 Hi, I'm Zor. Welcome to Unisor Education. We continue talking about phenomena of light and this lecture is about interference between different rays of light. Now this lecture is part of the course called Physics 14's presented on Unisor.com. If you found this lecture on YouTube or on any other source, I do suggest you to go to Unisor.com and through the corresponding menu go to this lecture and watch it from there and simultaneously you have notes for the lecture. Notes are actually like a textbook. So the advantage of watching it on Unisor.com is number one every lecture has its notes, number two all lectures are organized in menus basically. So there is a course called Physics 13. It's divided into into parts. Now this part is waves and parts are divided into topics like in this case it's phenomena of light and the topic contains certain lectures. All these phenomena which we are talking about each one is a lecture. 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The signing in is necessary for supervised studying. So if you would like to supervise or your student and you want to be supervised by somebody else then both supervisor or a parent actually and the student should register and they should establish the connection using the signing in and then the supervisor can actually do some kind of control over the educational process. Okay, back to light. And what's really important I think is to understand the wave theory, classical wave theory in more details and for this reason I think I would prefer to talk about waves on the water and interference of waves on the water before we talk about interference of light rays. Now why, well because it's visible, you know this kind of thing it's better to be able to touch it. So the waves on the water surface are more mechanical and you can see every wave actually with your eyes. You can touch it. So I think it's better to talk about the water waves first and about interference because with light it's exactly the same thing. So let me start with the water and we will introduce certain concepts etc. So interference starts when we have more than one source of oscillations. Now let's start with one source of oscillation. You have, let's say you are touching the water with certain periodicity and let's say your periodicity is always maintained and the interval between touching and the force you're touching is exactly the same. So what's happening is you will have concentric waves going from the point you're touching to all the different sides. So let's imagine that the water is an x, y plane coordinate plane. The surface of water horizontal surface is x, y plane and the z goes perpendicularly. Now every point on the surface of water has its x and y coordinates. Okay, so this is the point if you will look at this from the top. Now, so the waves as they are actually occurring on the water, we kind of think about this that every molecule goes up and down, up and down and every molecule when it goes up then the next one with a certain delay of time goes immediately after it and then another one and that's how the wave actually is propagating. So every molecule has x and y coordinates fixed because it goes up and down and only z coordinate is changing. Well, that's not exactly correct. As a matter of fact the movement is not exactly up and down because if one is up and another is down the difference, the distance between these molecules would be greater than when I, when they are on the horizontal level. So you cannot go this and this without basically like stretching something. So the movement is not exactly up and down. It's more like a ellipsoidal kind of thing. But again, we are in physics and in physics everything is approximate and approximately we imagine that every molecule goes up and down. Now, how can we describe this movement? Well, because it's an up-and-down periodic movement it reaches the maximum then goes down faster and faster, goes through the zero level and then goes down, slowing down. So it's basically very nicely described as a trigonometric function sine or cosine. So traditionally we use cosine of what? Okay, now this is important. First of all, it obviously depends on time. So there is a periodicity, right? So I'm touching the water with certain periodicity. Now, what can we say about the periodicity? Well, there is a time interval between my touching the water, right? So t is time period. Also, you can measure it with the frequency number of touching the water and correspondingly number of waves which are passing any particular point per unit of time. So this is frequency. Okay, what else? Well, since we are talking about cosine and and obviously as the time goes by, any particular molecule goes up and down, up and down, exactly like the cosine of some argument. Now, what is the argument? Well, the argument usually is represented as omega times t. Well, the t is time, that's obviously. Now, what is omega? Well, omega is actually very much related to this particular frequency. Let's think about it. If I have certain number of oscillations per second. Now, if I'm talking about cosine, it means that cosine goes up and down, which means it reaches the point where the cosine is equal to 1 and then it goes down minus 1 and from, let's say, from 0 to 1 to minus 1 to 0 is a period. Now, the period of a cosine is 2 pi. So as this thing is occurring certain number of times per second, this thing also omega times t. So the obvious relationship would be omega is equal to 2 pi f. Why? Well, let's consider we have f is equal to 1. We are touching 1s per second. Well, that means that omega would be equal to only 2 pi and now the t. If t is equal to 0, cosine is equal to 0 and this t is equal to 1. Cosine is equal to 1. 