Images by Reflection
From Lm
Optics by Benjamin Crowell
This is a version of the book that has been converted to wiki format from the original at lightandmatter.com, where it is available in PDF, HTML, and LaTeX formats. The version at lightandmatter.com is the one that is actively maintained by the author. Please see this page for information about the purpose of this wiki version.
Chapter 2 - Images by Reflection
- Narcissus, by Michelangelo Caravaggio, ca. 1598.
Images by Reflection
Infants are always fascinated by the antics of the Baby in
the Mirror. Now if you want to know something about mirror
images that most people don't understand, try this. First
bring this page closer and closer to your eyes, until you
can no longer focus on it without straining. Then go in the
bathroom and see how close you can get your face to the
surface of the mirror before you can no longer easily focus
on the image of your own eyes. You will find that the
shortest comfortable eye-mirror distance is much less than
the shortest comfortable eye-paper distance. This demonstrates
that the image of your face in the mirror acts as if it had
depth and existed in the space behind the mirror. If
the image was like a flat picture in a book, then you
wouldn't be able to focus on it from such a short distance.
In this chapter we will study the images formed by flat and
curved mirrors on a qualitative, conceptual basis. Although
this type of image is not as commonly encountered in
everyday life as images formed by lenses, images formed by
reflection are simpler to understand, so we discuss them
first. In chapter 3 we will turn to a more mathematical
treatment of images made by reflection. Surprisingly, the
same equations can also be applied to lenses, which are
the topic of chapter 4.
a / An image formed by a mirror.
c / The praxinoscope.
Contents |
A Virtual Image
We can understand a mirror image using a ray diagram. Figure a shows several light rays, 1, that originated by diffuse reflection at the person's nose. They bounce off the mirror, producing new rays, 2. To anyone whose eye is in the right position to get one of these rays, they appear to have come from a behind the mirror, 3, where they would have originated from a single point. This point is where the tip of the image-person's nose appears to be. A similar analysis applies to every other point on the person's face, so it looks as though there was an entire face behind the mirror. The customary way of describing the situation requires some explanation:
- Customary description in physics: There is an image of the face behind the mirror.
- Translation: The pattern of rays coming from the mirror is
exactly the same as it would be if there were a face behind the mirror. Nothing is really behind the mirror.
This is referred to as a virtual image, because the
rays do not actually cross at the point behind the mirror.
They only appear to have originated there.
self-check:
Imagine that the person in figure a moves his face down
quite a bit --- a couple of feet in real life, or a few
inches on this scale drawing. The mirror stays where it is. Draw a new ray diagram. Will
there still be an image? If so, where is it visible from?
(answer in the back of the PDF version of the book)
The geometry of specular reflection tells us that rays 1 and
2 are at equal angles to the normal (the imaginary
perpendicular line piercing the mirror at the point of
reflection). This means that ray 2's imaginary continuation,
3, forms the same angle with the mirror as ray 3. Since
each ray of type 3 forms the same angles with the mirror
as its partner of type 1, we see that the distance of the
image from the mirror is the same as that of the actual face from
the mirror, and it lies directly across from it. The image
therefore appears to be the same size as the actual face.
b / Example 1.
Example 1: An eye exam
Figure b shows a typical setup in an optometrist's examination room. The patient's vision is supposed to be tested at a distance of 6 meters (20 feet in the U.S.), but this distance is larger than the amount of space available in the room. Therefore a mirror is used to create an image of the eye chart behind the wall.
Example 2: The Praxinoscope
Figure c shows an old-fashioned device called
a praxinoscope, which displays an animated picture when spun.
The removable strip of paper with the pictures printed on it has twice
the radius of the inner circle made of flat mirrors, so each picture's
virtual image is at the center. As the wheel spins, each picture's image
is replaced by the next, and so on.
”Discussion Question”
◊
The figure shows an object that is off to one side of a
mirror. Draw a ray diagram. Is an image formed? If so, where
is it, and from which directions would it be visible?
-
d / An image formed by a curved mirror.
e / The image is magnified by the same factor in depth and in its other dimensions.
