 In this lecture, we're going to begin to explore the concept of images, and we're going to use as the vessel in which we explore images, the device known as a mirror. I'll introduce the terminology associated with both of these ideas, and we'll see examples as we go through the lecture of how to understand mirrors and the images that they produce. The mirror with which we're most familiar is likely the plain mirror. A plain mirror is just a flat piece of reflective material, and if you stand in front of it and you look into the surface of this flat piece of material, if it's highly reflective, that is, for instance, if 100% of the light that leaves the surface of your body strikes the mirror and reflects in a specular way off of the surface of the mirror, if 100% of that light does that, you have what's called a highly reflective or perfectly reflective surface. Mirrors are very good reflectors, they're never perfect, but they're highly reflective, they reflect much of the light that strikes their surface, and as you can see from this picture here, there is the possibility, under these conditions of nice specular reflection, for an image to form. If we were this vessel that's sitting here on the ground looking at ourselves in the mirror, we would see an image of ourselves that moves when we move. It steps forward closer to the surface of the mirror when we step forward closer to the surface of the mirror, but what's interesting about the image is that it appears that our twin is on the other side of the surface of the material. Now we know that physically that's not possible. It can't be that our mirror image is physically, literally on the other side of the material looking back at us. Rather, what we're seeing is an optical effect created by where the rays of light reflecting off the surface appear to converge in space. So a plain mirror offers us a doorway into a larger discussion of reflection and mirror surfaces and how different surfaces will cause different kinds of images to form. When we begin to talk about images, it helps to define some basic terminology. For instance, what is an image? Well, an image is merely a place in space where your eye perceives light rays to be emanating from. This point in space represents the location where light rays are focused by the mirror. So taking the example of the plain mirror, which we'll explore with some drawing in a moment using the law of specular reflection, you can already pick a point on your body like the tip of your nose and look at it inside of the mirror. And of course you'll see the tip of your nose reflected in the mirror. Well, light rays are bouncing off the tip of your nose and they're scattering off your nose in all directions, striking the surface of the mirror and then of course scattering again, they're reflecting off the surface of the mirror in many directions. But what matters is that if you were to pick a bunch of those reflected rays that all started on the tip of your nose, emanated out to the surface of the mirror, then bounced off the surface of the mirror. If you pick a bunch of those rays and trace them back behind the mirror where they appear to converge, that point of convergence is the location where the image of the tip of your nose will appear. We'll do this in a drawing way in a moment so that it better illustrates this point. But you get the basic idea. An image is merely a location in space from which reflected light rays appear to be emanating, or when we do lenses, refracted light rays appear to be emanating. What is a focal point? Well, the focal point is that point. It's the point in space where reflected light rays converge or more to the point appear to converge. Now, I keep saying actually converge versus appear to converge in a careful way because there are actually two kinds of images. There are real images and there are virtual images. Let's talk about the plane mirror for a second. When you look at your image in a plane mirror, your mirror twin appears to be standing inside of the flat surface of the mirror. Now, you and I both know that that's not physically possible. There's no way that this three-dimensional reconstruction of ourselves is quite literally inside the surface of this flat 2D plane off of which light is reflecting. And this gets to the heart of what is meant by virtual when you're talking about a mirror image. A virtual mirror image is an image that appears in a place where it physically cannot be. And for the case of a mirror, this is behind the surface of the mirror. Only your eye makes it real. You really think that your mirror twin is on the other side of this flat glass surface, but in reality they're not. It's not possible to place a screen or a camera, for instance, physically inside the surface of the mirror to take an image of your mirror twin. In order to take an image of yourself in a plane mirror, you have to put the camera outside the surface of the mirror. If you put it behind the mirror, you won't see anything because the mirror blocks light. Now this is to be compared and contrasted to a real image. A real image from a mirror is one that appears on the side of the optical instrument, the mirror in this case, where it could actually be physically located in space. And if you were to put a screen or a camera at that point, you could actually show the image clearly. Real images from mirrors appear to be located on your side of the mirror. You'll see this demonstrated either in the video or in class, but there are images formed by certain kinds of reflective surfaces, certain kinds of mirrors that appear to hover outside of