0 is cosine is 1, right? Cosine is 0. Then this t equals to 1. It will be 2 pi. Again, cosine would be equal to 1. So the distance between maximums of cosine would be exactly as it's supposed to be. It will be 1 per second. So that's very convenient angular representation of frequency. So omega is called angular frequency. So we have omega angular frequency. Okay, so we have period. We have frequency. We have the wavelength. Okay. We have angular frequency, right? Now, if I will close the parenthesis here, would that be correct? Well, it depends. Look, if at certain point, at point x, y, at certain time, cosine is equal to 1. If I will close this parenthesis, let's say again, let's say f is equal to 1, which means 1s per second. So this is equal to 2 pi. So I will have every second, I will be on the 1. But if my point x, y is further from the source, then it's not on every second. It will be on every second plus, let's say, 100s of a second, or maybe on every second plus half a second or something. So it depends on the distance, right? So there is always something which we call a phase shift. Phase shift is basically an angle we have to add to this one, which depends on the distance r, where r square is equal to x square plus y square. So it depends on the distance from the source. If the distance is equal number, not equal integer number of the waves, then it will be exactly the same as in the very beginning. So let's say in the very beginning, our water is at flat level, at zero, and then at moment t is equal to zero, and we touch it at this moment. So the t equals to zero equals to basically zero deviation from the level. And then the wave goes. Now if my integer number of waves, this is the one wave, so if integer number of waves fills the distance from source to my point on the water, then this phase would be exactly the same as this one. Right? If this is zero, this would be zero. Now, but if this is slightly off, well, it will end like this on the flat, and then it will go up or down or whatever. So I will not be exactly on the same level as this one. So there is always certain phase shift, which depends on what? Well, it depends on this distance and how many waves are in it. So every wave gives you basically a two-pi shift. One wave, angular frequency is omega, so it will be two-pi. So how many of these waves actually fit to this? Well, it depends on the distance. So if lambda is the wavelength, which means the distance between this, let's say, crest and this crest, or between this zero and this zero. That's the same thing. So that's the wavelengths. So how many wavelengths fit into this? Well, we have to divide r by lambda. And every wave, now that's not necessarily an integer number, right? So every wave gives you two-pi offset and angular displacement. So this would be a total displacement relative to the beginning of this. So instead of phi of r, you can say two-pi r divided by lambda. r is a distance, lambda is one wavelength, so r over lambda gives you how many waves, and every wave is basically in angular notation would be two-pi. Okay, there is one more detail. It depends on when I start it. If I start it, let's say, at level zero, I start at t is equal to zero. But it depends actually on how I basically, where exactly I start my timing. I can start timing when my water is already, let's say, in the upper position or the bottom position, on the crest or on the, how is it called, trough, trough, whatever. So there is always something like phi zero, which is initial phase of this. In most cases, we decided zero, but it depends on how we start timing, basically. But in any case, this is a relatively complete formula. Again, on the physical level of precision, let's say, because again, cosine is approximation, and periodicity is not an approximation, but it's always approximate, obviously, because you cannot exactly. But again, physics is all about approximation, the proper approximation. So this formula well describes the z component, how much up and down a molecule goes. Now, we are talking about so far only on case with only one particular source of oscillations. Okay. What else did I actually? Oh yes, how lambda related to frequency? Well, let's think about this way. If lambda is the wavelength and t period is the time it takes from one crest to another, so lambda is a distance between the two crests or two troughs or two zero points, doesn't matter. So if lambda is the distance and t is the time, then lambda divided by t is speed, right? The water has certain speed of propagating, and it's not dependent, actually, on the frequency. Because no matter how frequently you beat the water, the waves will go with its own speed, whatever the water properties actually are. So the waves will be shorter if you hit the water more frequently or longer, but it's still the same speed of propagation. So d is like a constant. For a water, it's a constant. So that's how they are related. So you can always say that since one over t, yes, one over t, what is one over t? That's the frequency, right? How many times? If t is the time it takes for one wave and you have f waves per second, then obviously one over t would be f. So you can always say that lambda times f is me. And since frequency can be expressed with angular frequency, you can replace it here. So it would be lambda times omega divided by 2 pi is equal to v. Or whatever else, like lambda is equal to 2 pi v divided by omega, etc. Now, all these formulas, I never remember them. And again, I suggest you to just think about these formulas. Logically, think about them. Let me just remind you again. Now, this is kind of obvious because lambda is lengths from one, let's say crest to another crest. And