Parabolic mirrors and real images

In this video, I want to expose you to a special class of mirrors called parabolic mirrors. Or sometimes called parabolic reflectors. And what's neat about parabolic mirrors-- and I'll draw a cross section of one right here. And if you're familiar with the algebra, they are essentially-- the cross section, especially, is in the shape of a parabola. So let me draw a parabola right here. So it's in the shape of a parabola. Just like that. And what's neat about a parabolic mirror-- and I'm not going to go into the math right here. I just want to give you the general idea. And let me just draw its principal axis. So this is the line of symmetry of the parabola. So this is its principal axis right over here. It divides it in two. This is just a cross section. You could imagine if this was spun around that principal axis, you would get something that would look like this. You would get something that would look like a bowl. But it's actually the shape of a parabola. It's not an actual sphere shape. So if you rotate this around, you would get a circle around the edge. So this would be a circle right over here. But this shape down here is not a hemisphere. It's not spherical. It's actually a parabola. And the reason why we care about a parabola, or what's neat about parabolic mirrors, is if I have parallel light rays coming into a parabolic mirror-- I'll do my best to draw a parallel light ray. So parallel to its central axis. So if I have a light ray that comes like that, it will reflect off of the-- it's parallel to this principal axis-- it will reflect like that. And I'll tell you what's neat about this in a second. Now let me draw another parallel ray. Let's say I have a parallel ray that's coming in right over there. So it hits the parabolic mirror at that point. It's going to reflect-- so it comes in like that. And if I have another ray that comes in like this, it will reflect so that the reflection goes right over there. So what's neat about this? Well, what's neat is any light ray that comes in parallel-- any incident light ray that's parallel to the principal axis of this parabolic mirror-- the reflected ray is going to go through the same point. I don't care where you hit the mirror. As long as it was parallel to the principal axis, the reflected ray is going to hit this point. And this point right here is the focus. This is the focus of the parabolic mirror. Now, what's neat about this? Well, let's say that you were trying to capture heat from the sun. You were trying to concentrate the electromagnetic radiation from the sun. So what you could imagine-- you could go to the middle of the desert-- and people do do this-- and you set up of parabolic mirrors like this that are pointed at the sun. And the sun's rays come in. And the sun is so far away, they're essentially just coming in parallel because they are radiating from the sun. But the sun is 93 million miles away. So the rays for our purposes are essentially coming in parallel. And what's neat about them is, is when they hit the surface of the parabolic mirror, they all get reflected to one point. So if you have a ray coming in there, it's going to get reflected there. If you have a ray coming in like that, it's going to get reflected like that. And so all of the energy can be focused on a point like that. And so could imagine you might have a water pipe running into the screen here. And so all of that light energy would be used to heat up that water pipe. So it's a pretty neat way to concentrate energy. Another thing you might want, maybe instead of taking in energy, maybe you want to give out energy so that all the beams of light are parallel. For example, let's say you have a light for a car. If you have a light, you could imagine if car headlights were just-- if I drew a car like this-- let me scroll down a little bit. If I drew a car like this-- let me draw-- have a reasonable attempt at a car. So let's say this is a car right over here. I think you get the idea. This is the wheel housing. That's the wheel. So forth and so on. This isn't about the drawing of the car. But you could imagine if we just stuck light bulbs at the front of cars. So you could imagine just a light bulb sitting at the front of a car. So that's a light bulb. And that would provide light but it would provide light in all directions radially outward. And it would be kind of useless. First of all, the way I drew it here, it would probably show up in the dude's eye who's trying to drive the car. But it's a lot of wasted energy. A lot of the light is coming back onto the car. And it's pointing in all sorts of random directions. It's not so useful. When you are driving a car, you want all of the light pointed at the road or maybe the stuff that's directly above the road. So how could you point the light? Well, you could use a parabolic mirror. And any car you look at will have a light inside of a parabolic mirror. And what does that do? Let's say instead of this situation that I just drew-- let me clear this out. And I'll draw it on a larger scale. Let's say I had a parabolic mirror here. So I have a parabolic mirror. Obviously, this looks more like a snow shovel or something. But I'm drawing it way huge just so you get the general idea. So this is a parabolic mirror. And let's say we put the light bulb now at the focal point. At the focus. At the focus of this parabolic mirror. Now what's going to happen? Well, light that has to go in this direction, that comes radially outward, that's good. Because