 Hi, I'm Zor. Welcome to Unisor Education. I would like to talk a little bit more about refraction of light, primarily from the viewpoint of Huygens' principle, which we have discussed in the previous lecture. Now, this lecture is part of the course called Physics for Teens, presented on Unisor.com. Well, actually this course is the second one. There is a mass for teens, which is definitely a mandatory knowledge to study physics. So, I do recommend you to familiarize yourself with whatever topics mass for teens contains before you go to physics for teens, or maybe from any other source. It doesn't really matter what the source of your knowledge is. Now, this is a lecture. It's part of the lectures dedicated to properties of light. In this particular case, we are talking about phenomena of light, if you wish. Well, just a nice word I would like to. Okay, so, we talk about refraction. It's not the first time we discuss this particular topic. And the previous time, talking about just general properties of light, we discuss something which is called the Fermat's principle of least time. So, when the ray of light goes to some kind of a substance from, let's say, from air to glass, it changes the direction. And according to Fermat's principle, this is the trajectory when the light can reach from this to this in least amount of time. Because if you will go along the straight line, you see this piece, which is in the glass, considering the speed of light is greater in the air than in the glass. So, look what happens in this particular case. We are spending less time in the air, but greater time in the glass when we go straight. And apparently, if you will do calculations, which we did in that particular lecture, you will find out that actually there is a connection between these two angles, angle of incident and angle of refraction. And the relationship is their sinuses are related as speeds of light in these corresponding substances. So, this angle is greater than this because the speed in the air is greater than speed in the glass. Or we introduce something which is called refraction index, which is the ratio of the speed of light in the vacuum divided by the speed of light in a particular substance. So, the less speed of light in the substance corresponds to a greater refraction index of refraction. So, in this particular case, that would be N2 divided by N1. Or, if you wish, N1 times sin theta1 is equal to N2 times sin theta2. So, these are the laws of refraction which we have derived from Fermat's principle of least time. So, this is the least time trajectory of light. And we came up to these formulas analytically just based on minimization of the time. Now, I would like to approach exactly the same problem using the Huygens principle. Now, here is kind of philosophical consideration. Fermat just came up with this principle. Now, why is it right? Well, obviously, we can say that experiments actually confirm that this is kind of correct. But if we have some other principle, also reasonable. Now, Fermat's principle is reasonable. That's why we accepted it. Now, the Huygens principle is also a principle which we just accept, because it looks like it corresponds to experiments. But it would be nice if the results are the same. And that's what this particular lecture is about. We will approach the same problem from the position of Huygens principle, and we will come up with the same result. Okay, we got our aim. Let's try to do it. Now, when we talk about Huygens principle, we usually talk about the wave front. So, what is the wave front? Well, let me just say it again. If you have certain rays of light which are coming from either flat surface, or if it's a point as a surface, then it would be spherical. So, these are surfaces which the light reaches at the same time. Because light is actually propagating. Now, in this particular case, it's a spherical propagation. In this particular case, it's a flat plane propagation. But in any case, there is certain speed of light. And the wave front is when the same crest, let's say, of the wave, which basically describes the light, it transfers oscillations of electromagnetic field. So, whenever, let's say, the crest is reaching certain point. So, basically, when we are talking about the surface to which all the rays are coming at the same time, that is the wave front. In this case, it's spherical. In this case, it's flat plane. So, we are talking about the wave fronts. Now, when we consider the refraction as before, we were talking about only one particular ray of light. Now, since we are talking about wave front, and the principle, Huygens principle actually is talking about how to derive the next wave of front, if you have the previous wave of front. Then we talk about more than one ray, actually, because it's the whole wave. Now, let me just remind how Huygens suggested that the propagation of light happens. Well, let's say we have a wave front which has reached this particular surface. But he's talking about that at time t plus delta t. Now, from each point where the light has reached on the wave front, in this case, it's a flat plane. Maybe it's spherical. Maybe it's something else. So, from each point, we have to really build some kind of a circle, the radius of which is equal to speed of light times the delta t time. So, and then we