t is the time the water passes one point, which means what doesn't mean passes. It means when the point is in the top position, when the water goes, the wave goes, and then the next wave comes until the time the next one. So that's exactly the time between two crests. So if you divide, that will be speed. That's one thing which should be remembered. Now, what's this? f is number of times the wave passes, number of waves per second. And every wave is 2 pi angular, right? Because the up and down, up and down. So the full cycle, we are all kind of gearing towards trigonometry here. So that's why if f is number of waves per second, then 2 pi times f would be the angular number of, number of regions, if you wish, per second. What else? Well, r is obvious, that's Pythagorean theorem. And well, that's it. I mean, all these formulas are logical. You do not have to remember them. All you do have to remember is there is a frequency, and there is an angular frequency, just the concept. There is a speed of propagation. And there is a wavelength between two crests or two troughs. Okay. So now we have finished with one particular source of oscillations. Now, let's think about interference, the purpose of this lecture. Well, the purpose of this lecture is when you have two sources and also waves here. So what happens in this particular case? Well, let's think about one particular molecule. Now, for simplicity, oh, I didn't really add one more thing. In this particular case, it all depends also on the amplitude, because it depends on how strongly we hit the water, right? So the stronger the bigger waves will be. So this is basically an amplitude. Okay. Now, let's talk about this particular point on the surface of water. Now, it actually is under influence of both waves propagating into this point from here and from there. So every molecule of water will participate in both of them. So if you have a point m and n, you have something called which is Zm of t, and you have Zn of t. So what happens if only one source would be that's this function? And what happens with this molecule if only this function? So if you will summarize them together, that's how you define the result. That's what result will be. It's like two forces actually acting on the same object. They are edging by vector addition. In this case, the waves are addition by basically using a regular addition sign. So you have to add this one for m and this one for n. Now for simplicity, it's usually assumed that the difference between these functions is only in distance. So we are assuming that these are synchronous oscillations of the source, which gives you exactly the same period, exactly the same frequency, obviously exactly the same angular frequency, the same amplitude. And the only difference is the difference between Rm and Rn. And the phase is also the same because they are synchronous. So the only difference is in this component. So what happens in this particular case? Let's just think logically. I mean, yes, obviously we can add one to another and we will get whatever the formula is, but that's not the purpose. The purpose is to understand what happens. If at this point this particular wave comes at a crest and this wave also comes at a crest, what happens? Well, two crests will increase each other and that would be a higher crest. So if the waves are coming in the same phase, they will increase each other. So both will be at the same time on the bottom, on the trough, and that would be even bigger trough. Or both will be on the crest, it will be bigger crest. Or both will be at zero and the result will be zero. So what I'm saying is that if two waves are coming at the same phase in this point, then at this point the oscillations would be, well, bigger than each one of them. Now, that's called in phase. If these two waves are coming, let's say, in opposite phase, which means whenever this goes all the way up to A, this goes by itself, would go to all the way down to minus A. And what happens in this particular case? Well, they will modify each other. So there are some points on the surface of the water which will be oscillating up and down greater than from one source. There are points which will be almost no oscillations at all if you have both sources because they would nullify each other. So that's exactly the picture which you will get on the surface of water. Instead of waves synchronously expanding and expanding as one particular source, you will have some kind of relatively chaotic picture of some pieces on the water are really oscillating up and down significantly, and other not at all. Well, obviously there are some in between cases when this comes, let's say, at the crest and this one becomes at half a crest. Well, then the corresponding characteristic would work out. So everything in between will basically oscillate in between, not as high as when both waves are in sync, in phase, and higher than if they are in opposite phases. Opposite phases sometimes called anti-phase. So there is in phase and in anti-phase. And everything else is in between. Now, is that everything I wanted? Yes, one little thing. So how can I basically know whether they are in phase or not in phase? Well, very simply. Let's talk about this. Delta is equal to Rm minus Rn, the difference between them. If the difference between them is integer times wave lengths, n is integer. If n is integer, and the difference between the distances is integer number of waves, both have the same lambda. We have agreed that these are exactly the same kind of oscillation just in different places. So lambda is the same, wave length is the same. So they are coming both either on the crest or both on the trough or both on the zero, whatever it is. But