Image Formation by Plane and Spherical Mirrors

Purpose

• •
  To become familiar with the nature of the images formed by plane and spherical mirrors.
• •
  To learn to distinguish between real and virtual images.
• •
  To discover the relationships among object position, image position, focal length, magnification, and the radius of curvature of plane, converging, and diverging mirrors.
• •
  To become familiar with sign conventions and how and why they're used.

Equipment

Simulation and Tools

Open the Virtual Optical Bench simulation to do this lab.

Explore the Apparatus/Theory

Open the Virtual Optical Bench simulation. The virtual Optical Bench Apparatus simulates the formation of images by plane mirrors and by converging and diverging mirrors and lenses. We'll focus on mirrors in this lab. Were it not for the development of eyes, there wouldn't be a lot to say about the interaction of light with reflective and refractive media except that it sometimes reflects and sometimes refracts, and sometimes does both. But because of the wondrous way that our eyes can take the light leaving an object and reconstruct a replica of that object on our retinas which can then inform the brain of the original object, we have developed the powerful study of optics. Everyone is very familiar with the plane mirror and has had at least some experience with curved mirrors and lenses in a number of settings. From our extensive experience with our daily primping we feel very confident with the use of plane mirrors. But when we first try to use a car's rear-view mirrors, we find that we a have a lot more to learn. When you start up the Optical Bench Apparatus, it should look like Figure 1. A dashed line has been added in Figure 1 that separates two parts of the apparatus. Below the line is the Ray Optics Tool, where light rays are used to show the geometry involved in the formation of images. Above the dashed line is the Optical Bench and its object and image tools and screens. The Ray Optics Tool serves another important function: it controls the Optical Bench. Let's try it with the concave (converging) mirror. Move your pointer over each of the five lenses and mirrors. The info box will identify each one and tell you how to switch choices. Click and drag the plane mirror until your pointer is over the concave mirror (below on the Ray Optics Tool) and release. The concave mirror should be replaced by the plane mirror. If not, try again. Remember, your pointer needs to be over the device being replaced when you release the button. Now swap them back. We want to work with the concave mirror. To the left of the mirror are two vertical arrows. The one pointing upward with the small light blue drag handle on the bottom is the Object Arrow. It represents whatever object we're working with. As suggested by Figure 2, it emits light in all directions. Actually light is emitted in this way from every point on the object. We display only two (sometimes three) of these rays. Drag the Object Arrow from side to side as far as you can. You'll see that the rays are continually redrawn, and the other arrow, the Image Arrow, can move, resize and flip vertically. There's also action on the Optical Bench. Notice how the little box with the cow on the front moves in sync with the Object Arrow on the Ray Optics Tool. That's the Optical Bench's object. It's a slide illuminated from behind. Thus it emits light rays in all directions. Leave the object somewhere near the back end of the Optical Bench. You've seen that the Image Arrow on the Ray Optics Tool moves, flips, and changes size. To let us see the image on the Optical Bench, we allow light from the object to illuminate a screen. To form an image on the translucent screen, it must be moved to where the image is located. You'll examine that concept in detail soon. The Screen Position Tool on the Ray Optics Tool is used to move the screen on the optical bench. It's a vertical gray line with a semicircular handle at one end. Drag it from side to side and notice how the screen moves on the Optical bench. Also notice that when it's to the left of the concave mirror a smudge of light appears on the screen. As you move the object, the smudge looks more or less like an inverted cow, depending on the location of the object. Leave the screen somewhere between the mirror and the object. If you have for some reason changed the Mirror/Lens Number to something other than 1, change it to 1 now. You'll want to leave it there until you start to work with unknowns in section IIA. Drag the object to the point labeled 2F on the left end of the ray tool. (You can also move the object with your left and right keyboard keys.) Drag the Screen Position Tool to the same 2F point. Ta da! A nicely focused, upside down cow appears on the frosted glass. An enlarged version also appears in the Image Close-up Screen next to the Object Close-up Screen. Each screen has a grid that can be moved around to make size measurements. Each grid square is 1 cm on a side. Also, try zooming in on the cow object and image on the bench. They should be side-by-side and equal in height. You can right click (Mac: CTRL click) as many as six times to zoom way in. "Show all" will take you back to 100%. You can trade in your cow for a Taj Mahal, a fast red car, or the moon, depending on your mood. The text of this lab generally refers to the cow as the object until the last part of the lab where we'll trade it in for the Taj Mahal. "Upside-down cow" carries a more serious tone than "upside-down moon."

