College Physics
11th Edition
ISBN: 9781305952300
Author: Raymond A. Serway, Chris Vuille
Publisher: Cengage Learning
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A beam of white light is incident on the surface of a diamond at an angle θa�a.(Figure 1) Since the index of refraction depends on the light's wavelength, the different colors that comprise white light will spread out as they pass through the diamond. The indices of refraction in diamond are nred=2.410�red=2.410 for red light and nblue=2.450�blue=2.450 for blue light. The surrounding air has nair=1.000�air=1.000. Note that the angles in the figure are not to scale.
![## Part C
**Objective:**
Derive a formula for \(\delta\), the angle between the red and blue refracted rays in the diamond.
**Instructions:**
Express the angle in terms of \(n_{\text{red}}, n_{\text{blue}},\) and \(\theta_a\). Use \(n_{\text{air}} = 1\). Note that any trig function entered in your answer must be followed by an argument in parentheses.
**Formula:**
\[
\delta = \sin^{-1}\left(\frac{\sin\theta_{\text{air}}}{n_{\text{red}}}\right) - \sin^{-1}\left(\frac{\sin\theta_{\text{air}}}{n_{\text{blue}}}\right)
\]
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Transcribed Image Text:## Part C
**Objective:**
Derive a formula for \(\delta\), the angle between the red and blue refracted rays in the diamond.
**Instructions:**
Express the angle in terms of \(n_{\text{red}}, n_{\text{blue}},\) and \(\theta_a\). Use \(n_{\text{air}} = 1\). Note that any trig function entered in your answer must be followed by an argument in parentheses.
**Formula:**
\[
\delta = \sin^{-1}\left(\frac{\sin\theta_{\text{air}}}{n_{\text{red}}}\right) - \sin^{-1}\left(\frac{\sin\theta_{\text{air}}}{n_{\text{blue}}}\right)
\]
**Interface Elements:**
- **Button Labels:**
- **Submit**: Finalize your answer.
- **Previous Answers**: Review past responses.
**Additional Resources:**
- **Hints:** Access via “View Available Hint(s)" for guidance.
- **Keyboard Shortcuts:** Optimize input efficiency.
- **Undo/Redo/Reset Options:** Easily correct or restart the input process.
Transcribed Image Text:### Figure Explanation
This diagram illustrates the refraction and dispersion of light as it passes through a prism-shaped object, commonly used to demonstrate the separation of light into its various color components (spectrum).
#### Components Explained:
- **Incident Ray**: The initial light ray entering the prism is represented by a pink arrow labeled with the angle \( \theta_a \). This angle is measured between the incident ray and the normal to the prism surface.
- **Refracted Ray**: As the light enters the prism, it refracts, bending towards the normal. The internal angles, \( \theta_{\text{blue}} \) and \( \theta_{\text{red}} \), are the angles of refraction for blue and red light, respectively. These are depicted as a blue and red arrow inside the prism.
- **Dispersion**: The separation of light into colors is shown by two distinct paths inside the prism:
- **Blue Light Path**: Refracts at a sharper angle (\( \theta_{\text{blue}} \)) due to having a shorter wavelength.
- **Red Light Path**: Refracts at a lesser angle (\( \theta_{\text{red}} \)) as it has a longer wavelength.
- **Critical Angle (\( \theta_c \))**: This is the angle of incidence above which total internal reflection occurs rather than refraction. It is marked near the largest angle inside the prism.
- **Angle of Deviation (\( \delta \))**: The change in direction of the light ray as it passes through the prism. This is the angle between the original path of the incoming beam and the path of the outgoing refracted beam.
- **Prism Angle (\( \alpha \))**: The angle of the prism itself, shown as a dotted line forming the apex of the triangular prism shape.
This diagram serves to explain how light behaves as it transitions between different media, specifically how it separates into various colors (dispersion) due to differences in wavelength. This principle is the basis for various optical instruments and phenomena.
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