/*! 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 41 A convex lens of diameter \(8 \c... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 17: Problem 41

A convex lens of diameter \(8 \cdot 0 \mathrm{~cm}\) is used to focus a parallel beam of light of wavelength \(620 \mathrm{~nm}\). If the light be focused at a distance of \(20 \mathrm{~cm}\) from the lens, what would be the diameter of the central bright spot formed?

Short Answer

Expert verified
The diameter of the central bright spot is approximately 1.89 mm.

Step by step solution

01

Understanding the Problem

We have a convex lens focusing parallel light beams of wavelength \(620 \text{ nm}\) onto a point \(20 \text{ cm}\) away. We need to find out the diameter of the central bright spot formed, which is the first diffraction minimum of the Airy disk pattern.
02

Applying the Diffraction Formula

The Airy disk pattern can be described by the formula for the angular radius of the first minimum: \( \theta = 1.22 \frac{\lambda}{D} \), where \(\lambda\) is the wavelength of light, and \(D\) is the diameter of the lens. Here, \(\lambda = 620 \text{ nm} = 620 \times 10^{-9} \text{ m}\) and \(D = 8 \text{ cm} = 0.08 \text{ m}\). Substituting these values gives:
03

Calculating the Angular Radius

Convert the diameter of the lens into meters: \(8 \text{ cm} = 0.08 \text{ m}\). Then, calculate \( \theta \) using the formula:\[ \theta = 1.22 \times \frac{620 \times 10^{-9}}{0.08} \].Perform the calculation to find \( \theta \).
04

Finding the Diameter of the Central Bright Spot

The actual diameter of the central bright spot (Airy disk) \(d\) at the focal plane can be found by: \( d = 2 \times f \times \theta \), where \(f = 20 \text{ cm} = 0.2 \text{ m}\).After calculating \(\theta\), substitute it back into this equation to find \(d\).
05

Performing Final Calculation and Conclusion

Now, substitute \(\theta\) into the formula for \(d\):\[ d = 2 \times 0.2 \times \theta \].Carrying out this multiplication gives the diameter of the central bright spot in meters.Converting to centimeters if necessary, we get the final answer.

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.

Convex Lens
A convex lens is a transparent optical device that bends light rays to converge or focus at a point. It has a thicker center and thinner edges, creating this focusing ability. When parallel rays of light pass through a convex lens, they refract towards its focal point. This makes them crucial in devices like cameras, telescopes, and glasses.

Some key attributes of convex lenses include:
  • Focal Length: The distance from the lens to the point where light rays meet.
  • Aperture: The diameter of the lens, which affects the amount of light it can gather.
  • Curvature: Determines lens power by altering how much it bends light.
Understanding these features helps in predicting how a lens will affect light passing through it.
Airy Disk
The Airy disk is a pattern of light found when light diffracts through an aperture like a lens. This diffraction creates circular patterns with bright and dark regions, termed rings. At the center of this pattern is the Airy disk鈥攁n intensely bright spot.

Why does the Airy disk form?
  • It鈥檚 due to the wave nature of light, which bends around edges.
  • Light waves interfer with each other to form distinctive images.
  • This effect becomes pronounced with smaller apertures or long wavelengths.
The Airy disk鈥檚 size reveals important details about the lens and light, such as resolution limits.
Central Bright Spot
The central bright spot is the brightest part of the Airy disk pattern. It emerges due to constructive interference of light waves near the center of the pattern creation.

Characteristics of the central bright spot:
  • Has the highest intensity in the diffraction pattern.
  • Its size increases with the light's wavelength or decreasing aperture.
  • Its diameter can be calculated to determine focusing effectiveness.
Calculating this spot鈥檚 diameter is vital in optical setups, especially where image clarity and resolution are essential. It's directly related to how well a system can function, whether in microscopes or cameras.
Wavelength of Light
The wavelength of light is the distance between consecutive crests of a light wave. It is crucial in determining light properties and is usually measured in nanometers (nm) for visible light.

Influences of wavelength on diffraction patterns:
  • Longer wavelengths produce larger central bright spots.
  • Wavelength affects the diffraction pattern style and interference.
  • Different colors (wavelengths) will spread differently in lenses.
Knowing the light鈥檚 wavelength is vital when working with lenses. It allows accurate calculation of the Airy disk and the performance of optical systems. In this exercise, wavelength (\(620\,\mathrm{ nm}\)) helped determine the central bright spot size.

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

Find the range of frequency of light that is visible to an average human being ( \(400 \mathrm{~nm}<\lambda<700 \mathrm{~nm}\) ).

The separation between the consecutive dark fringes in a Young's double slit experiment is \(1 \cdot 0 \mathrm{~mm}\). The screen is placed at a distance of \(2 \cdot 5 \mathrm{~m}\) from the slits and the separation between the slits is \(1 \cdot 0 \mathrm{~mm}\). Calculate the wavelength of light used for the experiment.

A parallel beam of monochromatic light is used in a Young's double slit experiment. The slits are separated by a distance \(d\) and the screen is placed parallel to the plane of the slits. Show that if the incident beam makes an angle \(\theta=\sin ^{-1}\left(\frac{\lambda}{2 d}\right)\) with the normal to the plane of the slits, there will be a dark fringe at the centre \(P_{0}\) of the pattern.

A source emitting light of wavelengths \(480 \mathrm{~nm}\) and \(600 \mathrm{~nm}\) is used in a double slit interference experiment. The separation between the slits is \(0-25 \mathrm{~mm}\) and the interference is observed on a screen placed at \(150 \mathrm{~cm}\) from the slits. Find the linear separation between the first maximum (next to the central maximum) corresponding to the two wavelengths.

Two narrow slits emitting light in phase are separated by a distance of \(1 \cdot 0 \mathrm{~cm}\). The wavelength of the light is \(5^{-0} \times 10^{-7} \mathrm{~m} .\) The interference pattern is observed on a screen placed at a distance of \(1 \cdot 0 \mathrm{~m}\). (a) Find the separation between the consecutive maxima. Can you
See all solutions

Recommended explanations on Physics Textbooks

Fields in Physics

Read Explanation

Turning Points in Physics

Read Explanation

Electric Charge Field and Potential

Read Explanation

Dynamics

Read Explanation

Radiation

Read Explanation

Linear Momentum

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.