Procedure

Please print the worksheet for this lab. You will need this sheet to record your data.

I. Plane Mirrors; Virtual and Real Images

We're going to look at the behaviors of our three mirrors to learn the nature of the images they form as well as the geometry behind the image formation. Then we'll look at the algebraic descriptions of this geometry. We'll start with the plane mirror. Drag the plane mirror in place on the Ray Optics Tool. It should look something like Figure 4 minus a few added details. The lower, green-colored rays illustrate this. A ray leaves the top of the Object Arrow, strikes the mirror at some point (its midpoint in Figure 4) and then reflects down to the left. The dashed line conveniently helps us see that the angles are indeed equal. There are two other ray pairs in Figure 4 that illustrate the same behavior. The orange rays at the top clearly show the same effect. But the purple rays in the middle are less obvious. A ray leaves the top of the object and hits the mirror normally, that is, at a right angle to the mirror. Thus it reflects straight back on itself. The angles of incidence and reflection are both zero in this case. Remember, there are many more rays from the object which hit the mirror. We just need at least two for our work. Behind the mirror (on the right), we see the image. Its height, h_i, is clearly equal to that of the object, h_o, and with the draggable ruler you can verify that the image is as far behind the mirror, d_i, as the object is in front, d_o. This becomes even more obvious when you move the object toward and away from the mirror. (We'll discuss the way that the light seems to bounce off the back surface of the mirror in the next section.) We clearly see our image move toward and away from our body when we move toward and away from a plane mirror. But there's really nothing there. An image is not something that has any solid reality in the physical world like the object it mimics. If this is so, how can it have a location? To understand the location of images, we first need to understand how we locate actual objects when we look at them. In Figure 5a we've added a pair of eyes. These belong to a person, seen from above, who is located between the object and the mirror. That is, you're looking down on the top of this person's head. This person is ignoring the mirror for now. Be sure to put yourself in these figures. For the plane mirrors you can actually reenact events by standing in front of a bathroom mirror. Note that the eyes are pointed toward the top of the object so that a different ray from the object enters each eye. Think about it. At any given instant you are actually just looking at one small region of space. For example, you can't look at a complete sheet of paper. Try it. Your eyes will start madly searching all over it. When you read this sentence you are only looking at a little bit at a time. The rest of it is handled very differently than the part you're actually focusing on. Consider how much work it would be to actually see all of a brick wall at once and have to keep actively seeing all of it to "keep it there." We actually see something more like "brickness." Sorry, I didn't mean to alarm you. Where were we? The two little eyes were pointed at the top of the object so that they can let the two rays from just that part of the object reach your retina. Most of the time, your eyes are pointed at the same thing. Your eyes can't look at two different things at the same time. Instead, at a given instant, they both "agree" to focus their attention on one thing, the tip of the Object Arrow in this case. In addition, based on the amount the eyes have to turn relative to straight ahead, we compute the distance to the object and adjust the focus of the eye's lenses to properly focus the light. Amazing. You probably remember learning to do that by examining your feet when you were lying on your back as a baby. Try it. Look at something far away. Now, focus on something close by. You can feel your muscles straining to turn your eyes away from their relaxed parallel alignment. (A good vacation spot always provides an ample supply of interesting things to look at with relaxed eyes. An ocean, for example!) Again, this is the process of locating an object, that is, assigning it a location. In Figure 5b, the person has turned around and is now facing the mirror. Now light is allowed to reflect from the mirror before entering the eyes. (Magically passing through the person's head.) Note that a different pair of rays, from the many available, is used this time. In Figure 5b, the reflected rays entering the eyes look no