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Magazine Chapter 27: Problem 2
In a Young's double-slit experiment, the wavelength of the light used is \(520 \mathrm{nm}\) (in vacuum), and the separation between the slits is \(1.4 \times 10^{-6} \mathrm{~m}\). Determine the angle that locates (a) the dark fringe for which \(m=0,\) (b) the bright fringe for which \(m=1,\) (c) the dark fringe for which \(m=1,\) and \((\mathrm{d})\) the bright fringe for which \(m=2\)
Short Answer
Expert verified
(a) \(10.70^\circ\), (b) \(21.82^\circ\), (c) \(33.87^\circ\), and (d) \(48.17^\circ\).
Step by step solution
01
Understanding the Problem
In a Young's double-slit experiment, the angle for dark and bright fringes are calculated using different formulas. For dark fringes, the formula is \( d \sin \theta = (m + 0.5) \lambda \), and for bright fringes, it is \( d \sin \theta = m \lambda \). Here, \( d = 1.4 \times 10^{-6} \text{ m} \), and \( \lambda = 520 \text{ nm} = 520 \times 10^{-9} \text{ m} \).
02
Calculate Angle for Dark Fringe with m=0
Using the formula for dark fringes: \( d \sin \theta = (m + 0.5) \lambda \). Substitute \( m = 0 \), \( d = 1.4 \times 10^{-6} \text{ m} \), and \( \lambda = 520 \times 10^{-9} \text{ m} \).\[ 1.4 \times 10^{-6} \sin \theta = (0 + 0.5) \times 520 \times 10^{-9} \]Solve for \( \sin \theta \):\[ \sin \theta = \frac{260 \times 10^{-9}}{1.4 \times 10^{-6}} \approx 0.1857 \]Calculate \( \theta \):\[ \theta = \arcsin(0.1857) \approx 10.70^\circ \]
03
Calculate Angle for Bright Fringe with m=1
Using the formula for bright fringes: \( d \sin \theta = m \lambda \). Substitute \( m = 1 \), \( d = 1.4 \times 10^{-6} \text{ m} \), and \( \lambda = 520 \times 10^{-9} \text{ m} \).\[ 1.4 \times 10^{-6} \sin \theta = 1 \times 520 \times 10^{-9} \]Solve for \( \sin \theta \):\[ \sin \theta = \frac{520 \times 10^{-9}}{1.4 \times 10^{-6}} \approx 0.3714 \]Calculate \( \theta \):\[ \theta = \arcsin(0.3714) \approx 21.82^\circ \]
04
Calculate Angle for Dark Fringe with m=1
Using the dark fringe formula again: \( d \sin \theta = (m + 0.5) \lambda \). Substitute \( m = 1 \), \( d = 1.4 \times 10^{-6} \text{ m} \), and \( \lambda = 520 \times 10^{-9} \text{ m} \).\[ 1.4 \times 10^{-6} \sin \theta = (1 + 0.5) \times 520 \times 10^{-9} \]Solve for \( \sin \theta \):\[ \sin \theta = \frac{780 \times 10^{-9}}{1.4 \times 10^{-6}} \approx 0.5571 \]Calculate \( \theta \):\[ \theta = \arcsin(0.5571) \approx 33.87^\circ \]
05
Calculate Angle for Bright Fringe with m=2
Use the bright fringe formula: \( d \sin \theta = m \lambda \). Substitute \( m = 2 \), \( d = 1.4 \times 10^{-6} \text{ m} \), and \( \lambda = 520 \times 10^{-9} \text{ m} \).\[ 1.4 \times 10^{-6} \sin \theta = 2 \times 520 \times 10^{-9} \]Solve for \( \sin \theta \):\[ \sin \theta = \frac{1040 \times 10^{-9}}{1.4 \times 10^{-6}} \approx 0.7429 \]Calculate \( \theta \):\[ \theta = \arcsin(0.7429) \approx 48.17^\circ \]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Interference Patterns
In Young's double-slit experiment, light waves pass through two closely spaced narrow slits, creating an interference pattern on a screen. This pattern consists of alternating bright and dark spots, known as fringes. These occur because light waves from the two slits overlap and either reinforce or cancel each other out depending on the path difference between them. The bright fringes occur where waves are in phase (constructive interference), leading to increased intensity. Conversely, dark fringes happen where waves are out of phase (destructive interference), resulting in lower or zero intensity.