will take the line which is tangent to each of these circles. Well, it's basically called envelope line, envelope surface in this particular case. So, we are talking about right now, this is a section, two-dimensional section of the three-dimensional picture. So, the surface which is tangent to all these circles. Now, if this is one particular kind of environment, then speed is the same for each point. Time is also the same, so the radius is the same. So, the surface which is enveloping all these circles, well, not circles, spheres I should really say, because it's three-dimensional world. So, the surface which envelopes all these spheres of the same radius with a center and the same flat plane will be another flat plane and the distance will be v times delta t. Now, that's the principle, Huygens principle, which we would like to approach when we are analyzing the refraction. But in this case it would be a refraction of the whole wave front. So, we have more than one ray of light. So, let's just consider that this is the first ray and this is the last ray from some kind of a, let's say we have a projector with a parabolic reflector. So, it's all parallel ray of lights and these are the boundaries. And it actually falls on incident angle to the surface which separates, let's say, hair and glass. Doesn't really matter what kind of substances, what's important is they are different. Let's say this one where speed would be faster and this one where the speed of light would be slower. So, what happens in this case? Well, obviously each ray will change the direction. I mean we will prove it, but basically we kind of assume that that's the case. Now, what's also important is if these are parallel and considering the condition this ray is falling on the surface, the same as this one, this one just be a little later, but the angle should be the same basically, right? So, if these are parallel, these will be parallel because the conditions of these two rays of lights are exactly the same. It's the boundary between two substances and the boundary is a flat plane we're assuming and this is the set of parallel lights. So, each ray should actually be refracted in the same way. But question is what kind of a way? So, whether this proportion between sinuses would be proportional to the speeds. Okay, that's what we are going to do. Now, at this particular moment, I would like to actually talk about one particular analogy. Let's consider you have a car, it has wheels and it comes to the boundary between, let's say, concrete and sand. Well, if it comes at an angle like the ray of light, what happens? This wheel will be on the sand earlier than this one. Now, whenever the wheel from concrete comes to sand, most likely it will slow down, right? So, what happens when this wheel will slow down at the point when it touches? This one is still going fast on the concrete and this one will go slower. Obviously, the car will turn a little bit and only at the moment when both wheels will be... Well, I should actually have four wheels, but it doesn't matter. When both sides will be completely on the sand it will be going along the straight line. But this straight line will not be the same angle at this surface, right? So, as the car is turning whenever one wheel is slowing down, same thing with this set of rays. This set of rays, this ray from this set is hitting before this one, this area where the light is propagating with a lesser speed. So, that's why we are just as an explanation why refraction actually happens. It's exactly the same as this. It's a very practical example. I think it's used in many different textbooks whenever they're talking about refraction. But what's important is to understand that this is natural, that the parallel rays, whenever one of them hits the surface where the light goes slower, will turn a little bit. Alright, so we understand kind of philosophical reason behind it. Let's talk about physics. Okay, for this we just have to start first with the front. What is the wave front? Well, whenever you have these parallel lights and we are assuming that they are coming from some kind of a projector, maybe with a parabolic reflector, so we have really parallel, synchronously going parallel lights. What is the wave front? Well, the wave front is perpendicular to their propagation. So I would like to draw the wave front which comes here. So it's perpendicular here and here, obviously. Alright? Okay, now, this is my TETA 1. This is my TETA 2. Alright, so right now we are talking only about pure physics and geometry. Okay? Now, let's call this point A and this point B. Now, I'm interested in two moments. Moment T A, let's call this ray A and this ray B. T A is a moment when the A ray touches the surface. T B is the moment when the B touches the surface. Obviously, it's later. T B is later than T A. Now, so delta T would be T B minus T A. So I'm interested in how the wave front will behave when the B will reach because this one will definitely be somewhere here by the time B will reach the surface the ray will be somewhere here already, right? So that's exactly my delta T is the time I need the time at T, this is T, initial time from T to delta T is the wave front will change the direction. T it was here. At T plus delta T it will be somewhere here where this is also at the right