Deriving prism formula

in a previous video we saw that when you take a prism and shoot a ray of white the ray of light undergoes a deviation it undergoes a deviation and this deviation has a minimum value and this really we're gonna figure out exactly what that minimum value of this deviation is to figure this out let's name some angles let's call this incident angle as I the ray of light got refracted so let's call this angle as our quad is r1 remember our angles are always with respect to the normal similarly over here again there's an incident angle we'll call this angle as r2 will call the angles inside the prism as our so let's call this angle as r2 and finally the ray of light is emerging out and we'll call that angle as the angle of emergence we'll call it angle is e so called this angle as e now if you if you see carefully will see that the ray of light is divided twice once I was here second time over here over here notice that this was the initial direction and the ray of light has bent downwards so this angle would represent the first angle of daeviation d1 let's call that and similarly over here the real flat was going this way but again it got deviated downwards and so this will be the second angle of daeviation let's call that as d2 so let's call this angle as t2 and so our total deviation d now becomes d1 plus d2 d the total deviation becomes d1 plus d2 and made to simplify this our first step will be to try and put D 1 and D 2 in terms of these angles I r1 r2 and let's see how we can do that if you look over here we can see that this angle is the same as this total angle I hope you can see that this total angle because they are vertically opposite and therefore we can see that d1 is I minus r1 so d1 is I minus minus r1 similarly what is d2 equal to again if you look at this carefully we can figure this out if you see this carefully we can see that this angle is the same as this angle so let me mark that this is the same as this angle they're both same because of again vertically opposite angles but this total angle is e and so d2 is e minus r2 so let's write that down this would be e minus minus r2 and so now we can write our angle of deviation s angle I plus angle e - I'm gonna take the minus common - r1 plus r2 now we cannot call this as our final equation for deviation because if you look carefully r1 is something we can calculate by using Snell's law by playing Snell slower here if you know I we can calculate R 1 similarly R 2 well that can be calculated once we know R 1 by doing some geometry over here and similarly once we know R - II can be calculated again by applying Snell's law and so these are numbers that can be calculated so this is not the final equation yet so we need to get rid of them the first thing we'll get rid of is R 1 and R 2 they're a little bit easy to do that and to do that we have to concentrate on this triangle look at this triangle if you look at this triangle carefully we can figure some kind of relationship between a R 1 and R 2 I want you to pause the video for a while and see if you can try this on yourself alright let's see well what we'll do is we'll set the angry angles of a triangle should add up and give us 180 degrees so angle a Plus this angle over here oh what's that angle equal to well if you look carefully this whole angle is 90 degrees because this is a normal this is a perpendicular drawn right so this is our 1 therefore this must be 90 minus R 1 so that is 90 minus R 1 and similarly this angle that is 90 minus r2 plus 90 minus r2 that should equal 180 degrees now 90 plus 90 is 180 that cancels and so what we see is that a should be equal to r1 plus r2 that is the relationship that we get between r1 and r2 and a and so notice we can now substitute this directly over here and our equation becomes d equals I plus E + E - - a alright one more thing we might have to do is eliminate e and guess what that turns out to be the most tedious part not the hardest part but the tedious part of this derivation because if we look carefully you can write e in terms of r2 by using Snell's law and then by using this equation R 2 can be written in terms of r1 and then again by using Snell's law r1 can be written in terms of I so if you think carefully e is actually a number that only depends on I and if you were to apply that you know what you'll end up with will get this scary-looking equation it only looks scary but it's not really and nobody we're not gonna analyze that don't worry at all but these signs and sine inverses are coming all because of Snell's law and this whole giant thing is actually e if you look carefully and you can see that the only thing it depends on is the angle of incidence I so if we were to use this equation if you look at this equation we finally walk got what we want we have angle of daeviation it only depends on the angle of incidence i and remember our goal is to figure out when this value becomes minimum and so now we have to ask a mathematical question how do you figure out in a given equation when the value of D becomes minimum and it turns out we have to use calculus for that in turns out that if you can differentiate this equation then we can figure out what the minimum value is and you can see that's absolutely Nightmares to differentiate this particular equation so this would be the logical step to do if you are doing it mathematically but since it's so tedious we are not going to do that take a slightly different approach that won't require any calculus so we will plot a graph of D versus I and then we'll try to figure out the minimum value directly from the graph of course you might be wondering oh how do we draw a graph of this manually drawing with that would be tedious well we can use computers to draw a graph for us and that graph pretty much looks like this graph you can see that as the angle of incidence increases the deviation actually decreases and then hits a minimum and then increases again so we are interested in what that minimum value is and the exact shape of this graph might vary a little bit depending upon what values of N and a we choose so don't worry too much about the how the graph exactly looks like but very interested in that minimum value so how do we figure this out the key is to understand this if you take up any particular value of daeviation so let's say you take this particular value of daeviation what you find is that that deviation can be obtained for two