Skip to main content
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
All textbooksGiancoli Douglas 5th editionCh. 32 - Light: Reflection and RefractionProblem 26
Chapter 31, Problem 26

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.

Verified step by step guidance
1
Start with the mirror equation for a convex mirror: \(\frac{1}{f}\) = \(\frac{1}{d_o}\) + \(\frac{1}{d_i}\), where f is the focal length, d_o is the object distance, and d_i is the image distance. For a convex mirror, f = -|f| because the focal length is negative.
Rearrange the mirror equation to solve for \(\frac{1}{d_i}\): \(\frac{1}{d_i}\) = \(\frac{1}{f}\) - \(\frac{1}{d_o}\). Substitute f = -|f| into the equation: \(\frac{1}{d_i}\) = \(\frac{1}{-|f|}\) - \(\frac{1}{d_o}\).
Simplify the equation: \(\frac{1}{d_i}\) = -\(\frac{1}{|f|}\) - \(\frac{1}{d_o}\). Combine the terms on the right-hand side: \(\frac{1}{d_i}\) = -\(\frac{d_o + |f|}{|f| \cdot d_o}\). Invert both sides to solve for d_i: d_i = -\(\frac{|f| \cdot d_o}{d_o + |f|}\).
The lateral magnification m is defined as m = -\(\frac{d_i}{d_o}\). Substitute the expression for d_i into the magnification formula: m = -\(\frac{-\frac{|f| \cdot d_o}{d_o + |f|}\)}{d_o}. Simplify the expression: m = \(\frac{|f|}{d_o + |f|}\).
To explain why your nose looks bigger in a convex mirror: The magnification m depends on the object distance d_o. Since your nose is closer to the mirror than the rest of your face, its d_o is smaller, resulting in a larger magnification m. This makes your nose appear larger relative to the rest of your face.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
6m
Was this helpful?

Key Concepts

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

Convex Mirror Properties

A convex mirror is a curved mirror that bulges outward, causing light rays to diverge. This type of mirror always produces virtual images that are upright and smaller than the object. The focal length of a convex mirror is considered negative, which is crucial for understanding image formation and magnification.
Recommended video:
07:49
Ray Diagrams for Convex Mirrors

Lateral Magnification

Lateral magnification (m) is the ratio of the height of the image to the height of the object. It can also be expressed in terms of distances: m = |ƒ| / (dₒ + |ƒ|), where |ƒ| is the absolute value of the focal length and dₒ is the object distance. This relationship helps determine how much larger or smaller an image appears compared to the actual object.
Recommended video:
09:03
Mirror Equation

Image Size Perception

When viewing oneself in a convex mirror, the nose appears larger due to the nature of the image produced. Since the convex mirror creates a virtual image that is upright and smaller, the closer parts of the face, like the nose, can seem disproportionately larger compared to the rest of the face, which is further away from the mirror, leading to a skewed perception of size.
Recommended video:
09:03
Mirror Equation
Related Practice
Textbook Question

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?

1
views
Textbook Question

An object is placed a distance r in front of a wall, where r exactly equals the radius of curvature of a certain concave mirror. At what distance from the wall should this mirror be placed so that a real image of the object is formed on the wall? What is the lateral magnification of the image?

2
views
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.

1
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.]

1
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.

(c) Calculate the required radius of curvature for the mirror.

2
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.]

1
views