What does a converging mirror do to parallel rays?

Curved mirrors can produce all sorts of images. We will restrict our attention to spherical mirrors. Mirrors that reflect on the inside of the spherical surface are called concave mirrors; they will cause parallel light to converge on a point. Mirrors that reflect on the outside of the spherical surface are called convex mirrors; they will cause parallel light to diverge as if coming from a common point. Figure 18.5 shows a cross sectional view of both a convex mirror and a concave mirror. The axis of symmetry is known as the optic axis; the axis of symmetry will pass through the center of curvature of the mirror. The optic axis will be a useful reference line throughout our study of image formation.




Figure 18.D Curved "fun house" mirrors produce strange and unusual images. Cylindrical mirrors can even "decode" strange pictures and turn them into recognizable figures.


Figure 18.5 A line perpendicular to a spherical mirror is called the optic axis. The optic axis passes through the center of curvature of the mirror and the focal point. The optic axis is an axis of symmetry.


Rays of light from an object that is infinitely far away are parallel by the time we see them. Such parallel rays, after reflecting from a concave (or converging) spherical mirror, are bent so they converge on a single point. They pass through that point and then diverge from that point. After reflecting from a convex (or diverging) spherical mirror, such parallel rays are bent so they diverge as if they had come from a single point. If our eyes intercept these rays after their reflection they will look exactly as if they had originated from this point. For both mirrors, this point from which the light seems to have originated is called the focal point and is labeled by a capital letter F. The distance from the mirror to the focal point is the focal length and is labeled with a small letter f. We will adopt the convention that the focal length is positive for a concave mirror (f > 0) and is negative for a convex mirror (f < 0). These ideas are illustrated in Figure 18.6.



Figure 18.6 Rays of light parallel to the optic axis are focused at a single point by a spherical mirror. This point is called the focal point of the mirror. The distance from the focal point to the mirror is the focal length.


One note of caution; this description is only a first approximation. All that we have said is true as long as the size of the mirror is small compared to its radius of curvature. Another way of saying this is to limit ourselves to rays of light that lie close to the optic axis.

The focal length of a spherical mirror is one half the radius of curvature of the mirror,

f = R / 2

This equation also holds for convex mirrors as well as concave mirrors. By convention, the radius R is considered positive for concave or converging mirrors and is considered negative for convex or diverging mirrors. This means the focal length f will also be positive for concave or converging mirrors and negative for convex or diverging mirrors.

Light from an object infinitely far away, after reflection from a spherical mirror, behaves as if it had originated from this point. We call this point the focal point of the mirror. And we can say that an infinitely distant object has an image formed at the focal point of the mirror. For a concave or converging mirror, the rays actually pass through this point so we say a real image is formed. For a convex or diverging mirror, the rays do not actually pass through this point-this point is behind the mirror-so we say a virtual image is formed.



Figure 18.E The focal length of a spherical mirror is one-half its radius. Triangle CFM is an isosceles triangle and, for rays near the optic axis, distances CF, FM, and FV are equal so the focal length f = FV = R / 2.


Q: How are you able to see a virtual image?

A: Virtual images are readily seen. An image is called virtual when it can not be projected on a screen. The light coming from a virtual image did not actually pass through the position of the image.


Q: How can the focal point for a concave mirror be located behind the mirror where no light can reach?

A: For a concave mirror, also called a diverging mirror, the focal point describes the point from which initially parallel light appears to come after it has been reflected by the mirror. The light does not need to actually pass through this focal point.

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In the previous section we have seen that an object at infinity has an image at the focal point after reflection in a curved mirror; light leaves the mirror as if it had originated there. But what happens if the object is not at infinity? In understanding image formation, ray diagrams will prove very useful. There are three rays of light whose reflections are especially easy to handle or to understand. We call these the principal rays and will make frequent use of them. They are shown in Figure 18.7. We have already seen that a ray of light that comes in parallel to the optic axis will strike a concave mirror and be reflected toward the focal point-that is the definition of the focal point. From a convex mirror it will be reflected as if it had come from the focal point. Both of these are illustrated in the figure by ray 1. And a ray is reversible. A ray of light which passes through the focal point on its way to a concave mirror leaves parallel to the optic axis. A ray of light that is headed toward the focal point of a convex mirror leaves parallel to the optic axis. Both of these are illustrated in the figure by ray 2. The third principal ray is ray 3 that strikes a mirror at the optic axis. The optic axis is normal to the mirror so that ray leaves with an angle of reflection equal to the angle of incidence as indicated in the figure.


