A diffraction grating has 6.5 x 10⁵ slits/m. Find the angular spread in the second-order spectrum between red light of wavelength 7.0 x 10⁻⁷ m and blue light of wavelength 4.5 x 10⁻⁷ m.

Verified step by step guidance

1. Determine the grating spacing (d) using the formula \( d = \frac{1}{N} \), where \( N \) is the number of slits per meter. Here, \( N = 6.5 \times 10^5 \) slits/m, so \( d = \frac{1}{6.5 \times 10^5} \) meters.
2. Use the diffraction grating equation \( m \lambda = d \sin \theta \) to calculate the diffraction angle \( \theta \) for each wavelength. For the second-order spectrum (\( m = 2 \)), substitute \( \lambda = 7.0 \times 10^{-7} \) m (red light) and \( \lambda = 4.5 \times 10^{-7} \) m (blue light) into the equation.
3. Solve for \( \sin \theta \) for each wavelength using \( \sin \theta = \frac{m \lambda}{d} \). For red light, \( \sin \theta_{\text{red}} = \frac{2 \times 7.0 \times 10^{-7}}{d} \). For blue light, \( \sin \theta_{\text{blue}} = \frac{2 \times 4.5 \times 10^{-7}}{d} \).
4. Find the angles \( \theta_{\text{red}} \) and \( \theta_{\text{blue}} \) by taking the inverse sine (arcsin) of the calculated \( \sin \theta \) values for each wavelength.
5. Calculate the angular spread by subtracting the angle for blue light from the angle for red light: \( \Delta \theta = \theta_{\text{red}} - \theta_{\text{blue}} \). This gives the angular spread in the second-order spectrum.

Key Concepts

Diffraction Grating

A diffraction grating is an optical component with a periodic structure that disperses light into its constituent wavelengths. It consists of multiple slits or grooves, which cause constructive and destructive interference of light waves. The angle at which light is diffracted depends on the wavelength and the spacing of the slits, allowing for the separation of different colors in a spectrum.

Wavelength

Wavelength is the distance between successive peaks of a wave, commonly measured in meters. In the context of light, different wavelengths correspond to different colors; for example, red light has a longer wavelength than blue light. The wavelength is crucial in determining how light interacts with materials and how it is diffracted by a grating.

Angular Spread

Angular spread refers to the range of angles over which light is dispersed when passing through a diffraction grating. It is calculated using the grating equation, which relates the angle of diffraction to the wavelength of light and the grating spacing. The angular spread between different wavelengths indicates how well a grating can separate colors in a spectrum, which is essential for applications in spectroscopy.