Light containing two different wavelengths passes through a...

Light containing two different wavelengths passes through a diffraction grating with $1.20 \times 10^{3}$ slits/ $\mathrm{cm} .$ On a screen 15.0 $\mathrm{cm}$ from the grating, the third-order maximum of the shorter wavelength falls midway between the central maximum and the first side maximum for the longer wavelength. If the neighboring maxima of the longer wavelength are 8.44 $\mathrm{mm}$ apart on the screen, what are the wavelengths in the light? Hint: Use the small-angle approximation.

Key Concepts

Diffraction Grating Equation

This equation, m? = d sin ?, relates the wavelength ? of light, the grating spacing d (the distance between adjacent slits), and the order m of the diffraction maximum that appears at an angle ?. It is the fundamental condition for constructive interference in a diffraction grating, enabling prediction of the positions of bright fringes in the diffraction pattern.

Grating Spacing

Grating spacing is the physical distance between adjacent slits in a diffraction grating and is determined from the number of slits per unit length. The reciprocal relation between the number of slits per unit length and the slit spacing is crucial for calculating the wavelength using the diffraction equation. Knowing this spacing allows conversion between interference angles and the light’s wavelength.

Order of Maximum

The order of maximum, indicated by the integer m in the diffraction grating equation, specifies the sequence of the bright fringes seen in the diffraction pattern. Each order corresponds to a set of angles at which light waves interfere constructively, with higher orders generally appearing at larger angles. The order directly scales the wavelength in the analysis of diffraction phenomena.

Small-Angle Approximation

The small-angle approximation is used when the angles in the diffracted light are small such that sin ? and tan ? can be approximated by ? when measured in radians. This simplification is particularly useful for relating the physical separation of fringes on a screen to the diffraction angle, and thereby to the wavelength of light, without the need for more complex trigonometric calculations.

Fringe Spacing on Screen

Fringe spacing refers to the distance between consecutive bright fringes (maxima) on the diffraction pattern as projected on a screen. It is connected to the diffraction angle through the geometry of the experimental setup, typically using the approximation that the distance from the grating to the screen multiplied by the angle gives the spacing. This relationship plays a key role in determining the wavelength when the distances between maxima are known.

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