Curved surface refraction formula

so far we looked at plain surface refraction but now we'll talk about curved surface reflection and this is where stuff gets real this will also help us understand properties of lenses later on because they're also curved surfaces so let's imagine we have some kind of medium over here which has a refractive index let's say in two and its boundary we'll assume it's a curved surface it's a spherically curved surface so it's a part of a sphere which has a center of curvature over here and that midpoint of that curvature is over here which we call is the pole is the principal axis let's also assume that the outside medium has a different refractive index let's call it a fret to index sn1 and the question we're going to tackle over here is if we keep an object somewhere over here let's say where will its image be after refraction that's what we're gonna try and find out so let's say here is the object to figure out the image we have to draw no surprise ray diagrams so let's draw a bunch of rays coming from this object towards this surface one ray of light will insert it along the principal axis and this ray will go undeviated because this is perpendicular to the surface so the angle of incidence is zero so there'll be no refraction and how do we know this is normal well any ray which passes through the center of curvature is going to be normal to the surface notice this ray passes through the center of curvature so it's normal it goes undeviated now to see where the image will be you have to draw at least another one more ray let's draw an oblique ray now this ray will bend over here it will refract and to figure out exactly how it reflects you have to draw a normal to at this point and again normal has to pass through the center of curvature and let's assume that this is a denser medium compared to this one and so this ray of light will not bend towards the normal and so this ray of light would bend towards normal and now really how much it bends really depends upon how dense this is compared to this let's assume it's dense enough that the two rays these two rays will eventually meet at some point over here creating a real image let's that's an assumption all right so we are assuming that n 2 is larger than N 1 it's large enough for the disarray to bend enough to meet at this point and that's where we are getting a real image now the big question is if we know what the object distance is can we figure out what the image distance will be what does it depend on that's what we need to figure out all right so let's do this but where do we even begin how do I bring my object distance and emails distance into the picture that's the question well there's only one thing we know when it comes to refraction Snell's law so that's probably what that's where we start all right so Snell's law connects the angle of incidence to the angle of refraction and this is the incident ray will apply Snell's law at this point so let's call it point as M let's say and this is the incident ray this is a normal so this would be the angle of incidence let's call it as I this is the refracted ray and this is the normal so this is the angle of refraction over here let's call it as R and now Snell's law tells us that N 1 Sinai and 1 Sinai that's going to be n 2 into sine R sine R and we're gonna make one up one approximation the approximation is we're going to choose we're going to assume that this point M is very close to point P in other words we're going to assume that MP is a very tiny value compared to the radius of curvature so it's much smaller than the radius of curvature which is PC okay the reason we're doing this is because what when M comes very close over here this incident ray will be actually somewhere along this line and as a result this angle of incidence I will be a very tiny value which means this angle R is also going to be very tiny and that means we can use small-angle approximations over here that's the whole reason we are doing that so that the mat becomes easier all right so small angle approximation is we can assume sine theta is just theta so sine I will assume it to be just I and this will be n 2 times sine R can be assumed to be just our so that's that comes from this approximation alright so what do we do next remember we need to connect the object distance and the image distance somehow we need to bring this into the picture what's the connection so obviously the next question we what's the connection between I R and these distances how do we bring them into the picture well we can't see the connection directly but if we define new angles so if we define an angle over here let's call it as a you'll see why we're defining that will define another angle called B here and let's define another angle over here called as C if we define these three angles we can somehow bring the object distance and image distance into the picture here let's see how first of all we can connect angle I and R with a B and C we can do that using the properties of a triangle so that's how we can introduce a B and C over here somehow and once we have brought these angles well we can define all these angles in terms of these lengths that's possible by looking at these triangles all right so that's the rough that's the rough flow that we're gonna go for all right so the first step is to somehow bring ABC into the picture can you see a connection between the angle I a and B I want you to pause the video and think about that connection I'll give you a clue look at this triangle giant triangle over here all right well angle is the exterior angle and a and B are the interior angles so you may already learnt that the sum of the xt sum of the interior angles must be equal to the exterior angle so i equals