1

00:00:00,090 --> 00:00:06,270

So let us now analyze the solution from the previous lecture in more detail, what we will establish



2

00:00:06,270 --> 00:00:08,370

here is the dispersion of relation.



3

00:00:08,760 --> 00:00:15,690

We will be able to relate these quantities, Omega and Kate, that I've introduced into previous lecture.



4

00:00:16,380 --> 00:00:20,400

And also in this lecture, we will talk briefly about WAF packet's.



5

00:00:21,930 --> 00:00:29,340

So the solution from the previous lecture was that the electric field and the magnetic field are both



6

00:00:29,340 --> 00:00:30,810

electromagnetic waves.



7

00:00:30,900 --> 00:00:37,980

So these are exponential functions with imaginary arguments and we have a fluctuation in time t and



8

00:00:37,980 --> 00:00:44,720

in the coordinate R and these equations where the solution of this differential equation here.



9

00:00:44,730 --> 00:00:48,880

So this is the wave equation for E and the wave equation for P.



10

00:00:49,950 --> 00:00:57,750

So what we must do to determine the relation of Omega and K, we must just take this equation, the



11

00:00:57,750 --> 00:01:00,240

solution here and put it into the equation.



12

00:01:00,840 --> 00:01:08,990

So what I've done is I've just given you the solution and basically we made an ANZAAS that solves this



13

00:01:09,000 --> 00:01:09,480

equation.



14

00:01:09,520 --> 00:01:13,310

So now let's check if it is really a proper solution.



15

00:01:13,320 --> 00:01:16,560

And also we will be able to determine these coefficients.



16

00:01:18,120 --> 00:01:20,910

So let's take this equation for E!



17

00:01:20,910 --> 00:01:28,200

And put it into this wave equation for E, what we will get is from this term we will get one over C



18

00:01:28,200 --> 00:01:28,800

Square.



19

00:01:28,800 --> 00:01:35,220

And then we have here the second derivative with respect to time, this gives us a square and omega's



20

00:01:35,220 --> 00:01:35,740

square.



21

00:01:36,330 --> 00:01:40,710

And then, of course, E zero times E to the I and so on.



22

00:01:41,370 --> 00:01:43,680

So this is this term here multiplied by this.



23

00:01:45,390 --> 00:01:49,420

And also we must of course, calculate here it is not class operator.



24

00:01:49,830 --> 00:01:56,940

So once again, it's maybe more easily to go to one dimension where this lipless operator just becomes



25

00:01:56,940 --> 00:02:02,870

the second derivative of respect to, let's say, X, and then we have here K times X.



26

00:02:03,300 --> 00:02:09,930

So what we will get as the the river Tife will get minus I times K Square.



27

00:02:10,170 --> 00:02:17,460

So this is this one here and then of course the whole term here, which stays because when you differentiate



28

00:02:17,460 --> 00:02:21,330

in the exponential function it will just stay the exponential function.



29

00:02:22,800 --> 00:02:27,690

So this is the what we get when we take this term and put it into the wave equation.



30

00:02:28,170 --> 00:02:30,530

And we know that this must be equal to zero.



31

00:02:31,050 --> 00:02:33,810

So we can just divide by this term here.



32

00:02:33,810 --> 00:02:38,040

And we now know that this bracket here must be equal to zero.



33

00:02:39,540 --> 00:02:45,750

Most of what you will see is you can also take the solution for B, put it in here and you will get



34

00:02:45,750 --> 00:02:50,720

a very similar relation where you can divide by zero times as exponential function.



35

00:02:50,740 --> 00:02:54,570

So you will get the very same relation we get.



36

00:02:54,570 --> 00:03:04,980

In both cases that I Omega's square of a C Square is equal to this term here, minus K Square.



37

00:03:05,940 --> 00:03:08,670

So of course we know what I square is.



38

00:03:08,670 --> 00:03:10,170

It is minus one.



39

00:03:10,590 --> 00:03:19,110

So let's simplify all of these minus signs here so we get minus a square of a C square and here we get



40

00:03:19,110 --> 00:03:23,570

a plus sign because here we have a minus and then we get another minus from the square.



41

00:03:24,330 --> 00:03:34,020

So we know that Omega is equal to plus or minus C times K, so Y plus or minus because we have this



42

00:03:34,020 --> 00:03:34,760

square here.



43

00:03:36,030 --> 00:03:40,740

So this is the dispersion relation of the electromagnetic wave.



44

00:03:41,310 --> 00:03:50,700

And as it turns out, we know that C is the velocity of light and we can interpret K as a wave vector.



