Suppose the angles measured in Problem 42 were produced when the spectrometer (but not the source) was submerged in water. What then would be the wavelengths (in air)?

Verified step by step guidance

1. Understand the problem: The spectrometer is submerged in water, which changes the effective wavelength of light due to the refractive index of water. The goal is to determine the wavelengths of light in air, given the angles measured in water.
2. Recall the relationship between wavelength in a medium and wavelength in air: \( \lambda_{\text{medium}} = \frac{\lambda_{\text{air}}}{n} \), where \( n \) is the refractive index of the medium (water in this case). Rearrange this equation to find \( \lambda_{\text{air}} = \lambda_{\text{medium}} \cdot n \).
3. Determine the refractive index of water: The refractive index of water is approximately \( n = 1.33 \). This value will be used to convert the wavelength from water to air.
4. Use the diffraction grating equation to relate the measured angles to the wavelength in water: \( d \sin \theta = m \lambda_{\text{medium}} \), where \( d \) is the grating spacing, \( \theta \) is the diffraction angle, \( m \) is the diffraction order, and \( \lambda_{\text{medium}} \) is the wavelength in water. Solve for \( \lambda_{\text{medium}} \) using the measured angles.
5. Substitute \( \lambda_{\text{medium}} \) into the equation \( \lambda_{\text{air}} = \lambda_{\text{medium}} \cdot n \) to calculate the wavelength in air. This will give the final result for the wavelengths in air.

Key Concepts

Refraction of Light

Refraction occurs when light passes from one medium to another, changing its speed and direction. In this case, when light travels from air into water, it slows down and bends, which affects the angles measured by the spectrometer. Understanding Snell's Law, which relates the angles of incidence and refraction to the indices of refraction of the two media, is crucial for calculating the new angles.

Wavelength in Different Media

The wavelength of light changes when it moves between different media due to variations in the speed of light. In water, the wavelength is shorter than in air because light travels slower in water. This concept is essential for determining the wavelengths in air based on the measurements taken in water, as the relationship between wavelength, frequency, and speed must be considered.

Spectrometer Functionality

A spectrometer is an instrument used to measure the properties of light over a specific portion of the electromagnetic spectrum. It can determine the wavelengths of light by analyzing the angles at which different wavelengths are refracted. Understanding how a spectrometer operates, including its calibration and the impact of environmental factors like the medium it is submerged in, is vital for interpreting the results accurately.