Figure 18.7 Principal rays will be of great use in understanding image formation.


Figure 18.8 shows an object placed some distance from a concave mirror. We have used an arrow for convenience. The tail of the arrow is on the optic axis. Simply from symmetry, we know that the image of this tail must also lie somewhere on the optic axis; whatever argument you can think of for the image's being above the axis is just as valid for its being below the axis. Therefore, all we need to do is to locate the image of the tip of the arrow. We can then draw in the rest of it just by dropping a line to the axis. While an infinite number of rays leave the tip of the arrow (or any other point on the arrow), we will concentrate on just the three principal rays shown in the first part of the figure. Having an image means that all of the rays that leave the object and are reflected from the mirror will pass through a single point that locates the image. If we construct a ray diagram using just two rays and find that they intersect we have determined the location of the image. However, it is prudent to confirm this with a third ray. If you draw three rays and they do not intersect at a single, common point then you know an error has occurred and you can track it down or begin again. Notice that the real image is inverted.




Figure 18.8 Ray diagrams locate a real image formed by a concave mirror. All the rays that leave a point on the object and are reflected from the mirror will pass through a common point if a real image is formed. For efficiency we concentrate on the three principal rays.


Figure 18.8 shows a real image produced. For a real image, the reflected light actually passes through the image. If a card or screen is placed at the location of the image an image will be projected on the card or screen. But a mirror can also produce a virtual image. Figure 18.9 shows additional examples of real images being produced by a concave mirror when the object is placed at various distances from the mirror. A far distant object produces a small, inverted, real image when reflected in a concave or converging mirror. Bringing the image in closer to the mirror enlarges the size of the image. When the object is at a distance of twice the focal length from the mirror, the image is the same size as the object (and still inverted). Moving the object in even closer makes the image larger than the object.






Figure 18.9 A far distant object produces a small, inverted, real image when reflected in a concave or converging mirror. Bringing the image in closer to the mirror enlarges the size of the image. When the object is at a distance of twice the focal length from the mirror, the image is the same size as the object (and still inverted). Moving the object in even closer makes the image larger than the object.


The magnification of a situation is the ratio of the image height to the object height. If the image is inverted, we will consider the image height negative so the magnification will be negative. If Figure 18.9, when the object is far away, the image is smaller in size and is upside down so the magnification is small and negative, like M = - 0.75 or M = - 0.50. When the object distance is two times the focal length, the image is the same size as the object and is upside down so the magnification is M = - 1.00. As the object moves closer than this, the image increases in size but remains upside down so the absolute value of the magnification continues to increase although the magnification is still negative.

Figure 18.10 shows two examples of producing a virtual image by a mirror. One is by a concave or converging mirror; the other, a convex or diverging mirror. Concave mirrors can produce either real or virtual images from a real object depending upon where the object is. If the object is beyond the focal point the mirror will produce a real image; if inside the focal point (between the mirror and the focal point), a virtual image. Convex mirrors produce only virtual images from real objects. A virtual image is very "real" in that you can see it quite clearly. But it can not be projected. If you place a card behind the mirror at the location of the image, there will be nothing projected upon it. Think again of your own virtual image in the bathroom mirror this morning. If you had held a card behind the mirror you would not have found your image projected upon it. While a virtual image can easily be seen, the light does not actually pass through the location of the image. That is precisely what is meant by a virtual image.



Figure 18.10 Ray diagrams locate a virtual image. All the rays that leave a point on the object and are reflected from the mirror leave as if they came from a common point when a virtual image is formed. We concentrate on the three principal rays because they are easy to handle.


Virtual images due to a reflection in a mirror will be right side up so the magnification will be positive in these cases. In Figure 18.10, the magnification is greater than one (M > 1.00) for the enlarged virtual image due to the concave mirror. The magnification is less than one (M < 1.00) for the reduced virtual image due to the convex mirror.

Ray diagrams are essential in understanding image formation. If they are carefully constructed all the dimensions can be accurately measured.



Figure 18.F A converging mirror can produce an upright and enlarged virtual image.


Figure 18.G A diverging mirror, such as the right outside rearview mirror on your car, produces an upright and reduced virtual image.


Q: What kind of images can a concave mirror produce?

A: Depending upon the distance the object is from the mirror, a concave mirror can produce a real or virtual image and the image can be enlarged or reduced in size. A shaving mirror or a make-up mirror is a good example of a concave mirror. The inside of a shiney spoon is another example of a concave mirror.


