13.6 Curved Mirrors

Ok, today we're going to look at how the images reflect off curved mirrors. So, straight mirrors, remember, flat mirrors, had same sides, same distance away. The image was the same distance away from the mirror, and the only thing that was flipped was left to right. Now, what's going to happen with curved mirrors is that the image is not necessarily going to be the same size. Quite often, it'll be smaller, and quite often the image also gets flipped upside down. So, we have two types of mirrors: a concave mirror and a convex mirror. So, think of if we had like a sphere and we cut it in half. Concave would be the inside of it, and convex then is looking at the outside. So, concave would be the inside of your spoon if you look at your reflection. Convex would be on the back side or the curvy side of the spoon. Concave mirrors actually converge. We call them converging mirrors. We're going to look at that in a second. A couple of key definitions we think of: Center Think of the center as being the center of the sphere. If we extended this mirror/sphere, making it a full circle, the center would be the center of the sphere. So, think of the center as being sort of like the radius of the sphere. The focus point ends up being halfway, so the focus is halfway between the center and the principal axis. We just draw a line that goes horizontally through the center and focus point, and we call the vertex where the mirror meets the principal axis. Those two aren't that big of a deal, but you have to be able to locate the focus point. To try to figure out how we're going to draw these, we're going to follow three rules. We're going to do these over and over again so you'll get lots of experience: 1. Make sure you have a ruler so you can draw the lines nice and straight. 2. Use graph paper if the questions we've given to you help you line them up properly. 3. Only worry about the tip of the objects. So, to draw the diagrams for these problems, remember to follow these rules and you'll do great! We're going to usually do is just draw the original object as an arrow pointing upwards. Those usually tell you where it's located with distance away and usually what height. And then from there, you can figure out what the image is going to be. So, the first rule that we do is we draw from the tip of the object, we go down through the focal point, and then we reflect back parallel. Any ray that goes through the focus point reflects back parallel. Our second rule then is basically just the opposite. If we go parallel to the mirror, it's going to reflect back through the focal point. So, where these two lines intersect, you can see that'll be where our images. So, you can see in this case, our intersection points below the principal axis. So, that means our image then is going to be upside down and you can see it's a little bit smaller. Okay, a couple of key definitions that we talked about. If it's bigger or smaller, sometimes we'll write it as. Enlarged or diminished, or the same size, if it's pointing upwards, we usually say it's erect. If it's pointing downwards, we sit down or inverted. And if it's a real image, what that means is it's in front of the mirror and it's reflected back in front of the mirror. If it ends up reflecting on the other side of the mirror, then we'd call it a virtual image. So, in my previous example, we would say it's smaller or diminished, inverted or upside down, and it's a real, in front of the mirror, image. The third rule that actually works the same and we don't need to use all three user to the trees to the mouth. But the third rule says if we go through the center of the image or center of the mirror, then it'll reflect back. So, if we extended this near, it would hit the mirror and then it would reflect back again, right back through the center. So, you can see that that third line ends up intersecting in. The same spot as the other two, so we still get the exact same location. So now let's see what happens if we do a convex mirror. A convex mirror, we often call them diverging. Let's go back to that. The reason we call this one a converging mirror is that you can see they all converge to a point. We go through the focus, we all converge to a point. That's why we call it diverging, because it's more spreading out, they won't converge. If we follow the same rules, we would go parallel and it would go through the focus. But you can't go through the mirror, so what we end up doing is we would reflect back along that dotted line. Our second rule would be going through the center, and once again it would reflect the mirror back. But if we continued it on through this as a dotted line, we'd hit the center. I can see these dotted lines intersect behind the mirror. So, in this case, we would say it is a right-side-up or erect image. It's smaller and virtual because it's behind the mirror. Instead of being in front of the mirror, where we want the image to be, it ends up showing up behind the mirror. So we call it a virtual image. Okay, there are a couple of formulas we can use. The first one involves using the focus and the distances to help you calculate. That means that anything in front of the mirror is called a positive distance, while anything behind the mirror is called a negative distance. The same goes for the focus point; anything in front is positive. You can see in this case, if we had a concave or converging mirror, our "F" or "C" and original distance are all positives. Because they're in front of the mirror, but if we had a diverging mirror, now you can see our objects on the front side of the diverging mirror. So it'll be positive, but our center and focus point would be negative because they're on the back side of the mirror. So that's the only thing you have to be careful of when you use the formula. Make sure you get your pluses and minuses in the right spot. So let's do this example. It says we have a diverging mirror with a radius of 20. So, in this situation, the radius is 20. That means our focus will be 10. So in our formula, we write it as -10 because it's on the back side of the mirror. It says the object is 30 centimeters in front, so that's a positive 30. The question is asking, what is the distance image? So di. We can use our calculator to solve for di. We just basically solve this by using fractions. So, if we go negative 1/10 (one tenth) - one thirtieth, then that'll be 1 over DI. So, in this case, 1 over DI works. So, two like negative zero point 1 3 repeating. Just go 1 divided by that, just do the reciprocal. So DI in this case is negative 7.5, okay? Which it should be if you look at the way we just did that last diagram. Our image was behind the mirror, pointing upwards. So, our distance would be negative because it's behind the mirror. The other formula we use is to determine how much bigger or smaller it is, so we can calculate the magnification. That should just be a number like 2, being twice as tall, 1/2 meaning it's reduced twice as short or whatever. Well, quite often what we do is we just use the heights and the... Distances can be measured as ratios. If we are above something, we call it a positive height. If it's downwards, we call it a negative height. This makes sense. Let's do this example: Example 3. It says, "For the same situation from Example 2, determine how tall the image would be if the original was 5." We are only going to use the second part of the formula. We don't care about the magnification. It asks, "How tall is the image?" We're looking for hi. Ho (the original height) was 5. Pointing upwards is a negative di, which we just calculated as 7.5. Our original object distance, ho, was 30. So, we cross multiply: 3/7.5 = 5/30. Solving this gives us a height. The magnification is 1.25, and it's positive, so that means it's pointing upwards. So, our original height was 5 centimeters and our image height at one point was -5. If we wanted to, we could actually calculate that magnification. So, we just go 1.25 divided by 5, and we see that our image is a quarter of the height. So, a couple of different ways to use these formulas, but they're pretty straightforward. And that's all.