the physical boundaries of the mirror itself on your side of the mirror. So if you put a camera there, you can take a clear, crisp picture of that real image. For lenses, real and virtual images are defined similarly, but there are some subtle differences with mirrors that we'll get to when we talk about thin lenses. So again, we can revisit the picture of the plane mirror. We have this clay jar and we see the reflection of the clay jar. That reflection appears to be located on the other side of the mirror. And if you were to put a camera behind the mirror, you'd see nothing. This is a virtual image. Your eye makes it real, but it's not really physically located on the other side of the mirror. So again, driving this home, because this is something where it's very easy to get caught up. What do you mean by real? What do you mean by virtual? Specifically for mirrors now, focusing on mirrors not generically on optical instruments. For mirrors, real images appear to be located on your side of the mirror. And it's actually possible, as I said, I will show this to you. Virtual images for mirrors are images that appear to be located behind the mirror, sort of inside the mirror world, even though there's no actual world there. It's not possible to place a camera or a screen behind the mirror and see the image. So we can begin to understand image formation geometrically using a plane mirror by doing something called ray tracing. Ray tracing takes advantage of the geometric properties of light rays. Remember, a light ray is merely an arrow that indicates the direction that wave fronts are traveling from an emitted wave of light. We can characterize the motion of the wave merely by looking along a direction, seeing which way the wave fronts are moving and pointing an arrow in that direction. So for instance, consider a simple object like this upright dark blue arrow, which I will denote as O, representing the object whose image will form on the other side of the plane mirror. Now we can see that the image will form there and not on the same side of the mirror as the object by doing ray tracing. Ray tracing is merely where we pick a number of representative light rays emanating from a common point on the object. We follow them as they strike the surface of the mirror, reflect according to the law of reflection, and then bounce away from the mirror. We then figure out where those reflected rays converge or appear to converge in space, and that point of convergence will represent the location on the image where the point on the object emitting the rays appears to be reflected. So let's consider the tippy top of the arrow as our point of emitting light. So we have light rays going off in all directions from the tippy top of the arrow, and some of these are going to strike the plane mirror. Let's look at a few of them and see what they do, and we'll use the law of reflection at each step. Consider a ray that leaves the top of the blue arrow and heads out parallel to this dark black line I've written and labeled as axis. That ray is following a line that is exactly parallel to a normal to the mirror surface. That is, the axis is at exactly a right angle to the mirror surface by construction in this image, and so when a ray travels along that line and strikes the surface of the mirror according to the law of reflection, its reflected ray will come out at the same angle on the other side of the normal. But here we're coming in along a normal, so the angle with respect to the normal is zero, and so the angle of reflection will also be zero. And so what we see happens to this ray that comes in parallel to the axis I've drawn here is it strikes the mirror surface and it bounces backward back toward the object, and I've indicated this as an arrow shooting off to the left side of the picture horizontally. Now we could look at another ray, a ray that rather than coming in at zero angle to a normal to the mirror surface, instead makes some angle theta 2 with respect to the normal to the mirror surface. So I've drawn a sort of dot dashed line here and labeled it normal to mirror surface, and I have placed that line perpendicular to the plain mirror surface at the point where the second ray strikes the mirror surface. And we see that comes in at a non-zero angle, and of course the law of reflection tells us that it will be reflected at an equal angle theta 2 on the other side of the normal and with a nice precision drawing program like that available in Libre office, I can go ahead and make these angles exact according to the law of reflection and draw the arrow coming out. We see that rather than coming out parallel to the axis, this one comes out at an angle theta 2 with respect to the axis, and it appears to diverge away from the first ray that reflected off the mirror. Let's look at a third ray. This ray comes off the top of the dark blue arrow and strikes a point on the mirror very close to the floor in this picture. That is very close to the line labeled axis that passes through the mirror itself. That, according to the law of reflection, reflects away at the same angle with respect to the normal that the incident ray came in, and we see that this one diverges even further away from the second and the first ray that were reflected. What we observe is on our side of the mirror, according to this geometric picture of optics. We have a situation where the rays do not converge at a point on our side of the mirror. That's okay. No need to panic. All you have to do is think about where in space these rays could have converged. And to do that, you can extrapolate them backward. So