 Hi, I'm Zor. Welcome to Unizor Education. Today we'll talk about a particular kind of refraction through a prismatic lens. Okay, now first of all, let me just make this standard introduction. This particular lecture is part of the course, which you can get from the Unizor.com website. The course is called Physics for Teens. There is a prerequisite course, which is called Math for Teens on the same website. The website is completely free. There are no ads, no strings attached and well, each course contains a set of lectures organized in certain parts, in certain topics, in certain individual lectures. There are some exams also on the website. What's very important, every lecture has a textual part. Basically, it's like technical notes, but it's like a textbook. So you basically have both. You have the video presentation and the piece of a textbook, which is related to this particular lecture. Now, yeah, that's it. So let's talk about refraction. We did define two very important properties of light, reflection and refraction. Reflection we started before. Now this lecture about refraction and the previous lecture basically defined what a refraction actually is and in two words, that's a change of direction, change of trajectory of the ray of light when it goes through a border between different media. What's very important is that the speed of light in one media is not equal to speed of light in another media, for instance, air and glass. So whenever the ray of light goes from the air to the glass and then from the glass to the air, it changes direction on each border and each boundary between two different media. Now, this is used very extensively in lenses. So everybody knows what lenses is and I will talk today about one particular type of lens. Well, probably the simplest one. It's a prismatic lens. So what is a prismatic lens, triangular prismatic lens? So let's just use this. Okay, let's say you have a right triangle and you have another right triangle and you have this. So this is a right triangular prism. Okay, now when I will talk about changing of the direction of the light propagation, I will use the ray of light which is perpendicular to one of the either the back side or the bottom side. Anyway, the side which actually is connected to one of the catheters. Not this one. But whenever the light goes, let's say through the back, from the back it's perpendicular to the plane and then whenever it reaches the front plane of the prism, that's actually when it goes at the angle to the surface of the prism. So the surface of the prism is inclined and the light goes straight. So there is an angle here. And if there is an angle which is not equal to 90 degrees, not perpendicular, then there will be a change of direction. Now we did talk about the perpendicular to a surface ray of light. It does not change the direction, just change the speed. But when there is an angle not equal to 90 degree between the surface and the ray of light, surface means the boundary between two different media like glass and air. Then the light actually is changing the direction. Now, whenever I will do all these calculations in this lecture, I will obviously use much, much easier presentation instead of three-dimensional one. I will just cut the whole thing which is by the plane which is parallel to the side. And if this is the point where the ray of light actually hits the back surface, I will use this particular section. So I will basically investigate only what's going on within this section because the light doesn't leave this section. And I will consider obviously the two-dimensional picture. So the next thing is to draw a nice picture. Now I did draw a nice picture on the website and I will use the same letters to the points. Okay, here it is. So let's assume this is my y-axis and this is my x-axis. So this is all. Now my prism would be positioned this way and the light goes down. This is the direction of the light. Now what happens with the light after it hits the plane which is perpendicular? It does not change the direction, just the speed. So now the light goes with the speed within, well, let's say this is a glass and this is air. So this is the light going through the air and its speed is, well, almost the same as in the vacuum. But then when it goes to the glass speed reduced. Okay, now here we have already a non-right angle between the light and the plane, between the two media, between the glass and the air. So it's supposed to change trajectory according to law of refraction. So you know the law of refraction, that was a previous lecture. Now i is incidence, r means refraction. Now n i is equal to c divided by vi, speed of light in the vacuum, speed of light in this particular medium. So this is air, this is air and this is glass and obviously nr is equal to similar thing. Now the speed of light here is speed of light in the air and that's basically vi. Now speed here is dr and speed in this air is again the same as before which is vi. But now what is theta i and theta r? Well these are angles with, angles which trajectory of the light makes with the normal to a surface it hits. So this is a normal to this particular surface and it's called mn. Okay, this is the point b, this is the point e and now what happens with the light? Well this is angle theta i. Now the light is supposed to make an angle theta r with this normal. Now let's just think about the speed of light in the air is greater than the speed of light in the glass which means that corresponding refraction index in the air would be less. So this one incident light would be the glass should be less than this one so this will be greater. So the trajectory of the light is here and here and this one will be theta r. So this is an incident and this is the refraction. Now I have two different tasks here. Number one I have to find out by how much the trajectory of the light is changing after refraction by how much angle is deviated. So this would be my delta. So delta is deviation from the original direction. So that's number one which I have to determine. Number two I would like to know what's this particular distance. Let's call it y and let's call it x. So I would like to know what's the dependency of the distance from the point when refracted light hits the y axis this distance from this distance. So I'm thinking about if I will change the location will this location change as well. Well the answer is yes questions by how much. So these are kind of calculations which I would like to basically make. And before I may call the calculations let's just think about the following. Now first of all we have to define our prism I define it by angle alpha. It's basically the property of the prism right. So alpha is given because the prism is given which means everything else is given. It's the right angle. So this is 90 degrees minus alpha. Now what's also given also given these guys these are characteristics of the environment. So I know what's the refraction index of air and what's the refraction index of glass. So whenever my incident is within the glass I know what the glass is and refraction is the refraction index of the air. So I know these characteristics it's physical characteristics. So based on this what's my variable. Well the only variable is x basically. So it depends on how far from the let's say base of the prism let's call it base. How far from the base of the prism the rail light hits this particular prism and how much location at this point depends on location of this point. Okay let's start with an angle. Now first of all first statement alpha is equal to data y. This is theta i this is alpha. Now this is perpendicular to this and this is perpendicular to a normal right it's normal it's perpendicular. So these are two angles with with mutually perpendicular angles that's why they're equal. Now the second observation this thing this angle is equal to qi as well. Why because they are vertical. This is continuation of the normal and this is continuation of the trajectory as it