 Hi, I'm Zor. Welcome to Indizor Education. Today I would like to talk about one specific effect of the light, which is called reflection. Well, we all know about what reflection actually is. Everybody saw the reflection in the mirror. But it's quite interesting actually to know about certain principles which are behind this particular effect of the light. And there are some interesting details, even historical details, which I'm going to share. This lecture is part of the course presented on Unizor.com and the course is called Physics for Teens. On the same website, Unizor.com, there is a prerequisite course called Maths for Teens. And whatever information is in that course is mandatory for studying physics for teens. There is a lot of math in physics, basically. That's my main point. Now, this site is completely free. There are no advertisements, no financial strings. You don't even have to sign in if you don't want to work with the course in some kind of supervisory mode. So, I will encourage you to use the website to watch these lectures of this course. Because every lecture has very detailed notes right near it on the website. And basically, the notes are like a textbook. So, you have both. You have the visual lecture and you have the textbook with basically the same information. Okay, back to reflection. Now, I would like to just talk about two distinct effects. One is called reflection and another is called refraction. So, reflection and refraction occur when the light is propagating somewhere near the boundary between two different media. Different environments, let's say the air and the glass or vacuum and water, whatever. Now, the boundary between these two substances is very important. Reflection means, so if you have one particular medium, boundary and another medium. So, reflection is when the light comes to the border, it goes back to the old medium it came from. That's a reflection. Now, refraction means that it's actually penetrating the border. This is a refraction. Now, in many cases, these both effects are occurring at the same time. Which means partially light is reflected and partially it penetrates through the border. But we're not talking about this mixed case. Our purpose for this particular lecture is reflection only. Another lecture will be about refraction. So, that's basically as far as terminology is concerned. So, reflection goes back into the medium it came from. Refraction means it penetrates like through a lens whenever the light goes through the lens and goes further. That's a refraction. The mirror is reflection. Now, in many cases, there are certain laws of reflection. Like the incidence, the angle of incidence is equal to angle of reflection. Many people know about this, but I'm not talking about this at this particular moment. I would like to actually understand why it happens. Why one angle is equal to another? So, basically, whenever you have a situation like this, if this is a mirror, the light goes this way. This is a perpendicular. So, these two angles are actually equal to each other. This is a direction. So, this is called incidence and this is reflection. So, these angles are equal. But I don't want to give it to you just like this basically. Just like a fact. Okay, just take it or leave it. That's the law of the nature or something like this. I would like to find some reasonable foundation behind this. And I was thinking about this and I came up with a certain explanation which happened to be basically invented by famous French mathematician Pierre Fermat in 1662, if I'm not mistaken, in the 17th century. So, it's known as Fermat's principle of the least time. Here is the brief explanation of this. Now, we all know that the light goes along the straight line in a uniform environment. So, if this is the source of light and this is the observer, the observer sees the light which goes along the straight line. These rays of lights are not visible by this particular observer. I mean, I assume that this observer is basically a point directed only in this particular direction. So, if you see the light, the light goes along the straight line. Well, the straight line, as we know, is the shortest distance, right? So, I suggest that whenever we are talking about the mirror, well, actually it's not me who suggested it. That's Pierre Fermat in 1662 suggested this. But the suggestion is that if this is the light and this is, let's say, observer. Now, there is a wall here, so observer doesn't really see the erect light. Observer can see only the reflected light. So, the principle of least time is that this trajectory, which basically depends on where exactly the point of reflection is. This trajectory is minimal as far as its length and therefore its time. Because this is the same medium. Since this is a reflection, the light is always within the same uniform medium, let's say, vacuum. So, the speed is the same. So, minimizing the time actually is minimizing the distance with a constant speed, right? So, again, my statement is that this particular trajectory is shorter than any other, let's say this one. So, that's a very important principle, which we are going to prove, obviously. We are going to prove that this is shorter. But the principle, I think, we can take intuitively as an axiom. So, again, the light goes along a trajectory which minimizes the time of travel. By the way, historically speaking, I read a little bit about this. When