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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 30
Chapter 31, Problem 30

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.

Verified step by step guidance
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Determine the type of mirror: Since the mirror magnifies the image (magnification > 1), it must be a concave mirror. Concave mirrors can produce magnified, upright images when the object is within the focal length.
Describe the type of image: The image produced by the mirror is virtual, upright, and magnified. This is because the object (your face) is placed closer to the mirror than its focal point, which is a characteristic of concave mirrors in this configuration.
Use the magnification formula to find the image distance: The magnification (M) is given by \( M = -\frac{d_i}{d_o} \), where \( d_i \) is the image distance and \( d_o \) is the object distance. Rearrange to solve for \( d_i \): \( d_i = -M \cdot d_o \). Substitute \( M = 1.8 \) and \( d_o = 20.0 \, \text{cm} \).
Apply the mirror equation to find the focal length: The mirror equation is \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \), where \( f \) is the focal length. Substitute the values of \( d_o \) and \( d_i \) (calculated in the previous step) into the equation and solve for \( f \).
Relate the focal length to the radius of curvature: The radius of curvature \( R \) is related to the focal length by \( R = 2f \). Use the focal length calculated in the previous step to find \( R \).

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Key Concepts

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

Types of Mirrors

Mirrors can be classified into two main types: concave and convex. A concave mirror curves inward and can produce magnified images, while a convex mirror curves outward and typically produces smaller images. In the context of the question, a concave mirror is used for magnification, making it suitable for shaving or makeup applications.
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Image Formation

The type of image formed by a mirror depends on the object's position relative to the mirror's focal point. A concave mirror can produce real or virtual images. When the object is within the focal length, the image is virtual, upright, and magnified, which is the case when the face is placed 20.0 cm in front of the mirror.
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Radius of Curvature

The radius of curvature (R) of a mirror is the distance from the mirror's surface to its center of curvature, which is the point where the mirror's surface would meet if extended. For concave mirrors, the relationship between the focal length (f) and the radius of curvature is given by the formula R = 2f. This relationship is essential for calculating the required radius of curvature based on the magnification and object distance.
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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?

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Textbook Question

A light beam strikes a 2.5-cm-thick piece of plastic with a refractive index of 1.62 at a 45° angle. The plastic is on top of a 3.8-cm-thick piece of glass for which n = 1.47. What is the distance D in Fig. 32–51?

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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?

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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.]

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Textbook Question

A flashlight beam strikes the surface of a pane of glass (n = 1.56) at a 69° angle to the normal. What is the angle of refraction?

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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.

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