that's light that's being useful to the driver. It's actually illuminating the road. But light that's going backwards-- light that's radiating outward from that focus of the parabola-- it's going to do the exact opposite of that solar energy collector. It's going to be reflected out parallely. Or a parallel way. And so all of the light-- because of this parabolic reflector, or parabolic mirror-- all of the light that this light source is generating, or most of it, is going to be emitted parallel to the principal axis of the parabola. And actually you could point the light. If you actually moved this parabola around, you can point which direction the light's in. So it's actually a pretty useful thing to have. Now the other thing about parabolic mirrors is that they actually form real images. In the last video, we talked about the notion of a virtual image. You think something is there because it looks like the light is converging at some point. But that point isn't even there. It's actually from some other point getting reflected. But a real image-- let me draw it over here. So let me draw a parabolic mirror. Let me draw big parabolic mirrors to make the diagram clear. And let me draw its principal axis. This is a side profile of it. Let me draw its principal axis, just like that. And let's put an object. So I'm going to define a couple of interesting points here. So first of all, we have our focal point. I'll call that F. And then there's something called the center of curvature. And the curvature I always imagine as a sphere. But for the center of curvature of a parabolic mirror, it's actually going to be two times the focal length of this distance right here. Let me make it clear. I'll call that-- this distance right here is F. Then this distance right here, to the center of curvature, we'll just call that point C. But this distance over here is going to be F as well. Or it's going to be 2F from-- you could imagine that vertex, or that minimum point of the parabola, depending on how you want to view it. Now, what I want to do is put a couple of objects in front of this parabolic mirror. And just think about what happens to the light rays of that object. So let's first put an object here. So I'm just going to draw the object as an arrow. And maybe some light is shining on it from who knows what direction. But it's going to reflect that light diffusely. Assuming it's not shiny. And I'm just going to pick points on this object to radially emit light outward from. Or reflect light outward from. And see what happens to those light rays. And for the sake of simplicity, whenever you do something with a parabolic mirror, it's good to emit one radial ray that's parallel and one that goes to the focus. Because we know what they're going to do after that. So let's do one that's parallel. And of course, these are just two of the gazillions of light rays that are being emitted from every point of this object. But we're just doing this to understand what will the image of this object actually look like. So let's do one parallel. It hits the surface of the parabolic mirror. And then it reflects and goes through the focus. We know that already. And then let's make another light ray go through the focus. Let me draw it a little bit better than that. Another light ray going through the focal point. Just like that. And then it reflects. And it'll be reflected in a parallel way. So what just happened here? Those two rays that were emitted by the same point on this arrow object, they radially emit outward. They reflect on this parabolic mirror at two different points, but then they converge again. They converge right over there. And actually if you put-- and we could do that with every point. If you did the stuff that leaves that point-- actually both of those are going to go and come back-- go through the focal point and then come back right over here. They'll keep going. But you could imagine, you could use with every point on this arrow. And what you're going to do is get an image that looks like this. This point up here corresponds to that point. This point corresponds to that point. And so if you were to put a screen right over here-- this is a screen. It could just be a, I don't know, white tablecloth. Or if there was a wall right over here. Then it would actually show the image. You would actually be projecting the image onto this wall right over here. It would actually be a projected image. And that projected image that we're talking about, where the light is converging-- so the light comes radially outward from each point of this arrow. And then it converges on a point on the screen. That image that gets formed, we call that a real image. It's real image. This is a real image. And you might want to compare that to what we call a virtual image. A virtual image is an image that looks like it's coming from someplace. Because it looks like things are diverging from some point. But they've really been reflected off of some surface. So what we think is there, really isn't there. A real image is an image that's actually projectable. We could put a screen right over here and then these guys are going to be hitting the screen and essentially defusing the exact same light as this point of the actual object. And because of that, the screen will look just like the object. This is a projectable image. Anyway, hopefully you found that useful. I realize I've gone longer than I like to with some of these videos. We'll talk a little bit more about parabolic mirrors in the next video.