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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: 9780137488179Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoGiancoli Douglas 5th editionCh. 32 - Light: Reflection and RefractionProblema 24
Capitolo 31, Problema 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?

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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.

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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.
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Ray Diagrams for Concave Mirrors

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.
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Mirror Equation

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.
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Mirror Equation
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