you can take the reflected rays and you can draw a line that points back along the trajectory of the reflected ray, even passing through the surface of the mirror into the mirror world, which isn't a real world. It appears to be a reflected world that lives inside the mirror, but there is nothing inside the mirror that's unphysical. We see that in the mirror world, on the other side of the mirror, the rays appear to converge at a single point. That point of convergence marks the location of the tippy top of the image of the blue arrow, and I've drawn that here in light blue to indicate that this is the reflection of the dark blue arrow. What you'll notice just by eye is that for a plain mirror, the distance of the object to the surface of the mirror is pretty much equal to the distance of the image to the surface of the mirror, and this sort of confirms using a geometric optical picture what we know from experience, which is that our image in a flat plain mirror appears to be equidistant from the surface of the mirror as we are to the surface of the mirror. And indeed we see that the image is expected to be upright. It has the same vertical orientation as the object, and it's the same size. You'll note that the horizontal ray passes right at the top of the image, just like it did on the object, marking the top of the image, and those are exactly the same heights. So we have an upright, virtual, and preserved image. It's not magnified in any way. They're in large nor reduced, and this is exactly what reproduces what we see in the natural world. If we make a plain mirror and we look at our image in it, we see these features, and we see that treating light waves as rays, which indicate the direction of the electromagnetic waves travel, allows us to reproduce that picture that we see in the world around us very quickly. And we didn't even have to pick multiple points on the arrow. We just picked a representative point and found out where that representative point was in the mirror world. As an exercise, why don't you go ahead and pick the bottom of the arrow, where it sort of stands on the ground labeled axis, that black line here. Why don't you pick that point as the point of emission of rays and see where that... We've explored a few concepts with the simple plain mirror, which is the simplest, specular, reflective surface that we can handle at this point in the discussion. So let me summarize what we've learned by ray tracing a few key beams of light emanating all from a common point in space and then being reflected in a specular way by the surface of a plain mirror. First of all, the image of the plain mirror is virtual. The rays that are reflected off the surface of the mirror appear to converge on the side of the mirror opposite where the object is actually located. That is the thing that's emanating the rays in the first place. Now, magnification. It's possible with different kinds of mirrors to make your image look bigger or smaller or the same size. The ratio of the height of the image to the height of the object, h' over h, is defined as the magnification factor. So for instance, in a plain mirror, you will find a perfectly flat plain mirror that the magnification factor is 1. Your height in the mirror world appears to be the same as your height in the real world. Now, this is not the case for other kinds of mirrors. If you've ever been in a fun house, for instance, at a carnival, if you've ever been in the house of mirrors or the fun house, you'll know that they have all kinds of crazy mirrors in there that not only confuse you, making you think you should head left when you should head right, but also that distort your image. They can make you look very fat or very thin, very tall or very small. And all you have to do to get those different kinds of images with different magnification factors is to change the reflecting surface's shape. And we'll explore a few very simple variations on the plain mirror in a moment. Let's look at spherical mirrors. A spherical mirror is most easily thought of if you imagine a big, perfect sphere of mirror material. So imagine taking that plain mirror and wrapping it around a basketball so that you had a perfectly reflective mirror surface basketball. That is a sphere. Now, if you were to cut a chunk of that basketball away, it would retain the curvature of the original sphere with the original radius of that sphere, which defines the curvature for that object. And yet it would only be part of it. So you see here in these pictures two examples of actual spherical reflectors. So for instance, in the right-hand side, we see what appears to be a giant ball whose outer surface is a mirror. In the left big image, the one that's tinted gold or orange, you see a picture of a person taking a photograph of their camera and the room behind them reflected on the surface of a holiday ornament, a Christmas ornament. So this is just a spherical reflector hanging on a Christmas tree. Christmas bulbs. Very common. You can buy them for fairly cheap. And they're a good example of a spherical reflector. They're almost completely a round sphere. They have a little hanger at the top, which distorts the shape a little bit. But they're a pretty good example of a perfectly round reflector. In the top left in the small inset image, you see an example of a spherical mirror that is not bowed out but bowed in. We'll explore the naming conventions of these two kinds of spherical mirrors in a moment. But unlike the other two that are pictured on this slide, which are bowed outward, this mirror bows inward. It's like the inner surface