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Monochromatic light of wavelength 592 nm from a distant source passes through a slit that is 0.0290 mm wide. In the resulting diffraction pattern, the intensity at the center of the central maximum (θ=0) is 4.00 × 105 W/m2. What is the intensity at a point on the screen that corresponds to θ = 1.20?

Short Answer

Expert verified
The intensity at θ=1.20 is approximately 1.33×108 W/m2.

Step by step solution

01

Understand the Diffraction Equation

The intensity of light in a single-slit diffraction pattern is given by the equation: I(θ)=I0(sin(β)β)2where I0 is the intensity at the center, β=πasin(θ)λ, and a is the slit width. We are tasked with finding I(1.20).
02

Convert Units and Calculate β

Convert wavelength λ=592 nm=592×109 m and slit width a=0.029 mm=0.029×103 m. For θ=1.20, sin(θ)sin(1.20)=0.0209. Calculate:β=π×0.029×103×0.0209592×109
03

Simplify and Insert Values

Calculate β using the given values:β=π×0.029×103×0.0209592×1093.20This will be used to find the new intensity at θ=1.20.
04

Compute the Intensity Ratio

Using the intensity equation:I(θ)I0=(sin(3.20)3.20)2Calculate sin(3.20) and divide by 3.20. Then square the result to find the intensity ratio.
05

Determine the Intensity at 1.20

Calculate sin(3.20)0.0584 (note the negative value, but the square is taken):I(1.20)4.00×105=(0.05843.20)2(0.01825)20.0003335Therefore, I(1.20)=4.00×105×0.00033351.33×108 W/m2.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Diffraction Pattern
When light encounters an obstacle, such as a slit, it tends to bend and spread out. This is known as diffraction. In a single-slit diffraction setup, when monochromatic light passes through a narrow slit, it produces a characteristic pattern of alternating bright and dark bands on a screen, known as a diffraction pattern.
This pattern results from the constructive and destructive interference of light waves. The central bright band, called the central maximum, is the brightest part of the pattern. Flanking it are alternating bands of light (maxima) and darkness (minima), whose intensity gradually decreases with distance from the center.
Intensity Calculation
The intensity of light in a diffraction pattern is not uniform. It varies based on the angle θ\, which measures the position on the screen relative to the central maximum. The intensity at any point in the pattern can be calculated using:
  • I(θ)=I0(sin(β)β)2
Where β=πasin(θ)λ\, I0 is the intensity at the central maximum, and λ is the wavelength. This equation accounts for the wave nature of light and demonstrates how intensity diminishes as one moves away from the central axis. Calculating β provides insight into intensity changes due to different angles and slit dimensions.
Monochromatic Light
Monochromatic light is light consisting of a single wavelength and frequency. This type of light is crucial for diffraction experiments because it ensures clear and distinguishable patterns.
In the given exercise, the light used has a wavelength of 592 nm. As all photons have the same energy and wavelength, the pattern produced is coherent and precise, aiding in accurate determination of diffraction characteristics such as maxima and minima positions.
Wavelength
The wavelength λ of light is the distance between consecutive peaks (or troughs) in a wave. It’s a vital parameter in understanding many optical phenomena, including diffraction. The wavelength determines the scale of the diffraction pattern.
  • Longer wavelengths create more spread-out patterns,
  • Shorter wavelengths yield narrower bands.
In our context, the light's wavelength is 592 nm, translating to 592 × 10^{-9}\ m. This small scale emphasizes the microscopic nature of diffraction patterns.
Slit Width
Slit width is the width (denoted as a) of the opening through which light passes. It significantly influences the diffraction pattern's appearance. As the width changes, so does the interference pattern's distribution:
  • Narrower slits cause more pronounced diffraction, leading to wider separation between maxima.
  • Wider slits result in less diffraction and narrower spacing of the bands.
In this exercise, the slit width is 0.0290 mm, which defines the initial condition for the diffraction phenomena observed.

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Most popular questions from this chapter

The VLBA (Very Long Baseline Array) uses a number of individual radio telescopes to make one unit having an equivalent diameter of about 8000 km. When this radio telescope is focusing radio waves of wavelength 2.0 cm, what would have to be the diameter of the mirror of a visible-light telescope focusing light of wavelength 550 nm so that the visible-light telescope has the same resolution as the radio telescope?

Coherent monochromatic light of wavelength l passes through a narrow slit of width a, and a diffraction pattern is observed on a screen that is a distance x from the slit. On the screen, the width w of the central diffraction maximum is twice the distance x. What is the ratio a/λ of the width of the slit to the wavelength of the light?

Monochromatic electromagnetic radiation with wavelength λ from a distant source passes through a slit. The diffraction pattern is observed on a screen 2.50 m from the slit. If the width of the central maximum is 6.00 mm, what is the slit width a if the wavelength is (a) 500 nm (visible light); (b) 50.0 μm (infrared radiation); (c) 0.500 nm (x rays)?

Light of wavelength 633 nm from a distant source is incident on a slit 0.750 mm wide, and the resulting diffraction pattern is observed on a screen 3.50 m away. What is the distance between the two dark fringes on either side of the central bright fringe?

Although we have discussed single-slit diffraction only for a slit, a similar result holds when light bends around a straight, thin object, such as a strand of hair. In that case, a is the width of the strand. From actual laboratory measurements on a human hair, it was found that when a beam of light of wavelength 632.8 nm was shone on a single strand of hair, and the diffracted light was viewed on a screen 1.25 m away, the first dark fringes on either side of the central bright spot were 5.22 cm apart. How thick was this strand of hair?

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