Show that the second- and third-order spectra of white light produced by a diffraction grating always overlap. What wavelengths overlap?

Name: Physics for Scientists and Engineers
Author: Giancoli Douglas
ISBN: 9780137488179

Verified step by step guidance

Step 1: Begin by understanding the diffraction grating equation, which is \( m \lambda = d \sin \theta \), where \( m \) is the order of the spectrum, \( \lambda \) is the wavelength of light, \( d \) is the spacing between adjacent slits in the grating, and \( \theta \) is the diffraction angle.

Step 2: To determine the overlap between the second-order (\( m = 2 \)) and third-order (\( m = 3 \)) spectra, consider the condition where a wavelength \( \lambda_2 \) in the second-order spectrum coincides with a wavelength \( \lambda_3 \) in the third-order spectrum. This means \( 2 \lambda_2 = 3 \lambda_3 \).

Step 3: Solve for \( \lambda_3 \) in terms of \( \lambda_2 \) using the relationship \( \lambda_3 = \frac{2}{3} \lambda_2 \). This indicates that wavelengths in the third-order spectrum are fractions of the wavelengths in the second-order spectrum.

Step 4: Recognize that white light contains a continuous range of wavelengths. For any wavelength \( \lambda_2 \) in the second-order spectrum, there exists a corresponding wavelength \( \lambda_3 \) in the third-order spectrum that satisfies the overlap condition. This overlap occurs because the diffraction grating equation is linear with respect to \( \lambda \).

Step 5: Conclude that the overlapping wavelengths are those that satisfy \( \lambda_3 = \frac{2}{3} \lambda_2 \). For example, if \( \lambda_2 \) is 600 nm in the second-order spectrum, \( \lambda_3 \) would be 400 nm in the third-order spectrum, demonstrating the overlap.

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Diffraction Grating

A diffraction grating is an optical component with a periodic structure that disperses light into its component wavelengths. When light passes through or reflects off the grating, it creates interference patterns due to the superposition of light waves. The angles at which constructive interference occurs depend on the wavelength of the light and the spacing of the grating lines, which is crucial for understanding the spectra produced.

Order of Spectrum

The order of a spectrum refers to the different sets of wavelengths produced by a diffraction grating. The first-order spectrum corresponds to the first set of angles where constructive interference occurs, while the second- and third-order spectra correspond to subsequent sets. These orders can overlap for certain wavelengths, leading to the same angle of diffraction for different orders, which is essential for analyzing the overlap in the question.

Wavelength Overlap

Wavelength overlap occurs when different orders of a diffraction grating produce the same angle of diffraction for specific wavelengths. This phenomenon can be mathematically described using the grating equation, which relates the angle of diffraction to the wavelength and the grating spacing. Understanding which wavelengths overlap between the second and third orders is key to solving the problem presented in the question.

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