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Magazine Chapter 35: Problem 37
(II) A diffraction grating has \(6.0 \times 10^{5}\) lines/m. Find the angular spread in the second-order spectrum between red light of wavelength \(7.0 \times 10^{-7} \mathrm{~m}\) and blue light of wavelength \(4.5 \times 10^{-7} \mathrm{~m}\)
Short Answer
Expert verified
The angular spread in the second-order spectrum is \(25.0^\circ\).
Step by step solution
01
Understand the Problem
We need to calculate the angular spread between the red and blue light in the second-order diffraction spectrum. We are given the number of lines per meter on the diffraction grating and the wavelengths of the red and blue light.
02
Write the Diffraction Grating Formula
The formula for the diffraction grating is given by \(d \sin \theta = m \lambda\), where \(d\) is the spacing between grating lines, \(\theta\) is the diffraction angle, \(m\) is the order of the spectrum, and \(\lambda\) is the wavelength of light.
03
Calculate the Line Spacing \(d\)
The line spacing \(d\) is the reciprocal of the number of lines per meter. Given \(6.0 \times 10^{5}\) lines/m, \(d = \frac{1}{6.0 \times 10^{5}} = 1.67 \times 10^{-6} \text{ m}\).
04
Calculate the Angle for Red Light
Using the formula, substitute \(\lambda = 7.0 \times 10^{-7} \text{ m}\) and \(m = 2\):\[1.67 \times 10^{-6} \sin \theta_\text{red} = 2 \times 7.0 \times 10^{-7}\]Solve for \(\sin \theta_\text{red}\):\[\sin \theta_\text{red} = \frac{1.4 \times 10^{-6}}{1.67 \times 10^{-6}}\]\[\theta_\text{red} = \sin^{-1}(0.839)\approx 57.6^\circ\]
05
Calculate the Angle for Blue Light
Similarly, for the blue light with \(\lambda = 4.5 \times 10^{-7} \text{ m}\) and \(m = 2\):\[1.67 \times 10^{-6} \sin \theta_\text{blue} = 2 \times 4.5 \times 10^{-7}\]Solve for \(\sin \theta_\text{blue}\):\[\sin \theta_\text{blue} = \frac{9.0 \times 10^{-7}}{1.67 \times 10^{-6}}\]\[\theta_\text{blue} = \sin^{-1}(0.539)\approx 32.6^\circ\]
06
Calculate Angular Spread
The angular spread \(\Delta \theta\) is the difference between \(\theta_\text{red}\) and \(\theta_\text{blue}\):\[\Delta \theta = \theta_\text{red} - \theta_\text{blue} = 57.6^\circ - 32.6^\circ = 25.0^\circ\]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Angular Spread
In a diffraction grating, the angular spread is an important concept. It refers to the angle difference between two colors of light as they emerge from the grating. The emergence angles differ due to the distinct wavelengths of light, which the grating separates. This spread, \( \Delta \theta \), tells us how much the colors diverge.
When calculating angular spread, we use the angles of diffraction for each color. For example, if the red light produces an angle \( \theta_\text{red} \) and blue light produces \( \theta_\text{blue} \), then the angular spread is \( \theta_\text{red} - \theta_\text{blue} \).
Main points to remember about angular spread:
When calculating angular spread, we use the angles of diffraction for each color. For example, if the red light produces an angle \( \theta_\text{red} \) and blue light produces \( \theta_\text{blue} \), then the angular spread is \( \theta_\text{red} - \theta_\text{blue} \).
Main points to remember about angular spread:
- It measures the divergence of different wavelengths.
- It depends on the grating's properties and light wavelengths.
- Larger spreads mean more noticeable separation of colors.
Second-Order Spectrum
A spectrum's order relates to the number of wavelengths that fit into the path difference created by the grating. The second-order spectrum means the path difference is two wavelengths. This order determines how the light is diffracted and the angles produced.
Using a diffraction formula, \( d \sin \theta = m \lambda \), where \( m \) represents the spectrum order, we can see how changing the order impacts angles. When we work with different orders, like the second one, the angles change, often increasing. This results in a larger angular spread compared to the first order.
Key facts about second-order spectra:
Using a diffraction formula, \( d \sin \theta = m \lambda \), where \( m \) represents the spectrum order, we can see how changing the order impacts angles. When we work with different orders, like the second one, the angles change, often increasing. This results in a larger angular spread compared to the first order.
Key facts about second-order spectra:
- Involves two wavelengths in path difference.
- Produces larger angles compared to first-order.
- Helps to separate colors more distinctly.
Wavelength
Wavelength is the distance between successive peaks of a light wave. It is crucial in determining how light behaves when it meets a diffraction grating. Different colors of light have different wavelengths; red light has a longer wavelength than blue light.
In our context, we see red light with \( 7.0 \times 10^{-7} \text{ m} \) and blue light with \( 4.5 \times 10^{-7} \text{ m} \). Due to these differences, light is diffracted at different angles, leading to an angular spread. The wavelength directly relates to the color perceived and the angle calculated.
Remember these points about wavelength in diffraction:
In our context, we see red light with \( 7.0 \times 10^{-7} \text{ m} \) and blue light with \( 4.5 \times 10^{-7} \text{ m} \). Due to these differences, light is diffracted at different angles, leading to an angular spread. The wavelength directly relates to the color perceived and the angle calculated.
Remember these points about wavelength in diffraction:
- Longer wavelengths diffract at larger angles.
- Critical for calculating diffracted angles with the formula \( d \sin \theta = m \lambda \).
- Helps distinguish different colors in light.
Grating Lines
The grating lines on a diffraction grating define how the light is spread out into its spectrum. These are essentially tiny, evenly spaced slits or grooves. The number of lines per meter indicates the grating's ability to separate wavelengths.
In this exercise, the grating has \( 6.0 \times 10^{5} \) lines per meter. More lines usually mean finer separation of light, producing a clearer spectrum. The distance between these lines is \( d \), which is the inverse of the number of lines. This spacing is crucial since it affects how the light waves interfere after passing through the slits.
Points about grating lines to keep in mind:
In this exercise, the grating has \( 6.0 \times 10^{5} \) lines per meter. More lines usually mean finer separation of light, producing a clearer spectrum. The distance between these lines is \( d \), which is the inverse of the number of lines. This spacing is crucial since it affects how the light waves interfere after passing through the slits.
Points about grating lines to keep in mind:
- More lines result in finer wavelength separation.
- Line spacing \( d \) is calculated as the reciprocal of lines density.
- Influences both the order and angular spread of the spectrum.