Pierre Fermat suggested this principle, it was met with some kind of a resistance, because it kind of assumed that the nature has certain intelligence, so to speak, and it chooses among all different trajectories, it chooses the one which minimizes the time of travel. Well, it's a philosophical kind of discussion, which I'm not going to go into right now. But it's an interesting thought, actually. In any case, the principle represents certain logic behind why the laws of reflection are such and such, and by the way, a reflection as well. And it seems to me, at least, that to accept this principle of minimizing the time is actually, as an axiom, it's easier than to say, basically, that the angle of incidence is equal to the angle of reflection as an axiom. I mean, the axiom about the principle of minimizing time seems to be reasonable to accept as an axiom rather than the quality of these angles, which natural question is why? Why we go this way instead of that way? So, if we are using, if we are accepting the principle of minimizing the time, then we can prove that this is exactly the right trajectory. And now here is the proof, very easy one, basically. Let's just consider a purely geometrical problem. You have a plane. Now let me just convert this into a plane instead of a line. So this is the plane. And you have two points on the same side of the plane. And we have to find such a point on the plane. Let's call it alpha. So this is a source of light. This is our observer. This is the point, this is the point. And we have to find the point on the plane alpha such that the distance, so this is, let's say, r. So the distance sr plus ra is minimally among all other points. Here is how we can do it. So let's forget about this trajectory for a while. We will use it later on. So we have to find this point. So here is a construction, basically, of this point. So I will really construct the point. I will give you the way how we can get this point very easily. Let's drop the perpendicular to the plane and move it further. So this is, let's say, b and this is a prime. So a prime and a are symmetrical relative to the plane alpha. Which means this is the perpendicular and a b is equal to b a prime. Now a b is perpendicular to alpha. That's a symmetrical point. Now if I will draw a line from s through the plane towards a and this r is where they intersect the line and the plane. I will get some point r. Now this is a straight line, sa prime, straight line. So the line intersects the plane at point r and I am actually stating that this particular point r is the one where the light should go towards the plane. Reflect to the point a. And the distance s r plus r a would be minimum in this particular case. Among all other points r. How can we do it? Again, very easily. Choose any other point, call it x. Now obviously, since a and a prime are symmetrical, x b belongs to the alpha. So these are two right angles. So the triangles are actually equal, a x b and a prime x b are equal to each other because a b is equal to a prime b. This is common and this is right triangle. So there is an angle. Which means a x is equal to x a prime. Now if I will add this, what will I have? Now s x plus x a is equal to s x plus x a prime, right? Because x a is equal to x a prime, x a and x prime. So s x plus x a is equal to s x plus x. Now s r a prime is a straight line, s x a prime is not a straight line. It's two different, it's two segments. Now sum of two segments s x plus x a prime is definitely greater than a straight line s a prime. So that's why any other point is not good because s r a prime is smaller. Very easy proof. It's a straight line and this is a broken line. So we have actually constructed the point r which gives us the minimum length of s r plus r a. Now minimum length means minimum time because the speed is constant. Now let me now give you some properties of this particular construction. So let me get rid of this x and everything related to x and I will leave only b and r now. So let's draw perpendicular here and let's connect this perpendicular. Let's say it's s prime, let's connect this to this. And I need another thing perpendicular from r, r prime. So r prime is perpendicular to alpha, r prime is perpendicular to alpha, s prime is perpendicular to alpha. Now a b is also perpendicular or a prime is perpendicular to alpha. So these are all perpendicular. Well, if these are all perpendicular, it means they are all parallel to each other, right? Now let's consider the plane, beta, the plane of how the light ray is actually traveling. s r and r a. So this is the plane, it's basically vertical, s, r and a. We are actually drawing the plane which goes through these three points. Now what else belongs to these three points? Let's just think about it. Well, now the s and r belong to this plane because that's how we have designed. That belongs to the plane. And r a belongs to the plane. Now since s r belongs to the plane, point a prime also belongs to the plane because it's in the same line, right? Two points belong to the plane, beta, so all the points on this plane. So a prime belongs to beta. Okay, now since a prime belongs to the beta and a belongs to the beta because again beta contains the whole r a. It means that the whole a a prime belongs to the beta, a a prime belongs to the beta and b as well, point b. So s a b a prime r, this is all one plane. Now it's s belongs to this plane. Now a b