of the sphere was coated in a mirror, and then you sliced off a chunk of the sphere and held it so that it curves away from you. This is a cosmetic mirror. Its job is to magnify your face so that you can put on makeup or do something like that, maybe shave a tricky location that you can't really see well in your normal plane mirror, but if you get a really close-up view of your face and you have to shave your face, then you can get a very good angle on the tricky spot that you have to shave. So whether it's putting on makeup or shaving, these cosmetic mirrors are very useful. They are spherical mirrors. They're shaped as if they were cut out of a much larger spherical ball, but of course they're not huge. They fit on your vanity in your bathroom. And their job is to magnify your face and show you very close-up details so you can do fine work on your face without injuring yourself in some way. So these are all examples of spherical mirrors. Cosmetic mirrors, they bow away from you. Ornaments on a Christmas tree, they bulge out toward you, and they have an outer reflecting surface. And then, for instance, this strange structure over here on the right, where you see people photographing themselves in it and reflected in it. They have a mirror surface, a spherical mirror that bows out toward you. A mirror that bows in away from you is known as a concave mirror. And you can remember this naming convention because it's like a cave, right? When you're looking into the mirror surface, it's like a hollow and it's aimed away from you. So you can remember concave mirrors because they're like little caves that you look into to see your reflection. Now, this picture illustrates a few key features of any curved reflector. And specifically, we're focusing on spherical curved reflectors. There is a point known as the center of curvature. Now, if you imagine extending the gray spherical concave reflecting surface I've drawn here, all the way around so it makes a circle and then extend that into a sphere. The point C, which is the center of curvature, lies dead center in the sphere. It is the location where all radii in the sphere converge. Now, this may not be the same as the location where light rays reflected off the surface of the concave mirror will focus. That point where reflected rays will meet is known as the focal point. And specifically, this is the point where rays that come in parallel to the long axis that's drawn here. So you see here this long horizontal line. This is known as the optical axis and it runs dead center through the curve of the mirror on the right-hand side. Any ray that enters parallel to the optical axis will be reflected through the focal point. And then finally, you see the surface of the mirror over here on the right. So to review the terminology for a concave mirror, really for any curved mirror, but focusing on this one for a moment, you have the mirror itself, which may only be part of a sphere. It has a central point around which it is symmetric. The axis that cuts through that symmetric point is known as the optical axis and is drawn here as a horizontal line. There is a point on that line inside the curvature of the mirror that represents the location where all spherical radii meet. That is the center of curvature. So a line drawn from C to the surface of the mirror will always have a length of the radius of the original sphere. And then finally, you have the focal point where incident parallel rays, that is parallel to the optical axis, will be focused by the mirror. Now, in a discussion of mirrors, especially as we're going to get into a more complex mirror like the spherical mirror, it's important to define some terminology for rays. And specifically I want to define the concept of principal rays. Now, you might think, oh gosh, if I have to ray trace a picture, if I put an object in the last slide and then try to figure out where its reflection might be, I have to trace all these rays. There could be, you know, millions of them or billions of them. Well, the good news is, is that you only have to pick a point on the object and then emanating from that point, you just have to pick a few principal rays that are fairly easy to remember once you get the hang of it. And you just always draw those and they will tell you where the corresponding point on the image is located. So principal rays are just these few special rays that when used together help you to quickly locate a point on an image that is equivalent to the original point where light started on the object. And they're much easier to use than other rays and they're representative enough to pinpoint the image. So, for instance, for spherical rays, you have... So, for instance, for spherical concave mirrors, we can talk about a parallel ray, which I've already mentioned. A parallel ray enters parallel to the axis through the middle of the sphere, the optical axis. All parallel rays meet at the focus of the spherical mirror. So wherever that point is, F, for a spherical mirror, that's where that ray will pass through. A focal ray is one that enters before striking the surface of the mirror by passing through the focal point. The focal ray will then bounce off parallel to the optical axis. So when a ray that comes in through the focus reflects off the spherical surface of the mirror, it will travel after reflection parallel to the optical axis. And then finally, I haven't written it here, but another really easy one to remember is a ray known as a central ray. A central ray is one that enters through the center of curvature of the mirror, and when it strikes the surface of