Q: What kind of images can a convex mirror produce?

A: For a real object, a convex mirror will always produce a virtual image that is reduced in size. The passenger-side rearview mirror on a car is a good example of a convex mirror. The outside of a shiney spoon is another example of a convex mirror.

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If you look at a swimmer standing in a clear pool, her legs may seem somehow bent compared to the rest of her body. Stick a pencil into a glass of water, as sketched in Figure 18.11, and the formerly straight pencil suddenly looks bent at the water's surface.




Figure 18.11 Long objects immersed in water seem to be bent at the water's surface.


A very striking example of this is shown in Figure 18.12 where a coin has been placed on the bottom of a bowl. The coin is not visible at first. But if water is poured into the cup the coin appears to be raised higher and, thus, is visible.



Figure 18.12 A coin on the bottom of this empty bowl can not be seen from this angle of view. Pouring water into the bowl makes an image of the coin appear above the actual coin and, thus, the coin's image is visible from this angle of view.


Another example of this which you may have already noticed is illustrated in Figure 18.13. There a fish is swimming in an aquarium. When viewed through the top, the fish seems closer to the top than it really is; when viewed through the front, the fish seems closer to the front than it really is. This means you can see two fish although there is really only one. If you position yourself carefully you may even be able to see three fish! You are seeing three images of the same fish.



Figure 18.13 Fish in an aquarium appear closer to the surface than they really are.


How does a plane surface like this produce an image? Figure 18.14 shows an object O located at a distance do below the surface of the water. One ray of light is shown going straight up. This ray of light strikes the water-air interface normal or perpendicular to the surface and passes into the air without being bent. A second ray strikes the water-air interface a horizontal distance x away with an angle of incidence of i and is refracted so that it leaves with an angle of refraction of r. This ray and the first one now appear to have originated from position I-the image-located at distance di below the surface. This image distance di is the apparent depth of the thing we are looking at.



Figure 18.14 The change of index of refraction in passing from water into air is responsible for the apparent depth.


As you can see from the figure, the apparent depth or image distance is going to be less than the actual depth or object distance. For rays at other angles or for other values of x, the image is not located at the same place. You can even notice this in looking at fish in an aquarium. At some angles the images of a fish received by your two eyes may not coincide precisely and the fish may appear blurred or simply "strange" or you may even feel slightly dizzy. This lack of a well-defined position for the image is an example of astigmatism.


Q: When you look at a fish in an aquarium, it is closer to the glass or farther from the glass than it appears to be?

A: The fish is farther away than it appears to be.

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Light that is refracted by a lens behaves similar to light that is reflected by a curved mirror. Figure 18.15 shows parallel rays of light coming into two lenses, very much as we earlier looked at parallel light coming into curved mirrors. A lens that is thicker in the middle is called a converging lens while a lens that is thinner in the middle is called a diverging lens; we will understand why momentarily. As with mirrors, the symmetry axis is called the optic axis; it is perpendicular to the lens. Again, for consistency, we will always start with light coming in from the left. Remember, parallel light is light that originated infinitely far away.


Figure 18.15 A converging lens refracts parallel light so that it converges on a point. A diverging lens refracts parallel light so that it appears to have come from a point. In both cases, that point is called the focal point and is labeled F in the diagram.


Light is bent at both surfaces of a lens. We will restrict our attention to thin lenses; lenses whose thickness is much smaller than any other dimension of interest in the problem. Then we may pretend the light undergoes a single refraction at the position of the lens. Rays of light parallel to the optic axis, as in Figure 18.15, strike a converging lens and are bent or refracted; these rays converge on a single, common point. By symmetry this point must lie on the optic axis. It is called the focal point and is labeled with a capital F. The distance between the focal point and the lens is called the focal length and is labeled with a lower case f; the focal length is positive for a converging lens (f > 0). These same rays of light parallel to the optic axis, as in Figure 18.15 again, strike a diverging lens and are bent or refracted so they appear to diverge from a single, common point. By symmetry this point must lie on the optic axis. It is called the focal point and is labeled with a capital F. The distance between the focal point and the lens is called the focal length and is labeled with a lower case f; it is negative for a diverging lens (f < 0). This should appear very similar to our earlier discussion of concave and convex mirrors. There is one difference. A lens can be turned around so there is really a focal point on both sides of a lens with the same value of f on either side.


Q: What kind of mirror corresponds to a converging lens?

A: A concave mirror is quite similar to a converging lens.


Q: What kind of mirror corresponds to a diverging lens?

