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.

Name: Physics for Scientists and Engineers
Author: Giancoli Douglas
ISBN: 9780137488179

Verified step by step guidance

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.

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 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.

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.

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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Related Practice

Textbook Question

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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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

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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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.]

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Textbook Question

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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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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