Interference is a fundamental phenomenon in wave optics and is a direct evidence of the wave nature of light. Understanding these patterns helps scientists and engineers to design precision optical instruments and technologies that utilize the principles of diffraction and interference.
In practical terms, elucidating interference patterns involves calculating specific angles or positions on the screen where these bright and dark fringes appear. Correctly predicting these positions requires a strong grasp of wave interactions and the mathematical formulas that describe them.
Interference is a fundamental phenomenon in wave optics and is a direct evidence of the wave nature of light. Understanding these patterns helps scientists and engineers to design precision optical instruments and technologies that utilize the principles of diffraction and interference.
In practical terms, elucidating interference patterns involves calculating specific angles or positions on the screen where these bright and dark fringes appear. Correctly predicting these positions requires a strong grasp of wave interactions and the mathematical formulas that describe them.
Wavelength
Wavelength is a crucial parameter in understanding light and optics. It is defined as the distance between consecutive peaks of a wave. In Young's double-slit experiment, wavelength (\(\lambda\)) influences the spacing and position of interference fringes. The experiment typically uses monochromatic light, meaning light of a single wavelength, to ensure clear and predictable patterns.
In a vacuum, light travels with a specific wavelength, denoted in nanometers (nm). For example, light with a wavelength of 520 nm appears green. This wavelength interacts with the slit separation and screen distance to determine the exact formation of the bright and dark fringes.
In practice, altering the wavelength changes the interference pattern. Increasing the wavelength generally results in more widely spaced fringes, as the path difference required for interference shifts. Consequently, understanding wavelength is fundamental for interpreting and predicting interference phenomena in double-slit experiments.
In a vacuum, light travels with a specific wavelength, denoted in nanometers (nm). For example, light with a wavelength of 520 nm appears green. This wavelength interacts with the slit separation and screen distance to determine the exact formation of the bright and dark fringes.
In practice, altering the wavelength changes the interference pattern. Increasing the wavelength generally results in more widely spaced fringes, as the path difference required for interference shifts. Consequently, understanding wavelength is fundamental for interpreting and predicting interference phenomena in double-slit experiments.
Fringe Angle Calculation
The angle at which the bright and dark fringes occur in a double-slit experiment can be calculated using established formulas. These calculations are critical to understanding the precise locations of fringes on a screen. For bright fringes, the formula used is:\[d \sin \theta = m \lambda\]where:
In this context, \(m\) is an integer representing the order of the fringe under consideration. Each specific value of \(m\) corresponds to either a crucial dark or bright fringe.
These formulas help determine how the wave crests align relative to the slits, allowing precise predictions of where each fringe will fall. Accurate angle calculations are vital for various optics applications, including those in engineering and scientific research.
- \(d\) is the distance between the slits.
- \(\theta\) is the angle made with the normal of the slits.
- \(m\) is the fringe order number.
In this context, \(m\) is an integer representing the order of the fringe under consideration. Each specific value of \(m\) corresponds to either a crucial dark or bright fringe.
These formulas help determine how the wave crests align relative to the slits, allowing precise predictions of where each fringe will fall. Accurate angle calculations are vital for various optics applications, including those in engineering and scientific research.
Optics
Optics is the branch of physics that deals with the study of light. It examines its properties, behavior, and interactions with different materials. Young's double-slit experiment is a classic optics experiment highlighting the wave nature of light.
Understanding optics helps in grasping concepts such as refraction, reflection, diffraction, and interference, all of which are crucial for a wide range of scientific and practical applications. Optics knowledge enables the design and development of lenses, microscopes, telescopes, and fiber optics communication systems.
Young's experiment demonstrates how light behaves similarly to other waves, providing insights into the principles of wave-particle duality. It is fundamental in advancing fields like quantum physics, where understanding the properties and behavior of light is essential.
Moreover, knowing optics allows for the creation of devices that alter and control light for various uses, from enhancing images to increasing security through advanced optical systems.
Understanding optics helps in grasping concepts such as refraction, reflection, diffraction, and interference, all of which are crucial for a wide range of scientific and practical applications. Optics knowledge enables the design and development of lenses, microscopes, telescopes, and fiber optics communication systems.
Young's experiment demonstrates how light behaves similarly to other waves, providing insights into the principles of wave-particle duality. It is fundamental in advancing fields like quantum physics, where understanding the properties and behavior of light is essential.
Moreover, knowing optics allows for the creation of devices that alter and control light for various uses, from enhancing images to increasing security through advanced optical systems.
