/*! 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 68 A concave mirror has a focal len... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 16: Problem 68

A concave mirror has a focal length of \(12 \mathrm{~cm}\). What is its radius of curvature?

Short Answer

Expert verified
The radius of curvature is 24 cm.

Step by step solution

01

Understanding the relationship

The formula relating the focal length \( f \) and the radius of curvature \( R \) for a concave mirror is given by \( f = \frac{R}{2} \). This formula states that the focal length is half of the radius of curvature.
02

Substitute the known values

We know that the focal length, \( f \), is \( 12 \text{ cm} \). According to the formula, \( f = \frac{R}{2} \). Thus, we substitute \( f = 12 \) into the equation: \( 12 = \frac{R}{2} \).
03

Solve for the radius of curvature

To find the radius of curvature \( R \), multiply both sides of the equation \( 12 = \frac{R}{2} \) by 2 to solve for \( R \). This gives us \( R = 24 \text{ cm} \).

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.

Understanding Focal Length
In the realm of optics, a focal length is a crucial concept, particularly when dealing with mirrors and lenses. The focal length of a concave mirror is the distance between the mirror's surface and its focal point, where parallel rays of light converge. This convergence is due to the way the mirror curves inward, reflecting the light rays towards a common point.
Understanding focal length can help you predict how the mirror will project images. It's important to remember that:
  • Shorter focal lengths imply stronger curvature, meaning light focuses closer to the mirror.
  • Longer focal lengths imply flatter mirrors, where light focuses farther away.
This concept is crucial in designing optical devices like telescopes and cameras, which rely on precise focus for clear images.
The Radius of Curvature
The radius of curvature is another fundamental concept when analyzing concave mirrors. It refers to the radius of the sphere from which the mirror segment is cut. All concave mirrors are part of a larger imaginary sphere, and the radius of this sphere is the radius of curvature.
It relates to how curved the mirror is:
  • A smaller radius of curvature indicates a more sharply curved mirror.
  • A larger radius of curvature indicates a mirror with gentler curvature.
In practical terms, by understanding the radius of curvature, one can manipulate how a mirror focuses light, which greatly affects its use in practical applications such as illuminating screens or focus light in devices.
The Mirror Formula: Unlocking Mirror Properties
The mirror formula is a key equation that links the focal length, radius of curvature, and how a mirror will focus light. For concave mirrors, the formula is expressed as:\[ f = \frac{R}{2} \]This simple yet powerful relationship tells you that the focal length, "f", is precisely half the radius of curvature, "R". This equation serves as a foundational stone in optics because it provides a straightforward way to assess the focal properties of any concave mirror.
Here鈥檚 why this formula is indispensable:
  • It allows easy calculation of one property if the other is known, making it invaluable for designing optical instruments.
  • It provides a way to predict how effectively a mirror will focus light, assisting in tasks ranging from precise scientific measurements to everyday uses like shaving mirrors or vehicle headlights.
By harnessing this relationship, engineers and designers can create devices that precisely control and utilize light, enhancing functionality across various fields.

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

Hale Telescope The \(5.08\)-m-diameter concave mirror of the Hale Telescope on Mount Palomar has a focal length of \(16.9 \mathrm{~m}\). An astronomer stands \(20.0 \mathrm{~m}\) in front of this mirror. (a) Where is her image located? Is it in front of or behind the mirror? (b) Is her image real or virtual? How do you know? (c) What is the magnification of her image?

An object is at the center of curvature of a concave mirror. (a) Is the resulting image real or virtual? (b) What is the location of the image? (c) Is the image upright or inverted? (d) What is the magnification of the image?

A beam of light is incident on a mirror. If the angle of incidence of this beam is increased, does the beam shift closer to the normal or farther from the normal? Explain.

How rapidly does the distance between you and your mirror image decrease if you walk directly toward a mirror with a speed of \(2.6 \mathrm{~m} / \mathrm{s}\) ?

What kind of mirror can produce a real image?
See all solutions

Recommended explanations on Physics Textbooks

Medical Physics

Read Explanation

Fields in Physics

Read Explanation

Linear Momentum

Read Explanation

Force

Read Explanation

Astrophysics

Read Explanation

Quantum 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.