/*! 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 44 In Young's double slit experimen... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 23: Problem 44

In Young's double slit experiment, the 7 th maximum with wavelength \(\lambda_{1}\) is at a distance \(d_{1}\) and that with wavelength \(\lambda_{2}\) is at a distance \(d_{2}\). Then \(\left(d_{1} / d_{2}\right)\) is (a) \(\left(\lambda_{1} / \lambda_{2}\right)\) (b) \(\left(\lambda_{2} / \lambda_{1}\right)\) (c) \(\left(\lambda_{1}^{2} / \lambda_{2}^{2}\right)\) (d) \(\left(\lambda_{2}^{2} / \lambda_{1}^{2}\right)\)

Short Answer

Expert verified
The ratio \(\frac{d_1}{d_2}\) is \(\frac{\lambda_1}{\lambda_2}\). (Option a)

Step by step solution

01

Understanding the Problem

In Young's double slit experiment, we have two different wavelengths, \(\lambda_1\) and \(\lambda_2\). We know that the 7th maximum (bright fringe) for each wavelength occurs at distances \(d_1\) and \(d_2\) from the central maximum. We need to find the ratio \(\frac{d_1}{d_2}\).
02

Formula for Fringe Position

The position of the m-th maximum (bright fringe) on the screen in the double slit experiment is given by the formula \(d = \frac{m \cdot \lambda \cdot L}{a}\), where \(m\) is the order of the maximum, \(\lambda\) is the wavelength, \(L\) is the distance between the slits and the screen, and \(a\) is the distance between the slits.
03

Apply the Formula for Two Wavelengths

For wavelength \(\lambda_1\), the position of the 7th maximum is given by \(d_1 = \frac{7 \cdot \lambda_1 \cdot L}{a}\). Similarly, for wavelength \(\lambda_2\), \(d_2 = \frac{7 \cdot \lambda_2 \cdot L}{a}\).
04

Derive the Ratio \(\frac{d_1}{d_2}\)

To find the ratio \(\frac{d_1}{d_2}\), we take the expressions from the previous step: \(\frac{d_1}{d_2} = \frac{\frac{7 \cdot \lambda_1 \cdot L}{a}}{\frac{7 \cdot \lambda_2 \cdot L}{a}}\). Simplifying this gives \(\frac{\lambda_1}{\lambda_2}\).
05

Conclusion

The ratio \(\frac{d_1}{d_2}\) simplifies to \(\frac{\lambda_1}{\lambda_2}\). Therefore, the correct answer is option (a).

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.

The Role of Wavelength in Young's Double Slit Experiment
Wavelength is a crucial aspect of Young's Double Slit Experiment as it directly influences where the bright and dark fringes appear on the screen. In simple terms, the wavelength of light is the distance between successive crests of a wave. It determines the color and energy of light. In the context of the experiment, the wavelength affects the path difference as light waves from the two slits meet on the screen, leading to constructive or destructive interference.

More specifically:
  • Constructive interference occurs when waves meet in phase. This results in a bright fringe or maximum, and it happens when the path difference is an integral multiple of the wavelength (\( m \cdot \lambda \)).
  • Destructive interference occurs when waves meet out of phase, resulting in a dark fringe or minimum. This happens when the path difference is a half-integral multiple of the wavelength.
By affecting how far apart these points of interference occur, the wavelength dictates both the color pattern and spacing of fringe maxima and minima.
Understanding Fringe Position
Fringe position refers to the location of bright and dark bands that appear on the screen during Young's Double Slit Experiment. These bands or fringes are the result of light waves from the slits meeting on the screen and interfering with one another.

The position of these fringes is determined by the formula: \[ d = \frac{m \cdot \lambda \cdot L}{a} \]Where:
  • \(d\) is the fringe position measured from the central maximum or center of the pattern.
  • \(m\) is the order number of the maximum (e.g., 1, 2, 3 ...).
  • \(\lambda\) represents the wavelength of the light used in the experiment.
  • \(L\) is the distance from the double slits to the screen.
  • \(a\) is the distance between the two slits.
The formula helps in calculating the exact point on the screen where these bright fringes will occur. This knowledge allows us to predict how the position shifts when various elements like wavelength or slit spacing change.
An Overview of Optics in the Experiment
Optics, the branch of physics that studies light and its properties, plays an important role in understanding Young's Double Slit Experiment. This experiment is a classic demonstration of one of the fundamental principles of optics: the wave nature of light.

Exploring optics in this context:
  • The wave behavior of light is confirmed by how it spreads out or diffracts as it passes through the slits.
  • When light waves overlap or superpose, they either amplify or cancel each other, leading to visible interference patterns.
  • The phenomena seen during the experiment highlight light鈥檚 dual nature, exhibiting both wave-like and particle-like properties.
By examining how light passes through the slits and forms patterns on the screen, more profound insights into the fundamental nature of light and its behavior can be gained, reinforcing the wave theory of light.
Interference Patterns: The Backbone of the Experiment
Interference patterns are at the heart of the Young's Double Slit Experiment. They are the series of alternating bright and dark bands visible on the screen as a direct result of light wave interference. The patterns reveal critical information about light's nature and the conditions affecting it.

To understand these patterns:
  • They result from constructive interference, which creates bright fringes, and destructive interference, creating dark fringes.
  • The regularity and spacing of the patterns provide insights into wavelengths and slit spacing, revealing information like color and energy levels of light.
  • Similar setups can be used to explore various light properties and test theories within physics.
These patterns do not just illustrate wave interference; they solidify foundational concepts of wave behavior and are a powerful investigative tool in optics and physics.

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

The concept of secondary wavelets from all points on a wavefront was first proposed by (a) Newton (b) Huygen (c) Faraday (d) Raman

The wave front of a distant source of unknown shape is approximately (a) spherical (b) cylindrical (c) elliptical (d) plane

Microwaves from a transmitter are directed normally towards a plane reflector. A detector moves along the normal to the reflector. Between positions of 14 successive maxima the detector travels a distance of \(0.14 \mathrm{~m}\). The frequency of transmitter is \(\left(c=3 \times 10^{8} \mathrm{~m} / \mathrm{s}\right)\) (a) \(1.5 \times 10^{10} \mathrm{~Hz}\) (b) \(10^{10} \mathrm{~Hz}\) (c) \(3 \times 10^{10} \mathrm{~Hz}\) (d) \(6 \times 10^{10} \mathrm{~Hz}\)

In Young's double slit experiment the separation between the slits is halved and the distance between the slits and screen is doubled. The fringe width is (a) unchanged (b) halved (c) doubled (d) quadrupled

The angular fringe width of fringes formed with sodium light of wavelength \(5890 \AA\) is \(0.20^{\circ}\). If the whole arrangement is immersed in water, then the angular width of the fringes will become (a) \(0.11^{\circ}\) (b) \(0.15^{\circ}\) (c) \(0.22^{\circ}\) (d) \(0.30^{\circ}\)
See all solutions

Recommended explanations on Physics Textbooks

Physical Quantities And Units

Read Explanation

Famous Physicists

Read Explanation

Atoms and Radioactivity

Read Explanation

Modern Physics

Read Explanation

Turning Points in Physics

Read Explanation

Solid State Physics

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.