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Magazine Chapter 26: Problem 34
A laser beam of wavelength \(600.0 \mathrm{nm}\) is incident normally on a transmission grating having 400.0 lines \(/ \mathrm{mm} .\) Find the angles of deviation in the first, second, and third orders of bright spots.
Short Answer
Expert verified
Angles are approximately \(13.9^\circ\), \(28.7^\circ\), and \(46.4^\circ\) for first, second, third orders.
Step by step solution
01
Identify Key Parameters
First, identify the key parameters of the problem: the wavelength of the laser beam \( \lambda = 600.0\, \text{nm} \), and the transmission grating with \( 400.0 \) lines\/mm. We must convert the wavelength to meters by noting that \( 1\, \text{nm} = 1 \times 10^{-9} \text{m} \), so \( \lambda = 600.0 \times 10^{-9} \text{m} \).
02
Determine Grating Spacing
The grating has \( 400.0 \) lines per millimeter, which means the spacing \( d \) between the lines is the reciprocal: \( d = \frac{1}{400.0\, \text{lines/mm}} = \frac{1}{400.0\times 10^3\, \text{lines/m}} = 2.5 \times 10^{-6} \text{m} \).
03
Use Grating Equation
The grating equation is \( d \sin \theta = n \lambda \), where \( n \) is the order number. We need to find \( \theta \) for the first, second, and third orders (\( n = 1, 2, 3 \)).
04
Calculate First Order Deviation
For \( n = 1 \), the equation is \( 2.5 \times 10^{-6} \sin \theta = 1 \times 600.0 \times 10^{-9} \). Solving for \( \sin \theta \), we get \( \sin \theta = \frac{600.0 \times 10^{-9}}{2.5 \times 10^{-6}} = 0.24 \). Therefore, \( \theta_1 = \arcsin(0.24) \approx 13.9^\circ \).
05
Calculate Second Order Deviation
For \( n = 2 \), the equation is \( 2.5 \times 10^{-6} \sin \theta = 2 \times 600.0 \times 10^{-9} \). Solving for \( \sin \theta \), we get \( \sin \theta = \frac{1200.0 \times 10^{-9}}{2.5 \times 10^{-6}} = 0.48 \). Therefore, \( \theta_2 = \arcsin(0.48) \approx 28.7^\circ \).
06
Calculate Third Order Deviation
For \( n = 3 \), the equation is \( 2.5 \times 10^{-6} \sin \theta = 3 \times 600.0 \times 10^{-9} \). Solving for \( \sin \theta \), we get \( \sin \theta = \frac{1800.0 \times 10^{-9}}{2.5 \times 10^{-6}} = 0.72 \). Therefore, \( \theta_3 = \arcsin(0.72) \approx 46.4^\circ \).
07
Conclusion: Summary of Results
The angles of deviation for the first three orders of bright spots are approximately \( 13.9^\circ \), \( 28.7^\circ \), and \( 46.4^\circ \) respectively.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding Wavelength
Wavelength is a crucial concept in the study of waves. It's the distance between two successive points that are in phase, which means they have the same pattern of motion.
In this context, wavelength is often denoted by the Greek letter lambda (\(\lambda\)). For our laser beam, the wavelength is given as \(600.0\, \text{nm}\), which is a unit called nanometers. To convert to meters, which is more useful in calculations, we remember that \(1\, \text{nm} = 1 \times 10^{-9} \text{m}\). Thus, our wavelength is \(600.0 \times 10^{-9}\, \text{m}\).
Understanding wavelength impacts how we analyze patterns produced by diffracting lights, such as the colors seen when light passes through a prism or a grating.
In this context, wavelength is often denoted by the Greek letter lambda (\(\lambda\)). For our laser beam, the wavelength is given as \(600.0\, \text{nm}\), which is a unit called nanometers. To convert to meters, which is more useful in calculations, we remember that \(1\, \text{nm} = 1 \times 10^{-9} \text{m}\). Thus, our wavelength is \(600.0 \times 10^{-9}\, \text{m}\).
Understanding wavelength impacts how we analyze patterns produced by diffracting lights, such as the colors seen when light passes through a prism or a grating.
Characteristics of a Laser Beam
A laser beam is a stream of coherent light where the waves are in phase and have a constant amplitude. Lasers are prized for their precision and intensity, mainly due to this coherence.
In diffraction experiments, such as those involving a grating, the laser's consistent wavelength leads to clear patterns, making lasers an excellent choice for investigating wave properties.
The properties of the laser are essential to predict how it will interact with a diffraction grating, and what kind of diffraction pattern will emerge.
In diffraction experiments, such as those involving a grating, the laser's consistent wavelength leads to clear patterns, making lasers an excellent choice for investigating wave properties.
The properties of the laser are essential to predict how it will interact with a diffraction grating, and what kind of diffraction pattern will emerge.
Angles of Deviation in Diffraction
When light passes through a diffraction grating, it bends, and this bending is termed as 'deviation.' The angles at which the light is bent are the angles of deviation.
In our problem, these angles determine where bright spots appear on the screen in various orders (first, second, third, etc.). Each order corresponds to a specific angle, which can be calculated using known formulas.
These angles are critical in applications such as spectrometers, where light needs to be split into its component colors. The distinct deviation angles help researchers and scientists identify substances based on their spectral lines.
In our problem, these angles determine where bright spots appear on the screen in various orders (first, second, third, etc.). Each order corresponds to a specific angle, which can be calculated using known formulas.
These angles are critical in applications such as spectrometers, where light needs to be split into its component colors. The distinct deviation angles help researchers and scientists identify substances based on their spectral lines.
The Grating Equation
The grating equation is essential for calculating the angles of deviation. It is given by \(d \sin \theta = n \lambda\), where:
This equation encapsulates how coherent light of a specific wavelength will divide into patterns when encountering a structured grating. Spacing (\(d\)) and the wavelength are fundamental in determining different harmonic orders, making the calculation straightforward to understand.
- \(d\) is the spacing between lines on the grating
- \(\theta\) is the angle of deviation
- \(n\) is the order of the bright fringe
- \(\lambda\) is the wavelength of the light
This equation encapsulates how coherent light of a specific wavelength will divide into patterns when encountering a structured grating. Spacing (\(d\)) and the wavelength are fundamental in determining different harmonic orders, making the calculation straightforward to understand.