different than the direct rays used in Figure 5a. They're all just rays of light coming from a specific point. The mind does the same triangulation. But this time it notices that your eyes don't have to point inward so much and concludes that the more parallel rays are coming from a source farther away and behind the mirror. All the information that's needed for the mind to conjure up an impression of the existence of the object is available. We say that the person is now looking at (locating) an image of the object. When the rays entering the eye are in any way redirected from their straight line path away from the object, we say that the observer is looking at an image. (Even mirages are images.) In this particular case, if you extrapolate back to the apparent source of the light, the image, you find that the rays never passed through that location (behind the mirror). We call this a virtual image. Virtual means "in effect, but not in reality." Plane mirrors always form virtual images—the image will always be behind the mirror where the light never goes. And since there's no light present at the image, you can't form a virtual image on a screen there. You'll see that our Optical Bench is similarly limited. The Ray Optics Tool will always show where the image is located because it just uses geometry to locate it. But the Optical Bench realistically creates the image using light coming from the object. If no light strikes the screen, no image can be formed on it. Thus the screen can only display real images. Let's take a closer look at these two types of images. Replace the plane mirror with the converging (concave) mirror. Move the object to the 2F point. Move the Screen Position Tool to the 2F point, which is where the Image Arrow is located. You'll see the real image on the screen of the Optical Bench and on the Image Close-up Screen. The light is actually hitting the screen and point by point creating the full image. Our Ray Optics Tool just follows a pair of rays from the top-most point on the object. But if you added rays from other points you'd find that they will produce their images at the proper places necessary to build up the full image. Does this mean that you have to have a screen to see a real image? No. Consider Figure 6 with the frosted image screen at the location of the inverted image shown. With the screen in place, the light arriving at the tip of the Image Arrow would be scattered in all directions. Some of this light would travel into the eyes. So the eyes would see light coming from the screen. If the tip of the arrow was pink, the eyes would see pink light coming from that point. Now move the screen away. The light no longer scatters from the image screen. But if you were to put your eyes in the location shown in Figure 6, the rays will continue on into your eyes just as if there was an actual (pink) object at the tip of the inverted image shown. You would then see the virtual image and locate it sort of hanging in space at that point. The images below illustrate a great example of this. In the first image you see a push pin at the bottom of a concave mirror. In the second image you see a second concave mirror with a hole in its center sitting on top of the first one. But this one's mirrored surfaces faces downward. Note also that it has a circular hole in the top. An image of the push pin and part of the lower mirror is formed by the two mirrors. The image is created at the location of the hole in the top mirror. The push pin appears to be sitting on a mirror at that location. When you study corrective lenses later in this course, you'll learn how they take light from an object at a location where you can't see things clearly and produce an image of it where you can. For example, if you're nearsighted, you may only be able to see things clearly out to 1 meter from your face. Your lenses will take light from a distant mountain and form an image at approximately 1 meter from your eyes! We've seen two different types of image—real and virtual. Let's summarize our key observations about them.

The Nature of an Image

Categorizing an image as real or virtual is a part of the process of determining the nature of an image. The nature of an image is determined based on three criteria. As noted earlier, for a plane mirror, the image is as far behind the mirror as the object is in front and the size of the image and object are always the same. One further measurement that will be introduced in more detail soon is the magnification, M. The magnification is the ratio of the image height, h_i, over the object height, h_o. So for a plane mirror this will always be equal to 1.

