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Sign convention for mirrors (& lenses)

Let's explore the sign conventions (rules to assign +ve and -ve values) we use for mirrors and lenses. These rules are called Cartesian sign conventions. Created by Mahesh Shenoy.

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Video transcript

- Let's explore Sign Convention in this video. Basically we're gonna talk about the rules that we're going to apply positive and negative signs to measurements. Now you may be having a lot of questions in your mind. As in what measurements are we talking about? Why are we applying signs here all of a sudden? And where are we going to use them? So what we'll do is, we'll first look at what the rules are. Then we'll look at some of the examples. And then, hopefully, you'll kind of, sort of make sense why are we talking about it and where it's going to be useful. And one last thing before we begin. Although in this video we are only going to talk about mirrors. If you are watching this video while studying about lenses. Then the same rules are gonna be applied even for lenses as well. Just wanted to clear that out. And, by the way if you are not studying lenses or you haven't studied them yet don't worry about it. So let's start with the rules, and to understand these rules let's take one example. Let's say we have a concave mirror with us, and by the way this is called a concave mirror because the reflecting side is forming a cave. So what are the rules now? Rule number one, is that all the distances are measured from the pole. We call what pole is. Pole is the center of this mirror. Now when I say all distances, honestly there are only three distances. One is the focal length, how far the focus is. Then object distance, how far the object is. And the image distance, how far the image is. All the three distances are measured always from the pole. That's rule number one which I will not write down. Now rule number two is, when you measure these distances, these distances can be measured either in front of the mirror or it can be measured behind the mirror. To give you an example: Images. Sometimes the images will be formed in front of the mirror. And you may have seen that sometimes it can be formed behind the mirror. So to differentiate between the two distances. We're gonna call one of this measurement as positive. And the other measurement as negative. So the second rule is to figure out which distance then should we call positive. Alright. And let me write this down. The rule is we always chose incident direction as positive. Incident direction is always chosen as positive. So what do we mean by this? Well if we come back over here. Notice that because this is the reflecting side over here, the incident light will always be to the right side. Because, you know you'd keep an object over here and notice where the incident ray would be. Incident ray, if you we're to draw any would always be to the right. And so since incident direction is positive. In this example, right side is positive. And what do we mean by right side? The right side of the pole, because all measurements are done from the pole. So any measurement which is towards the right side of the pole, in this particular example will be positive measurement. And any measurement which you'll do to the left side of the pole will be negative measurement. So to take an example, imagine we are given the focal length of this mirror as two centimeters, everything is in centimeters let's say. There's an object kept at 3.5 centimeters from the pole. And let's say, once we draw ray diagrams we figure out that the image is formed at five centimeters from the pole. These numbers may not be accurate, I don't know. But just take an example. Now let's assign signs to them. Let's start with the object distance. Notice that the object is lying on the negative side. Therefore, this object distance would be negative. Similarly look at the focus. The focus lies on the negative side of the pole. Therefore, this focal length would be negative. Similarly look at the image. The image lies also on the negative side. So here the image distance would be negative. Alright, let's take one more example. And it would be great if you could pause the video, and see if you can try this yourself. So just pause the video and apply signs to these three distances. Remember, incident direction is always chosen as positive. Alright. So incident direction is taken as positive. Over here this is the reflecting side, and our object is over here. So the incident rays will be towards the left. Therefore, you start from the pole and all the measurements to the left are considered as positive this time. Does that make sense? Because you are choosing, because our incident ray is to the left. And all the measurements to the right side now will be negative. And so the object is on the negative side, so it'll get a minus sign. The focal length color you'll notice is on the positive side. The focus is on the positive side, so this focal length is positive. And similarly if you look at the image it's on the positive side. So that image distance will be positive. Alright now there's one more rule. And that rule has something to do with heights. Sometimes when you consider objects and images, the heights will be above the principal axis and sometimes the height will be below the principal axis. Again we'll differentiate between them. We're gonna call as one, one of them as positive. The other one as negative. So again the rule that we're gonna follow with here is you're gonna choose above principal axis or (muffled speaking) I'd like to just say, up is positive. Alright? So if there's any height which is above the principal axis, we'll call it as a positive height. And if there's any height below the principal axis we'll call it as a negative height. So if we knew the height of this object. I don't know, maybe say the height of this object was five centimeters, then that would be a positive height. Alright. And whatever is the height of this image, that will be a negative height. I'm pretty sure You can understand this. Since this object height is above the principal axis it will be a positive height, whatever the height is. And look at the image, again the height is above the principal axis. This will be a positive height as well. And that's it for Sign Conventions. So we have three rules. All the distances are measured from the pole. Then you start from the pole and you go in the incident direction that's chosen as positive. And when it comes to heights. Any height above the principal axis is chosen as positive. And by the way, this rule is given a name, this particular convention that we are following. It's called as the Cartesian Sign Convention. Cartesian Signs. And the word Cartesian might remind you of a graph. And the reason behind it is because that's what we do in graphs as well, right. In graphs we start from the origin. And we choose the right side usually as positive, left side is negative, upwards is positive, downwards is negative. That's basically what we're doing over here as well. We're treating the pole os the origin because you're measuring all the distances from the pole. And instead of choosing right side as positive, the only difference is you are choosing incident direction as positive. That's it. But pretty much everything else is the same as graph. That's why it's called the Cartesian Signs. Now one small thing is that in some textbooks they follow this a little differently. It's pretty much the same so the only different thing that they would do is while drawing the diagrams they will always make sure that the incident direction is drawn to the right. Okay. They will always make sure this is true. Then, automatically the right side will always be positive. Alright. For example they would recommend never to draw like this. Because over here incident direction is to the left. They will always recommend you make sure that your object lies on the left side of the mirror. So we have to flip this whole image. So basically they say that you always draw in such a way that the incident direction is to the right. Alright. And then we can always choose the right side as positive, which is what we do usually in any graph. So it's more aligned to the graphical system. Okay. Now you can follow that, obviously, but in general what I like to remember is you draw however you want. Just take incident direction as positive. So anything you can follow it's pretty much the same thing. So the last thing we'll talk about is, why do we even consider signs? And for that, let's take an example where we've already done this before. So an example of things falling under a gravity. You may recall that whenever things are falling under gravity, usually when an object is falling down we would say it's velocity is negative. And when it's going upwards you would say it's velocity is positive. If we didn't use that sign we might have to explicitly say in English the velocity is downwards, the velocity is upwards. But we don't like that. We want to reduce English and use math as much as possible. So Sign tell us what direction the ball is traveling mathematically. And the idea with here is exactly the same. Without signs we have to explicitly mention we're dealing with a concave mirror, or a convex mirror, or the images in front of the mirror, or the images behind the mirror. But with Signs these things are done mathematically. For example, if I just say negative focal length it automatically means it's a concave mirror. Positive focal length, convex mirror. If I say the object distance, or let's say the image distance is negative automatically it means it's in front of the mirror. It's positive automatically means it's behind the mirror. So long story short, Signs help us differentiate between the directions in which you are doing the measurement mathematically.