a plus b so instead of i we just write this as a plus b and that will be equal to n 2 + 2 into R again I want you to pause the video and see if you can find a connection this time between R and B and C again the trick is the same look at this big triangle now alright this time R and C are the interior angles so if we add R plus C we must get B and so R is just B minus C so we can put that over here our is B - C and now since MP is a very tiny patch on this entire sphere we can assume it to be flat just like when you take a tiny patch on earth we assume it to be flat and now we can use this right angle triangle with a very tiny angle and figure out what a is so let's write that down so this equation is now going to be + 1 times what is angle a well let's look at this triangle we can use small angle approximation a is the same as tani right and what is tan a equal to well that's opposite side MP divided by the object distance Opie Opie plus what is B equal to well can you look at this figure and similarly figure out which triangle we should use and again use the same approximation I want you to pause the video and see if you can fill in these now figure out what B is and what angle C is all right let's see do figure an angle B let's look at this triangle and do the same thing small angle approximation so B is the same as can be and tan B is going to be the opposite side MP divided by the edges inside the edges inside PC is just the radius of curvature PC that equals n/2 times again B is the same as MP / PC - C or what is angle C well look at this big triangle now again small angle approximation C is the same as tan see that's going to be the opposite side divided by the hidden side which is the image distance so that's going to be P I and if you look at this equation carefully we have found what we wanted because we have the relationship between the object and the image distances all you have to do is make this expression a little bit more pretty alright so MP is common so you can take it out from here we can take it out from here and we can divide so the MP cancels out second of all this need not be a general formula because we are deriving it for this specific case we have zoomed over here that the ray of light is going to bend enough to come and meet over here but in general it need not for example if enter alley was a little lower then maybe this ray of light would have bent only so much now see these two Ray's appear to be coming from somewhere over there and so we would have gotten a different formula maybe if this was curved the other way around again you would have gotten a different situation so there are so many cases available over here so we might start panicking now but we don't have to because we have a super power with us sign conventions we've seen in mirrors how by using sign conventions we can make a general formula guess what we'll do the same thing over here if you use sign conventions and substitute then we'll end up getting the general formula again will not dealer I will not prove that but it turns out to be true and the sign convention is the same we're going to treat Paul as our origin and incident direction is going to be positive so everything on the right side of the pole is going to be positive positions and then to the left side of the pole are going to be negative positions and now we can substitute so let's make a little bit more room over here and see what we get so if we substitute we get n 1 into 1 over object distance Oh P but that's a negative so we'll call is minus u plus 1 over PC PC is the radius of curvature and that's positive because the C is on the positive side so that's going to be R plus R that's going to be n 2 times 1 or PC which is R minus 1 or PI again P I is positive because eyes on the positive side so that's going to be V image distance is V and now all we have to do is simplify this so pretty much physics is almost done not almost physics is done we just have to simplify this now so let's make some more room now and let's simplify this so let's do this over here we get minus N 1 over u plus N 1 over R which is multiplying multiplying equals n 2 or R minus n 2 or V and the last step which is going to put U and V on one side and the Artem's on the other side so that will give us we'll add into Y V on both sides so we'll get like that over here and 2 over V minus N 1 over u minus N 1 over u that will be equal to n 2 minus N 1 I'm going to subtract and 1 by R on both sides so that's going to be n n 2 minus N 1 divided by r just check that all I've done is rearrange this and this will be our equation for curved surfaces so this is a general formula that will work for any case for curved surface refraction so pretty big deal we might have to remember this force numericals and stuff so the last thing i'll do is tell you how I love to remember this formula so you see if you look at this refractive index n 2 that's really the refractive index of the medium that contains the refracted ray this is the refracted ray and it's divided by the image distance that's nice for me because after refraction comes the image distance so I remember it as the refracted medium this is the refracted medium divided by the image distance minus N 1 well that's the refractive index that contains the incident ray so I have to think of this as the incident medium and incident where is incident ray come from that comes from the object so incident medium divided by the object distance that's how I remember this and I don't need confuse so I always remember refracted medium by image distance minus incident medium divided by the object distance that equals this minus this say it's n 2 minus N 1 we have to be very careful with the order n 2 minus N 1 divided by the radius of curvature