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

Giancoli Douglas 5th edition

Ch. 32 - Light: Reflection and Refraction

Problem 19b

Chapter 31, Problem 19b

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.

Verified step by step guidance

Step 1: Understand the problem. We are tasked with showing that the image distance can be computed using the mirror equation, Eq. 32–2, with the given focal length of -9.0 cm. The mirror equation is:

\frac{1}{d} = \frac{1}{f} - \frac{1}{d}

, where

d o

is the object distance,

d

is the image distance, and

f

is the focal length.

Step 2: Identify the given values. The object height is 4.0 mm (not directly relevant for this part of the problem), the object distance

d o

is 18 cm, and the radius of curvature of the convex mirror is 18 cm. The focal length

f

for a mirror is related to the radius of curvature

R

by the formula:

f = \frac{R}{2}

. For a convex mirror, the focal length is negative.

Step 3: Calculate the focal length. Using the formula

f = \frac{R}{2}

, substitute

R

= 18 cm. Since the mirror is convex, the focal length is negative:

f = - \frac{18}{2} = - 9 cm

Step 4: Apply the mirror equation. Substitute the values

f = - 9 cm

and

d o = 18 cm

into the mirror equation:

\frac{1}{d} = \frac{1}{-} - \frac{1}{18}

. Rearrange the equation to solve for

d

, the image distance.

Step 5: Simplify the equation. Combine the fractions on the right-hand side:

\frac{1}{d} = - \frac{1}{9} - \frac{1}{18}

. Find a common denominator and simplify further to compute

d

. This will yield the negative image distance, confirming the computation.

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Key Concepts

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

Convex Mirror and Focal Length

A convex mirror is a spherical mirror that curves outward, causing light rays to diverge. The focal length of a convex mirror is considered negative because the focus is virtual, located behind the mirror. The focal length (f) is half the radius of curvature (R), so for a radius of curvature of 18 cm, the focal length is -9 cm.

Mirror Formula

The mirror formula relates the object distance (u), image distance (v), and focal length (f) of a mirror. It is expressed as 1/f = 1/v + 1/u. For convex mirrors, the object distance is taken as negative (since it is measured against the direction of incident light), while the image distance is also negative, indicating that the image is virtual and located behind the mirror.

Image Formation in Convex Mirrors

In convex mirrors, images formed are always virtual, upright, and reduced in size compared to the object. The image distance is negative, indicating that the image appears behind the mirror. This characteristic is crucial for understanding how convex mirrors are used in applications like vehicle side mirrors, where a wider field of view is needed.

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Ray Diagrams for Convex Mirrors

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. Compute the image size, using Eq. 32–3.

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Textbook Question

You look at yourself in a shiny 8.4-cm-diameter Christmas tree ball. If your face is 25.0 cm away from the ball’s front surface, where is your image? Is it real or virtual? Is it upright or inverted?

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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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