 In optics, power is defined as the converging or the diverging capacity of your lenses or mirrors. For example, consider these two lenses. Just by using this definition, can you predict which of these two lenses have more power? Just think about it. All right, the way to do this is if you look at how much the rays are bending, you can see that the ray of light over here has bent a lot more compared to the ray of light over here. Look, look at the bending angle over here. It's less compared to over here. Therefore, we would say this lens has more converging power and this lens has less converging power. Does that make sense? Similarly, if you consider diverging lenses, you can clearly see over here, look, the ray has bent more, look at the angle, compared to over here, look, the angle is smaller. And so we would again say this has this time more diverging power and this has less diverging power. Makes sense, right? So that's basically how we think about power. But notice something very interesting. You see that if the converging power is more, you see you get a smaller focal length. If you have a smaller converging power, you get a bigger focal length. You see the same thing over here as well. And that makes sense, right? If you have a bigger converging power, they focus much more quickly. They focus much more nearer to the lens and therefore you'll have a shorter focal length. So shorter the focal length, more the power. And therefore mathematically, we define power as the reciprocal of the focal length. That's how mathematically we define it and what will be the unit of it? Well, since focal length has a unit of meter, one over focal length becomes meter inverse. And meter inverse in optics, for power, we call it diopters. So that's the standard unit for power. So let's take an example. So imagine the focal length of this was 10 centimeters. What would be the optical power of this lens? Well, because power is one over focal length, we would substitute and we'd put it as one over. We need to be substituting in meters. So 10 centimeter becomes 0.1 meters. To convert into meters, you divide by 100. 10 divided by 100 is 0.1, so you get 0.1 meters. And now what is one divided by 0.1? Well, that's 10. And so you'll get 10 diopters. 10 meter inverse, 10 diopters, that's the power. That's the converging power of this lens. All right, now can you pause the video and find the power of this lens? Pause and try yourself. All right, okay? Well, power equals one over F. That gives you one over this time, 20 centimeters. Again, you need to convert that into meters. So 0.2 meters it becomes. What is one more divided by 0.2? You get five. So the power over here is five diopters. And as you predicted, less power. You can see that, right? Smaller power, bigger the focal length, smaller the power. Smaller the focal length, quickly the convergence happening, bigger the power. But wait a second, how do we differentiate between converging power and diverging power? Well, remember for diverging lenses, their focal length should be negative. So over here, if the focal length is negative 10 centimeter, what will be the power? Well, it will be one over negative 0.1. So it'll be minus 10 diopters. Similarly, if the focal length is minus 20 for a diverging lens, you'll get minus five diopters. And so you can now see that for converging lenses you'll get positive power. For diverging lenses, we get negative power. And that's basically how we differentiate between converging and diverging lenses. But finally, that brings us to the question, why should I care about power? Isn't focal length enough? Why define a new thing? Well, for that, let's look at a problem like this. Imagine we have two lenses combined. And our goal is to find the total focal length of this combination. What would the answer be? Well, my first instinct is if I know the focal length of this lens as 10 and this one as 20, I might think that the total focal length should be just f1 plus f2, which is 30 centimeters. So the total focal length of this lens system must be 30 centimeters, just add them. But that cannot be true. That's wrong. Why? Well, think about this. If you're to draw the ray diagram, you can see that this lens also converges and this lens also converges. If both the lens converge together, then all of that, the focal point should be much closer than before. That means the focal length should be much shorter, isn't it? So it can't be 30 centimeters. So how do I figure it out? Focal length doesn't add up. But wait, because the convergence is happening together, this is being converged and this also gets converged, the convergence adds up, which means the power adds up. And so power will be useful over here. So what we can do is we can first find the powers, which we already did in the previous slide. So we've already found the power here is 10 and the power here is five. Now I can say the total power of this system, that should be P1 plus P2. The total power adds up and that would be 15 diopters. So I now found the total power of this system. But you might say, well, but I don't want total power. I want to find the total focal length. But we know how to do that. If I know the power, I can find the focal length by using the same formula. Power is one by F. They should also be applicable for the total power over here. So from this, I can find out what the focal length is. It's gonna be one over P, which is going to be one over 15 diopters. And if I substitute, if I figure this, multiply, divide, sorry, I'll get 0.066 meters. It's in meters because remember, this is in diopters. And if I convert it into centimeters, I'll get 6.6 centimeters. So that's now the focal length of the combined system. And you can see, as you predicted, it's smaller. It's smaller than 10 centimeters, as you predicted, because it's having a much more converging power, so much smaller focal length. And so because powers add up, when you keep lenses in contact, that's why we absolutely love power. And even when you're using contact lenses, for example, or when your doctor is deciding the power of your spectacles, that we're talking about the same optical power because it's much easier number to deal with.