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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 19c
Capitolo 31, Problema 19c

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.
Mirror equation formula showing the relationship between image height and object height.

Guida verificata passo dopo passo
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Step 1: Identify the given values. The object height \( h_o \) is 4.0 mm, the object distance \( d_o \) is 18 cm, and the radius of curvature \( R \) of the convex mirror is 18 cm. Note that the focal length \( f \) of a convex mirror is \( f = \frac{R}{2} \), and since it is convex, \( f \) is positive.
Step 2: Calculate the focal length \( f \) using \( f = \frac{R}{2} \). Substituting \( R = 18 \) cm, we find \( f = 9 \) cm.
Step 3: Use the mirror equation \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \) to solve for the image distance \( d_i \). Rearrange the equation to \( \frac{1}{d_i} = \frac{1}{f} - \frac{1}{d_o} \). Substitute \( f = 9 \) cm and \( d_o = 18 \) cm into the equation.
Step 4: Once \( d_i \) is calculated, use the magnification formula \( m = \frac{h_i}{h_o} = -\frac{d_i}{d_o} \) to find the image height \( h_i \). Rearrange the formula to \( h_i = m \cdot h_o \). Substitute \( h_o = 4.0 \) mm and the values of \( m \) obtained from \( -\frac{d_i}{d_o} \).
Step 5: Interpret the result. Since the mirror is convex, the image will be virtual, upright, and smaller than the object. The calculated \( h_i \) will represent the size of the image.

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

Convex Mirror Properties

A convex mirror is a spherical mirror that curves outward, causing light rays to diverge. This type of mirror always produces virtual images that are upright and smaller than the object. The image distance is negative in the mirror equation, reflecting the virtual nature of the image formed by convex mirrors.
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Ray Diagrams for Convex Mirrors

Mirror Equation

The mirror equation relates the object distance (do), image distance (di), and focal length (f) of a mirror. For a convex mirror, the equation is given by 1/f = 1/do + 1/di. The focal length is positive for convex mirrors, and this equation is essential for calculating the position and size of the image formed.
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Mirror Equation

Magnification

Magnification (m) is the ratio of the height of the image (hi) to the height of the object (ho) and is also related to the distances of the image and object. It is expressed as m = hi/ho = -di/do. For convex mirrors, magnification is always less than one, indicating that the image is smaller than the object, and the negative sign indicates that the image is virtual and upright.
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Percorso guidato
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Mirror Equation
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A dentist wants a small mirror that, when 2.00 cm from a tooth, will produce a 3.0 x upright image. What kind of mirror must be used and what must its radius of curvature be?

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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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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.

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(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.]

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An object 4.0 mm high is placed 18 cm from a convex mirror of radius of curvature 18 cm. Show by ray tracing that the image is virtual, and estimate the image distance.

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