Problem 18
Question
In double-slit experiment using light of wavelength \(600 \mathrm{~nm}\), the angular width of a fringe formed on a distant screen is \(0.1^{\circ} .\) What is the spacing between the two slits? (a) \(3.44 \times 10^{-4} \mathrm{~m}\) (b) \(3.03 \times 10^{-4} \mathrm{~m}\) (c) \(4.03 \times 10^{-4} \mathrm{~m}\) (d) \(2.68 \times 10^{-4} \mathrm{~m}\)
Step-by-Step Solution
Verified Answer
The spacing between the two slits is \(3.44 \times 10^{-4} \mathrm{~m}\). (a)
1Step 1: Understanding the Problem
We have a double-slit experiment where we need to find the slit separation (d). We are given the wavelength \( \lambda = 600 \text{ nm} \) and the angular width of a fringe is \( \theta = 0.1^{\circ} \).
2Step 2: Convert Units
Convert the wavelength from nanometers to meters: \( \lambda = 600 \text{ nm} = 600 \times 10^{-9} \text{ m} \). Convert the angle to radians: \( \theta = 0.1^{\circ} \times \frac{\pi}{180} = 1.745 \times 10^{-3} \text{ rad} \).
3Step 3: Use the Fringe Width Formula
The fringe width in a double-slit experiment is given by \( \Delta \theta = \frac{\lambda}{d} \). We rearrange to find the slit separation \( d = \frac{\lambda}{\Delta \theta} \).
4Step 4: Calculate the Slit Separation
Substitute the values we have: \( d = \frac{600 \times 10^{-9} \text{ m}}{1.745 \times 10^{-3}} = 3.44 \times 10^{-4} \text{ m} \).
5Step 5: Find the Correct Answer
The calculated slit separation is \( d = 3.44 \times 10^{-4} \text{ m} \). This matches option (a).
Key Concepts
Fringe WidthSlit SeparationWavelength
Fringe Width
In the double-slit experiment, one of the most intriguing observations is the formation of consecutive bands of light and dark regions on a screen. These bands are known as fringes. Each light fringe represents constructive interference, where the light waves from the two slits enhance each other, leading to bright spots.
The fringe width, denoted as \( \Delta y \), is the distance between two consecutive bright (or dark) fringes on the screen. It depends on several factors, which include the wavelength of the light used, the separation between the slits, and the distance from the slits to the screen.
In essence, the formula for the angular width of a fringe in the double-slit experiment is given as \( \Delta \theta = \frac{\lambda}{d} \), where \( \lambda \) is the wavelength, and \( d \) is the distance between the slits. Understanding this relationship highlights how a smaller slit separation or a larger wavelength would result in a larger fringe width, allowing for more spread-out patterns on the screen.
The fringe width, denoted as \( \Delta y \), is the distance between two consecutive bright (or dark) fringes on the screen. It depends on several factors, which include the wavelength of the light used, the separation between the slits, and the distance from the slits to the screen.
In essence, the formula for the angular width of a fringe in the double-slit experiment is given as \( \Delta \theta = \frac{\lambda}{d} \), where \( \lambda \) is the wavelength, and \( d \) is the distance between the slits. Understanding this relationship highlights how a smaller slit separation or a larger wavelength would result in a larger fringe width, allowing for more spread-out patterns on the screen.
Slit Separation
The slit separation in a double-slit experiment is a critical factor that influences the interference pattern observed on the screen. It is the distance, \( d \), between the two slits through which light passes and subsequently interferes.
To determine the slit separation given the angular width \( \Delta \theta \) and the wavelength \( \lambda \) of the light, we use the formula \( d = \frac{\lambda}{\Delta \theta} \). By rearranging this expression, we can solve for \( d \) when both \( \lambda \) and \( \Delta \theta \) are known.
The value of slit separation affects the spacing of the fringes. A smaller slit separation will result in larger spacing between the fringes because the entire pattern will spread out more. Conversely, wider slits will produce fringes that are closer together. This delicate balance demonstrates the wave nature of light and is essential in applications like diffraction grating and holography, where manipulating fringe width through slit separation is crucial.
To determine the slit separation given the angular width \( \Delta \theta \) and the wavelength \( \lambda \) of the light, we use the formula \( d = \frac{\lambda}{\Delta \theta} \). By rearranging this expression, we can solve for \( d \) when both \( \lambda \) and \( \Delta \theta \) are known.
The value of slit separation affects the spacing of the fringes. A smaller slit separation will result in larger spacing between the fringes because the entire pattern will spread out more. Conversely, wider slits will produce fringes that are closer together. This delicate balance demonstrates the wave nature of light and is essential in applications like diffraction grating and holography, where manipulating fringe width through slit separation is crucial.
Wavelength
The wavelength of light \( \lambda \) plays a vital role in determining the characteristics of the interference pattern in a double-slit experiment. Wavelength is the distance between successive crests or troughs in a wave.
In our problem, the given wavelength is \( 600 \text{ nm} \) or \( 600 \times 10^{-9} \text{ m} \). This specific value falls within the visible spectrum, explaining why we can see the fringes formed on the screen.
In our problem, the given wavelength is \( 600 \text{ nm} \) or \( 600 \times 10^{-9} \text{ m} \). This specific value falls within the visible spectrum, explaining why we can see the fringes formed on the screen.
- Wavelength affects both the position and brightness of the fringes.
- In the equation \( \Delta \theta = \frac{\lambda}{d} \), altering the wavelength while keeping \( d \) constant will change \( \Delta \theta \), the fringe width. This means longer wavelengths lead to a broader fringe pattern, while shorter wavelengths result in a tighter pattern of fringes.
Understanding the concept of wavelength is fundamental, as it resonates in broader contexts like optical instruments and communication technologies. The principles observed in the double-slit experiment with respect to wavelength highlight the wave properties of light extensively studied in physics.
Other exercises in this chapter
Problem 16
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