Curved Mirrors
An image in a flat mirror is a pretechnological example: even animals can look at their reflections in a calm pond. We now pass to our first nontrivial example of the manipulation of an image by technology: an image in a curved mirror. Before we dive in, let's consider why this is an important example. If it was just a question of memorizing a bunch of facts about curved mirrors, then you would rightly rebel against an effort to spoil the beauty of your liberally educated brain by force-feeding you technological trivia. The reason this is an important example is not that curved mirrors are so important in and of themselves, but that the results we derive for curved bowl-shaped mirrors turn out to be true for a large class of other optical devices, including mirrors that bulge outward rather than inward, and lenses as well. A microscope or a telescope is simply a combination of lenses or mirrors or both. What you're really learning about here is the basic building block of all optical devices from movie projectors to octopus eyes. Because the mirror in figure d is curved, it bends the rays back closer together than a flat mirror would: we describe it as converging. Note that the term refers to what it does to the light rays, not to the physical shape of the mirror's surface . (The surface itself would be described as concave. The term is not all that hard to remember, because the hollowed-out interior of the mirror is like a cave.) It is surprising but true that all the rays like 3 really do converge on a point, forming a good image. We will not prove this fact, but it is true for any mirror whose curvature is gentle enough and that is symmetric with respect to rotation about the perpendicular line passing through its center (not asymmetric like a potato chip). The old-fashioned method of making mirrors and lenses is by grinding them in grit by hand, and this automatically tends to produce an almost perfect spherical surface. Bending a ray like 2 inward implies bending its imaginary continuation 3 outward, in the same way that raising one end of a seesaw causes the other end to go down. The image therefore forms deeper behind the mirror. This doesn't just show that there is extra distance between the image-nose and the mirror; it also implies that the image itself is bigger from front to back. It has been magnified in the front-to-back direction. It is easy to prove that the same magnification also applies to the image's other dimensions. Consider a point like E in figure e. The trick is that out of all the rays diffusely reflected by E, we pick the one that happens to head for the mirror's center, C. The equal-angle property of specular reflection plus a little straightforward geometry easily leads us to the conclusion that triangles ABC and CDE are the same shape, with ABC being simply a scaled-up version of CDE. The magnification of depth equals the ratio BC/CD, and the up-down magnification is AB/DE. A repetition of the same proof shows that the magnification in the third dimension (out of the page) is also the same. This means that the image-head is simply a larger version of the real one, without any distortion. The scaling factor is called the magnification, M. The image in the figure is magnified by a factor M=1.9. Note that we did not explicitly specify whether the mirror was a sphere, a paraboloid, or some other shape. However, we assumed that a focused image would be formed, which would not necessarily be true, for instance, for a mirror that was asymmetric or very deeply curved.
A Real Image
If we start by placing an object very close to the mirror,
f/1, and then move it farther and farther away, the image at
first behaves as we would expect from our everyday
experience with flat mirrors, receding deeper and deeper
behind the mirror. At a certain point, however, a dramatic
change occurs. When the object is more than a certain
distance from the mirror, f/2, the image appears upside-down
and in front of the mirror.
f / 1. A virtual image. 2. A real image. As you'll verify in homework problem 6, the image is upside-down
Here's what's happened. The mirror bends light rays inward, but when the object is very close to it, as in f/1, the rays coming from a given point on the object are too strongly diverging (spreading) for the mirror to bring them back together. On reflection, the rays are still diverging, just not as strongly diverging. But when the object is sufficiently far away, f/2, the mirror is only intercepting the rays that came out in a narrow cone, and it is able to bend these enough so that they will reconverge. Note that the rays shown in the figure, which both originated at the same point on the object, reunite when they cross. The point where they cross is the image of the point on the original object. This type of image is called a real image, in contradistinction to the virtual images we've studied before. The use of the word “real” is perhaps unfortunate. It sounds as though we are saying the image was an actual material object, which of course it is not. The distinction between a real image and a virtual image is an important one, because a real image can projected onto a screen or photographic film. If a piece of paper is inserted in figure f/2 at the location of the image, the image will be visible on the paper (provided the object is bright and the room is dark). Your eye uses a lens to make a real image on the retina. self-check: Sketch another copy of the face in figure f/1, even farther from the mirror, and draw a ray diagram. What has happened to the location of the image? (answer in the back of the PDF version of the book)
g / A Newtonian telescope being used with a camera.
h / A Newtonian telescope being used for visual rather than photographic observing. In real life, an eyepiece lens is normally used for additional magnification, but this simpler setup will also work.
