Power of lens

suppose we have a lens of focal length 20 centimeters then if we incident parallel rays of light it's going to get focused 20 centimeters in front of it but what if we keep another lens of focal length say 50 centimeters in contact with the first lens and again if we incident parallel rays of light now where will the rays of light get focused to answer questions like these conveniently we introduce a new quantity called power so usually power is defined as the converging or the diverging capacity of a lens and sometimes books define power mathematically so they would directly write power as the reciprocal of the focal length 1 divided by the focal length and we'll see in this video that these two definitions are very very similar to each other they're pretty equivalent to each other and I think so the best way to understand power would be to start with an example so let's say we have two converging lenses convex lenses one which has a focal length of 20 centimeters the other one which has a focal length of 50 centimeters and let's say I were to ask you which of the two has more power what would you say and by the way I don't want you to look at this definition as of now so let's let's not look at that let's not concentrate on that let's only concentrate on the first definition so which of the two lenses has more converging power I think it's easier to answer this question if we could actually see the Rays of light being converged by these two lenses so let's throw some parallel rays of light now could we tell which of the two lenses is converging more here let me help you a little bit more let's only concentrate on the topmost ring if we could see only the topmost ring and if we looked at how much the Ray got bent can you see that in the first case the lens is bending it more compared to the second one so similarly all the rays of light are being bent more by the first lens compared to the second lens and as a result these rays of light are getting focused much closer to the lens compared to the second one so doesn't miss clear state that the first lens has a higher converging property it's converging it's bending the rays of light more it's converging it more and as a result it's able to focus it closer compared to the second one and so now we can confidently say that it's the first lens that has more power and indeed it does so this lens has a higher converging power higher capacity to converge rays of light compared to the second lens so this has more power compared to the second lens and now if you look in terms of the focal length that is that more power automatically means a shorter focal length hopefully that makes sense higher the power more the convergence means the rays of light it converged closer to the lens and as a result the focal and gets shorter so shorter the focal length more the power so can you see that power and focal length have inverse relationship and as a result I'm pretty sure now we can understand the second definition then the reason why we define power mathematically as the reciprocal of the focal length is because of this as the converging power increases the focal length gets shorter and now that we know how to calculate power let's just go ahead and calculate the power for these two examples so over here the power for this particular lens would be 1 over the focal length and the focal length is 20 centimeters and notice since 1 over 20 is going to be a really tiny number let's keep this in terms of meters so that the denominator will be smaller so 20 centimeters is 0.2 meters and this has point two meters and one over point two is five so this would be five and we will get the unit as meter inverse and so you see the power has a unit of meter inverse and usually we like to call that as D let me just write that down so usually we write meter inverse as five D and this D stands for diopters die up terse we don't have to worry about what that name means all that matters is diopters is just meter inverse so let's go ahead and write the same thing for this one you can pause the video and see if you can find how many diopters the power for this one is alright let's do this so power will be 1 over 0.5 meters and so this would be 1 10.5 is 2 so 2 meter inverse which we'll call this 2 diopters again the same thing diopters alright before we continue just I want to talk about a couple of places where I would get confused and get wrong answers so one is when power is asked in diopters remember we should always substitute the value of focal length in meters in a hurry sometimes I would substitute directly in terms of centimeters and I would get wrong answer if you remember diopters is meter inverse we will not go wrong all right the second thing is sometimes I would get questions where only the focal lengths are mentioned no drawings whatsoever and I would be asked which has more power now without thinking much I would say whichever has the larger focal length has more power you see that's wrong remember that power is inversely related to focal length so larger focal length means smaller power and if you ever get confused about this always draw it and you'll make sense so as discussed we see that the first lens has a higher power compared to the second one now another important thing is power is sign-sensitive for converging lenses power always turns out to be a positive number why well remember sign conventions for lenses you start from the optic center and you call all the positions to in the incident direction as positive positions and the opposite directions are negative positions so that means over here right side is positive and left side is negative so notice for converging lenses the focal focal length lies on the positive side and as a result the focal length is positive power also ends up becoming positive but if we were to have a diverging lens so let's say we had a concave lens and let's say we had a diverging lens of this focal length 20 centimeters we can't quit powerful diverging lenss is exactly the same way now the only difference over here is that the focal length is measured on the negative side and as a result this focal length will be negative and so the power will also be negative so diverging lenses end up having negative power converging lenses end up having positive power so now we can go back and answer the question that we asked in the beginning if we have those two lenses in contact what will be the focal length of this combined lens well the answer is not 70 centimeters because the focal lengths don't add up and here's the way to see why if we didn't have the second lens and if we just threw parallel rays of light on this first one then these rays would get focused 20 centimeters in front of it because it has a focal length of 20 centimeters but now when we keep the second lens these rays of light are converged further by the second lens and as a result the due to the second convergence notice the rays of light will now get focused even closer than 20 centimeters so we would expect the total focal length to be less than 20 centimeters so the focal lengths don't add up but maybe you can guess that although the focal lengths don't add up they're converging property is adding up because there are two times convergence happening and as a result we may guess that their powers end up adding up and it turns out that if you do the detailed analysis which we won't do over here that it's indeed right their powers do get added up that's why we even like to talk about power so instead of looking at focal lengths if we talk in terms of their powers and this is the power we just calculated then it turns out that their combined power the combined power or power of the combined lens will equal the sum of their individual powers you just add them up so it'll be five plus two that's just going to be seven diopters and if you want to now find out what is the combined focal length well we can think that's just the reciprocal of the combined power so we can just use the same formula this is the reciprocal of the combined focal length combined focal length and so the focal length the combined focal length or the total focal length will be just one over seven and afters I'm gonna put back as a meter inverse and if you calculate this you get roughly fourteen centimeters you can just pause and check that so these rays of light are going to get focused at a distance of 14 centimeters from the lens and by the way it's a small thing these lenses we are assuming them to be pretty thin compared to their focal length so we don't have to worry about where we are calculating this measurement from but anyways think about it could you have guessed then when you take 20 centimeter focal length and put a 50 centimeter focal length lens in front of it the total focal line becomes 14 centimeters I don't know about you but that doesn't seem intuitive to me but if you look in terms of power of five doctors and the Studer afters lens combines to give me a seven doctors power that seems very intuitive and one last thing this is super useful for our eye doctors turns out our eyes also converge rays of light to a single point so let's say that our eyes need a 60 doctor power to see things clearly but for some reason let's say you're not able to see things clearly so you go to an eye doctor and your dog figures out that the problem with your eye is that your eye is only able to provide you a pro power of 58 doctors but we need 60 diopters power and since powers add up it's easy to recommend the solution to get 60 doctors all we need is a contact lens of two doctors so your dog is going to recommend you to wear a lens which has the power of two doctors and we are done just like that on the other hand if your eyes had more than 60 diopters power then we need to reduce that power in such cases our dog is going to prescribe you a lens of negative power a diverging lens and although in reality things are way complicated than what's discussed over here this gives us some insights on things work over here and so if you ever seen an eye prescription then you would see the powers of the required lenses are mentioned over here powers in diopters not focal lengths so to quickly summarize power is the converging or the diverging capacity of events and we saw that shorter the focal length more is the power and as a result mathematically we calculate power as just the reciprocal of the focal length and therefore it's unit becomes reciprocal of meters that is meter inverse which we call as diopters and converging lenses have positive power diverging lenses have negative power