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Chapter 25: Problem 25

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

Short Answer

Expert verified
(a) Image distance: approx. 19.23 m (b) Magnification: approx. -1.75

Step by step solution

01

Understand the Mirror Equation

The mirror equation relates the object distance \( d_o \), image distance \( d_i \), and the focal length \( f \) of a mirror. It is given by \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \). We will use this equation to find the image distance.
02

Substitute Known Values

We know the focal length \( f = 7.0 \ m \) and the object distance \( d_o = 11 \ m \). Substitute these into the mirror equation: \( \frac{1}{7.0} = \frac{1}{11} + \frac{1}{d_i} \).
03

Solve for Image Distance

Rearrange the equation to solve for \( \frac{1}{d_i} \): \( \frac{1}{d_i} = \frac{1}{7.0} - \frac{1}{11} \). Calculate this: \( \frac{1}{d_i} = 0.1429 - 0.0909 = 0.052 \). Thus, \( d_i = \frac{1}{0.052} \approx 19.23 \ m \).
04

Calculate Magnification

The magnification \( m \) of a mirror is given by \( m = -\frac{d_i}{d_o} \). Substitute the values \( d_i = 19.23 \ m \) and \( d_o = 11 \ m \) into the equation: \( m = -\frac{19.23}{11} \approx -1.75 \).
05

Interpret the Results

The negative sign of the magnification indicates that the image is inverted. The magnitude tells us the size is reduced by a factor of approximately 1.75.

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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 concept used in optics to relate different dimensions involved when light interacts with mirrors. Specifically, it shows how the object distance, image distance, and focal length are connected.

The equation is:
  • \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \)
Here, \( f \) is the focal length of the mirror, \( d_o \) is the distance from the object to the mirror, and \( d_i \) is the distance from the image to the mirror.
Convex mirrors, like the one in a car's passenger side rearview, have a focal point behind the mirror. Thus, their focal length is negative. This equation helps determine how and where images are formed using these distances.
Object Distance
The object distance \( d_o \) refers to how far away the actual object is from the mirror. In the mirror equation, it is a crucial input value.

When you know the object distance and the focal length of a mirror, you can predict where the image will form by solving for the image distance.
The object distance in a convex mirror is always considered as positive in calculations.
  • In our example, the truck is located 11 meters away from the mirror, so \( d_o = 11 \, m \).
Knowing this, we can use the mirror equation to find out more about the image, such as its distance and magnification.
Image Distance
The image distance \( d_i \) is the distance from the mirror to the point where the image appears. In the mirror equation, solving for \( d_i \) tells us where that image is located.

In convex mirrors, the image appears behind the mirror, making the image distance negative once calculated.
To find \( d_i \), we rearrange our mirror equation for \( \frac{1}{d_i} \) and solve:
  • \( \frac{1}{d_i} = \frac{1}{f} - \frac{1}{d_o} \)
We place the known values into the equation:
  • \( \frac{1}{7.0} = \frac{1}{11} + \frac{1}{d_i} \)
Solving, we find \( d_i \approx 19.23 \, m \), but this will be a virtual image situated 19.23 meters "inside" or to the back of the mirror.
Focal Length
The focal length, represented by \( f \), represents the distance between the mirror's surface and its focal point. For convex mirrors, this focal point is behind the mirror, meaning the focal length is negative.

Understanding the focal length is key to predicting how a mirror will reflect light and form images.
  • In the exercise, \( f = 7.0 \, m \).
When using the mirror equation, the focal length affects all other related measurements.
A shorter focal length results in stronger curvature of the mirror, impacting the size and position of images formed. This is why knowing the focal length is central to determining image properties accurately.

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Most popular questions from this chapter

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.

In an experiment designed to measure the speed of light, a laser is aimed at a mirror that is 50.0 km due north. A detector is placed 117 m due east of the laser. The mirror is to be aligned so that light from the laser reflects into the detector. (a) When properly aligned, what angle should the normal to the surface of the mirror make with due south? (b) Suppose the mirror is misaligned, so that the actual angle between the normal to the surface and due south is too large by 0.004. By how many meters (due east) will the reflected ray miss the detector?

A convex mirror has a focal length of 27.0 cm. Find the magnification produced by the mirror when the object distance is 9.0 cm and 18.0 cm.

An object is placed in front of a convex mirror. Draw the convex mirror (radius of curvature 15 cm) to scale, and place the object 25 cm in front of it. Make the object height 4 cm. Using a ray diagram, locate the image and measure its height. Now move the object closer to the mirror, so the object distance is 5 cm. Again, locate its image using a ray diagram. As the object moves closer to the mirror, (a) does the magnitude of the image distance become larger or smaller, and (b) does the magnitude of the image height become larger or smaller? (c) What is the ratio of the image height when the object distance is 5 cm to its height when the object distance is 25 cm? Give your answer to one significant figure.

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