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Chapter 26: Problem 10

Coherent light with wavelength \(600 \mathrm{nm}\) passes through two very narrow slits, and the interference pattern is observed on a screen \(3.00 \mathrm{~m}\) from the slits. The first-order bright fringe is at \(4.84 \mathrm{~mm}\) from the center of the central bright fringe. For what wavelength of light will the first-order dark fringe be observed at this same point on the screen?

Short Answer

Expert verified
The wavelength for the dark fringe is 1200 nm.

Step by step solution

01

Understand the relationship for bright fringes

For bright fringes in a double-slit interference pattern, the path difference is given by \(d \sin \theta = m\lambda\), where \(d\) is the slit separation, \(\theta\) is the angle of deviation, \(m\) is the order of the fringe, and \(\lambda\) is the wavelength of the incident light. Since we are looking for the first-order bright fringe with \(m = 1\), we have: \(d \sin \theta = \lambda\).
02

Calculate angle 胃 for bright fringe

Using small angle approximation, \( \sin \theta \approx \tan \theta = \frac{y}{L} \), where \(y = 4.84 \text{ mm} = 4.84 \times 10^{-3} \text{ m}\), and \(L = 3\text{ m}\). Thus, \(\theta = \frac{4.84 \times 10^{-3}}{3}\).
03

Solve for the angle 胃

Calculate \(\theta\): \(\sin \theta \approx \frac{4.84 \times 10^{-3}}{3} = 1.6133 \times 10^{-3}\).
04

Determine condition for dark fringes

For dark fringes, the path difference is given by \(d \sin \theta = (m + \frac{1}{2}) \lambda\). For the first-order dark fringe \(m = 0\), therefore: \(d \sin \theta = \frac{1}{2} \lambda\).
05

Relate dark and bright fringe equations

From the bright fringe \(d \sin \theta = \lambda_{600}\) and dark fringe \(d \sin \theta = \frac{1}{2} \lambda_{\text{dark}}\). If the first-order bright fringe corresponds with the first-order dark fringe's position, then \(\lambda_{\text{dark}} = 2 \lambda_{\text{600}}\).
06

Substitute the given wavelength to find the dark fringe wavelength

Given \(\lambda = 600 \text{ nm}\), the wavelength for which the dark fringe is at the same position is \(2 \times 600 = 1200 \text{ nm}\), since it is twice the first one.

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Key Concepts

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

Coherent Light
Coherent light refers to light waves that are in phase and have a constant phase difference. This is crucial in experiments like the double-slit experiment because coherent waves interfere predictably, leading to visible interference patterns on a screen. The coherence of light ensures that the waves are stable over a considerable distance, allowing them to superpose effectively. With coherent light, beams of equal wavelengths and consistent phase differences meet, forming patterns of bright and dark fringes.

In practical terms, coherent light is commonly produced by lasers, which emit beams where all the light waves have the same frequency, making them perfectly in phase with each other. This differs from incoherent light, such as that from a light bulb, where the phase of the waves is random and changes rapidly, resulting in no clear interference patterns.
Double-slit Experiment
The double-slit experiment is a foundational physics phenomenon that demonstrates the wave nature of light. When coherent light passes through two closely spaced narrow slits, it divides into two separate waves that overlap on the other side. As these waves interact, they create an interference pattern of alternating bright (constructive interference) and dark (destructive interference) fringes on a screen.

This experiment illustrates several important concepts:
  • Wave Interference: Where the peaks of two waves align (constructive interference), they reinforce each other, creating bright fringes.
  • Path Difference: Since the light waves travel different distances from each slit to a point on the screen, the path difference leads to different types of interference.
This visual demonstration supports the wave theory of light, suggesting that light behaves as a wave under certain conditions.
Wavelength Calculation
Wavelength calculation in the context of interference patterns deals with determining the specific wavelengths of light that result in particular patterns. In any double-slit setup, the wavelength is vital for predicting and analyzing the spacing of interference fringes.

Formulas derived from experimental setups help in understanding these patterns:
  • The formula for bright fringes is given by: \(d \sin \theta = m\lambda\), where \(d\) is the distance between the slits, \(\theta\) is the angle of the fringe relative to the central axis, \(m\) is the order of the fringe, and \(\lambda\) is the wavelength of light.
  • Knowing the geometric parameters and fringe order allows for calculating the wavelengths of light responsible for different patterns. This process facilitates determining which wavelengths will result in a desired pattern, such as specific dark or bright fringes observed in an experiment.
Path Difference
Path difference in the double-slit experiment is the difference in distance traveled by the two coherent light waves from the slits to a given point on the screen. This difference is crucial in determining whether the ensuing interference is constructive (bright fringes) or destructive (dark fringes).

To delve deeper, let's explore the conditions:
  • Constructive Interference: Occurs when the path difference is an integral multiple of the wavelength, maxing out at bright fringes.
  • Destructive Interference: Happens when the path difference is a half-integral multiple of the wavelength, resulting in dark fringes.
Understanding path difference is fundamental in predicting and analyzing the locations of bright and dark fringes, which helps in precisely calculating light's wavelength needed for a particular interference pattern to occur.

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Most popular questions from this chapter

Coherent light that contains two wavelengths, \(660 \mathrm{nm}\) (red) and \(470 \mathrm{nm}\) (blue), passes through two narrow slits separated by \(0.300 \mathrm{~mm},\) and the interference pattern is observed on a screen \(5.00 \mathrm{~m}\) from the slits. What is the distance on the screen between the first-order bright fringes for the two wavelengths?

Two thin parallel slits that are \(0.0116 \mathrm{~mm}\) apart are illuminated by a laser beam of wavelength \(585 \mathrm{nm}\). (a) How many bright fringes are there in the angular range of \(0<\theta<20^{\circ} ?\) (b) How many dark fringes are there in this range?

You have just put some medical eyedrops on your sensitive eyes. The cornea (the front part of the eye) has an index of refraction of \(1.38,\) while the eyedrops have a refractive index of \(1.45 .\) After you put in the drops, your friends notice that your eyes look red, because red light of wavelength \(600 \mathrm{nm}\) has been reinforced in the reflected light. (a) What is the minimum thickness of the film of eyedrops on your cornea? (b) Will any other wavelengths of visible light be reinforced in the reflected light? Will any be canceled? (c) Suppose you had contact lenses, so that the eyedrops went on them instead of on your corneas. If the refractive index of the lens material is 1.50 and the layer of eyedrops has the same thickness as in part (a), what wavelengths of visible light will be reinforced? What wavelengths will be canceled?

Parallel light rays with a wavelength of \(600 \mathrm{nm}\) fall on a single slit. On a screen \(3.00 \mathrm{~m}\) away, the distance between the first dark fringes on either side of the central maximum is \(4.50 \mathrm{~mm}\). What is the width of the slit?

A diffraction grating has 5580 lines \(/ \mathrm{cm}\). When a beam of monochromatic light goes through it, the second pair of bright spots occurs at \(\pm 26.3 \mathrm{~cm}\) from the central spot on a screen \(42.5 \mathrm{~cm}\) past the grating. (a) What is the wavelength of this light? (b) How far from the central spot does the next pair of bright spots occur on the screen?
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