II. Concave (Converging) Mirrors

In the first question asking you to make these choices, you were describing the "nature of the image formed by a plane mirror." For any plane mirror, there's exactly one answer for each category. This time the question was not so general. You were asked about "this particular image." As you'll soon see, the nature of the image produced by a spherical mirror changes according to the object's distance from the mirror. We'll pay a lot of attention to the nature of images in this lab. It's important. Here's an example of why it matters.

A. Focal Length, f, Object Distance, d_o, and Image Distance, d_i

The nature of the images formed by a concave mirror is not as neat and tidy as what we found with plane mirrors. The nature of the image and its location depend on the particular curvature of the mirror and the location of the object relative to the mirror. As shown in Figure 9a, the object distance, d_o, is the distance from the center of the reflective surface of the mirror out to the object, O. Similarly, the image, I, is located a distance d_i from the center of the reflective surface of the mirror. (Just as with plane mirrors we draw the rays and make our distance measurements as if they reflect off the back surface of the mirror.) The property of the mirror that determines the nature of the image is its focal length, which is determined by the curvature of the mirror. The focal point, or focus, F, from the Latin word for "hearth," is the point where rays of light traveling parallel to the principle axis of the mirror come together after reflecting from the mirror. See Figure 9b. (With convex mirrors the definition of the focal length is somewhat different as we'll see.) Incidentally, the reverse behavior is also true. If you place a light source, say a small bulb, at the focal point, the rays will reflect off the mirror parallel to the axis. This is the principle of a flashlight, searchlight, etc. The focal length is not an arbitrary distance. A spherical mirror is actually a small section of sphere. Thus it has a certain radius of curvature, R. The focal point is half way between the mirror and its center, C. That is For convenience we label the center point 2F.

B. Images Formed by a Concave Mirror

With our plane mirror, we found that the image and object are always equidistant from the mirror and that the nature of the image is that it's always upright, virtual, and the same size as the object. We'd like to be able to determine all the same information about images formed by concave and convex mirrors. We'll develop methods for determining this information geometrically with drawings and algebraically with equations based on the geometry. With concave mirrors, there are a number of different outcomes all depending on the relative values of the focal length and the object distance. They're different in that the nature of the image is different for each. This is important if you want an enlarged (relative to the object) image, a reduced image, etc., for a given situation. Although there are several outcomes, actually five, there is a logical pattern. We specify these as cases I–V. Yes, you do need to know these cases and be able to draw them! Case I: Object distance, d_o > 2f    (d_o > R) For case I, the object is at any distance greater than two times the focal length from the mirror. That is, the object is beyond the 2F point. Drag the object back and forth between 2F and the left end of the principle axis to see its range of motion. Also notice the range of motion of the image. Here's how the geometry works to form an image for this case. If we can find the image of one point on our object, all other points can be similarly found. We'll start with the tip of the object arrow. All rays leaving the tip of the object, after hitting the mirror, will pass through the matching point on the image and then continue. There is no limit to the number of rays that you could draw from the tip of the arrow to its image. And any two will do since all reflected rays converge at the same point. If you picked another point on the object, say at the middle of the arrow, you'd find that the rays would converge at another point on the image. It would be at its midpoint. These rays are very busy! Our apparatus draws what's called the parallel ray in magenta. This ray is easy to draw since it leaves the object traveling parallel to the axis and then reflects back through the focal point. Our apparatus draws a second, cyan ray, which strikes the center of the mirror and reflects back at an equal angle below the axis. It's not reliable in hand drawings since an "equal angle" is not easy to draw. So we won't use it in our hand drawn figures. Instead we'll choose from two others for our hand drawings. Here are the three to choose from. Are you clear on what these drawings mean? We want to know where the image is as well as its nature. We said earlier that when rays of light from an object are redirected before they enter your eyes, the light will appear to come from a different location. The three rays in Figure 11, along with all the others not shown, diverge from the object and then converge at the image. But they then diverge apart just as they do from an object. If any two of these diverging, reflected rays enter your eyes, you'll be able locate the image. To emphasize this, you should also include a pair of eyes at the end of your two rays just as we did in Figures 10 and 11. Case II: Object Distance, d_o = 2f    (d_o = R) We'll repeat most of what was done in case I now for case II. The instructions will be repeated but streamlined. In case I, the object and image both had a range of possible positions. In case II, the object can be at exactly one point—the 2F point. Not surprisingly the image has only one possible location. Case III: Object Distance, f < d_o < 2f Now we return to a situation where the object has a range of positions. Drag the object back and forth between F and 2F. Notice the range of motion of the image. Case IV: Object Distance, d_o = f Similar to case II, we again have a situation with just one position for the object. We'd also assume that we'd again have just one position for the image. As you'll see, that would in fact be wrong! How can that be? Case V: Object Distance, d_o < f Now we return again to a situation where the object has a range of positions. Drag the object back and forth between F and the mirror. Notice the range of motion of the image.