is the line within the plane, beta. s s prime is parallel, right? Because both are perpendicular to alpha. S belongs, so let me just draw it again from this. If you have one line and a point, so this is a b and this point is s. Now if this is parallel to this, this line belongs to the plane, this one is on the plane and it's parallel to this. Obviously this s s prime also belongs to the plane. So s s prime also belongs to the beta. Absolutely the same thing with r r prime because r belongs to the plane and r r prime is parallel to a b. a b belongs to the plane so r r prime should belong to the plane. So this is all plane geometrical properties of the lines, points and planes. And by the way, all these topics are definitely presented in the math 14 course which is prerequisite for this. So now what we have? This line, this line, this line, these two lines, this line all belongs to the same plane beta which actually goes like this. Okay, so now we basically are ready to do something with angles. Look, this is the line, now let's consider only the plane beta with vertical plane. Now within this line s a prime is a line and s prime b is a line and these are vertical angles, correct? So they are equal to each other. Now the triangle a b r and a prime b r are equal or congruent as we call it right now. Well because these categories are equal to each other, a b and b a prime. This is the common side and obviously these are right angles. So the angles, so the triangles are equal so these angles are equal. So we have all three angles are equal. Now s prime b is a straight line basically. Now why is it a straight line? Because these are two perpendicular and yes and these angles are equal, right? So if these two angles are equal now these two are complementary because this is 90 degree, this is perpendicular. So these two with two arcs are complementary to a single arc angle so they are equal as well. And these are exactly the angles which the laws of reflection actually is talking about. This is an incidence angle and this is a reflection angle. These are angles with the perpendicular to the plane of reflection. Okay so what else? I think these are all properties which I wanted to talk about and now let's talk about the laws of reflection. So we have the first law is that I have already stated that the incident ray, reflected ray and the perpendicular to a point of reflection they all belong to the same plane and this is the plane beta which we were talking about. So that's the first law of reflection. Both rays incident and reflection and perpendicular to the plane at a reflecting plane at the point of reflection is called normal actually, this is called normal. Our prime is a normal to a plane at point R. So these all belong to the same plane. So we have already discussed that. Now the second law of reflection that the angle of incidence is equal to angle of reflection. Again we have already addressed that. We derived all these from the main principle we started from in the very beginning. The principle of the least time. Okay so we've got that. And the third one is that within this plane which beta plane S and A the source and the recipient of the light are on different side from the normal from R, R prime. Again that's how we basically constructed the whole thing. Because we have this line and this point is the intersection it's in between these. So that's why we have this property. So all these laws of reflection, three laws of reflection that two rays and normal are in the same plane. The angles are equal incidence and reflection and both points are on different sides of the normal. Now these were actually derived using this principle of the least time. And I think it's easier to accept principle of the least time rather than just, you know, these are properties of reflection, memorize it. Okay. I would rather prefer to derive it from something more fundamental like principle of the least time. Okay so this is kind of purely mathematical approach to reflection using this principle of the least time. I would like to also add to this certain physical confirmation so to speak of these principles. And the physical confirmation is basically quite easy. Let me just talk about this now. So let's assume that we have this plane and plane would be, this would be x, this would be y, and this would be z axis. Okay so we have introduced the frame of reference. So the mirror, the plane of reflecting is x, y plane in this system of coordinates. And z would be at the point of reflection basically. The origin of co-ordinates will be the point of reflection. So we are observing this particular process. Light is emitted from some point. And let me just turn the whole thing in such a way that initial position of the light is within zy, sorry, within, no, not zy, zx. So zx, this is a plane where my initial source of light actually is. So the s has certain coordinates. At time is equal to zero. Our ray, well we are talking about particles. So it's kind of a corpuscular theory is using. This type of thing. So at point time is equal to zero. s has coordinates, certain x which is negative, some negative x0, where x0 is greater than, y is zero, and z0, these are coordinates. So this would be minus x0 and this would be z0. This is coordinates of the source of light. At moment time is equal to zero. It issues the particle of light. Particle of light goes towards this particular point. Now, because that's