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Anderson Video - Spherical Mirrors

Professor Anderson
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 So mirrors, of course, don't have to be flat. They can be spherical. And you're all familiar with these. You have them on your car of course on the, on the right passenger side of your car, that mirror is not flat, it's curved. And it says on the car door you know, "objects in mirror are closer than they appear." So when we think about mirrors, we can use the same idea of the law of reflection. But let's take a curved mirror. Okay, so a mirror curve like this is what's called a concave mirror. And light that comes in to this mirror will still bounce off according to the law of reflection but it's the local angle that's important. So that ray will bounce off like so. But let's say we have another ray that's coming in near the bottom. It sees a different surface. It's going to bounce off like so. This is called a concave mirror because it's like a cave, you can crawl into it. Okay, and it has this interesting property that rays that come in parallel are suddenly focused towards each other. If the mirror is bent the other way then it's called a convex mirror. And this is the one on your passenger side mirror of your car. And now a light ray that comes in near the top will be bent up at a big angle. A light ray that comes in near the bottom will bit be bent down at a big angle. Okay, so this is a primary difference between these things. Obviously a convex mirror, if you just flip it around and it's reflective on the other side, it would behave just like a concave mirror, okay? So it's really sort of the same element. It's just what side are you hitting. Are you hitting something that looks like a cave? Or are you hitting something that is bent back the other way called convex? Alright, so let's think about our spherical mirror. There it is. It was made from some sort of spherical object, right you cut it out with a big mil or something like that. There is therefore some center to this thing. Right, if I have this radius R and I sweep it out I'm going to mimic this arc of this spherical mirror. All right, what does this mean for the rays that are coming in? Well, when a ray comes in here it is of course going to hit the mirror and bounce off at the exact same angle as it came in at. But that angle is relative to the local surface. And so we have to worry about this angle right here. This is your theta I. If that's your theta I then this is your theta R . Okay, and so the ray takes off at that angle: theta R. And we know that those have to be exactly equal. So if those are equal that means that it crosses this thing which is called the optic axis or the optical axis It means that it crosses the axis at a very particular point. And that point is called the focal point of the mirror. Okay? Likewise, if a ray comes in on the other side it is also going to go through the focal point. This F is called the focal point. There is a focal distance associated with that, which is this distance right here, F. And that F has a very special relationship to the radius of curvature of the mirror R. And it's the following. F is equal to R over 2. So what do we mean by focal point? Focal point means where all parallel rays incoming converge to. All these parallel rays that came in bounced off the mirror, and they converge right at the focal point. Now, this sort of mirror is of course what you use in telescopes. Right, if I think about a distant star a distant star looks like a point. It's so far away that the rays coming in from that star are all parallel. When they hit the mirror, I want them to go back to a single point. Put my CCD or film right there. That will generate a point, an image of that star. If the mirror behaves really well, right, then all those rays will converge at that point. But if the mirror is not quite perfect if the mirror in fact is made from a sphere not a parabola then it suffers from something called spherical aberration. Spheres are really easy to make. Why are they easy to make? Because when you go to the machine shop everything works on rotational principles, right you go to the lathe, it has a big cutter on it that rotates in an arc. And so it's very easy to make circles. It's hard to make parabolas. And that's why typical astronomy telescopes cheap astronomy telescopes have spherical mirrors. Because a sphere is pretty close to a parabola until you get very far out to the edge. And when you get out far to the edge then you suffer from something called, "spherical aberration." And in fact, you're all familiar with this idea of spherical aberration because that's what plagued the Hubble telescope initially. When they first built the Hubble telescope and they sent it up? You were very young, of course, but the pictures that came back from the Hubble telescope were totally blurry. All right, it was supposed to be nice sharp pictures of stars and they were just these blurry spots. And it's because they had spherical aberration in the Hubble telescope. A few years later, they went up with some corrective optics, literally a pair of eyeglasses to put on the Hubble, and they fixed the problem. And the Hubble vision went back to good 20/20 vision and now I can see into the farthest reaches of the universe. So, it's important to understand that spheres and parabolas are the same near the axis, they get worse as it goes farther out.