the mirror, it's reflected directly back along its original path. And you can figure that one out pretty quickly from the law of reflection. We know that if a ray enters... strikes a flat surface along the normal to the surface, it will be reflected back out exactly along the path along which it came. Because the law of reflection says that the entering angle is equal to the exiting angle on the other side of the normal. Well, if you enter with an angle of zero relative to the normal, you must exit with an angle of zero relative to the normal. So any ray that comes in on a line that's perpendicular to the reflecting surface will go back out along that same line. Since a ray that passes through the center of curvature of a spherical concave mirror is by definition traveling along a radial line, and all radial lines are perpendicular to the surface of the sphere where they touch, the ray that enters by passing through the center of curvature will necessarily be reflected back out along a path that takes it back through the center of curvature. Now, if you remember your geometry, you know that you only need to find out where two lines cross in order to locate a point in space. The point of intersection can be determined merely by drawing two. You're welcome to draw all three principal rays, parallel rays, focal rays, and central rays if you choose. But you only need two of these in principle to locate a point on an image that corresponds to the point on the object where light originated from. So for instance, um, so let's draw as our object a vertical blue arrow shown here and labeled as O for object. It's perfectly straight. It's perpendicular to the optical axis. So this could be a person standing there or it could be a pencil that's being reflected in the mirror or a candle that's standing upright on a table being reflected in a concave spherical mirror. And we can try to figure out where a corresponding point on the image will be located. So for instance, if we imagine rays of light emanating outward from the tip of the arrow, we can follow just a couple of principal rays. We can trace them, and we can see where they converge in space or where they appear to converge in space if they don't do so on our side of the mirror. So in this case, we have our object located to the left of the center of curvature of the mirror. We can follow a parallel ray, a ray that starts out by being emanated parallel to the optical axis. It travels toward the surface of the mirror. It strikes the mirror, and according to the rules I told you a moment ago, it's reflected, and it passes through the focal point of the concave mirror. We can also think about an incident focal ray. This is a ray which starts its journey from the tip of the arrow, but it passes not parallel to the axis, but crosses through the optical axis through the focal point. It then strikes the surface of the mirror, and it bounces off parallel to the optical axis when it leaves. And where those two lines converge, the incident parallel ray reflected and the incident focal ray reflected, locates where the top of the arrow image will be located. We see interestingly that the point of convergence is not above the optical axis, which is where the object pointed, but below the optical axis, which means that the image formed by this concave mirror, putting the object to the left of the center of curvature, will be upside down. This is known as an inverted image. It is real because this image appears to form on the side of the mirror where the object is located, so that defines it as real. It is inverted because it is upside down. Its orientation is opposite what the orientation of the object was. And it is a reduced image. We see that by carefully ray tracing using a computer program, we very clearly get an image which is reduced in size compared to the original object. And so we refer to this as a real inverted reduced image of the original object, O. Now we can look at another case with the same concave mirror. Here we could locate the object between the center of curvature and the focal point. So now let's trace the parallel ray and the focal ray. The parallel ray strikes the surface of the mirror and then passes through the focal point. The focal ray passes through the focal point, strikes the surface of the mirror and comes out parallel to the optical axis. And now we can figure out where these two rays converge. The reflected rays converge at a point behind where the object was located. It is still inverted. It is still real, but now the image appears to be larger than the original object. And so we refer to this as a real inverted enlarged image. And this is all determined simply by doing very basic ray tracing. Here's an example of a candle which is our object and a concave spherical mirror which you can see in the background here. The candle is sitting on a table and if we look at where the image of the candle appears to be located the candle's image appears to be floating outside of the surface of this concave mirror. Notice how the image of the candle appears to bow out. Whereas we know the surface of the mirror bows inward away from us. Whereas the candle appears to bow outward toward us. It appears to be hovering above the surface of this mirror. Weird, right? But this is a real image. You don't get these from plane mirrors, concave spherical mirrors. This candle happens to have been placed exactly at the center of curvature and when you place a candle or an object exactly at the center of curvature, you'll get a real inverted image with the same height as the original object. If we move this