III. Convex (Diverging) Mirrors

By bulging outward in the center rather than inward a convex spherical mirror has a very different effect than a concave mirror. Each type of mirror has a focal length and a radius of curvature, but these two points, F and 2F, are on opposite sides of these two types of mirrors. Figure 12 illustrates the difference in the behavior of these two types of mirrors. You may have noticed that we have two names for each of these mirror types. One pair of names describes their shapes—concave and convex. But when we begin discussing their behaviors we often switch to the terms describing how they bend light—converging and diverging. One advantage of using the latter terms is that converging mirrors and lenses act similarly as do diverging mirrors and lenses. While concave mirrors act similarly to convex lenses! Not so good. Let's quickly look at this. Put the object between F and 2F. Now swap back and forth between the converging mirror and the converging lens. (The one that bulges in the middle.) They work exactly alike. The images are just on different sides of the devices. Note that their case behaviors are identical, too. So when you understand converging mirrors, you get converging lenses for free! Try the diverging devices, and you'll see the same 2 for 1 special. When you study lenses, you'll see that we will use the F and 2F points on both sides of the lenses. Can you see why? So just how do converging and diverging devices bend light? Let's now compare the way that these two types of mirrors produce images. In the box below for the concave mirror, you see what you learned earlier about three rays often used to locate images produced by a concave mirror.

IV. Equations and Sign Conventions for Spherical Mirrors

From our diagrams summarized in Figure 14, we can clearly see that the focal length and object distance geometrically determine the nature and location of any image. It should be very simple to produce equations to describe this geometry algebraically, and it is. There's one problem, however. For cases I–III, you get one pair of equations, then a slightly different pair for case V, and another slightly different pair for the diverging mirror. If we look at lenses while we're at it, we find a similar situation. We could just memorize several pairs of equations, but the equations only differ by the signs of the different terms. Further examination reveals that the different signs are connected to the different natures of the images. So since we already need to know about the natures of the images it makes sense to just use a pair of equations and a set of sign conventions that adjust the signs of f, d_o, d_i, h_o, h_i, and M according to the situation. What follows is the system that we'll use.

A. Sign Conventions for Mirrors and Thin Lenses for Use with the Mirror or Thin Lens Equation and the Magnification Equation

The following two equations can be used to describe any of the five converging mirror or lens cases as well as diverging mirror or lenses. The equations will always take the same form but the signs of the values inserted into the equations are determined by the sign conventions. For example, if an image is formed 12 cm behind a converging mirror, you'll insert a value of –12 cm for d_i. And if you were solving for that particular image distance based on a given f and d_o, the resulting d_i will be found to have a negative value. The magnification, M, is a measure of the degree of enlargement or reduction of the image relative to the object. It is a ratio of the images height, h_i, to the object height, h_o, and thus a dimensionless, signed, number. * When more than one device (mirror or lens) is present, one device will form an image which becomes the object for the second device. If that virtual object is on the "wrong" side of the second device, that is, not on the side from which the light actually arrives, this virtual object's object distance is negative. In Figure 15, the image, I, formed by the large mirror could become the object for either of the two smaller mirrors shown. For mirror #1 the initial image acts as a real object since rays actually diverge away from it and then arrive at the mirror. For mirror #2 the initial image is behind the second mirror and acts as a virtual object. We will not see virtual objects in this lab or consider them any further. Solution: Known:

• do = +15.0 cm	The object is placed in front of the mirror where the light comes from.
  di = −5.0 cm	The image is behind the mirror, which is not where the light travels after reflection.
  hi = +2.5 cm	We know that an object in front of a diverging mirror produces upright images.
  f = ?, M = ?, ho = ?	The signs of these unknowns will be determined as described in the following note.

The focal length: If we use Equation 2 to solve for f, the algebra will produce a negative value for f. Solving Equation 2

1

f

=

1

do

+

1

di

        Mirror or Thin Lens Equation

for f, and substituting in our given values gives us: The negative sign indicates that this is a diverging mirror. This focal length was selected for this example because it's the default for our apparatus. With your apparatus, select mirror #1. Drag and drop the diverging mirror onto the the mirror or lens currently at the center of the Ray Optics Tool. Measure the focal length to see that it is indeed –7.5 cm. That is, 7.5 cm behind the mirror. Move the object to 15 cm in front of (to the left of) the mirror. You'll see that it's at the 2F point which makes sense. As you'd expect, the image can't be displayed on the screens since it's a virtual image. Measure, with the Ray Optics Tool, to confirm that the image is 5.0 cm behind the mirror. The magnification: Equation 3

M = − 

di

do

=

hi

ho

        Magnification Equation

describes the relative sizes of the object and image. We can solve the left side of the magnification equation for M to find the magnification. Note the negative value inserted for d_i from our sign conventions. The positive sign corresponds to upright images formed by a diverging mirror. The fact that the absolute value of M is less than one reflects the reduced size of the image. We can't measure the image size since we can only measure images formed on the screen. But, using the Ray Optics Tool, you should be able to see that the image is about 1/3 the height of the object. The height of the object: We can solve the right side of the magnification equation for h_o. This is three times the image height and positive as expected. Notice that the negative sign before 2.5 cm was part of the original equation. h_i is positive for this case. So by including the proper signs on the known values, we get a result with signs that can be interpreted to identify the type of mirror, object, image, etc. Knowing the five cases for a convex device and the single case for a diverging device lets us provide the proper signs and interpret the ones we compute.

B. Determining Magnification and Image Location with the Mirror and Magnification Equations

The example problem we've just explored should give you a good idea of the skills you need to work with image formation using mirrors. We'll finish this lab by doing similar explorations of each image case and both mirror types. For each of these cases, you'll find a table similar to the sample table below, which you will complete using the following procedure. Let's use the Taj Mahal for the rest of the lab since it's a more regular shape than a cow. The central building, the tomb, is at the center of four 130-foot-tall minarets, one at each corner of a square. The two minarets in the foreground, of course, look taller. For our object, we'll use the shorter-looking minaret to the left of the tomb. This minaret is on the right in the Object picture in Figure 16. The small, inverted minaret shown in the Image picture is its image. (Remember, the image is inverted and reversed left to right.) We won't use the actual 130-ft heights of the minarets. We'll use the measured sizes of the objects and images on our screens instead. Both Screenshots in Figure 16 were taken with double magnification. (Zoomed in twice.) You might want to do the same. You can click and drag the grid to line up the fine scale lines with the pictures. Each grid square is 1.0 cm on a side, so the fine scale markings are 0.10 cm apart. The minaret (object) looks to be about 2.3 cm tall to the top of its spire. Let's all use that value for the experimental object height, h_o. For the 7.5-cm converging mirror used to make these images, the image in the right figure looks to be about 1.4 cm tall for this case. You'll make and record your own h_i measurement for your assigned mirror.

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