Images of Images
If you are wearing glasses right now, then the light rays
from the page are being manipulated first by your glasses
and then by the lens of your eye. You might think that it
would be extremely difficult to analyze this, but in fact it
is quite easy. In any series of optical elements (mirrors or
lenses or both), each element works on the rays furnished by
the previous element in exactly the same manner as if the
image formed by the previous element was an actual object.
Figure g shows an example involving only mirrors. The
Newtonian telescope, invented by Isaac
Newton, consists of a large curved mirror, plus a
second, flat mirror that brings the light out of the tube.
(In very large telescopes, there may be enough room to put a
camera or even a person inside the tube, in which case the
second mirror is not needed.) The tube of the telescope is
not vital; it is mainly a structural element, although it
can also be helpful for blocking out stray light. The lens
has been removed from the front of the camera body, and is
not needed for this setup. Note that the two sample rays
have been drawn parallel, because an astronomical telescope
is used for viewing objects that are extremely far away.
These two “parallel” lines actually meet at a certain
point, say a crater on the moon, so they can't actually be
perfectly parallel, but they are parallel for all practical
purposes since we would have to follow them upward for a
quarter of a million miles to get to the point where they intersect.
The large curved mirror by itself would form an image I, but
the small flat mirror creates an image of the image, I'. The
relationship between I and I' is exactly the same as it
would be if I was an actual object rather than an image: I
and I' are at equal distances from the plane of the mirror,
and the line between them is perpendicular to the plane of the mirror.
One surprising wrinkle is that whereas a flat mirror used by
itself forms a virtual image of an object that is real, here
the mirror is forming a real image of virtual image I. This
shows how pointless it would be to try to memorize lists of
facts about what kinds of images are formed by various
optical elements under various circumstances. You are better
off simply drawing a ray diagram.
Although the main point here was to give an example of an
image of an image, figure h also shows an interesting case where we
need to make the distinction between magnification
and angular magnification.
If you are looking at the moon through this telescope, then
the images I and I' are much smaller than the actual
moon. Otherwise, for example, image I would not fit inside
the telescope! However, these images are very close to your
eye compared to the actual moon. The small size of the image
has been more than compensated for by the shorter distance.
The important thing here is the amount of angle
within your field of view that the image covers, and it is
this angle that has been increased. The factor by which it
is increased is called the angular magnification, Ma.
i / The angular size of the flower depends on its distance from the eye.
”Discussion Questions” ◊ Locate the images of you that will be formed if you stand between two parallel mirrors.
-
◊
Locate the images formed by two perpendicular mirrors, as
in the figure. What happens if the mirrors are not
perfectly perpendicular?
-
◊
Locate the images formed by the periscope.
-
Summary
Vocabulary
real image — a place where an object appears to be, because the rays diffusely reflected from any given point on the object have been bent so that they come back together and then spread out again from the new point virtual image — like a real image, but the rays don't actually cross again; they only appear to have come from the point on the image converging — describes an optical device that brings light rays closer to the optical axis diverging — bends light rays farther from the optical axis magnification — the factor by which an image's linear size is increased (or decreased) angular magnification — the factor by which an image's apparent angular size is increased (or decreased) concave — describes a surface that is hollowed out like a cave convex — describes a surface that bulges outward
Notation
M — the magnification of an image Ma — the angular magnification of an image
Summary
A large class of optical devices, including lenses and flat and curved mirrors, operates by bending light rays to form an image. A real image is one for which the rays actually cross at each point of the image. A virtual image, such as the one formed behind a flat mirror, is one for which the rays only appear to have crossed at a point on the image. A real image can be projected onto a screen; a virtual one cannot. Mirrors and lenses will generally make an image that is either smaller than or larger than the original object. The scaling factor is called the magnification. In many situations, the angular magnification is more important than the actual magnification.
j / Problem 7.
k / Problem 9.
Homework Problems
1. (answer check available at lightandmatter.com) A man is walking at 1.0 m/s directly towards a flat mirror. At what speed is his separation from his image decreasing?
2. If a mirror on a wall is only big enough for you to see yourself from your head down to your waist, can you see your entire body by backing up? Test this experimentally and come up with an explanation for your observations, including a ray diagram. Note that when you do the experiment, it's easy to confuse yourself if the mirror is even a tiny bit off of vertical. One way to check yourself is to artificially lower the top of the mirror by putting a piece of tape or a post-it note where it blocks your view of the top of your head. You can then check whether you are able to see more of yourself both above and below by backing up.
