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Chapter 25: Problem 29

Using the mirror equation and the magnification equation, show that for a convex mirror the image is always (a) virtual (i.e., \(d_{\mathrm{i}}\) is always negative) and (b) upright and smaller, relative to the object (i.e., \(m\) is positive and less than one).

Short Answer

Expert verified
For a convex mirror, the image is always virtual, upright, and smaller.

Step by step solution

01

Understand the Mirror Equation

The mirror equation is given by \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \), where \( f \) is the focal length, \( d_o \) is the object distance, and \( d_i \) is the image distance. For convex mirrors, the focal length \( f \) is negative.
02

Explore the Sign of Image Distance

Assume \( d_o > 0 \) because the object is in front of the mirror. Since \( f \) is negative for a convex mirror, applying the mirror equation results in \( \frac{1}{d_i} = \frac{1}{f} - \frac{1}{d_o} \). This equation implies \( d_i \) must be negative, since subtracting a positive \( \frac{1}{d_o} \) from a negative \( \frac{1}{f} \) results in a negative value for \( \frac{1}{d_i} \). Thus, the image distance \( d_i \) is always negative, indicating a virtual image.
03

Use the Magnification Equation

The magnification equation is \( m = -\frac{d_i}{d_o} \). For convex mirrors, since \( d_i \) is negative and \( d_o \) is positive, the negative sign cancels out, making magnification \( m \) positive.
04

Determine Image Size Relative to Object

According to the magnification equation \( m = \frac{|d_i|}{d_o} \), where \( |d_i| \) is less than \( d_o \) because \( d_i \) is found from \( \frac{1}{d_i} \) which becomes a larger negative fraction compared to \( \frac{1}{d_o} \). Therefore, \( m \) is a positive value less than 1, indicating the image is upright and smaller than the object.

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Key Concepts

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

Mirror Equation
The mirror equation is a fundamental formula in optics. It helps us understand how mirrors form images. The equation is written as:\[ \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \]Where:- \( f \) is the focal length- \( d_o \) is the object distance- \( d_i \) is the image distanceIn the context of a convex mirror, it's important to note that the focal length \( f \) is negative. This is because convex mirrors spread out light rays, which causes the virtual focus point to be behind the mirror. This equation helps us find the position of the image created by the mirror by knowing its focal length and how far the object is placed from it.
Magnification Equation
The magnification equation tells us how the size of the image compares to the size of the object. It's given by:\[ m = -\frac{d_i}{d_o} \]In this equation:- \( m \) is the magnification- \( d_i \) is the image distance- \( d_o \) is the object distanceThe negative sign indicates the image orientation. However, in the case of a convex mirror, the image distance \( d_i \) is negative while \( d_o \) is positive, which makes \( m \) positive. A positive magnification suggests that the image is upright compared to the object.
Virtual Image
A virtual image occurs when the image formed by a mirror cannot be projected on a screen. In the case of a convex mirror, the image is always virtual. Since convex mirrors diverge light rays, the image appears to be located behind the mirror.
  • The image distance \( d_i \) is negative in the mirror equation.
  • This negative value indicates a virtual placement, as real images have positive image distances.
The key idea here is that although the image forms a clear representation, it doesn't exist in physical space in front of the mirror.
Image Distance
The image distance \( d_i \) is a crucial component of understanding mirror images. It's derived from the mirror equation and tells us where the image is located in relation to the mirror. For convex mirrors:
  • \( d_i \) is always negative, indicating the image is virtual and appears behind the mirror.
  • The location of the image depends on both the focal length \( f \) and the object distance \( d_o \).
This negative image distance reflects the nature of the convex mirror's diverging surface, which cannot focus the light at a single point but instead creates a virtual area at which the image appears to originate.
Focal Length
The focal length \( f \) of a mirror is the distance from the mirror's surface to the focal point. In convex mirrors, the focal length is unique:
  • It is always negative.
  • This negative value means the focal point lies behind the mirror's surface.
This characteristic of convex mirrors is essential for understanding how they create images. The negative focal length results in virtual, upright, and smaller images relative to the object. It signifies how a convex mirror diverges light and influences the overall geometry of reflected light paths.

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Most popular questions from this chapter

A ray of light strikes a plane mirror at a \(45^{\circ}\) angle of incidence. The mirror is then rotated by \(15^{\circ}\) into the position shown in red in the drawing, while the incident ray is kept fixed. (a) Through what angle \(\phi\) does the reflected ray rotate? (b) What is the answer to part (a) if the angle of incidence is \(60^{\circ}\) instead of \(45^{\circ} ?\)

A small postage stamp is placed in front of a concave mirror (radius \(=R\) ), such that the image distance equals the object distance. (a) In terms of \(R\), what is the object distance? (b) What is the magnification of the mirror? (c) State whether the image is upright or inverted relative to the object. Draw a ray diagram to guide your thinking.

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.

A concave mirror has a focal length of \(30.0 \mathrm{~cm} .\) The distance between an object and its image is \(45.0 \mathrm{~cm} .\) Find the object and image distances, assuming that (a) the object lies beyond the center of curvature and (b) the object lies within the focal point.

The outside mirror on the passenger side of a car is convex and has a focal length of \(-7.0 \mathrm{~m}\). Relative to this mirror, a truck traveling in the rear has an object distance of \(11 \mathrm{~m}\). Find (a) the image distance of the truck and (b) the magnification of the mirror.
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