candle back a little bit we'll get a real inverted reduced image and if we move the candle forward a little bit between the focal point and the center of curvature we'll get a real inverted enlarged image. Now what happens if you move that object deep inside the focal point of the concave mirror? So that's what I've sketched here. We now move our object O inside the focal point. It's now between the surface of the mirror and the focal point of the mirror. We can do our ray tracing again. So now we have a ray out of the tippy top of the blue arrow parallel to the optical axis. It doesn't have to travel very far and we know that when it strikes the surface of the mirror it will be reflected in such a way that it will want to pass through the focal point of the mirror and so I've drawn that here. Now in this case of course we know that the object is in the way of that ray and we'll come back to that consequence and we know, in order to determine where the image is, is where these reflected rays appear to converge in space. Let's look at a focal ray. Well, this is where things get a little difficult but you just have to remember the rule and you'll be okay. A focal ray can be one that passes directly through the focal point and then strikes the mirror surface. We've been looking at those. But in this particular case there's no mirror back there for it to strike. A focal ray however can be one which could have originated from a point at the focal point. Then travels in a direction on a line between the focal point and the surface of the mirror. Then strikes the mirror and then bounces. So that's what I've drawn here. I have a focal ray which you can trace back to the focal point. It appears to have originated along a line between the surface of the mirror and the focal point. But of course it started its journey on the top of the object. It just appears to project back along that line. It strikes the surface of the mirror and like all other focal rays that we've been looking at so far, it reflects backward parallel to the optical axis. Now you see we have a problem here. It's clear that these two reflected rays do not converge anywhere on the real side of the mirror. So rather we have to think about how these reflected rays appear to have originated from after they've reflected. And if we trace them backward in space, we find out that they appear to have converged on a point on the opposite side of the mirror. That is the image of this object will appear to form on the other side of the mirror. It's virtual. We notice it's upright so it has the same orientation as the original object. We also notice that it's larger than the object. It's enlarged. So when you move inside the focal point of a concave mirror and look at the image that results, what you will find is that light rays appear to originate from the other side of the mirror even though physically they can't have come from here that's where they appear to have originated from. And so you get a virtual upright and it turns out enlarged image. No matter where you pass the focal point, you'll get an enlarged image. Well, here's an example of that. The mirror has a little groove in the bottom. This lets you move an object within the focal length of the mirror and that's exactly what has been done with the candle on the table here. What's kind of cool about this is that you can see the difference between images of objects that are behind the focal point and objects that are within the focal point. Look at the room reflected in the mirror. It's still a reduced inverted real image. You have the chalkboards, the camera, even the table top surface that we can see reflected in the mirror. All of that is clearly outside of the focal point of the mirror. It's all inverted. It all appears to bow outward from the surface of the mirror so it's real. It appears to be on our side of the mirror and it's all inverted and it's all reduced. But the candle, which has been moved inside the focal length of the mirror, it appears now, its image appears to be on the other side of the mirror. Notice how it curves. It curves following the curvature of the concave mirror. It bows with the mirror. The candle is huge. Its image is greatly enlarged over the original object. It's upright. The orientation of the candle's image is the original candle and we see the flame is magnified and everything. So this beautifully illustrates what it means to form images from objects that are beyond the focal point and within the focal point of a concave spherical mirror all in one picture. Now, what if we flip that mirror surface around so that it's not bowing away from us but bowing out toward us? This is known as convex. This is a little harder to remember. I find it easier to remember concave because it's like a cave that bows away from me and I have to look into the cave to see my image. Convex bows out toward you. And so we can think again about ray tracing. Now, we flip the sphere around. So the focal point is on the other side of the mirror and the center of curvature is on the other side of the mirror from where we're putting the object. The object is on the left, the focal point and center of curvature on the other side of the mirror. And so now we can try to send a ray parallel to the optical axis and it will be bounced off of the convex surface of the mirror. It will appear to track back through the focal point. So you see I've traced that ray, that reflected ray that came in parallel. It bounces back on the same side of the mirror as the object but it appears to have originated from the focal point on the other side of the mirror. Its