they are always enhancing each other. So this is in phase. If, however, is equal to n plus 0.5 times lambda, which means certain number of full waves plus half a wave. Well, that actually, this half makes it opposite. So if one is in this particular case, another is in this particular case. And when we will add them together, they will nullify each other. That's what means, because if you will shift it by half, it will be exactly the same as here. Right? This is half a wavelength shift. So this is condition of in empty phase. So and everything in between obviously would be in between. So there are points where waves enhance each other and that would be a larger deviation from zero level from XY plane when the water is still. And this would be a condition when the waves would nullify each other, which means that the water will be still at this point, because the waves will completely nullify each other. If you will add this to this, you will have zero. And that's my end of the water thing. Now let's talk about life. So basically, what I would like to say is that there is absolutely nothing new. I can tell you about the light, but I just wanted to use the water as kind of a more tangible example of this. So what can we do about light? And we can actually try to simulate this type of situation. Well, let's do it this way. Let's say we have flat wave front of monochromatic light. Now why is monochromatic? Well, you remember that the color is related to the wavelengths. Since we are talking about this example of interference on the water with two different oscillators of the same wavelength, etc., so everything is exactly the same. Now the white light is the combination of different frequencies, different wavelengths. So we are not talking about white light right now. White light is much more complex situation. Let's talk about monochromatic. So let's say it's a red light, whatever the color, whatever your favorite color is. Now let's say we have some kind of a wall here with two small slots. Well, as we have a synchronous rays of light, which are coming into two slots, slits, I would probably should say slit, because it's like a wall. We are looking from the top and this is the wall and there is a vertical slit on it. So this is the screen. So what kind of a picture we will have here? So we will have two basically sources of light according to the Higgins principle. Whenever we reach with our waterfront certain point, this certain point can be considered as a source of additional light if you wish. And that's how the light is propagating. So every slit has basically been a source of light and these two are in complete sync because they are coming from the same flat wave front of light. So basically from this we will have light going this way. And in a way it's similar to whatever we saw on the surface of water, where we had two oscillators, oscillating with exactly the same frequencies, synchronously, etc. That's exactly how we built this model with light. So what shall we see here? Well, first of all we will not see oscillations like we saw it on the water, right? Because on the water we really see how the molecules are moving up and down. Here we don't really see it. So what exactly will we see? Here it is. Every point here has certain distance from this and from this. So this is m, this is n. So this point has rn and rn. Now if this delta is equal to n times lambda, exactly the same as for the water, where lambda is the wavelengths of this favorite color of light you have, let's say it's red, what happens then? Well, then the waves from this guy and the waves from this guy will enhance each other. Which means, how should I say it, per unit of lengths of this screen we will have more light energy coming into it. Now we don't see the oscillations of light. What we see is a brighter spot because more energy are coming into this particular spot and it actually is viewed by our eye. Now our eye has certain nerves, etc. How do we see it? Basically we see it as a bright spot. Now if this delta is equal to n plus 0.5 lambda, the waves will nullify each other and we will see it as a dark spot. So my point is that the picture on the screen would be basically spots of different brightness or it's not really spots because it's gradually comes from one to another. Now the brightest spot will be in the middle because the distances are the same, right? Distances are the same so the difference is zero so it's definitely in sync, in phase and that would be the brightest spot. Now if brightness I will display as a curve then it will be brighter here and then diminishing and at some point it will be almost zero, then there will be more and more and more here and more and more waves here. So this is a graph of brightness of whatever picture we have on the screen. Now you can always go to internet and search for interference of light and look at images which it will display. It will actually be exactly the picture like this of different brightness. So if these are slits then these will be the lines, brighter lines. We're looking from the top it's a section, right? So that will be brighter lines and darker lines and the brightest will be in the middle and then the brightness would be diminishing and why the brightness is diminishing? Well obviously because the angle is more and more the incident angle if you will take this point for instance. Obviously per unit of of the unit of lengths on the screen the amount of energy would be obviously less because we are at greater and greater incident angle. The farther it is the less energy would feel would fall and that's why you will have these bright spots really diminishing of the brightness. Okay so that's the main thing I wanted to talk