/*! 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 97 A spy satellite orbiting at \(16... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 36: Problem 97

A spy satellite orbiting at \(160 \mathrm{~km}\) above Earth's surface has a lens with a focal length of \(3.6 \mathrm{~m}\) and can resolve objects on the ground as small as \(30 \mathrm{~cm}\). For example, it can easily measure the size of an aircraft's air intake port. What is the effective diameter of the lens as determined by diffraction consideration alone? Assume \(\lambda=550 \mathrm{~nm}\).

Short Answer

Expert verified
The effective diameter of the lens is approximately 14.3 cm.

Step by step solution

01

Determine the Necessary Formula

To calculate the effective diameter of the lens based on diffraction, we use the formula for the diffraction limit, which is given by the Rayleigh criterion: \( \theta = 1.22 \frac{\lambda}{D} \). Here, \(\theta\) is the angular resolution, \(\lambda\) is the wavelength, and \(D\) is the diameter of the lens.
02

Find the Angular Resolution

First, calculate the angular resolution \( \theta \) using the given information about the satellite's ability to resolve objects on the ground. The formula \( \theta = \frac{d}{h} \) can be used, where \(d = 0.30 \mathrm{~m}\) (the smallest resolvable object size) and \(h = 160000 \mathrm{~m} + R_E\) is the operational height above Earth's surface, including Earth's radius \(R_E = 6371000 \mathrm{~m}\). So, \( \theta = \frac{0.30}{160000 + 6371000} \approx 4.688 \times 10^{-8} \text{ radians}. \)
03

Substitute Values into the Formula

With \( \theta \) calculated, substitute \( \theta = 4.688 \times 10^{-8} \mathrm{~radians}\), \(\lambda = 550 \times 10^{-9} \mathrm{~m}\), and rearrange the Rayleigh criterion formula to solve for \(D\): \( D = \frac{1.22 \lambda}{\theta} \). This gives \( D = \frac{1.22 \times 550 \times 10^{-9}}{4.688 \times 10^{-8}} \mathrm{~m}. \)
04

Calculate Diameter

Perform the calculations to find \(D\). Using the rearranged formula we get: \( D \approx \frac{670 \times 10^{-9}}{4.688 \times 10^{-8}} \approx 0.0143 \mathrm{~m} \) or \(14.3 \mathrm{~cm}\).

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.

Rayleigh criterion
The Rayleigh criterion is a fundamental principle in optics that helps determine the limit at which two points of light can be distinguished as separate.
Simply put, it's a measure of an optical system's ability to resolve detail.
The criterion is expressed through the formula:
  • \( \theta = 1.22 \frac{\lambda}{D} \)
Where \( \theta \) is the angular resolution, \( \lambda \) is the wavelength of light, and \( D \) is the diameter of the lens or the aperture.
In the context of a satellite, using a wavelength of 550 nm, the Rayleigh criterion allows us to calculate the smallest angle at which the satellite can differentiate between two close objects on Earth's surface.
This formula takes into account the effects of diffraction, which is the bending of light around the edges of the lens.
Understanding the Rayleigh criterion is crucial for designing optical systems where clear and precise imaging is necessary, such as photography, telescopes, and satellite cameras.
Angular resolution
Angular resolution is another key concept in understanding how optical systems operate.
It describes the smallest angle between two objects that allows them to be individually distinguished.
Angular resolution depends on several factors:
  • The wavelength of the light being used
  • The size of the lens or aperture
  • The distance from the lens to the objects being observed
In our satellite example, the angular resolution is significant because it defines the level of detail the satellite's camera can capture from its orbit 160 kilometers above Earth.
Mathematically, it can be expressed as \( \theta = \frac{d}{h} \), where \( d \) is the size of the smallest resolvable object, and \( h \) is the height of the satellite.
This measurement is critical for tasks like Earth observation and reconnaissance, as better resolution means more detailed and useful images.
Satellite optics
Satellite optics refers to the sophisticated optical systems and designs used in satellites to capture images and gather data from space or Earth's surface.
These systems are tailored to provide high-resolution images, often utilizing lenses of large diameters to maximize resolution, according to the principles of diffraction. In the context of spy satellites, like the one described in our exercise, the optics must be incredibly precise to allow for detailed surveillance applications.
This involves calculating the effective diameter of the lens that would provide the needed clarity to resolve objects as small as 30 cm from high altitudes.
The main goals of satellite optics include:
  • Capturing high-resolution, clear images
  • Minimizing the effects of atmospheric distortion
  • Maximizing the amount of light captured
Effective satellite optics combine advanced materials, engineering, and cutting-edge optical science to achieve the stringent requirements necessary for modern satellite missions.

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

A diffraction grating has 200 lines \(/ \mathrm{mm}\). Light consisting of a continuous range of wavelengths between \(550 \mathrm{~nm}\) and \(700 \mathrm{~nm}\) is incident perpendicularly on the grating. (a) What is the lowest order that is overlapped by another order? (b) What is the highest order for which the complete spectrum is present?

A diffraction grating is made up of slits of width \(300 \mathrm{~nm}\) with separation \(900 \mathrm{~nm}\). The grating is illuminated by monochromatic plane waves of wavelength \(\lambda=600 \mathrm{~nm}\) at normal incidence. (a) How many maxima are there in the full diffraction pattern? (b) What is the angular width of a spectral line observed in the first order if the grating has 1000 slits?

X rays of wavelength \(0.12 \mathrm{~nm}\) are found to undergo secondorder reflection at a Bragg angle of \(28^{\circ}\) from a lithium fluoride crystal. What is the interplanar spacing of the reflecting planes in the crystal?

A plane wave of wavelength \(590 \mathrm{~nm}\) is incident on a slit with a width of \(a=0.40 \mathrm{~mm}\). A thin converging lens of focal length \(+70\) \(\mathrm{cm}\) is placed between the slit and a viewing screen and focuses the light on the screen. (a) How far is the screen from the lens? (b) What is the distance on the screen from the center of the diffraction pattern to the first minimum?

Assume that Rayleigh's criterion gives the limit of resolution of an astronaut's eye looking down on Earth's surface from a typical space shuttle altitude of \(400 \mathrm{~km}\). (a) Under that idealized assumption, estimate the smallest linear width on Earth's surface that the astronaut can resolve. Take the astronaut's pupil diameter to be \(5 \mathrm{~mm}\) and the wavelength of visible light to be \(550 \mathrm{~nm}\). (b) Can the astronaut resolve the Great Wall of China (Fig. 36-40), which is more than \(3000 \mathrm{~km}\) long, 5 to \(10 \mathrm{~m}\) thick at its base, \(4 \mathrm{~m}\) thick at its top, and \(8 \mathrm{~m}\) in height? (c) Would the astronaut be able to resolve any unmistakable sign of intelligent life on Earth's surface?
See all solutions

Recommended explanations on Physics Textbooks

Physical Quantities And Units

Read Explanation

Measurements

Read Explanation

Kinematics Physics

Read Explanation

Radiation

Read Explanation

Particle Model of Matter

Read Explanation

Astrophysics

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.