trajectory is literally if I send a ray along a line toward the focus on the other side of the mirror when it strikes the mirror surface well before reaching the focus it will bounce off parallel to the optical axis. And again we see a situation where these rays on our side of the mirror they don't converge at a point. They're happily traveling away from one another never to meet again. But if we track them back, if we take the reflected rays and we figure out well where did they appear to originate from a point on the other side of the mirror. So this will be a virtual image. That image is upright and it's reduced in size compared to the original object. So in fact it turns out that these are the only kinds of images that convex mirrors that is bowing out toward you. Convex spherical mirrors can only ever make virtual upright reduced images. So here's an excellent example of that. Flip that concave reflector around. Now you have a convex reflector and we see that no matter the distance of the object from the mirror all reflections all images are virtual upright and reduced. The camera in the picture appears to be much smaller than the camera in reality. The blackboards certainly look far tinier than the actual blackboards. The candle is reduced in size. It's all upright and it all appears to be located on the other side of the mirror surface. Virtual. So virtual, upright and reduced. Here's an excellent example of a convex mirror in the wild. You've probably seen these before and never thought about what they were. These are special mirrors placed at dangerous places where you have to leave a driveway or maybe at a sharp corner where there might be a driveway located or where you can't see cars around the corner of a road. Often cities, towns other municipalities they'll put up these convex reflectors. These are convex spherical mirrors so they look like pieces of a larger spherical mirror that have been cut to size. They bow outward toward you. They're highly reflective on the outside and what they do is they show you images of objects that may not be visible to you where you're sitting but light rays can reflect off at extreme angles from objects outside your immediate field of vision and be made visible to you. The danger of this though is that all of these images appear to thankfully they're upright and they're virtual but they appear to be smaller than they actually are. So you need to be very careful with mirrors like this because you may misjudge distances and sizes as a result of the fact that you have a reduced image in the mirror. You can't really judge well distances looking in this mirror. Cars may in fact actually be closer than they appear objects traveling toward you may be larger than they appear in the mirror and so you may misjudge the danger you have to be very careful when using these mirrors but nonetheless it's better than not being able to see anything coming around a corner at all. So you'll often find these at driveways and other dangerous places where visibility may be compromised and drivers need to see objects that are not in their immediate field of vision. What they fail to give to us is any numerical results at the end. We certainly don't want to have to bust out a ruler and attempt to measure distances and heights and things like that. Instead we can employ something called the mirror equation. It's based on the geometric principles in fact that's where it's derived from but it's a nice compact simple equation that you can whip out, give an information about an object or an image or a mirror for instance radius of curvature or its focal length things like that and you can actually solve numerically questions of image formation, object distances and so forth. The mirror equation is very simple. Keeping the conventions in mind is something that you'll have to practice but the equation itself is fairly straightforward. If we denote the distance of the object from the surface of the mirror, specifically from the point where the mirror touches the optical axis as a distance p and we denote the object that the image forms from the point of contact with the optical axis and the mirror denote that as i we can relate the focal length of the mirror which is fixed by the geometry of the mirror to the object in image distances using this mirror equation. 1 over p plus 1 over i equals 1 over f that is 1 over the distance to the object plus 1 over the distance to the image equals 1 over the focal length of the mirror. Now as I said the distance p is the distance of the object from the point of contact with the optical axis which is the mirror center for the spherical mirrors we've been considering. This is by definition a positive number. Object distances are positive. The side of the mirror on which the object is located defines the location of all positive distances. So your coordinate system for optics problems is defined not by positive increasing numbers on an x-axis and negative decreasing numbers on an x-axis, but rather positive numbers defined by whether or not those distances occur on the same side of the mirror as the object, negative distances occur on the side of the mirror opposite where the object is located. The distance of the image from the mirror center is written as i. This can be a positive number. If it's positive it means you have a real image. Or it can be a negative number in which case you have a virtual image. Real images form on the same side as the object, virtual ones form on the opposite side from the object. An image will have a negative image distance if it forms on the opposite side of the