3. In this chapter we've only done examples of mirrors with hollowed-out shapes (called concave mirrors). Now draw a ray diagram for a curved mirror that has a bulging outward shape (called a convex mirror). (a) How does the image's distance from the mirror compare with the actual object's distance from the mirror? From this comparison, determine whether the magnification is greater than or less than one. (b) Is the image real, or virtual? Could this mirror ever make the other type of image?
4. As discussed in question 3, there are two types of curved mirrors, concave and convex. Make a list of all the possible combinations of types of images (virtual or real) with types of mirrors (concave and convex). (Not all of the four combinations are physically possible.) Now for each one, use ray diagrams to determine whether increasing the distance of the object from the mirror leads to an increase or a decrease in the distance of the image from the mirror. Draw BIG ray diagrams! Each diagram should use up about half a page of paper. Some tips: To draw a ray diagram, you need two rays. For one of these, pick the ray that comes straight along the mirror's axis, since its reflection is easy to draw. After you draw the two rays and locate the image for the original object position, pick a new object position that results in the same type of image, and start a new ray diagram, in a different color of pen, right on top of the first one. For the two new rays, pick the ones that just happen to hit the mirror at the same two places; this makes it much easier to get the result right without depending on extreme accuracy in your ability to draw the reflected rays.
5. If the user of an astronomical telescope moves her head closer to or farther away from the image she is looking at, does the magnification change? Does the angular magnification change? Explain. (For simplicity, assume that no eyepiece is being used.)
6. In figure f in on page 34, only the image of my forehead was located by drawing rays. Either photocopy the figure or download the book and print out the relevant page. On this copy of the figure, make a new set of rays coming from my chin, and locate its image. To make it easier to judge the angles accurately, draw rays from the chin that happen to hit the mirror at the same points where the two rays from the forehead were shown hitting it. By comparing the locations of the chin's image and the forehead's image, verify that the image is actually upside-down, as shown in the original figure.
7. The figure shows four points where rays cross. Of these, which are image points? Explain.
8. Here's a game my kids like to play. I sit next to a sunny window, and the sun reflects from the glass on my watch, making a disk of light on the wall or floor, which they pretend to chase as I move it around. Is the spot a disk because that's the shape of the sun, or because it's the shape of my watch? In other words, would a square watch make a square spot, or do we just have a circular image of the circular sun, which will be circular no matter what?
9. Suppose we have a polygonal room whose walls are mirrors, and there a pointlike
light source in the room. In most such examples, every point in the room ends up
being illuminated by the light source after some finite number of reflections.
A difficult mathematical question, first posed in the middle of the last century,
is whether it is ever possible to have an example in which the whole room is
not illuminated. (Rays are assumed to be absorbed if they strike exactly at a vertex of the polygon,
or if they pass exactly through the plane of a mirror.)
The problem was finally solved in 1995 by G.W. Tokarsky,
who found an example of a room that was not illuminable from a certain point.
Figure 9 shows a slightly simpler example found two years later
by D. Castro. If a light source is placed at either of the locations shown with
dots, the other dot remains illuminated, although every other point is lit up.
It is not straightforward to prove rigorously that Castro's solution has this property.
However, the plausibility of the solution can be demonstrated as follows.
Suppose the light source is placed at the right-hand dot. Locate all the images
formed by single reflections. Note that they form a regular pattern. Convince yourself
that none of these images illuminates the left-hand dot. Because of the regular
pattern, it becomes plausible that even if we form images of images, images of images
of images, etc., none of them will ever illuminate the other dot.
There are various other versions of the problem, some of which remain unsolved.
The book by Klee and Wagon gives a good introduction to the topic, although it
predates Tokarsky and Castro's work.
References:
G.W. Tokarsky, “Polygonal Rooms Not Illuminable from Every Point.” Amer. Math. Monthly 102, 867-879, 1995.
D. Castro, “Corrections.” Quantum 7, 42, Jan. 1997.
V. Klee and S. Wagon, Old and New Unsolved Problems in Plane Geometry and Number Theory. Mathematical
Association of America, 1991.