angle, obviously. So during this time it will... certain rays will be refracted but by the time the B will touch the surface all rays will be already in this direction. So that's why I need the wave front at the very end of this period. Okay, so let's say that during this period, during delta T period, so B will be at point B prime and A will be at point A prime. Now, let's say this distance is D. It's a fixed distance. This is the width of our set of rays. Now, what I would also like to do, I would like to find out how all intermediary rays between A and B will behave. So let's call another C and this would be letter C on the front at point T and this would be deviated. Okay, and this would be, let's say C prime and this would be C double prime. C prime is on the surface and C double prime on the new wave front. So obviously all the angles of refraction will be the same because as I was saying, all the conditions are the same. These rays are falling exactly the same way on the surface. The question is whether the line C prime, C double prime will be of the length proportional to A A prime because if all these are proportional then it will be a straight line which I probably assume it will be and that's what I would like to prove actually that that would be a straight line. So the wave front after refraction would be also flat basically perpendicular to the rays of light but we have to provide some kind of a proof that these are proportional to the distance. So let's assume this distance is X. So the whole distance is AB is D, AB is equal to B and BC is equal to X where X might be variable. So D is a fixed, that's the width of our set of rays and the X can be anything in between obviously from 0 to D. 0 when this ray is coincided with B and D if ray C coincides with ray A. Okay, so now we go to just play in arithmetic. So first let's just evaluate what is the distance from B to B prime? B B prime is equal to... Okay, now this is the distance the light goes during this time with the speed B1. So it's B1 times TV minus TA. Now in terms of incident angle TETA. Now this is TETA, now this is obviously TETA as well. TETA1. Why? Because these are perpendicular, these perpendicular to this and this is perpendicular to this. Yes, should be the same, right? Let me check. No, it's this one would be TETA1. This is 90 degree minus TETA1. But this one is also TETA1. Okay. Yes, this perpendicular to this and this perpendicular to this. Okay, so ABB prime is a triangle and we can say that BB prime is equal to what? AB times tangent of TETA1, right? Tangent is opposite to this calculus. So knowing that AB is equal to D, so that equals to D times tangent TETA1. From which we can derive TB minus TA is equal to D tangent TETA1 divided by B1. Okay. So these are known variables. We know the incident angle, we know the widths of the set of rays and we know the speed. So we know the time. During this time, exactly the same time, the ray A will go from A to A prime. So A A prime is equal to V2, the speed in the glass times the same time. D tangent TETA1 divided by V1. Okay. We have counted, we have calculated basically the length of A prime. Now, what is this ray? What is this particular distance? According to the Huygens principle, remember what we have to do. As soon as we reach certain point of the wave front, from this point we have to really make a sphere with the radius equal to speed times delta T. Okay. Delta T is basically this time by which B point will become B prime and A becomes A prime. So this is the radius of the sphere around point A where our proposed new wave front surface should actually touch. So whatever the surface, new wave front surface will be, it should be enveloping this sphere as well and all spheres in between. So let's just also make a similar calculation for a point C, which is on the distance X from B. And let's see what happens in this particular case. Well, let's find out the time Tc minus Ta. Tc is the time when ray C touches this particular thing. Okay. Now, the distance, this distance, can actually be calculated from the same similar triangle, ACC prime. And considering AC is equal to D minus X and from ACC prime, we see that Cc prime is equal to AC times tangent, which is D minus X times tangent delta 1 from which Tc minus Ta, which is this one, the distance between time of touching. So this is the time during which this particular ray C was travelling before hitting the point. Now, we would like to calculate again the radius. Well, this radius is the time times speed. Now, what is the time? The time should be equal to... One second, let me just think about it. So during the time, ray goes from point B to B prime. This ray goes from A to A prime. But this ray has two components. The first component, it went in the air with the speed V1. And the second component, it was with the speed V2. So we have to calculate them separately. So first, we know the distance between C to C prime. And we know the time from C to C double prime. So let's just calculate this distance. So first period of time from Tc to Ta, this time I have actually just determined. Now what I do need is the additional time, which is the whole delta T minus Tc minus Ta. It was travelling here, which is equal to Tb minus Tta minus Tc minus Ta, which is Tb minus Tc. Actually, I'm not exactly right. Yes, Tb