mirror inside the mirror world where it physically can't be located, and it will have a positive distance if it forms on the same side of the mirror as where the object is located. I find it helpful to keep a little table of information like this in my notes so that as I'm working problems I can refer to it and check my answers and make sure that everything makes sense. The focal length of the mirror is written as f. It is designed into the construction of the mirror, and in fact for a spherical mirror the focal point of the mirror is one half of the radius of curvature of the mirror. The radius of curvature, remember, is the radius of the sphere from which the spherical mirror was cut effectively to make the spherical mirror you're looking at. If you have a concave cosmetic mirror it looks like it was cut from the inside of a sphere with a big radius of curvature so that you are put firmly inside the focal length of the cosmetic mirror and you get an enlarged, upright image of your face in the mirror. So for a spherical mirror the focal length is related directly to the geometry of the sphere from which the mirror might have been cut and the focal length is one half of curvature. This will come in handy in case you ever need to solve for the focal length given the geometric properties of the mirror itself. Now the reality or virtuality of an image is determined by the sign positive or negative of the distance i. You can also determine things like the magnification factor and whether or not the image is upright or inverted when it is produced by the mirror. Magnification is a simple equation the magnitude of the magnification factor is merely given by the ratios of the height of the image and the height of the object. So if you have an object that's one meter tall and you have an image that's two meters tall the magnification factor is two. If you have an object which is one meter tall and an image that is one half meter tall then the magnification is one half. One means there's no magnification. A number greater than one means there has been an increase in the size of the image relative to the object and you get positive magnification. You get enlarged images as a result. If you have a magnification factor that's less than one you get a reduced image so it's an image smaller than the original object that it is a reflection of. Now for spherical mirrors using geometry we can derive we're not going to do that here but we can derive a very nice relationship between the magnification factor and the image and object distances. So m now not absolute value of m but m itself which can be a signed quantity is equal to negative i over p. So for instance magnification can be negative what does that mean? Magnification factor of negative one half. Well all that means is if i is a positive number and the image forms on the same side of the object you get an inverted image. That's all that means a negative magnification factor gives you an inverted image. So by calculating m you can figure out right away whether or not you're dealing with an inverted image. So if you want to know if the image is real or virtual look at the sign of i, the image distance. Positive is real, negative is virtual. If you want to know if the image is enlarged or reduced in size compared to the object look at the magnitude of the magnification factor. That is equal to h prime over h. That is also equal to the magnitude of negative i over p. That number greater than one means enlarged, less than one means reduced and one means the same size. Finally if you want to know if you have an upright or an inverted image look at the sign of the magnification. If it's positive you have an upright image and if it's negative it means you have an inverted image. All this information can be determined from quantities that go into the mirror equation. So finally let me close with some tricks of the trade regarding mirrors and it boils down to these sign conventions that I've been mentioning. Objects are at a positive distance p from the center of the mirror. Period. End of sentence. There are no negative object distances for the problems we're going to be dealing with. Images can form either on the same side as the object in which case i will be greater than zero and you have a real image. Or they can form on the opposite side of the mirror from the object in which case i will be a negative number less than zero and that means a virtual image. Focal points for mirrors can be on the same side as the object such as in concave spherical mirrors that are the focal point is on the same side of the object in that case f is a positive number. So for concave mirrors the focal length is a positive number. But if the focal point is on the opposite side of the mirror from where the object is located then you have a negative focal length and convex mirrors have negative focal lengths. Finally magnifications can be positive numbers meaning that you have an upright image that is formed or negative numbers meaning that you have an inverted image that has formed. So here are the helpful sign conventions that will aid you in waiting through mathematical problems involving mirrors, images and all the details of the formation of those images. This concludes the lecture on mirrors. In the next lecture we will move into thin lenses so rather than looking at the effects of reflection on rays and ray tracing we will look at the effects of refraction on rays and ray tracing we will develop a very simple lens system known as the thin lens and we will look at the implications of image formation for different kinds of thin lenses.