Mirror Formula and Magnification
What did we see till now? If I’ve given you a mirror, we found out if we keep an object somewhere, where is the image going to form? And why did we need to do that? All needed was a scale, maybe a pencil, protractor also not required right? All we can do is just take 2 rays, extend them, reflect them back and find out where the object image forms, from where the object forms. And we saw that in different cases the image forms in different places. Pretty simple. But what if you’re even more lazy? We don’t want to use even pencil, scale, nothing. Given where the object forms we should be able to tell where the image will form. Just like that. Could we do that? Wouldn’t it be nice if we were able to do that? Now till now what did we need? Draw a diagram and figure out. Now, forget the diagram. We can do the same thing just by using some math and let’s see what we can do. But the first thing we need before we talk any math is to define something called a convention. What’s a convention? Lots of people can do things in many, many ways. Let’s say we all drive, we all in India have decided to drive on the left side of the road, right? It’s a convention, in one sense, because it’s going to be a little irritating if each of us decides in our own way and then it’s going to be very difficult to communicate. It’s going to be pretty irritating if you think about it. So what we do? All of us get together and say, okay fine. We’ll drive on the left side of the road. Now there are some countries where they do it in the right side of the road. In fact most of the countries in the world do it in the right side. But this is one example of a convention. See most of us sit together and decide okay this is how we’ll do it. We will all write A this way and not in some other way. It would be pretty pissing off if we all wrote A’s in our own ways and then we had to translate stuff. So that’s one of the things we’re going to do here right now. What we are going to do is figure out how exactly do we measure these lengths? Because the object is at some length from the mirror. Right? Now, at some length and the image might be at some other distance. So all these distances are what we are going to try and connect because that’s what we mean by finding out where the image is. So how do we measure where the image is? By giving it some distance from some known fixed point. So let’s do something very, very simple. Let’s define what’s called a convention. Now, we all know that there is a mirror. It could either be this way or it could be that way. Concave or convex. Then we’ll take the pole of the mirror as what we call our origin. Right? And what is the origin? It’s the place from which we start all out measurements. Pretty simple right? And if you have already been accustomed to what’s called the Cartesian coordinates, then we know something. From the origin, which is the pole of our mirror, everything else becomes very simple and it’s just this. Right? Anything right is positive, anything left is negative. Anything down is negative, and anything up is positive. So right and up – positive. Left and down – negative. So if you were to stick to that in any length we measure, let’s put the signs appropriately, we’re good. Because before this there was a pretty difficult convention where you measured this way in a particular manner, that way, no. It’s very simple right now. No matter what it is, right is positive and up is positive. Left’s negative and down is negative. So with that in mind the other thing we want to say is all distances are measured from the origin or the pole. It could be a mirror or as you would see in some time it could also be a lens. It doesn’t really matter. So now with this knowledge in our hand, we can jump into and try to find out where image is formed even without any pencil, paper, drawing, nothing. Right? So now that we have the sign convention in place what do we do to use it? So what do we need though? We’re going to drive something that’s very favorite you know, it’s pretty weird to most of the people who’ll ask you a few questions. It’s called the mirror formula. So we all love formulas don’t we? Have to remember this mirror formula. But we’re going to make it so that remembering it won’t be so mundane to you because you’ll also understand where it comes from. So that’s what we are going to do today, or in this couple of minutes that we’re going to spend. We’re going to understand what this mirror formula is. And first the purpose. Why do we need it? That’s the first question to ask right? Anything you want to know, why do I need it? And why do we need it? To avoid all the other drawings and everything. Just calculating out of pure math. Just one formula it tells you, if you know the object, where’s the object it will tell you where the image is. If you know where the image is it will tell you where the object is. Clean. So now let us see what we can do? Right? Let’s take one particular specific case and let you generalize to the other cases. Let’s take the case of a concave mirror and you’ve your principal axis, you’ve your pole and you’ve an object over here. The object is beyond 2f. Or let us see what happens and where the image forms by our known method. What do we do? Take one ray, and draw, that’s right, the parallel ray is going to go through the focus. And the other ray that’s going through the center of curvature is going to come back that way. So where is the image forming? Aah there. And that’s the top part of our image, this is the bottom part, so that’s where the image is going to be. Great. So now what we are going to do is just define some distances. The distance from the object to the mirror is called u. We’ll call it u, the object distance. The distance from the image to the mirror is called v. Yeah. Just giving them some names. And the focal length, let’s call it f. Pretty simple right? We’re doing this so that we can connect these 3 quantities. What are we connecting? The image distance, the object distance and the focal length. So once we’re done with these 3 we’ll see how they are related for mirrors. So let’s move this aside right now and let’s kind of take it up here and let’s see how we reason this whole thing out. So let’s take, now you can watch there and you can see, I want you to notice that there are two triangles there that have a very special property. And what are those two triangles? Yeah, those are the two. Now look at that. What do you know about those two triangles? Okay they have one set of opposite angles, the other one is a right angle, both of them. What strikes you? Two angles same for two triangles, what does that mean about those two triangles? Yeah, AA similarity, if you want to call it by a particular name, it’s called AA similarity. Those two are similar triangles and what do we know about similar triangles? All their angles are equal, but also all their sides are proportional. So let’s use our knowledge from some of the triangles. Of course you’re going to use math and physics right? Whatever we learn in math is going to get connected to physics and vice versa, it keeps happening that way. So let’s see what happens now. Take some of the triangles, take the ratio of the sides and what happens? That side by this side equals this side by that side. Great. So write it down in mathematical form that way and let’s now look again. Why are we doing this by the way? What struck us? Why did we feel at picking up similar triangles? Yeah. Because we know we have to connect some quantities. We’re looking for something that relates to those quantities. Right? We know u, we know v, we know f. How can we relate them? So we found one particular set of similar triangles. Let’s look for one more. Yeah where do you find it? Look around. There’s one more there. Two triangles that seem to be similar. Right? Because they have an opposite angle and they have right angles. Great. Now write the same very similar equation for them and if that’s the object size this is also going to be equal to the object size. So write that equation. It looks like this and what do you observe? Yes, that’s right. Those two quantities are going to be equal. And now those two being equal, just cross multiply them and see what happens. Right, becomes a very simple thing. One term cancels. Bring it down there. We’ve a very simple looking expression right now and divide the whole thing by a product. Now why we are doing this don’t ask me okay. It just makes it look more beautiful. See all arguments in physics need not just be for utility, it can also sometimes be for beauty right? It can be artistic sometimes. So I am going to do something pretty sneaky here. Divide the whole thing, the entire equation by uvf. u into v into f. Why do I do that? Because something that looks like that now begins to look like this. 1 by v, plus 1 by u equals 1 by f. Right? That’s how it’s going to look. No matter how you want to rearrange it make it look that way. What is even more important is that 1, you don’t have to remember this in a very, you know, mugging up fashion because you already know how we get it. Two pairs of similar triangles equate. That’s how it will be stored in your mind. And you’ll learn this. And what do you know about this even more? You know that this will connect, image distance u, object distance v, and focal length f. Which means that if you know any two of these, you can get the third one. And you’ll have plenty of examples where you understand how this is really useful. Makes it so that our old method is too long. So we use this from now on. So that for you is the mirror formula. (1/v) + (1/u) = (1/f). And I hope you understand how it is true as well.