/*! 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 49 Light of wavelength \(600 \mathr... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 36: Problem 49

Light of wavelength \(600 \mathrm{~nm}\) is incident normally on a diffraction grating. Two adjacent maxima occur at angles given by \(\sin \theta=0.2\) and \(\sin \theta=0.3 .\) The fourth-order maxima are missing. (a) What is the separation between adjacent slits? (b) What is the smallest slit width this grating can have? For that slit width, what are the (c) largest, (d) second largest, and (e) third largest values of the order number \(m\) of the maxima produced by the grating?

Short Answer

Expert verified
(a) \(6 \times 10^{-6}\) m; (b) \(1.2 \times 10^{-5}\) m; (c) 10; (d) 9; (e) 8.

Step by step solution

01

Identify Given Information

You are given the wavelength of light, \(\lambda = 600\,\text{nm} = 600 \times 10^{-9} \text{m}\), and the two adjacent maxima appear at \(\sin \theta_1 = 0.2\) and \(\sin \theta_2 = 0.3\). The fourth-order maxima are missing.
02

Calculate Slit Separation

For diffraction gratings, the formula for maxima is \(d \sin \theta = m \lambda\). The difference in \(\sin \theta\) is due to the change in order number, \(\Delta m = 1\). Thus, \(d(0.3 - 0.2) = \lambda\), so \(d = \frac{600 \times 10^{-9}}{0.1}\) meters.
03

Express d

Compute \(d\) from the previous step: \(d = \frac{600 \times 10^{-9}}{0.1} = 6 \times 10^{-6} \) meters. This is the separation between the slits.
04

Determine Slit Width

For the fourth-order maxima to be missing, the condition for destructive interference \(a \sin \theta = n\lambda\) must be satisfied. We know \(a \sin \theta = 4 \lambda\). Substitute for \(\sin \theta = 0.2\): \(a(0.2) = 4 \times 600 \times 10^{-9}\), so \(a = \frac{2.4 \times 10^{-6}}{0.2} = 1.2 \times 10^{-5}\) meters.
05

Calculate Largest Order Number

The largest \(m\) occurs when \(\sin \theta = 1\). Thus, using \(d \sin \theta = m \lambda\), we find \(m = \frac{d}{\lambda} = \frac{6 \times 10^{-6}}{600 \times 10^{-9}} = 10\).
06

Calculate Second Largest Order Number

The second largest order number must leave one order out, that is 9 because the fourth order is already missing.
07

Calculate Third Largest Order Number

The third largest order that is not missing, after eliminating the fourth and tenth order, is 8.

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.

Understanding Wavelength in Diffraction
Wavelength refers to the distance between successive peaks of a wave. In the context of diffraction grating, it is the key parameter that influences the position and intensity of the interference pattern formed on a screen. When light passes through a diffraction grating, it is split into various colors, each corresponding to different wavelengths.
For our exercise, we work with a wavelength (\( \lambda \)) of 600 nm. Converting this to meters, it becomes 600 x 10^-9 meters. This value is crucial in determining other factors like slit separation and order number.
Understanding how wavelength affects light behavior in diffraction helps in predicting where the light maxima and minima will appear on the screen, which is important in applications like spectrometry.
Slit Separation in Diffraction Gratings
In a diffraction grating, slit separation (\( d \)) is the distance between two adjacent slits. This distance is a determining factor for the angle at which different wavelengths of light are diffracted. In our problem, using the diffraction formula \( d \sin \theta = m \lambda \), where \( m \) is the order number, we can determine the slit separation. By substituting the given values, including the difference in sin(胃) from 0.2 to 0.3, we calculate \( d \) to be \( 6 \times 10^{-6} \) meters.
This slit separation ensures that the setup can resolve different orders of maxima, allowing observation of distinct interference patterns.
Exploring Order Number in Diffraction
The order number (\( m \)) in diffraction refers to the number representing the position of a maximum (bright fringe) in the interference pattern. It indicates how many wavelengths fit into the path difference between two adjacent slits.
For our problem, the missing fourth-order maximum signifies destructive interference at this point. The largest order number for this setup, calculated when \( \sin \theta = 1 \), is found to be 10. However, the largest observable order is adjusted to m = 9, skipping the missing fourth order.
Understanding how different order numbers relate to path differences and positions of maxima can aid in applications such as measuring the spectral characteristics of light.
Destructive Interference in Slit Systems
Destructive interference occurs when waves combine to cancel each other out, resulting in dark spots in a diffraction pattern. This happens when the path difference between two waves is a half-integral number of wavelengths, contributing to the phenomenon of missing orders, such as the fourth order in our problem.
The condition for destructive interference in a grating occurs when \( a \sin \theta = n \lambda \), where \( a \) is the slit width and \( n \) is an integer. In our scenario, for \( \sin \theta = 0.2 \) and the given wavelength, the calculated slit width that results in missing the fourth order is \( a = 1.2 \times 10^{-5} \) meters.
This understanding can help in designing optical instruments where certain wavelengths need to be minimized or eliminated from the observation spectrum, aiding the clarity and precision of the output.

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

Monochromatic light with wavelength \(538 \mathrm{~nm}\) is incident on a slit with width \(0.025 \mathrm{~mm}\). The distance from the slit to a screen is \(3.5 \mathrm{~m}\). Consider a point on the screen \(1.1 \mathrm{~cm}\) from the central maximum. Calculate (a) \(\theta\) for that point, (b) \(\alpha\), and (c) the ratio of the intensity at that point to the intensity at the central maximum.

A source containing a mixture of hydrogen and deuterium atoms emits red light at two wavelengths whose mean is \(656.3 \mathrm{~nm}\) and whose separation is \(0.180 \mathrm{~nm} .\) Find the minimum number of lines needed in a diffraction grating that can resolve these lines in the first order.

Visible light is incident perpendicularly on a grating with 315 rulings/mm. What is the longest wavelength that can be seen in the fifth-order diffraction?

Nuclear-pumped x-ray lasers are seen as a possible weapon to destroy ICBM booster rockets at ranges up to \(2000 \mathrm{~km}\). One limitation on such a device is the spreading of the beam due to diffraction, with resulting dilution of beam intensity. Consider such a laser operating at a wavelength of \(1.40 \mathrm{~nm} .\) The element that emits light is the end of a wire with diameter \(0.200 \mathrm{~mm}\). (a) Calculate the diameter of the central beam at a target \(2000 \mathrm{~km}\) away from the beam source. (b) By what factor is the beam intensity reduced in transit to the target? (The laser is fired from space, so that atmospheric absorption can be ignored.)

The distance between the first and fifth minima of a single-slit diffraction pattern is \(0.35 \mathrm{~mm}\) with the screen \(40 \mathrm{~cm}\) away from the slit, when light of wavelength \(550 \mathrm{~nm}\) is used. (a) Find the slit width. (b) Calculate the angle \(\theta\) of the first diffraction minimum.
See all solutions

Recommended explanations on Physics Textbooks

Atoms and Radioactivity

Read Explanation

Rotational Dynamics

Read Explanation

Force

Read Explanation

Fluids

Read Explanation

Astrophysics

Read Explanation

Electromagnetism

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.