would go without refraction. So that's why I immediately see that my delta is equal to theta r minus theta i delta is equal to this angle which is theta r minus this angle which is equal to qi which is equal to alpha. All I have to know to find delta is to find qr and qr I can find from here knowing qi and knowing and one and r and qi not q that theta i sorry theta i I know what it is. So from this particular equation I see that theta r this one is equal to well not theta r sine of theta r is equal to ni divided by nr theta i or ni divided by nr alpha because alpha and theta i are the same thing alpha is given because the prism is given and ni and nr in this particular case ni would be the incident with the refraction index of the glass and and r would be refraction index of the air. This is the this is the glass this is the r. Now the glass has by the way has a refraction index something like I don't know 1.7 or something like this. So it's 1.7 times slower than in the vacuum air on the other hand is almost the same as 1. I mean it's greater than 1 but just a little bit. So that's why this is greater than 1 and that's why this particular angle is greater. Now I made a mistake here it's not again it's not theta it's sine of theta sorry about that sine of theta i is equal to ni divided by nr sine alpha. Okay so that's how we get the qr now qr is equal to arc sine of ni divided by nr of sine of alpha. So knowing ni and nr and knowing alpha I know theta r and that's why I have the delta. Now what's very important right now to notice that this delta which is this minus alpha so let me just change this delta deviation minus alpha so deviation from the original direction. So if this is the original direction this is continuation where the light is not going light goes this way so delta is deviation of the angle. It does not depend on the position of this particular point x. So all the parallel lines of trajectories of light propagation if they are parallel they will be parallel here because the angle of deviation will be the same so parallel lights are reflected refracted to parallel rays of light. Okay so this one will go straight and then parallel to this one but obviously further down the further down I go along the x along this particular side of the prism further from the from the base the further from this point will be the intersection of my ray with y axis. So parallel lights are refracted by the prism this particular prism into parallel. Now let's define actually what let's determine what exactly this value y is. Well basically everything it depends on the fact that I know delta I know this angle right. Now this which is called let's put it letter c here and d here. Now what's obvious is that oc is equal to this particular what's the letter which I'm using e okay this is letter e this point. So we can say that oc is equal to ed oe is equal to x angle is equal to alpha because these are two parallels and this is the intersecting one so it's opposite angles with two parallel and and intersection. So I can determine lengths of oc which is y as a difference between bd and ed. Now cd is also equal to x obviously right this is all x oe is x cd is x because this is perpendicular to this parallel to y. So this segment be I can find from this triangle obe knowing it's a right triangle so I know one cactus and the angle and this lengths bd I can find from bcg triangle again knowing angle and this side and the difference between bd and be will give me y okay so let me just do this way so y is equal to oc is equal to bd minus ed now bd from this triangle bcd okay what is bd divided by cd is cotangent of delta so that's why bd is equal to cd which is x times cotangent of delta now be now that's my bd right now um bd minus no I'm sorry that's again that's wrong be so bd I have determined now be from this triangle that's x this is opposite category so it's tangent so it's x times tangent alpha from which follows y is equal to x times cotangent delta minus tangent alpha alpha is known this is the angle and delta is related to alpha and and the indices of refraction like this so this is the final formula now as you see it's proportional to x which means if I will double the x my y will be double so that's why actually it represents the parallelism preservation of parallelism of the rays of lights which are coming into this type of prison perpendicular to this particular surface after they refracted by that surface so y is proportional to x this is coefficient of proportionality rather complex coefficient I mean just imagine cotangent of this minus tangent of alpha it looks ugly but nevertheless it's some kind of a number number which depends only on the qualities of the glass air and angle of the prison now obviously you understand that the direction would change exactly the same way regardless of the fact how this part of the prison looks so what's important is this angle and this angle so as long as it's perpendicular to this one then we can count only on this one if the original light is not perpendicular then what well it's very easy actually if the angle is something like this what we do is first we can do proportional to we can do parallel okay let's do this I can always split a prison in two if I have something like this and the light goes at the angle I can always split the prison in two right so first it would be one particular this is from air to the prison which means from faster to slower and the angle would be less then it goes within the glass this way and perpendicular to this it would be even greater increasing so the light will go so it's basically two different prisms right triangular prisms attach to each other and that's why you can just calculate first the deviation here using this formula and then deviation here and then the sum of these two angles would be the total deviation of the angle by this particular prism if the light is at the angle but what I did was kind of easier just I just considered only one particular prism only this one and did it once so for different kind of angles which are following on the surface of the prism not perpendicular to this surface you just have to split prism in two and calculate twice the deviation on one particular surface and then deviation on another surface exactly using exactly the same formulas obviously but obviously you will have incident and refraction reversed the first time it would be from air to to the glass so it will be air and glass and secondly it will be glass to the air so this will be glass this will be air so that's why the change of the okay that's it basically that's all I wanted to talk about prisms now what's interesting about the prism is that it retains the parallelism now what we don't actually need well in most of the cases we need to focus the lights the ray of lights into one particular point so I would like to change the geometry so instead of having this prism when this light goes to this and this goes to this different points I would like somehow to make it the same point how can I make it the same point well I'll probably have to have different I have to have different angle here and then it will be this way so if this would be let me just make it a little bit better so if this is one angle and it goes to this point and this is a different angle which actually makes the angle with the with the normal this is normal so this angle will be greater than than than this one so deviation would be greater so this deviation would be greater than this deviation and that's why I can expect they will focus in the same point so that's why we have uh convex lenses but that would be the subject of the next lecture okay thank you very much I suggest you to read the notes for this lecture the notes have much better picture than I was just trying to to draw and in colors and all the formulas are basically there obviously so it's like a textbook very useful so read that thing and the next lecture would be about the curved lenses convex lenses thank you very much and good luck