Ch. 32 - Light: Reflection and Refraction

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook

Giancoli Douglas 5th edition

Ch. 32 - Light: Reflection and Refraction

Problem 24

Chapter 31, Problem 24

In Example 32–4, show that if the object is moved 10.0 cm farther from the concave mirror, the object’s image size will equal the object’s actual size. Stated as a multiple of the focal length, what is the object distance for this “actual-sized image” situation?

Verified step by step guidance

Understand the problem: The goal is to determine the object distance at which the image size equals the object size for a concave mirror. This occurs when the magnification (M) is 1. Recall that magnification is given by \( M = -\frac{d_i}{d_o} \), where \( d_i \) is the image distance and \( d_o \) is the object distance.

Recall the mirror equation: \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \), where \( f \) is the focal length of the mirror. This equation relates the focal length, object distance, and image distance.

Set the magnification to 1: For the image size to equal the object size, \( M = 1 \), which implies \( d_i = d_o \). Substitute \( d_i = d_o \) into the mirror equation: \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_o} \). Simplify this to \( \frac{1}{f} = \frac{2}{d_o} \).

Solve for \( d_o \): Rearrange the equation \( \frac{1}{f} = \frac{2}{d_o} \) to find \( d_o \). This gives \( d_o = 2f \). Thus, the object distance must be twice the focal length for the image size to equal the object size.

Account for the 10.0 cm shift: The problem states that the object is moved 10.0 cm farther from the mirror. Therefore, the new object distance is \( d_o = 2f + 10.0 \; \text{cm} \). Use this relationship to express the object distance in terms of the focal length.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Concave Mirror

A concave mirror is a spherical mirror that curves inward, resembling a portion of a sphere. It can converge light rays that strike its surface, allowing for the formation of real or virtual images depending on the object's distance from the mirror. The behavior of light with concave mirrors is governed by the mirror equation and the principles of reflection.

Mirror Equation

The mirror equation relates the object distance (d_o), the image distance (d_i), and the focal length (f) of a mirror. It is expressed as 1/f = 1/d_o + 1/d_i. This equation is essential for determining the position and size of the image formed by the mirror, allowing us to analyze how changes in object distance affect image characteristics.

Magnification

Magnification is the ratio of the height of the image to the height of the object, and it can also be expressed as the negative ratio of the image distance to the object distance (M = -d_i/d_o). When the magnification equals 1, the image size is the same as the object size. Understanding magnification is crucial for determining the conditions under which an image appears the same size as the object.

Recommended video:

09:03

Mirror Equation

Related Practice

Textbook Question

An object 4.0 mm high is placed 18 cm from a convex mirror of radius of curvature 18 cm. Compute the image size, using Eq. 32–3.

views

Textbook Question

An object 4.0 mm high is placed 18 cm from a convex mirror of radius of curvature 18 cm. Show that the (negative) image distance can be computed from Eq. 32–2 using a focal length of -9.0 cm.

views

Textbook Question

When walking toward a concave mirror you notice that your image flips at a distance of 0.80 m from the mirror. What is the radius of curvature of the mirror? [Hint: Carefully examine Section 32–4.]

views

Textbook Question

A shaving or makeup mirror is designed to magnify your face by a factor of 1.8 (when compared to a flat mirror) when your face is placed 20.0 cm in front of it.

(a) What type of mirror is it?

(b) Describe the type of image that it makes of your face.

views

Textbook Question

(II) Show, using a ray diagram, that the lateral magnification m of a convex mirror is m = -dᵢ/dₒ , just as for a concave mirror. [Hint: Consider a ray from the top of the object that reflects at the center of the mirror.]

views

Textbook Question

Let the focal length of a convex mirror be written as ƒ = ―|ƒ|. Show that the lateral magnification m of an object a distance dₒ from this mirror is given by m = |ƒ| / (dₒ +|ƒ| ). Based on this relation, explain why your nose looks bigger than the rest of your face when looking into a convex mirror.

views