Parabolic mirrors 2

Let's draw a bunch of parabolic mirrors. And what I want to do in this video is, do a bunch of examples of objects in front of parabolic mirrors. And think about what the images of those objects will be based on how far those objects are. And besides just giving us a better understanding of parabolic mirrors, this will also, it 'll hopefully also give us a sense of how do we manipulate or how can we conceptualize these light rays? Which will be a pretty useful tool when we tackle other types of reflective or refractive devices like lenses. So that's a parabolic mirror. I've drawn its principal axis right over here. And let me just copy and paste this just so that I-- actually, let me draw the focal point, too. So this is the focus right over here. So let me just draw the focus right over there. And this is the center of curvature. It's twice the distance from this point as the focus. So that is the-- let me make it as close as possible. So that is the center of curvature right over there. And let me copy and paste this. So we can reuse this later on in the video. I don't have to keep drawing it. So copy it. So I've copied that. Now, let's put an object-- and I think we did this in the last video. Let's put an object beyond the center of curvature. So let's put an object here. And the convention is to use an upward pointing arrow. This isn't a light ray. It's used to show an object. And we use the tip of the arrow to really show the top of the object. And that's usually where we trace our light rays from. But it doesn't have to be there. We could do the middle. You could do the bottom. And you could figure out what the image of the object's going to be. So let's do that. And when you're dealing with parabolic mirrors, it's easiest to have to show just two light rays. One that goes parallel to the principal axis. And one that goes through the focus. Because you know what's going to happen to each of those when they reflect. And you don't have to do any math there. So let's have the parallel ray. That's the parallel ray. When it reflects-- the parallel incident ray, when it reflects, it will go through the focus. And then, let's have an incident ray that goes through the focus. And when it reflects, it will be parallel. And this is the example we actually saw in the last video. And so whatever light is being emitted from this point over here in that direction, it will come back and converge again at this point. And we could actually do it at every point along the-- we could do it halfway. This halfway point of the arrow, just to make it clear. This halfway point of the arrow, same thing. You have a parallel. Something parallel will go, will reflect a parallel incident ray will reflect through the focus. And I'll just do it for this one right over here. It will reflect through the focus. And then, if you have something that goes through the focus, an incident ray that goes through a focus, it will reflect parallel. So this point will correspond to this point over here. I think that makes it clearer, that this image, the image of this object when it's reflected by this parabolic mirror will look just like that. So it will actually form a real image. It'll form a real image that is smaller than this original image. It's not so clear, the way I did it over here. But you could push this even further back out. And it'll be clear that this is going to be a smaller real image than that right over there. Now, let's do a couple more examples. So let me just paste my drawing. So I don't have to redraw it. Let's see what happens. Let me write it. So here the image, just so we can keep track of things. Here, the image is real and smaller than the actual object when the actual object is beyond the center of curvature. And actually, let me make it a little bit clearer by drawing. Let me do another example like that where I do something big way out here just to make it clear. So once again, we go parallel, reflect through the focus. And then, we can go through the focus. We go through the focus and then reflect out like that. And there you see, now, it's much clearer that the image is going to be much smaller. And, of course, inverted relative to the actual object. Now, let's do this again. But this time, let's place the object at the center of curvature. So let's place the object right over here. So right at that distance that twice the distance from the vertex of the parabola to the focus. So we do a parallel line. These lines are the hardest thing to do. So parallel incident ray, parallel to the principal axis. This is the principal axis right here. Principal axis-- that's what this line is right over here. It's kind of the line of symmetry of the parabola. When it reflects, it will reflect through the focus. And then, let's take another ray that goes, the incident ray goes through the focus. And when it reflects, it will reflect parallel. And my drawing isn't the neatest drawing on the planet. And actually, let me draw it a little bit better than that. Well, that's pretty good. Let me just. So that's the incident ray that was parallel. And then, an incident ray that goes through-- let me-- I'm having trouble drawing these. An incident ray that goes through the focus will then come out and reflect right over there. And they'll converge. And the way I've drawn it, my drawing isn't ideal. But the way, but the reality is, is that they'll converge so that the image will just be in an inverted same size version as this thing up here. Because it's symmetric. Let me see if I can redraw this whole thing, so it comes out neater. So far, that looks good. So then, you want to reflect like that. You got-- you're coming through the focus. And then, you have another ray that goes through the focus. This whole thing should be symmetric. And then, when it reflects, it comes back out like that. So that makes a little bit clearer. So this is the object. And now, its image is just an inverted version of this object. The image comes into focus, or the rays converge at the same