how we have chosen it, right? We have chosen system of coordinates in such a way that the light behaves this particular way. It goes along the straight line, which means it's always within xz plane, this one. It's y coordinates, it's zero, and there is certain speed the light travels. The particle of light, if you wish, is traveling. Well, the speed is c. So if the distance, for instance, from s to zero is something, let's say d, then the time would be d divided by c. Distance divided by speed, and that would be the time when it will reach point of reflection. Okay, fine. Now, the speed is constant. Now, let's assume that this angle is theta i. i means incidence, so it's theta i incidence. Now, if the speed is c, linear speed is c, then the x component would be c times sin of theta i. Now, y component is not changing because it's always within xyz, so y coordinate is not changing. So, if this is a vector of speed, it would have this zero as the y, speed within the y coordinate. And the z coordinate, it goes down, so it's diminishing. Now, that would be c times cosine, but with a minus sign, right? c cosine theta i. Now, this is the speed at minus because it goes down along the z. So, it goes towards positive direction of the x, but negative direction towards i, so that would be the speed. Okay. Now, let's assume that it hits the surface of the xy plane, which is supposed to do reflection. Well, reflection can be, from the corpuscular theory, reflection can be viewed as basically elastic hit. So, let's assume it's ideal elastic hit. And it means that this elasticity will affect every coordinate separately. Now, the y coordinate is zero, so there is no change in the y direction. Now, the plane xy, if it's elastic, the force of elasticity goes straight up. Now, the x component, again, has no effect because the force goes up. So, it's only the z component which will be affected. So, the z component which goes straight down, it will feel this elasticity of the surface. And if it's ideal, then considering we have to preserve the certain laws of momentum and energy, the conservation, etc. So, the z component will just change the direction to the opposite. It's like you've dropped some kind of metal ball on a very springy kind of surface. It goes, again, vertically up to the same height if we would like to preserve certain laws of conservation of energy. So, that happens exactly the same way. So, x component will not be affected, y component will not be affected, but z component will be affected. So, this trajectory is the result of this elastic hit. And this is a vector of incidence now. So, a vector of reflection would have no effect on the x speed, no effect on the y, because it's zero always. And the z component will change to opposite. So, it was minus c cosine theta i, it will be plus c. So, that would be calculated speed. Now, what is the speed if this angle is theta r, the reflection angle? Well, the speed is the same, absolute linear speed is the same, it's c. So, again, the x component would be c times sine. So, from another perspective, the same vector should have c sine qr, also zero as y component. And c cosine r, it goes to the positive direction now, right? So, that's why it's plus here. So, we have the same vector. One is basically calculated based on this angle, and another calculated based on elasticity of the mirror. Elasticity in this particular case means that if we are shooting the particles of light, they are just elastically reflected. So, these are two the same thing. So, we have the sine of theta i is equal to sine of theta r. This is theta i and this is theta r, incidence and reflecting. And the cosine also supposed to be equal. So, if sine of the angle is equal to cosine of one angle is equal to sine of another angle. And cosine is equal to another cosine. Angles are the same, again, plane trigonometry. And that proves this second law of reflection. Now, the first law of reflection is basically, we have already satisfied it. Because in the very beginning, we have chosen the xz plane as the plane of action. So, obviously the z axis and s, this is zero, origin of coordinate. And the reflected light, they all belong to this xz plane. So, the first law that raise of incidence of reflection and the normal are in the same plane. So, that's an xz plane. And the third one, these two are on opposite side. Well, it's basically because this is a negative x and this is a positive x. That's why they are both on different sides. You see, negative x in the beginning and after that it was positive because the speed is positive. From point zero it goes to the positive x. So, they are on opposite side. So, this is the third one. Again, I'm not saying that this is like very mathematically correct proof of anything. Now, everything I spoke about today is, I would say, explanation of something which we observe and an attempt to give some kind of a logical foundation behind it. Foundation either, well, kind of mathematical from the position of minimizing the time of traveling and more physical explanation based on corpuscular theory of light and elasticity of the reflection mechanism. Well, I suggested to read the notes for this lecture. So, you go to Unisort.com, Physics 14's course. This is the part which is called waves. And among waves you will see the reflection. I think it's properties of light and then reflection. Alright, that's it. Thank you very much and good luck.