/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 14 Coherent light that contains two... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 35: Problem 14

Coherent light that contains two wavelengths, \(660 \mathrm{nm}\) (red) and \(470 \mathrm{nm}\) (blue), passes through two narrow slits that are separated by \(0.300 \mathrm{~mm}\). Their interference pattern is observed on a screen \(4.00 \mathrm{~m}\) from the slits. What is the distance on the screen between the first-order bright fringes for the two wavelengths?

Short Answer

Expert verified
The fringe separation between red and blue first-order bright fringes is obtained by the difference of \( y_{red} \) and \( y_{blue} \).

Step by step solution

01

Calculation of Fringe Position for Red Light

Insert the given values into the formula for the red light, \(y_{red} = Lm\lambda_{red}/d \), with \(L = 4 m, m = 1 \) (first order), \( \lambda_{red} = 660 nm = 660 x 10^{-9} m \), and \( d = 0.300 mm = 0.300 x 10^{-3} m \). Solve for \( y_{red} \).
02

Calculation of Fringe Position for Blue Light

Do the same calculation for the blue light, \(y_{blue} = Lm\lambda_{blue}/d\), with \(L = 4 m, m = 1 \) (first order), \( \lambda_{blue} = 470 nm = 470 x 10^{-9} m \), and \( d = 0.300 mm = 0.300 x 10^{-3} m \). Solve for \( y_{blue} \).
03

Calculation of Fringe Separation Between the Wavelengths

To find the distance between the first-order bright fringes for the two wavelengths, subtract \( y_{blue} \) from \( y_{red} \) which gives us the distance between the fringes: \( Separation = y_{red} - y_{blue} \).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Coherent Light
Coherent light is a fundamental concept in the studying of wave interference patterns. It occurs when light waves maintain a constant phase relationship. This special characteristic is essential when it comes to producing clear and stable interference patterns.
Understanding coherent light helps us figure out how waves interact and combine:
  • Light waves from coherent sources have the same frequency and phase difference which remains constant over time.
  • This steady phase relationship is crucial for the interference pattern to remain stable.
  • Coherent light is what allows us to observe distinct fringe patterns on the screen.
Lasers are a common source of coherent light because they generate light waves that are synchronized, much like a group of synchronized swimmers performing in harmony.
Wavelength
Wavelength is a pivotal concept when it comes to understanding light interference. It determines how light behaves and how interference patterns emerge on the screen. In the given problem, two different wavelengths are at play: 660 nm (red light) and 470 nm (blue light).
Here's how wavelength plays a role:
  • Wavelength is the distance between successive crests of a light wave.
  • It influences the position and spacing of bright and dark fringes on the screen.
  • Different wavelengths will produce fringes at different positions. That's why the red and blue light fringes do not overlap completely.
In the interference of two different wavelengths, the fringe distance reflects light鈥檚 color properties, making the concept of wavelength indispensable for understanding and calculating the details of interference patterns.
Fringe Separation
Fringe separation is the distance between corresponding positions of fringes (like bright fringes) in an interference pattern. It results from different wavelengths of coherent light passing through the same apparatus.
Let's dive into how fringe separation is calculated:
  • The formula for fringe position is given by: \( y = \frac{Lm\lambda}{d} \).
  • Here, \( L \) is the distance from the slits to the screen, \( m \) indicates the order of the fringe, \( \lambda \) is the wavelength, and \( d \) is the slit separation.
  • After calculating the fringe positions for both red and blue light with their respective wavelengths, the difference between these fringe positions gives us the fringe separation.
Understanding fringe separation helps us appreciate how different colors of light can exhibit unique behaviors when undergoing interference, a fascinating phenomenon observed in experiments like this one.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

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?

Two light sources can be adjusted to emit monochromatic light of any visible wavelength. The two sources are coherent, \(2.04 \mu \mathrm{m}\) apart, and in line with an observer, so that one source is \(2.04 \mu \mathrm{m}\) farther from the observer than the other. (a) For what visible wavelengths \((380\) to \(750 \mathrm{nm})\) will the observer see the brightest light, owing to constructive interference? (b) How would your answers to part (a) be affected if the two sources were not in line with the observer, but were still arranged so that one source is \(2.04 \mu \mathrm{m}\) farther away from the observer than the other? (c) For what visible wavelengths will there be destructive interference at the location of the observer?

A researcher measures the thickness of a layer of benzene \((n=1.50)\) floating on water by shining monochromatic light onto the film and varying the wavelength of the light. She finds that light of wavelength \(575 \mathrm{nm}\) is reflected most strongly from the film. What does she calculate for the minimum thickness of the film?

The index of refraction of a glass rod is 1.48 at \(T=20.0^{\circ} \mathrm{C}\) and varies linearly with temperature, with a coefficient of \(2.50 \times 10^{-5} / \mathrm{C}^{\circ} .\) The coefficient of linear expansion of the glass is \(5.00 \times 10^{-6} / \mathrm{C}^{\circ} .\) At \(20.0^{\circ} \mathrm{C}\) the length of the rod is \(3.00 \mathrm{~cm}\) A Michelson interferometer has this glass rod in one arm, and the rod is being heated so that its temperature increases at a rate of \(5.00 \mathrm{C}^{\circ} / \mathrm{min} .\) The light source has wavelength \(\lambda=589 \mathrm{nm},\) and the rod initially is at \(T=20.0^{\circ} \mathrm{C}\). How many fringes cross the field of view each minute?

Monochromatic light with wavelength \(\lambda\) is incident on a screen with three narrow slits with separation \(d\), as shown in Fig. P35.54. Light from the middle slit reaches point \(P\) with electric field \(E \cos (\omega t) .\) From the small-angle approximation, light from the upper and lower slits reaches point \(P\) with electric fields \(E \cos (\omega t+\phi)\) and \(E \cos (\omega t-\phi),\) respectively, where \(\phi=(2 \pi d \sin \theta) / \lambda\) is the phase lag and phase lead associated with the different path lengths. (a) Using either a phasor analysis similar to Fig. 35.9 or trigonometric identities, determine the electric-field amplitude \(E_{P}\) associated with the net field at point \(P,\) in terms of \(\phi\). (b) Determine the intensity \(I\) at point \(P\) in terms of the maximum net intensity \(I_{0}\) and the phase angle \(\phi\). (c) There are two sets of relative maxima: one with intensity \(I_{0}\) when \(\phi=2 \pi m,\) where \(m\) is an integer, and another with a smaller intensity at other values of \(\phi .\) What values of \(\phi\) exhibit the "lesser" maxima? (d) What is the intensity at the lesser maxima, in terms of \(I_{0} ?\) (e) What values of \(\phi\) correspond to the dark fringes closest to the center? (f) If the incident light has wavelength \(650 \mathrm{nm},\) the slits are separated by \(d=0.200 \mathrm{~mm},\) and the distance to the far screen is \(R=1.00 \mathrm{~m},\) what is the distance from the central maximum to the first lesser maximum? (g) What is the distance from the central maximum to the closest absolute maximum?
See all solutions

Recommended explanations on Physics Textbooks

Dynamics

Read Explanation

Turning Points in Physics

Read Explanation

Medical Physics

Read Explanation

Torque and Rotational Motion

Read Explanation

Magnetism and Electromagnetic Induction

Read Explanation

Rotational Dynamics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.