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Chapter 35: Problem 20

Two slits spaced \(0.0720 \mathrm{~mm}\) apart are \(0.800 \mathrm{~m}\) from a screen. Coherent light of wavelength \(\lambda\) passes through the two slits. In their interference pattern on the screen, the distance from the center of the central maximum to the first minimum is \(3.00 \mathrm{~mm}\). If the intensity at the peak of the central maximum is \(0.0600 \mathrm{~W} / \mathrm{m}^{2},\) what is the intensity at points on the screen that are (a) \(2.00 \mathrm{~mm}\) and (b) \(1.50 \mathrm{~mm}\) from the center of the central maximum?

Short Answer

Expert verified
The intensity of light at points 2.00 mm and 1.50 mm away from the center of the central maximum can be found by substituting the given values into the formula for intensity in a double-slit experiment. Remember to use the calculated wavelength from step 1. The actual values depend on the specifics of the problem and will require computation to determine.

Step by step solution

01

Calculate the Wavelength of the Light

From the distance of the first minimum, we know that \(\sin (\theta)=\frac{\lambda}{d}\), where \(\theta\) is the angle to the first minimum. Therefore, we can re-arrange this to find \( \lambda = d \sin (\theta)\) which gives: \(\lambda = d \sin (arctan(\frac{y_{min}}{D}))\)
02

Calculate Intensity at 2.00 mm

Next, substitute the given values into the formula for Intensity. Here, \(y = 2.00 mm\)\(I=I_{\mathrm{max}} \cos ^{2}(\frac{\pi d y}{\lambda D})\)
03

Calculate Intensity at 1.50 mm

Similarly, substitute the given values into the formula for Intensity. Here, \(y = 1.50 mm\)\(I=I_{\mathrm{max}} \cos ^{2}(\frac{\pi d y}{\lambda D})\)

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Key Concepts

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

Wave Interference
Interference is a phenomenon where two or more waves superpose to form a combined wave with a new waveform. When waves meet, they can interfere constructively or destructively, depending on their phase relationship.

In the case of constructive interference, the waves are in phase, which means their peaks align with each other, resulting in a higher amplitude of the combined wave. Conversely, if the waves are out of phase (the peak of one wave aligns with the trough of another), they cancel each other out partially or entirely; this is known as destructive interference.

The principles of wave interference are pivotal in understanding various phenomena in optical physics, such as the patterns observed in the double-slit experiment. These patterns are visible evidence of interference between light waves, providing valuable insight into the nature and behavior of light.
  • Constructive interference leads to bright areas on a screen due to higher amplitude.
  • Destructive interference causes dark areas due to wave cancellation.
  • The overall interference pattern can be predicted using principles of wave superposition.
Double-slit Experiment
The double-slit experiment, famously performed by Thomas Young, is a demonstration of the wave-like nature of light and provides solid evidence for the concept of wave interference. When coherent light passes through two closely spaced slits, the light waves originating from each slit overlap and interfere with each other.

The result is a series of bright and dark fringes on a screen behind the slits, known as an interference pattern. The bright fringes occur where the waves from the slits constructively interfere, while the dark fringes represent areas of destructive interference.
  • Coherent light is essential to maintaining consistent phase relationships between the waves.
  • The fringes can be used to calculate features of the light, such as wavelength.
  • The experiment illustrates the principles of optical physics and quantum mechanics.
This experiment is fundamental for understanding the dual nature of light, behaving both as a wave and as a particle, which is a core idea in quantum mechanics.
Optical Physics
Optical physics is a branch of physics that deals with the study of light and its interactions with matter. This field encompasses both classical and quantum properties of light, ways that light can be produced, manipulated and detected.

Key concepts in optical physics include refraction, reflection, diffraction, and interference of light. Understanding these concepts allows physicists to explain optical phenomena and develop applications ranging from optical communication systems to medical imaging.
  • Refraction is the bending of light when it passes through different media.
  • Reflection is the bouncing back of light from a surface.
  • Diffraction is the spreading of light waves around obstacles.
  • Interference is the combination of two or more light waves, leading to variations in intensity.
Interference, specifically, is the mechanism behind the patterns observed in the double-slit experiment and numerous other optical devices and technologies.
Light Intensity Distribution
Light intensity distribution refers to the way light's power, or brightness, is distributed across a particular space or over a surface area. In optical physics, this is especially important in the context of interference patterns, where the intensity varies along the screen.

The intensity is typically highest at the center of the pattern, known as the central maximum, and diminishes towards the edges due to the nature of wave interference. As per the double-slit experiment, the light intensity can be mathematically described by certain formulas, taking into account factors such as the slit separation, distance to the screen, and the wavelength of the light.
  • Intensity can be calculated using the formula: \( I = I_{\text{max}} \cos^2(\frac{\pi d y}{\lambda D}) \).
  • The variables in the formula help predict the intensity at various points on the screen accurately.
  • These calculations allow scientists and engineers to design optical systems with desired light intensity distributions for various applications.
Understanding light intensity distribution is crucial for various fields, including photography, lighting design, and laser technology.

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Most popular questions from this chapter

Figure \(\mathbf{P} 35.56\) shows an interferometer known as Fresnel's biprism. The magnitude of the prism angle \(A\) is extremely small. (a) If \(S_{0}\) is a very narrow source slit, show that the separation of the two virtual coherent sources \(S_{1}\) and \(S_{2}\) is given by \(d=2 a A(n-1)\), where \(n\) is the index of refraction of the material of the prism. (b) Calculate the spacing of the fringes of green light with wavelength \(500 \mathrm{nm}\) on a screen \(2.00 \mathrm{~m}\) from the biprism. Take \(a=0.200 \mathrm{~m}\) \(A=3.50 \mathrm{mrad},\) and \(n=1.50\)

Coherent light with wavelength \(450 \mathrm{nm}\) falls on a pair of slits. On a screen \(1.80 \mathrm{~m}\) away, the distance between dark fringes is \(3.90 \mathrm{~mm} .\) What is the slit separation?

Red light with wavelength \(700 \mathrm{nm}\) is passed through a two-slit apparatus. At the same time, monochromatic visible light with another wavelength passes through the same apparatus. As a result, most of the pattern that appears on the screen is a mixture of two colors; however, the center of the third bright fringe \((m=3)\) of the red light appears pure red, with none of the other color. What are the possible wavelengths of the second type of visible light? Do you need to know the slit spacing to answer this question? Why or why not?

Two slits spaced \(0.450 \mathrm{~mm}\) apart are placed \(75.0 \mathrm{~cm}\) from a screen. What is the distance between the second and third dark lines of the interference pattern on the screen when the slits are illuminated with coherent light with a wavelength of \(500 \mathrm{nm} ?\)

After a laser beam passes through two thin parallel slits, the first completely dark fringes occur at \(\pm 19.0^{\circ}\) with the original direction of the beam, as viewed on a screen far from the slits. (a) What is the ratio of the distance between the slits to the wavelength of the light illuminating the slits? (b) What is the smallest angle, relative to the original direction of the laser beam, at which the intensity of the light is \(\frac{1}{10}\) the maximum intensity on the screen?
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