45

00:03:50,940 --> 00:04:01,020

So this is like the two pi over the wavelength of the electromagnetic wave and we have Omega, which



46

00:04:01,020 --> 00:04:09,910

is essentially two pi over the period of this fluctuation of the electromagnetic wave.



47

00:04:10,860 --> 00:04:17,820

So we have the frequency that is linearly related to the wave factor and this proportionality constant



48

00:04:17,820 --> 00:04:20,190

here is the velocity of light.



49

00:04:21,150 --> 00:04:25,810

And also we know that we get a positive and a negative sign here.



50

00:04:26,550 --> 00:04:31,620

So it means that also we can write it in a different way.



51

00:04:31,620 --> 00:04:35,510

We can write down that K is equal to plus minus Omega oversea.



52

00:04:36,030 --> 00:04:42,440

So it means that the key vector can be oriented along two different directions, you could say.



53

00:04:43,260 --> 00:04:47,360

So it means the wave can propagate forwards and backwards.



54

00:04:48,960 --> 00:04:54,300

And in fact, this relation here is not restricted to only fixed.



55

00:04:54,300 --> 00:04:59,390

Constant is C, but you can just scale Omega and you can scale K.



56

00:05:00,030 --> 00:05:10,650

So when you take some, for example, light from the sun and you analyse it, you will see that for



57

00:05:10,650 --> 00:05:12,900

every portion of this light.



58

00:05:13,560 --> 00:05:16,980

So to see you, this relation is true.



59

00:05:16,980 --> 00:05:26,310

Omega is always linearly related to the way Vector K, but the light consists of many, many waves that



60

00:05:26,310 --> 00:05:30,350

all have different frequencies and have different weigh factors.



61

00:05:31,620 --> 00:05:33,570

These are the different colors of light.



62

00:05:33,990 --> 00:05:39,540

So for example, red light has different by a factor than blue light.



63

00:05:39,540 --> 00:05:41,790

And also they have different frequencies.



64

00:05:42,310 --> 00:05:45,380

But of course they always have the same velocity.



65

00:05:46,410 --> 00:05:53,790

But what this means is that the most general solution of the wave equation is not just a single wave,



66

00:05:53,790 --> 00:05:57,810

as I've written down here, but it's superposition of many waves.



67

00:05:59,170 --> 00:06:05,560

So let's take these waves here, which are, for example, for electric field zero times this exponential



68

00:06:05,560 --> 00:06:10,330

function and and add them up for different characters.



69

00:06:11,280 --> 00:06:18,960

And this is what I have done here, so adding up different characters since K is a is not a discrete



70

00:06:19,440 --> 00:06:21,300

quantity, but a continuous quantity.



71

00:06:21,770 --> 00:06:24,360

It's not a sum, but it's an integral part.



72

00:06:24,400 --> 00:06:30,960

We well, we do not add up different waves, but where we integrate over these waves with respect to



73

00:06:30,960 --> 00:06:39,060

K and then we have here another function which tells us by which amount the different waves with a different



74

00:06:39,060 --> 00:06:42,180

way factors contribute to our total wave.



75

00:06:42,930 --> 00:06:53,040

So in general, the light is a wave packet's that is the superposition of many different individual



76

00:06:53,040 --> 00:06:59,670

waves that are characterized by one particular K vector and by one particular frequency Omega.



77

00:07:01,190 --> 00:07:09,230

And this distribution function or test is a weight function, F of K can be considered the full year



78

00:07:09,230 --> 00:07:12,140

transform of this function here.



79

00:07:12,980 --> 00:07:21,340

So what is what this means is that when you know the different contributions in the way of packages,



80

00:07:21,350 --> 00:07:30,590

for example, you know that when you analyze light from the sun and tells you it consists to five percent



81

00:07:30,980 --> 00:07:37,760

out of light with a way effect or K one and five percent out of line with the character K two and so



82

00:07:37,760 --> 00:07:41,480

on, then you know exactly what your total wave looks like.



83

00:07:42,170 --> 00:07:47,660

So the information about this distribution function is sufficient for knowing the whole function because



84

00:07:47,660 --> 00:07:49,570

it is just a different representation.



85

00:07:49,580 --> 00:07:52,430

It is the four year transform of this function here.



86

00:07:55,190 --> 00:08:02,690

Now we know about this person relation of light and also about WAF packet's, so please don't worry



87

00:08:02,690 --> 00:08:08,840

if this last part was a bit too theoretical for you in the next lecture, things will become more simple



88

00:08:08,840 --> 00:08:09,290

again.



89

00:08:09,290 --> 00:08:17,450

And I want to show you what an electromagnetic wave looks like and how the wave vector K is oriented



90

00:08:17,450 --> 00:08:20,540

with respect to E and also to be.