Minimum deviation in prism

in this picture you see the Sun with some kind of a halo around it what's causing this phenomena turns out that it's got something to do with the fact when you shoot a light through a paddle so a glass slab the reemerge is out without any deviation parallel to the incident ray but if we shot the same ray on a prism where the two sides are not parallel to each other then the ray of light does get deviated compared to the incident ray so let's say in this video how this mere fact that the ray of light through a prism gets deviated results in something so beautiful like this so let's say we have a prism in which the refracting surfaces are making an angle a and the material is made up of refractive index n so if you shoot a ray of light on the first surface it would have gone this way but it's going to bend over here to do that bending we're gonna draw in normal now if you consider the outside medium to be just air and the Ray is moving from a rarer medium into a denser medium so it's gonna bend towards the normal downwards and so the ray of light has bent downwards let's call that angle as d1 now again if it wasn't for refraction over here it would have gone this way but again it's going to bend over here so we're gonna draw another normal and now this ray is moving from a denser to rarer medium as a result it's gonna Bend away from the normal again downwards and if you call this second angle as d2 then this total deviation the total deviation that our incident ray has suffered turns out to be d1 plus d2 so the total deviation D is d1 plus d2 and if we were to call the angle of incidence over here as I and say we were to analyze this whole situation carefully then it turns out that these angles d1 and d2 really only depend on the angle of incidence I the refractive index N and the angle of the prism a now in this video we're not going to do that analysis we'll do it in another video but if we do that analysis then we would see that the equation for D the total deviation turns out to be this the equation seems extremely scary it's popping out of nowhere but like I said this equation really comes if we were to analyze this properly and use Snell's law in some geometry which we will do in another video so don't worry about where this equation comes from we don't even have to remember this equation all that matters is look at that angle of daeviation as mentioned it only depends on the angle of incidence I the refractive index N and the angle of the prism a that's all that matters and for a given prism if you have a particular prism then the value of a and n are fixed numbers which means for a given prism the angle of daeviation only depends on the angle of incidence that's one of the key takeaways of this video the deviation only depends on the angle of incidence but of course just by looking at this equation at least for me it's almost impossible to understand how D varies with eye for example if I were to increase if the angle of incidence were to increase would do D increase or would it decrease that's something that we cannot figure out just by looking at this equation because it's an eye here as well I don't know whether this is going to be a positive number or a negative number so you know what people do usually people will draw a graph of D versus I and analyze that instead again plotting that graph manually is going to be very difficult but we have computers to do that so if you were to put this equation and plot a graph out of it we would get this and you gotta admit this is very easier to analyze compared to that scary equation and it's probably why those equations are not usually included in the textbook but they draw this graph but of course the graph really comes from that equation anyways if you were to look at this graph carefully let's see how this angle of daeviation this is the angle of daeviation changes with the angle of incidence if you start all the way from zero angle of incidence is zero then P will have some value but now as we increase the angle of incidence notice the angle of daeviation starts decreasing look at that angle of daeviation starts decreasing decreases decreases decreases hits same minimum value and then further increasing the angle of incidence the deviation starts increase can you see that this is the key takeaway of this entire video that the angle of daeviation has a minimum value so what does that mean well let's take an example if that minimum value was say as an example 15 degrees which means if we take our prism and shoot a ray of light then regardless of the angle of incidence meaning regardless of how this prism is oriented we know for sure that the angle of daeviation this angle has to be larger than or equal to 15 degrees it cannot be smaller than 15 degrees it can be larger but it cannot be smaller than 15 degrees 15 is just an example that is the key takeaway over here and guess what this is the very reason we see that halo around the Sun so let's understand how say he was looking at the sky say pointing towards the Sun and this is the sunlight that's reaching towards you and let's say there are some prisms suspended in the air somehow now these prisms are all oriented randomly let's say so the angle of incidence everywhere is different but one thing's for sure the emergent ray is deviated by at least 15 degrees or more but not less now if the rays of light from these prisms were to reach your eye then this prisms will start glowing so let's consider one ray of light that reaches your eye now the important thing over here is that the angle of daeviation this angle of daeviation that that ray has suffered is the same angle that is subtended right at your eye can you see that this angle is the same as this angle so we can say even this angle is the angle of daeviation and as a result if this angle is greater than 15 degrees then the ray of light can reach your eye but if there but that if that angle is smaller than 15 degrees that ray cannot reach your eye so for example if you're a draw a ray of light from this prism to from your eye and let's say this angle turns out to be less than 15 degrees then that is not possible that means the ray of light from this prism cannot reach your eye because you do that the angle of daeviation has to be smaller than 15 degrees so what will happen is that the ray of light from here will miss your eye similarly the ray of light from here will miss your eye they cannot reach your eye so only those rays of light will reach your eye where the angle is greater than 15 degrees something like this so if this angle is 15 degrees all these prisms can shoot the rays of light right to your eyes but any prisms over here cannot shoot there is to your eyes and as a result none of these prisms will glow for you you only see these prisms glowing so far so good the hard part is done but this was in 2d when our world is in 3d so let's take this one step further here we have drawn the same situation but in three dimension it's the same thing if you were to take rays of light from here this angle is more than 15 degrees if I take rays of light from here this angle has to be greater than 15 degrees and that's why these prisms inside are not glowing but guess what there are prisms above you and below you as well so the same thing holds for those prisms as well for any ray of light from the above prisms to reach your eye this angle again has to be larger than 15 degrees so only those prisms about that 15 degrees angle you can go for you and the britons below 15 degrees will not go for you and the same thing we can say below 15 degrees and below the prisms will glow and we can do this in all directions in any direction as long as the angle is more than 15 degrees those prisms will glue the ones small in different degrees won't glow and when you put all those glowing prisms together you end up with this circle we can also imagine these rays of light which are reaching your eyes are forming a cone and the base of that cone will glow if that angle is greater than 15 degrees if it is smaller than 15 degrees the base won't glow now of course remember 15 degrees was just an example if the minimum deviation of this prism value was say I don't know maybe 22 degrees then we would see us charcoal that was obtained an angle of at least 22 degrees and that's exactly what's happening over here there are a lot of prisms suspended in the air and it turns out that the minimum daeviation angle of all those prisms they turn out to be about 22 degrees so this is called as a 22 degree halo any prism inside the 22 degrees will not glow and that's why this is appearing dark for us at 20 degrees it's growing nicely about 20 degrees is also glowing and that's the whole reason for this 22 degrees glow and these prisms are formed by the ice crystals suspended in the air so only when you have these ice crystals you see this amazing glow so to quickly summarize any ray of light that is shot through a prism will suffer a deviation and that angle of daeviation has some minimum value that's the key takeaway now what's mind-boggling for me is that this fact that rays of light have a minimum deviation value which sounds like some boring textbook II kind of fact directly results in this beautiful phenomenon that is truly amazing