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Chapter 25: Problem 17

The image behind a convex mirror (radius of curvature \(=68 \mathrm{~cm}\) ) is located \(22 \mathrm{~cm}\) from the mirror. (a) Where is the object located and (b) what is the magnification of the mirror? Determine whether the image is (c) upright or inverted and (d) larger or smaller than the object.

Short Answer

Expert verified
(a) Object is at 62.33 cm, (b) magnification is 0.353, (c) image is upright, (d) image is smaller than the object.

Step by step solution

01

Understand the Problem

We have a convex mirror with a radius of curvature of \( R = 68 \text{ cm} \). The image distance from the mirror is given as \( v = -22 \text{ cm} \) (negative because it's a virtual image behind the mirror). We need to find the object distance \( u \), the magnification \( m \), and determine the orientation and size comparison of the image.
02

Calculate the Focal Length

The focal length \( f \) of a mirror is half its radius of curvature, but negative for a convex mirror. Hence, \( f = -\frac{R}{2} = -\frac{68}{2} = -34 \text{ cm} \).
03

Use the Mirror Formula to Find Object Distance

The mirror formula is \( \frac{1}{f} = \frac{1}{u} + \frac{1}{v} \). Plugging in \( f = -34 \text{ cm} \) and \( v = -22 \text{ cm} \), we get:\[ \frac{1}{-34} = \frac{1}{u} - \frac{1}{22} \].Solving for \( \frac{1}{u} \), we get:\[ \frac{1}{u} = \frac{1}{-34} + \frac{1}{22} = \frac{22 - 34}{748} = \frac{-12}{748} \],Thus, \( u = \frac{748}{12} = 62.33 \text{ cm} \).
04

Calculate the Magnification

The magnification \( m \) is given by the formula \( m = -\frac{v}{u} \). Substitute \( v = -22 \text{ cm} \) and \( u = 62.33 \text{ cm} \):\[ m = -\left(\frac{-22}{62.33}\right) \approx 0.353 \].
05

Determine the Image Orientation and Size

Since the magnification \( m \) is positive, the image is upright. Magnitude of \( m \) is \( 0.353 \), which is less than 1, indicating the image is smaller than the object.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Radius of Curvature
The radius of curvature plays a vital role in understanding mirrors, especially convex mirrors. It represents the radius of the sphere from which the mirror segment is derived. In simpler terms, imagine the mirror as part of a large, hollow sphere. The radius of this sphere is the radius of curvature (\( R \)).
For this convex mirror example, the given radius of curvature is \( R = 68 ext{ cm} \). This value is crucial for determining the focal length of the mirror, which is half of this curvature.
It's important to note that for mirrors,
  • Convex mirrors have a positive \( R \) for the geometrical sense, but we treat \( R \) as negative when calculating the focal length.
  • The radius also influences the mirror formula, which helps in determining various distances related to the mirror.
Understanding this concept helps solve complex problems involving object and image distances.
Focal Length
The focal length of a mirror is the distance from the mirror to its focal point. It is a crucial factor in determining how light converges or diverges after reflecting from the mirror's surface. For any mirror, the focal length \( f \) relates directly to the radius of curvature \( R \).
Specifically, the formula is:
  • For a spherical mirror: \( f = \frac{R}{2} \)
  • For a convex mirror, the focal length is considered negative (\( f = -\frac{R}{2} \)).
In our example with \( R = 68 ext{ cm} \), the focal length is \( f = -34 ext{ cm} \). This negative sign indicates that a convex mirror diverges light rays. Understanding focal length helps in solving problems using the mirror formula, which ultimately relates object and image distances to the mirror.
Mirror Formula
The mirror formula connects the object distance (\( u \)), the image distance (\( v \)), and the focal length (\( f \)) of a mirror in a single equation:\[\frac{1}{f} = \frac{1}{u} + \frac{1}{v}\]This formula is fundamental in optics to predict where an image will form in relation to a mirror, allowing calculations of either the object or image distance if the other is known.
For a convex mirror in our scenario:
  • Given \( f = -34 \text{ cm} \) and \( v = -22 \text{ cm} \)
  • We solve the formula: \( \frac{1}{u} = \frac{1}{-34} + \frac{1}{22} \)
Finding \( u \) helps us understand how far an object needs to be placed from the mirror to form a particular type of image.
Image Magnification
Image magnification describes the ratio between the size of the image and the size of the object. In optics, it's a measure of how much larger or smaller the image is compared to the actual object. The magnification (\( m \)) of a mirror is given by the formula:\[ m = -\frac{v}{u} \]Here, \( v \) is the image distance, and \( u \) is the object distance.
In the problem, using \( v = -22 \text{ cm} \) and \( u = 62.33 \text{ cm} \):
  • The magnification is \( m = 0.353 \)
  • The positive value of \( m \) indicates the image is upright.
  • The magnitude less than 1 (\( < 1 \)) suggests the image is smaller than the object.
Understanding image magnification helps us not only determine an image's size but also gives insight into its orientation, like determining if it's upright or inverted.

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Most popular questions from this chapter

A tall tree is growing across a river from you. You would like to know the distance between yourself and the tree, as well as its height, but are unable to make the measurements directly. However, by using a mirror to form an image of the tree, and then measuring the image distance and the image height, you can calculate the distance to the tree, as well as its height. (a) What kind of mirror, concave or convex, must you use? Why? (b) You will need to know the focal length of the mirror. The sun is shining. You aim the mirror at the sun and form an image of it. How is the image distance of the sun related to the focal length of the mirror? (c) Having measured the image distance \(d_{\mathrm{i}}\) and the image height \(h_{\mathrm{i}}\) of the tree, as well as the image distance of the sun, describe how you would use these numbers to determine the distance and height of the tree. Problem A mirror produces an image of the sun, and the image is located \(0.9000 \mathrm{~m}\) from the mirror. The same mirror is then used to produce an image of the tree. The image of the tree is \(0.9100 \mathrm{~m}\) from the mirror. (a) How far away is the tree? (b) The image height of the tree has a magnitude of \(0.12 \mathrm{~m}\). How tall is the tree?

When viewed in a spherical mirror, the image of a setting sun is a virtual image. The image lies \(12.0 \mathrm{~cm}\) behind the mirror. (a) Is the mirror concave or convex? Why? (b) What is the radius of curvature of the mirror?

At illustrates the concepts pertinent to this problem. A \(2.0-\mathrm{cm}-\) high object is situated \(15.0 \mathrm{~cm}\) in front of a concave mirror that has a radius of curvature of \(10.0 \mathrm{~cm}\). Using a ray diagram drawn to scale, measure (a) the location and (b) the height of the image. The mirror must be drawn to scale.

The image produced by a concave mirror is located \(26 \mathrm{~cm}\) in front of the mirror. The focal length of the mirror is \(12 \mathrm{~cm}\). How far in front of the mirror is the object located?

A clown is using a concave makeup mirror to get ready for a show and is \(27 \mathrm{~cm}\) in front of the mirror. The image is \(65 \mathrm{~cm}\) behind the mirror. Find (a) the focal length of the mirror and (b) the magnification.
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