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Chapter 23: Problem 77

In many cars the passenger's side mirror says: "Objects in the mirror are closer than they appear." (a) Does this mirror form real or virtual images? (b) since the image is diminished in size, is the mirror concave or convex? Why? (c) Show that the image must actually be closer to the mirror than is the object. (d) How then can the image seem to be farther away?

Short Answer

Expert verified
Answer: Passenger's side mirrors in cars form virtual images, and these mirrors are convex.

Step by step solution

01

Identify the type of images formed by the mirror

According to the question, the passenger's side mirror makes objects appear further away than they actually are. In this case, a virtual image is formed. This is because a virtual image is formed when light rays diverge after reflecting off the mirror and cannot converge to form a real image.
02

Determine whether the mirror is concave or convex

As the mirror forms virtual and diminished images, the mirror must be convex. Convex mirrors cause light rays to diverge, making objects appear smaller and further away, while concave mirrors cause light rays to converge, making objects appear larger and closer.
03

Prove that the image is closer to the mirror than the object

Let's consider convex mirror's equation: \[\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}\] Where f is the focal length, \(d_o\) is the distance from the object to the mirror, and \(d_i\) is the distance from the image to the mirror. A convex mirror has a negative focal length. Since the image appears closer to the mirror, \(d_i\) must be less than \(d_o\). Therefore, when we plug in the values, we will get a negative result for f, confirming that the mirror is convex and also that the image is closer to the mirror than the object: \[ \frac{1}{d_o} > \frac{1}{d_i}\] \[ \frac{1}{f} < 0\]
04

Explain why the image appears to be farther away

The image appears to be farther away because the convex mirror causes the light rays to diverge, and the image is formed where these diverged light rays appear to come from. Our eyes perceive the virtual image to be farther away than it actually is, as the brain interprets the angles at which the light rays enter the eye, making it seem as if the reflected objects are further away. This illusion is useful for increasing the driver's field of view and providing a wider perspective of traffic behind the car.

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