distance from the actual mirror as the actual object. And it's going to be the same size, just inverted. So that's-- so here the image is real and the same size as the object. Let's do a couple more of these. I think you get the hang of it. And you might want to try them out. You might want to pause the video and try it on paper because really nothing beats practice. So let's stick our object between the center of curvature and the focus and the focal point. So if we put our object there, we could have a light ray that goes parallel to the principal axis. And then, it will reflect out through the focus. And then, you could have another point. It goes through another ray. It goes through the focus. And then, it reflects. And then, it will reflect out. And let me draw it better than that. Actually, this is-- I should probably have used a more precise tool when I did all of this. So let me draw it right over here. So I have the parallel one. And then, it goes through the focus just like that when it gets reflected. I really should have had a line tool for this to have neater drawings. And then, a ray that goes through the focus will be reflected out parallel. It would be reflected out parallel. And at least for the light that comes from that tip, they will re-converge at that tip. And if you did it for every point on this arrow, the image would be an inverted arrow that is bigger than the original. And it is beyond. It's almost the opposite of the first example that we showed. And so now the image is bigger than the original. So the image is real. And it is bigger. And the image will, where it converges is going to be beyond the center of curvature. And you could imagine. If this was the object right here, then this would be the image over here. If you just trace the lines backwards. So there's a symmetry here between this example and the first one we did up here. Now, let's just do a couple more. Let's imagine if the object is actually at the focal point. It's actually at the focus. So let's draw an object there. Let's think about what would happen. So if we're at the focus, a ray that comes out parallel will go through the focal point and come out just like that. And then, here we're going to-- you can't-- we can't have a ray that just goes. Well, actually we could have a ray that goes through. Well, that-- you can't go into the object. So here, I'll do a slightly different ray. I'll do a ray that intersects the parabolic mirror right over there. And the reason why I want to do there is because there the parabolic mirror is essentially flat and essentially vertical. So you can imagine that the incident ray is going to be the same thing as the reflected ray. So you could draw a ray that comes in like that. So this is a departure from what we did before. And then the reflected ray will come out like that. So what happens when an object is at the focal point is that all of the light that's coming off of this object in any direction will all be made, it will all be made parallel. And so it won't converge. So it won't converge. So it won't be able to form a real image. And it's not, it doesn't look like it's diverging from some point in the mirror. So it won't even form a virtual image. So here, there will actually be no image when the object is actually at the focal point. And then, the last case, as you can imagine, is if an object is closer than the focal point. So let's draw that. So let's put an object at the focal point. So right over here. And here, just for the sake of argument, one I can draw. I can always draw a parallel. And anything that, any light that goes parallel will then look to come out in a direction that would go through the focal point. So it would go through the focal point. It would come out in that direction. Although the object itself is blocking. But it would look to go in that direction. It would be reflected in that direction. And then, you could imagine a ray of light that would have been coming from the focal point, or would have been coming from the same direction. So you could imagine coming from the same direction as the focal point would be reflected parallel, in a parallel direction to the principal axis. Now, these two light rays are not converging. But they look like they're diverging from some point behind the mirror. They'll look like they're diverging from some point behind the mirror. So in this case, we are forming a virtual image. And the virtual image will actually look something like this. And so it'll be larger than the original virtual image. So it's kind of a magnifying. If you were to go to-- what are called? --a fun house at the circus or the amusement park, whatever. And if you were to get close enough to parabolas mirror, it would show a magnified version of you. A virtual version of you. And actually, let me draw that a little bit bigger just because it might not be clear. So let me draw. So if this is the mirror, that is the focal point. This right here is the principal axis. This is maybe you, maybe whatever object. You could draw a ray that goes parallel. It will reflect in the direction of the focal point. So it'll reflect out like that. It'll be blocked by the object, though. And then, something that looks like it would have come from the focal point, from the same direction as the focal point would then be reflected parallel to the principal axis. So these two rays, once again, there are diverging. But they look like, to the human brain, to the human eye, they look like they came from that point over there. And so this would correspond to that point on the virtual image. So hopefully, that gives you some practice. But the most important thing, it gives you some practice dealing with these arbitrary rays that we're showing emanating from the tip of this arrow. And we could do it for the whole arrow. But the reason why we're picking these rays in these directions is that they're easy to work with. They go through the focus, they'll come out parallel. If the incident ray is parallel, it'll come out through the focus.