \mathrm{nm})$ incident on a 600 line/mm diffraction grating produces rainbows of diffracted light. What is the width of the first-order rainbow on a screen 2.0 m behind the grating?

Key Concepts

Geometric Projection

When the diffracted light rays hit a screen placed at some distance from the grating, the angles are converted into spatial separation. The position of each wavelength on the screen is determined by the tangent of the diffraction angle multiplied by the distance between the grating and the screen. This geometric relationship is used to calculate the physical width of the diffracted spectrum.

Angular Dispersion

Angular dispersion refers to the phenomenon where different wavelengths of light are diffracted at different angles. In a diffraction grating, shorter wavelengths (such as blue light) typically bend less than longer wavelengths (such as red light) for the same diffraction order. This difference in diffraction angles is what produces the spread or width of the rainbow seen on a screen.

Grating Equation

The grating equation relates the diffraction angle to the wavelength of the incident light and the spacing of the grating. It is expressed as d sin(?) = m?, where d is the distance between adjacent lines, ? is the diffraction angle, m is the order of the diffraction, and ? is the wavelength. This formula is fundamental in analyzing how light is dispersed by a grating.

Diffraction Grating

A diffraction grating is an optical component with a regularly spaced series of lines or slits that diffract light into several beams. The pattern is produced by the interference of light waves, and it allows different wavelengths to deviate by different angles, creating a spectrum. This underlying principle is key to many optical devices such as spectrometers.

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