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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 22
Chapter 31, Problem 22

(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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Draw a ray diagram with a convex mirror, an object, and the principal axis. Place the object in front of the mirror on the principal axis.
Draw a ray from the top of the object traveling parallel to the principal axis towards the mirror. After reflecting off the mirror, this ray should diverge as if it is coming from a point behind the mirror (the virtual image).
Draw another ray from the top of the object heading towards the center of the mirror. After reflecting, this ray should follow the law of reflection, where the angle of incidence equals the angle of reflection, and it should appear to diverge from the same point behind the mirror as the first ray.
Identify the point where the extended reflected rays converge behind the mirror; this point represents the top of the virtual image formed by the mirror.
Using the ray diagram, measure the distances from the mirror to the object (dₒ) and from the mirror to the virtual image (dᵢ). The lateral magnification (m) can be calculated using the formula m = -dᵢ/dₒ, where the negative sign indicates that the image is virtual and upright.

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Key Concepts

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

Ray Diagrams

Ray diagrams are graphical representations used in optics to illustrate the path of light rays as they interact with mirrors and lenses. They help visualize how light reflects off surfaces, allowing us to determine the characteristics of the image formed, such as its position, size, and orientation. In the context of mirrors, specific rays are traced to show how they converge or diverge, leading to the formation of images.
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Ray Diagrams for Converging Lenses

Lateral Magnification

Lateral magnification (m) is a measure of how much larger or smaller an image is compared to the object. It is defined as the ratio of the height of the image (hᵢ) to the height of the object (hₒ), and can also be expressed in terms of distances: m = -dᵢ/dₒ, where dᵢ is the image distance and dₒ is the object distance. The negative sign indicates that the image formed by a convex mirror is virtual and upright, contrasting with the real and inverted images produced by concave mirrors.
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Mirror Equation

Convex Mirrors

Convex mirrors are curved mirrors that bulge outward, causing parallel incoming light rays to diverge after reflection. They always produce virtual images that are upright and smaller than the object, regardless of the object's distance from the mirror. The properties of convex mirrors make them useful for applications such as vehicle side mirrors, where a wider field of view is needed, despite the smaller image size.
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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

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.

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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

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

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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