minus Tc. So Tc is earlier and Tb is later. So that's okay. So this time it was actually travelling with the speed of V2. So that means that this particular distance, C prime C double prime is equal to, sorry, my voice, V2 times Tb minus Tc, which is equal to, okay, now since I know the Cc1, I know this is equal to Cc prime divided by V1, which is equal to D minus X tangent Tata1 So we know this and we know this. So if I will subtract from this, I will subtract this, what happens? From this I subtract this, so D tangent will be cancelled out and the result will be Tb minus Tc is equal to X tangent Tata1 divided by V1. If you subtract from this, you subtract this, D and tangent, denominator is the same and in denominator, tangent goes out so D will cancel out and only X will remain. So this is the time during which the light C, ray C, will travel this. So if I will multiply it by speed, it would be this, which is equal to X times tangent Tata1 divided by V1 and multiplied by V2. So this is the radius of the sphere from the point C prime. Now, let's just think about this formula. If my C coincides with A, X is equal to D and you see this formula and this formula are exactly the same. If X is equal to D, we have V2 multiplication, V1 is denominator and tangent Tata1. If C coincides with B, my X is equal to 0 and obviously this sphere is 0 because this is the point where we know that the wave front should go through. So as you see, it's proportional to the distance from ray C to ray B. So C can be like a variable. So whenever the variable is, whenever the variable X actually is, we have this radius proportional to this distance. Now, what does it mean? Well, it means that our line, which is surface would be a flat surface because if the radius of these spheres are proportional to the distance, to this distance or this distance, it doesn't really matter because it's all parallel lines. So proportionality gives us exactly the result that it would be in a section, it would be a straight line. Okay, so that's the most important part of it. Now, let's talk about the law of refraction. If we know this, we have one little trigonometric trick. What is, now this is my theta 2. Okay, fine. So what else is theta 2? This one? Yes, this one is also theta 2 because this is perpendicular to the ray and this is perpendicular to normal. And from the triangle A B prime, A prime, A B prime, A prime, what can we see? That the sinus, sin of theta 2, sin of this angle, is equal to opposite derivatives A A prime divided by A B prime. Right? The casualties divided by hypotenuse. Now, A A prime we know, which is this. So sin of theta 2 is equal to V2 D tangent theta 1 divided by V1 divided by A B prime. What is A B prime? A B prime is a hypotenuse. I know this distance, these casualties and an angle. So this angle divided by hypotenuse is equal to cosine. So hypotenuse is equal to... So instead of A B prime, I will put hypotenuse, which is D, divided by cosine of angle. So I will multiply it. Okay, divided in the denominator, I will put it in the numerator, which is equal to what? D is cancelling out. Tangent is sin divided by cosine and multiplied by cosine. So it would be only sin theta 1. From which, as you see, you have exactly the same fraction law that V2 divided by V1 is equal to sin theta 1 divided by sin. Exactly the same law. Or again, you can rewrite it as V1 divided by V2 is equal to sin theta 1 divided by sin theta 2. Or again, if you use the refraction index, index of refraction, refractive, whatever it's called, that would be, again, instead of V1, you can put C divided by N1. Instead of V2, you can put C divided by N2. And that would be sin divided by sin. Or 2 times sin theta 2 would be N1 times sin theta 1. In any case, all these are exactly equivalent and what's important is we have proven the refraction law of refraction, law of refraction, which we have derived using the Fermat's principle of least time, using the Huygens principle. So it actually, again, means that considering it's all actually corresponding to experimental facts, so it all actually brings us to the idea that those guys were smart. They came up with really good principles and obviously every principle has its draw points, has its negative sides. But nevertheless, whatever we know about universe is always an approximation. We don't know everything. But whatever we did know at the time when these two principles were introduced were really very, very good ideas and they allowed basically to theoretically understand certain things about light. Okay, so that's it. That's all I wanted to talk about today. Don't forget to read the notes for this lecture. Every lecture on this website has very detailed notes because it's like a textbook basically. But piece of the textbook, which is dedicated to the same material the lecture is. So lecture and notes are always together. Plus I put some better pictures than this one in the notes. So whenever you will read the notes you might actually have a better understanding of what's going on. And don't forget that analogy with a car. When the car goes with one wheel actually is reaching the sand earlier than another wheel, then the car will turn. I think it's very important to understand the principle of refraction. Other than that, you got it. Thank you very much and good luck.