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# Spherical mirrors, radius of curvature & focal length

## Video transcript

in a previous video we saw that if you take a mirror which is a part of a sphere then we call such mirrors as spherical mirrors and if this mirror is small enough meaning if it's a small enough part of the sphere then if you incident parallel rays of light it will almost get focused at a single point in this video we'll talk about some details about this we'll talk about exactly where do these patterns of light after reflection get focused exactly where is this point and then we had seen that if the mirror is too big if if it's too big a part of a sphere then it won't be able to focus rays of light to a single point so exactly how small should the mirror be to be able to achieve this and we will also introduce some technical terms because we will be using those terms a lot in all the future videos so let's just familiarize ourselves with those so let's start with some technical names this point where all the rays of light all parallel rays of light eventually meet up it's called as focus because the rays of light are getting focused and is represented by capital F and if you go back to the mirror then the center part of that mirror the central part of that mirror the center of this would be somewhere over here that point is often called as the pole and it's represented by P pole P and this distance the distance from the pole to the focus that distance is also given a name it's called and this is something that we keep using all the time this is called the focal length all right and it's represented by small F so small F represents focal length for cooling it's basically number that tells us how far the rays of light are being focused from the center of the mirror and focal length is an important number and we would like to know that because if you want to use this mirror for anything then we need to know exactly where the rays of light would get focused isn't it and we can figure out where the focus would be or how big the focal length is by doing some math by using the rules of reflection so basically we'll have to use some geometry some approximations and if we do all of that which will not do in this video but if we do all of that then it turns out that this point F lies pretty much not exactly pretty much because everything is approximation but this point pretty much lies in between the pole and the center of the sphere all right so this distance the focal length is pretty much half this distance the radius of the sphere so let me just write that down so F turns out to be half the radius of the sphere so let's use our for radius R divided by 2 okay and that means if we know what the radius of this sphere is then immediately we know what the focal length is so if the radius of the sphere is say 20 centimeters then it immediately means the focal length is 10 centimeters because it's half of that and by the way this R the radius of this sphere is also given a name it's called the radius of curvature radius of curvature okay it's called so because it's literally the radius of this curve we can't say radius of sphere because the sphere is imaginary and it's a sphere doesn't exist so that's why we say the radius of this curve and I used to always get confused with this especially when we used to go to labs let me tell you why so when we go to labs we we get to experiment with this mirror so if you hold the mirror in our hand and if you look at from the front this is what it might look like and you know in this picture there is a pen and it's reflection ignore those for a while let's not worry about that let's only look at the mirror now when someone says the word radius you know what would eventually what would come to my mind I would just think about this distance the radius of this circle because that's what radius is isn't it but when we say the word radius of curvature notice we're not talking about the radius of the mirror we're talking about the radius of the sphere of which the mirror form say part so to do this what we need to do is we need to take this mirror and look at from the side so we need to tilt its side worse somewhat like this and then imagine a sphere that completes that particular mirror and then imagine the radius of that sphere that would be the radius of curvature and so whenever we used to do a wish to go to the lab and figure out the radius of curvature and say radius of curvature of this turns out to be our number could be like 25 30 centimeters I would always be thinking like how can the radius be 30 centimeters but you get the mistake that I was making right and this point the center of the sphere is also your name it's called the centre of curvature and again if we are given a mirror in our hand the center that we see that's not the center of curvature that's the center of the mirror and that's the pore okay so this point represents the pole the centre of the mirror - do we visualize the center of curvature again we have to imagine the sphere of which the mirror forms apart all right one more technical term is related to the width of this mirror the width of the diameter of this mirror if you look at this diameter from here to here this is often referred to as the aperture of the mirror aperture of the mirror and that's important because this diameter tells us how much how big the mirror is and that will tell us how much light it can gather so you can pretty much see from this figure let me just make this a little bit transparent so if the aperture or the width of the mirror was bigger then this mirror would be a bigger part of this sphere and as a result it would gather more light and focus it at this point and so this brought this spot would be brighter however remember in previous videos we've seen that if we make the mirror too big a part of the sphere then all the Rays will not get focused at a single point then this point will start getting spread out you've studied that in in in previous videos isn't it so usually we like to keep the aperture small and what we mean by small is we like to keep it about two or three times at least smaller than the focal length if it is that small then the Rays of light almost get focus at a single point if the aperture size becomes too big the spot becomes bright true but it also tends to spread out more and one last thing usually when we draw these spherical mirrors we like to draw a line that passes through the pole and the center of curvature all right so we like to draw on straight line like this and that line will also have F on it and this line is often called this line is often called as the principal axis principal axis and this is important because only those rays of light which are parallel to our principal axis only those rays after reflection will pass through this point F so if we have rays of light which are not palette of principal axis so let's look at an example so if we have let's say rays of light which are somewhat like this notice these rays are parallel to each other but they are not pilot to principal axes look at what happens to these rays after reflection they don't get focused at that point they get focused somewhere below it all right now that's not all that important what's important is they don't get focused over here and so the important thing and by the way this is an important fact because we'll be using this all the time in the future videos the important fact is any ray of light which is parallel to this principal axis that ray after reflection will pass through this point f and so whenever we're gonna draw ray diagrams for convex or concave mirrors as we will do in future videos the first thing we'll always do are for drawing the mirror is will draw a principal axis which passes through the pole and the center of curvature such that if there is any ray of light which is parallel to it we know immediately after reflection it has to go through the focus and that's pretty much it so let's summarize what we learned we learned a bunch of technical terms that we'll be using in the future and we saw that the important result that we saw over here is whenever we have spherical mirrors of small width or small apertures when approaches are about 2 or 3 times at least smaller than the focal length then the focal length turns out to be roughly half the radius of curvature