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Optics Using Mirrors

The methods and formulas for mirrors are almost identical to those for lenses, except that a mirror has only one focus. There are three kinds of mirrors: convex, plane, and concave (Figure 13-19). For a convex mirror, incoming parallel rays diverge after reflection, so the focal length is negative. For a plane mirror, the focal length is infinity. And for a concave mirror, incoming parallel rays converge after reflection, so the focal length is positive.
 

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Figure 13-19
 

For mirrors, you use the same equations, but you have to be careful, because the meaning of the sign for di is different from that for lens. A positive sign for di means that the image is on the same side of the mirror as the object, that is, in front. You use the same kinds of ray diagrams, but since there is only one focus, you use that same focus in steps 2 and 3.
 

Rav-tracing method for a converging mirror

  1. Draw the mirror, object, and focus.
  2. Draw a ray parallel to the principal axis which reflects from the mirror and passes through the focus.
  3. Draw a ray from the object passing through the focus and reflecting off the mirror to become parallel to the axis.
  4. The intersecting point is the location of the image. If the rays do not intersect, extend the rays behind the mirror.
 

Ray-tracing method for a diverging mirror

  1. Draw the mirror, object, and focus (behind mirror).
  2. Draw a ray parallel to the principal axis which reflects and goes up. as if it came from the focus. Extend the ray behind the mirror.
  3. Draw a ray going toward the focus of the mirror and reflecting as a horizontal ray. Extend the horizontal ray behind the mirror.
  4. The intersection of rays behind the mirror is the location of the image.

 

Example-1

The passenger mirror in Larry's car is a diverging mirror with focal length 0.8 meters. A car is 10.0 meters away from the mirror.

  1. Where is the image of the car?
  2. What is the magnification of the image?
  3. Is the image real or virtual?
  4. Is the image upright or inverted?
Solution

Figure 13-20 shows the ray diagram.

 

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Figure 13-20

 

Note that the second ray uses the same focus as the first ray. The image of the car is behind the mirror and is virtual and tiny. The exact location is given by

 

 

where the negative sign indicates the image is behind the mirror. The magnification is given by

 

 

where the positive sign indicates the image is upright. You do not need to pay attention to the sign conventions if you get the diagram right.
 

So why is there a warning "Objects in mirror are closer than they appear"? There are two things going on. The first is that the image is much closer to Larry than the object itself, and the second is that the image is smaller than the image Larry would see if he turned around and looked. Larry's brain does not care about where the image is and does not notice from which point the light rays appear to be diverging. Larry's brain compares the size of the image to what it knows is the size of a car in order to obtain a distance to the car. The distance thus calculated is about a factor of two too far away.
 

 

Example-2
Alice looks at herself in a plane mirror, standing 4 meters away.
  1. Where is her image?
  2. What is the magnification?
  3. Is the image real or virtual?
  4. Is the image upright or inverted?
Solution

Figure 13-21 shows a ray diagram, with which we can be a bit creative, since the focuses are an infinite distance away.

 

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Figure 13-21

 
The image is 4 meters on the other side of the mirror, the magnification is 1, and the image is virtual and upright. To see this in the equations, we calculate

 


Also
 


 

 

Example-3

The image of a candle lies 10.0 meters behind a converging mirror (focal length 5.0 m). Where is the object?

 

Solution

Figure 13-22 shows the ray diagram. Treat the image as the object and thus the origin of light rays. The hard part of this problem (if you have to do the calculation) is remembering the sign di = –10 m. Then

 


 
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Figure 13-22

 





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