A: A convex mirror is quite similar to a diverging lens.

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As with mirrors, we will find it extremely useful to look at three principal rays because it is easy to predict their behavior. They are sketched in Figure 18.16 and are very similar to the principal rays we have already seen in our study of mirrors. Ray 1 is an incoming ray parallel to the optic axis. For a converging lens, ray 1 is bent or refracted and passes through the focal point on the other side of the lens as shown. For a diverging lens, ray 1 is bent or refracted away from the optic axis and leaves the lens as though it originated from the focal point on the incoming side of the lens. Ray 2 is just the reversal of this ray. For a converging lens, ray 2 has passed through the focal point in front of the lens. It is bent or refracted, leaving the lens parallel to the optic axis. For a diverging lens, ray 2 is headed toward the focal point on the opposite side of the lens. It, too, is bent or refracted, leaving the lens parallel to the optic axis. Ray 3 passes through the center of the lens undeviated. The reason is the two surfaces of the lens at the optic axis are parallel!


Figure 18.16 Three principal rays will be very useful in understanding the formation of images.


We will now do with lenses what we earlier did with mirrors-their role in image formation is similar. Figure 18.17 shows an object placed some distance from a converging lens. We have again used a simple arrow. The tail of the arrow is again on the optic axis. An infinite number of rays leave the tip of the arrow but we will concentrate on only the three principal rays we have just defined. Having a real image means that all of the rays that leave the object and go through the lens will pass through a single point that locates the image. Figure 18.18 shows more ray diagrams for real images produced by various object distances.



Figure 18.17 Ray diagrams locate a real image. All the rays that leave a point on the object and go through the lens will pass through a common point if a real image is formed. For efficiency and convenience we concentrate on the three principal rays.





Figure 18.18 As the object is moved in from far away, the real image produced becomes larger, and is upside down. When the object distance is twice the focal length, the size of the image is the same as the size of the object; the image is still real and upside down. As the object moves closer to the focal point, the image size becomes larger than the object size.


Just as with mirrors, the magnification is the ratio of the image height to the object height. If Figures 18.17 and 18.18, when the object is far away, the image is smaller in size and is upside down so the magnification is small and negative, like M = - 0.50 or M = - 0.75. When the object distance is twice the focal length, the image is the same size as the object and is upside down so the magnification is M = - 1.00. As the object moves closer than this, the image increases in size but remains upside down so the absolute value of the magnification continues to increase although the magnification is still negative.

Figures 18.17 and 18 show a real image produced as long as the object distance is greater than the focal length. For a real image, the light actually passes through the image. If a card or screen is placed at the location of the image, an image will be projected on the card or screen. But a lens can also produce a virtual image. Figure 18.20 shows two examples of a lens producing a virtual image. One is with a converging lens; the other, a diverging lens. Converging lenses can produce either real or virtual images from a real object depending upon where the object is. If the object is beyond the focal point the converging lens will produce a real image; if the object is inside the focal point, a virtual image will be produced. Diverging lenses produce only virtual images from real objects. The virtual image, as before, can be seen quite clearly and looks like the object. But it can not be projected; that is what a virtual image means.

Figure 18.19 When the object is between the converging lens and the focal point, a virtual image is produced.


Figure 18.20 Ray diagrams locate a virtual image. All the rays that leave a point on the object and go through the lens leave as if they came from a common point when a virtual image is formed. We concentrate on the three principal rays because they are easy to handle. The dotted lines indicate where the light rays appear to have come from.


For virtual images formed by a single lens, the virtual image will always be right side up so the magnification will be positive. In the first examples of Figure 18.20, with a virtual image formed by a converging lens, the virtual image is larger than the object so the magnification is greater than one (M > 1.00). In the second example, with a virtual image formed by a diverging lens, the virtual image is smaller than the object so the magnification is smaller than one (M < 1.00).




Figure 18.H Converging lenses can produce enlarged or reduced images, depending upon the distance of the object from the lens.


Q: What kind of images can be produced by a converging lens?

A: Figure 18.18 illustrates several real images produced by a converging lens. By varying the object distance, the image may be reduced or enlarged in size. Figure 18.19 illustrates a virtual image produced by a converging lens.


Q: What kind of images can be produced by a diverging lens?

A: For a real object, a diverging lens produces only a virtual image that is reduced in size; this is illustrated in Figure 18.20.

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Lens and mirror aberrations refer to defects or failure to obey the simple image formation rules we have outlined here. What we have discussed may be called thin lens theory or small mirror theory. Small lenses and mirrors may be made with spherical surfaces which are easy to produce. As lenses or mirrors get larger, image formation is not as clear. We will describe some of the things that can be done to improve the image.