values of AI can you see that we can obtain that for two values so for example for this value of AI and for these this value of I so these two values of incidences gives you the same value of d why is that happening why are we getting the same value of daeviation for two values of AI to see why let's bring back our prism so here it is it's pretty much the same thing but we're taking an example let's say the angle of incidence was I don't know maybe 20 degrees and let's say we calculate the angle of emergence it turns out to be 50 degrees now the deviation you can see this is the incident direction this is the emergent direction so this is the angle of daeviation so it will get some value of daeviation let's say that is d and let's say this is that angle 20 degrees now what if we were to incident the same ray backwards meaning we made this as our incident ray and so this would be our new angle of incidence well if you think carefully geometry and Snell's law doesn't care about the direction of the incident ray we would end up with the same result and it's for that reason we can argue now the ray of light will just trace backwards and so then this would be the emergent ray this would be the emergent angle so now notice this is the incident direction this is the emergent direction so this would be a new angle of daeviation but guess what this angle is the same as this angle which means 450 degrees incidence we get the same deviation as what we got for 20 degrees incidence so we can think of it as if one of this angle is what we call as I then the other angle at which we get the same deviation is the angle e hopefully that makes sense but you may be wondering well what's the big deal so for two angles of incidences we get the same deviation why should we care about that well because if you look now at the minimum value of daeviation if you look at this case when this value of daeviation is minimum so I'm just gonna call em for minimum we see that that happens only at one one particular angle of incidences and not two what could that mean I want you to pause the video and just think about this for a while what could that mean oh that could only mean one thing that at this point the angle of incidence must be equal to the angle of emergence I mean think about it if they were not equal to each other at this point then you would have still gotten two values at which the deviation should have been same just like what we had over here the very fact that there is only one value gives us the clue that at minimum daeviation when the angle of daeviation is minimum these two have to be equal to each other and that's the clue that we need to solve this without doing any calculus so let's go back to our equations here it is I've rearranged a little bit to make more space and so what we found is that when when our angle of daeviation hits the minimum value the angle of incidence has to be equal to the angle of emergence and let's call that special angle let's call it as I naught and if you think carefully if these two angles are equal to each other then from Snell's law we can argue when these two angles must be equal to each other think about that for a while so that also means that our 1 should be equal to our - let's call that angle as are not so let's make some space and what I love about this approach is we don't have to look at the big scary equation at all alright so here is the situation that I've drawn at minimum and so the next step would probably be well we can just now go ahead and substitute this over here let's see what happens so now D becomes D minimum so we get at minimum daeviation we get I plus E but they're both equal to I naught so we can say that's equal to 2 I naught 2 I naught minus a but we still haven't found what our minimum daeviation is because we don't know what I naught is we still don't know but we have another equation so maybe we can go ahead and substitute that so the next thing we can do is we could say well a equals R 1 plus R 2 so that's - or not - or not and now here comes the tricky thing what do you think we should do next I mean we still have a lot of variables R naught and I naught are variables we don't want them in our equations what do you think we should do next to figure out exactly what D minimum is I want you to pause the video and think about this well we can connect I naught and R naught by using Snell's law over here and that is the last step for us so it's not very obvious a little tricky but that's what we'll do so connect so apply Snell's law at this point so let's see what we do that ok we'll do that here itself so if we apply Snell's law we can say 1 times sine I naught which is the same as sine I naught that should be equal to n times sine R naught n times sine R naught and now we can find what I naught is from this equation we'll do this mentally what is I not equal to well we'll add a on both sides so I'll get D minimum plus a and we'll divide by 2 so if you look carefully we'll get a a Plus D minimum plus D minimum divided by 2 can you see that that is I naught and that equals n times sine of r not well our R is a divided by and if you look at this equation carefully that is the final equation and notice in this equation we know what is we know what n is that means we can calculate what the minimum deviation is of course it's it's not very easy calculation but you can calculate so to summarize in this derivation the first thing we do is we write down the angle of daeviation in terms of the angle of incidence and the angle of emergence and then we see that the deviation becomes minimum this is the key takeaway that the angle of incidence equals the angle of emergence and then we eventually substitute that's what we come over here and we use Snell's law to finally get this expression and one last super quick thing is that in a previous video we saw this 22° halo around the Sun is caused sometimes due to the suspended ice crystals acting like a prism but why is it 22 degrees or because these prisms have a minimum daeviation angle of 22 degrees and we can test that you see we know the refractive index of this prism they're ice crystals so that's pretty much water then we also found that these ice crystals have an a value of about 60 degrees and if you plug that in you'll actually see that the minimum radiation value is roughly around 22 degrees that is amazing if you think about it and that's the whole reason we get a 20 degrees hero