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Chapter 25: Problem 24

A concave mirror \((f=45 \mathrm{cm})\) produces an image whose distance from the mirror is one-third the object distance. Determine (a) the object distance and (b) the (positive) image distance.

Short Answer

Expert verified
(a) Object distance: 180 cm. (b) Image distance: 60 cm.

Step by step solution

01

Understand the Given Information

We know that the focal length \( f \) of the concave mirror is 45 cm. We need to find the object distance \( d_o \) and the image distance \( d_i \). Also, it is given that the image distance is one-third the object distance, thus \( d_i = \frac{1}{3}d_o \).
02

Apply the Mirror Equation

The mirror equation relates the object distance \( d_o \), the image distance \( d_i \), and the focal length \( f \):\[ \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \] Substitute \( f = 45 \) cm and \( d_i = \frac{1}{3}d_o \) into the equation to get \[ \frac{1}{45} = \frac{1}{d_o} + \frac{1}{\frac{1}{3}d_o} \]
03

Simplify and Solve the Equation

Substituting \( d_i = \frac{1}{3}d_o \) into the equation simplifies it to:\[ \frac{1}{45} = \frac{1}{d_o} + \frac{3}{d_o} \] \[ \frac{1}{45} = \frac{4}{d_o} \] We rearrange to find \( d_o \):\[ d_o = 4 \times 45 = 180 \text{ cm} \]
04

Calculate the Image Distance

Now that we have \( d_o = 180 \) cm, use \( d_i = \frac{1}{3}d_o \) to find \( d_i \): \[ d_i = \frac{1}{3} \times 180 = 60 \text{ cm} \]

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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 crucial formula in optics, especially when dealing with mirrors like concave mirrors. It provides the relationship between three vital parameters: the focal length \( f \), the object distance \( d_o \), and the image distance \( d_i \). The equation is expressed as:
  • \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \)
Using this equation, you can determine one of the three variables if the other two are known. This powerful equation explains how the light converges or diverges after reflecting from the surface of a mirror. In the given exercise, knowing the focal length and the relationship between the object and image distances allowed us to compute the exact position of both the object and the image.
Focal Length
The focal length \( f \) is a fundamental property of a concave mirror that determines how it focuses light. It鈥檚 the distance from the mirror to the focal point, where parallel rays of light either converge (in the case of mirrors) or appear to diverge from. The focal length is related to the mirror's curvature; a more curved mirror will have a shorter focal length.
  • In the scenario given, the focal length is 45 cm, a crucial piece of information necessary for applying the mirror equation.
By knowing the focal length, you can predict how the mirror will project an image, whether magnified or reduced, and calculate other essential properties like image distance or object distance, as seen in the exercise.
Object Distance
Object distance \( d_o \) is the distance between the object and the mirror. It plays a vital role in determining how the image is formed and where it will appear. In practical applications and exercises, it's often given or deduced using other known values.
  • In the exercise, \( d_o \) was calculated using the mirror equation and the given ratio between \( d_o \) and \( d_i \).
The calculation is straightforward once the mirror equation is rearranged to isolate \( d_o \). For the given concave mirror, the object was placed at a distance of 180 cm from the mirror to achieve an image distance that was one-third of that, reflecting the geometric properties of the mirror.
Image Distance
Image distance \( d_i \) is the distance from the mirror to the point where the image is formed. It can either be in front of or behind the mirror depending on the mirror type and the object's position.
  • For concave mirrors, the image distance can be positive (real and inverted image) or negative (virtual and erect image).
In the example, \( d_i \) was found using the relationship \( d_i = \frac{1}{3}d_o \) once \( d_o \) was determined. This led to discovering \( d_i = 60 \) cm, showing the image is real and inverted, positioned opposite the object, further emphasizing the effectiveness of using the mirror equation to predict and understand image formation.

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Most popular questions from this chapter

An object is located 14.0 cm in front of a convex mirror, the image being 7.00 cm behind the mirror. A second object, twice as tall as the first one, is placed in front of the mirror, but at a different location. The image of this second object has the same height as the other image. How far in front of the mirror is the second object located?

A small postage stamp is placed in front of a concave mirror (radius R) so 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.

Suppose that you are walking perpendicularly with a velocity of \(+0 . \overline{90} \mathrm{m} / \mathrm{s} \mathrm{t}\) toward a stationary plane mirror. What is the velocity of your image relative to you? The direction in which you walk is the positive direction.

An object is located 14.0 cm in front of a convex mirror, the image being 7.00 cm behind the mirror. A second object, twice as tall as the first one, is placed in front of the mirror, but at a different location. The image of this second object has the same height as the other image. How far in front of the mirror is the second object located?

A drop of water on a countertop reflects light from a flower held 3.0 cm directly above it. The flower鈥檚 diameter is 2.0 cm, and the diameter of the flower鈥檚 image is 0.10 cm. What is the focal length of the water drop, assuming that it may be treated as a convex spherical mirror?
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