You will remember the spectrum that is produced when light goes through a prism. The prism bends or refracts the light-but bends or refracts light of different colors different amounts. This is fine if you want the colors of a spectrum. In that case, the more dispersion there is the better things are. But if you are trying to produce a sharp image, any dispersion at all will spoil it.

Figure 18.21 shows parallel white light being focused by a lens. Just as with a prism, violet light will be bent more than red light-and all the other colors of the spectrum will lie between these two extremes. This means the violet light will be focused closer to the lens and red light will be focused farther away. If we place a card at the location of the violet focus, we will see colored shadows or halos around it with red on the outside. If we place a card farther away, at the location of the red focus, we will see colored shadows or halos around the central red dot with violet on the outside. This problem with image formation is called chromatic aberration. We can correct for chromatic aberration by replacing a simple, single-element lens with a lens made of two or more pieces of glass that have different dispersion characteristics.



Figure 18.21 White light focused by a lens will not focus to a tiny white image but will have halos or colored shadows around the image due to the dispersion of colors in the lens. This is an example of chromatic aberration.


We can correct for chromatic aberration by replacing a simple, single-element lens with a lens made of two or more pieces of glass that have different dispersion characteristics. Modern camera lenses are always multi-element designs to correct for this and other aberrations. Mirrors do not have chromatic aberration; light of different colors still behave exactly the same under reflection. For several reasons, most astronomical telescopes use mirrors instead of lenses. One very important reason for this is that reflecting telescopes do not need to be corrected for color.

As larger and larger spherical lenses or larger and larger spherical mirrors are used, we find that light that comes in farther and farther from the optic axis is bent more and focused closer to the lens or mirror. This is shown in Figures 18.22 and 18.23. This is known as spherical aberration and is simply the result of geometry. For a mirror, this aberration may be corrected if the mirror is ground to be part of a parabola of revolution instead of a sphere. Such mirrors are called parabolic mirrors and all reflecting astronomical telescopes will have parabolic mirrors. The problem is the same for spherical lenses-lenses whose surfaces are parts of spheres. The problem may be corrected in lens design by going to a multi-element lens with different elements made of glass with differing characteristics. Another solution with lenses, is to go to parabolic lens elements which are usually known as aspheric lenses. These problems become greater as the lens gets larger and this explains why large aperture lenses which are great for low-light conditions are so very much more expensive than small aperture lenses which take great pictures outdoors in the Sun.




Figure 18.21 Light near the optic axis is brought to a single, common focus. But as light comes into a spherical mirror farther from the optic axis, it is bent more and crosses the optic axis closer to the mirror. This spherical aberration can be corrected by using a parabolic mirror.


Figure 18.22 Just as with a mirror, a lens will bend light more as the light arrives farther from the optic axis. This, too, is spherical aberration.


Q: Almost all large, modern astronomical telescopes are reflectors that use mirrors rather than refractors that use lenses. Why might that be?

A: Mirrors do not have chromatic aberration. A mirrors has only a single surface to grind instead of two for a lens. Mirrors can also be supported from the back by something like a steel frame.

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We see an object by the light that comes from that object. An image is produced when the light appears to come from some other place. If the light actually passes through the position of the image, we call it a real image. A real image may be projected onto a screen or a card. If the light does not pass through the position of the image, we call it a virtual image.

A plane mirror produces a virtual image located as far behind the mirror as the object is in front of the mirror. The image is the same size as the object so the magnification of a plane mirror is 1.0 .

Incoming, parallel rays of light will be reflected by a spherical, concave mirror so they pass through a point called the focal point. Such light reflected by a spherical, convex mirror will appear to have originated at a point called the focal point.

Image formation may be understood by close observation of three principal rays that are easy to follow or to draw. All other rays will be bent (reflected by a mirror or refracted by a lens) to pass through the image position established by these principal rays.

Incoming, parallel rays of light will be refracted by a converging lens so they pass through a point called the focal point. Such light reflected by a diverging lens will appear to have originated at a point called the focal point.

The distance from the mirror or lens to the focal point is the focal length.

A concave mirror or a converging lens can produce either a real image or a virtual image depending upon the position of the object. The real image will be upside down and the virtual image will be right side up. The size of the image may be smaller or larger than the size of the object according to the position of the object.

A convex mirror or a diverging lens can produce only a right side up, virtual image of reduced size.