about. So it all depends on this. Now how about some formulas? Well we are still kind of approaching physics from a little bit of mathematics here. So there is some very little amount of mathematics here. How can I find out where exactly are the bright spots and where are the dark spots? So let's have this is at zero and this is an x. So let's say my point is at x. Now what can I say about this particular brightness level? Well let's find out what is the difference between the distances right? So the distance rm, rm. So let's say this is l, this is d between the slits and this is x. Okay I think it's a very bad feature. Let me just do it better. So we have two slits at distance d. I have this distance l and this is x. So what is this? Well this is d over 2 and this is g over 2 right? It's the middle point between two slits. So this is l, this is m. So rm is equal to square root of l square plus x plus d over 2 square. That's a Pythagorean theorem right? x plus d over 2 rn is equal to square root of l square plus x minus d over 2 square. Okay? Okay that's it. Now what is the difference between them? rm minus rn is equal to well that would be one square root minus another square root right? Now how can I simplify it? Well it's equal to one square root minus another square root. I will multiply it by some of these square roots. Okay? Now minus and plus would be square of this and minus square of that right? a minus b times a plus b is a square minus b square. So square of this would be whatever is under this root under this square root and square of this would be whatever under this root and if I will subtract them so what would be remaining? l square would be nullified, x square would be nullified, d square over 4 would be nullified and the only thing which would remain is 2 times x times d over 2 which is x d and then plus another x d. So it would be 2 x d right? So what will be on the bottom? Some of these two. Here we go to a physicist's level of precision. What they're saying is that this is approximately, some of these is approximately. This is a little bit more and this is a little bit less. So if l is relatively big relative to x and g then it's approximately 2 square root of l square plus x square. So I will just diminish slightly this one, increase slightly this one. Obviously if x and d are really small and l is really large then it's a decent approximation. So that's what they're using. This is also cancelling and what remains is this. I better put it differently. Better put it this way. It's my mistake. Now what is this? Well, l x. Well this is, if this is an angle theta, it's approximately g times sine of theta and again it depends what exactly is theta. In this particular case I'm talking about the ray which goes from the middle between the two slits to this point of interest. Again it all depends on how small are these x and g relatively to l. So but in any case this is what we have. X is cosine, not sine. Now, okay, that's not theta. I'm sorry. Theta we are talking about incident angle, right? Theta. So incident angle is angle with perpendicular. So this theta and this is theta. Okay, sorry. So if we divide this to this we will have sine. So this is basically the condition on having a bright or light line on the screen. If this is an integer number of wavelengths, so this is rm minus rn, right? So by the way it doesn't depend on l. So it all depends on the incident angle into point of interest. So if this quantity is integer number of wavelengths of light, of red light or whatever your monochromatic light here, then we will have a bright line at that point. If this is n plus half of the wavelengths, then we will have a dark spot and everything in between will be correspondingly less bright or greater etc. Okay, and very small note to this. How many bright lines we will have? So bright line is n times lambda is equal to d times sine theta, right? Lambda is the wavelengths, n is some kind of a number. So how big number n can be? Well, obviously if n is equal to zero, we will have the perpendicular here. Sine will be equal to zero, so that would be 90 degree incident angle. Now as n is increasing, we have a certain limit, right? Sine cannot be greater than one. So n times lambda should always be less than equal to d, actually even less, which means number of lines should be less than d divided by lambda. That's a very interesting formula. So if we know the distance between the slits and the wavelengths of the color whatever of the light which comes down, we can calculate how many maximum number of lines we can see. And that's the end of this lecture. It's a little longer, but in any case, I think what's important is to basically having in mind that it all depends on whether the two rays of light are coming in phase or not in phase, maybe in anti-phase, something like this. And that's the reason for having bright and dark lines of light on the screen. And the number of lines is restricted basically by this very interesting ratio. If this distance is significant, let's say a centimeter, it's a significant relative to the wavelengths of the light. And that's why you will have a very big number of lines, but you will not see them basically, all of them obviously, because the intensity is definitely diminishing. It's like the sun, when sun goes down perpendicular to the earth, you feel the heat. If sun is above the horizon at the angle, it's much less energy falls on the unit of square foot or square meter or whatever. Okay, thank you very much. And I do suggest you to read the notes for this lecture. There is a nicer picture over there. And again, there are formulas, etc. It will give you another look at this material. Thank you very much and good luck.