1

00:00:00,600 --> 00:00:08,010

So far, I've told you that the Maxwell's equation are the four main equations of electrodynamics and



2

00:00:08,010 --> 00:00:14,640

that we only need these four Maxwell's equations to explain all of the phenomena in this course.



3

00:00:15,810 --> 00:00:23,250

But this is not the exact truth, because we need a fifth law of electrodynamics and this is a law that



4

00:00:23,250 --> 00:00:24,970

you probably know already about.



5

00:00:24,990 --> 00:00:26,490

This is the Lauryn's force.



6

00:00:27,780 --> 00:00:29,880

So why do we need the Lawrence force?



7

00:00:30,390 --> 00:00:36,440

So here you can see again the full Maxwell equations and these relate the charges and the currents.



8

00:00:36,540 --> 00:00:47,130

So, for example, Roe and J with the fields, E and B, however, how are these charges and currents



9

00:00:47,130 --> 00:00:48,630

interacting with the fields?



10

00:00:49,140 --> 00:00:55,200

It's not really clear, not really intuitive, but of course you would say these charges and currents



11

00:00:55,200 --> 00:00:57,840

interact via forces with these fields.



12

00:00:58,560 --> 00:01:03,240

But these terms here in these equations, they don't tell us anything about forces.



13

00:01:04,350 --> 00:01:08,660

The only force that we know and that we must know is the Lawrence force.



14

00:01:09,180 --> 00:01:16,520

And this is given by the discharge queue times e so this is something we have considered already.



15

00:01:17,100 --> 00:01:21,330

Plus, there is another term which occurs when there is a moving charge.



16

00:01:21,690 --> 00:01:26,940

Then we have queue times the velocity of this charge and the vector product with the magnetic field.



17

00:01:27,840 --> 00:01:34,680

So this explains as how the fields act on the currents and charges and this also makes them physically



18

00:01:34,680 --> 00:01:35,220

relevant.



19

00:01:35,220 --> 00:01:40,880

Or you could even say the Lawrence forces define what these fields even mean.



20

00:01:41,370 --> 00:01:44,310

So far they were just abstract vectors.



21

00:01:45,660 --> 00:01:51,630

So let's start from this Lawrence force, and I want to explain to you again how this Loren's force



22

00:01:51,630 --> 00:01:52,790

acts on electrons.



23

00:01:53,880 --> 00:01:57,630

So this is something I have already shown you and one of the optional lectures.



24

00:01:57,990 --> 00:01:59,370

But I want to tell to you again.



25

00:01:59,520 --> 00:02:05,460

So consider we have an electron here and it moves towards a magnetic field.



26

00:02:06,240 --> 00:02:11,100

And let's forget about the electric field so far, because this is really trivial and we have discussed



27

00:02:11,100 --> 00:02:11,700

this already.



28

00:02:12,390 --> 00:02:16,890

Now, if the electron is outside of the magnetic field, then the force is zero.



29

00:02:17,670 --> 00:02:23,580

However, once it enters the region where the magnetic field is present, then there is a force that



30

00:02:23,580 --> 00:02:28,190

is given by the vector product of the velocity and the magnetic field.



31

00:02:28,950 --> 00:02:37,080

So the velocity is pointing into this direction here and the magnetic field is pointing into the plain



32

00:02:37,080 --> 00:02:37,970

of your screen.



33

00:02:38,850 --> 00:02:43,010

So the cross product is pointing along this direction.



34

00:02:43,170 --> 00:02:49,060

If you assume such a negative charge here, because for an electron the charge is negative and there's



35

00:02:49,060 --> 00:02:58,790

the reason why this electron is forced onto such a spherical trajectory here through this magnetic field.



36

00:02:59,430 --> 00:03:05,640

Then at one point it exits the magnetic field and will then not feel any force anymore.



37

00:03:05,640 --> 00:03:08,940

And it will move along this direction here.



38

00:03:11,640 --> 00:03:20,880

Now, what's really important to realize here is that you cannot really drive this force in terms of



39

00:03:20,970 --> 00:03:28,410

classical physics, it looks quite simple and you probably know it since school, but it is actually



40

00:03:28,410 --> 00:03:33,630

really difficult to derive this equation and you need special relativity.



41

00:03:34,410 --> 00:03:37,290

So I do not want to explain this here.



42

00:03:37,290 --> 00:03:39,120

And I do not want to go into detail.



43

00:03:39,120 --> 00:03:46,020

I just want to make sure that you understand that here, even though it looks quite simple, it's really



44

00:03:46,020 --> 00:03:46,600

difficult.



45

00:03:47,590 --> 00:03:55,500

So the starting point of special relativity or the typical starting point of special relativity is a



46

00:03:55,500 --> 00:03:56,630

thought experiment.



47

00:03:57,270 --> 00:03:59,160

You probably heard of this already.



48

00:03:59,490 --> 00:04:06,990

So you can consider, for example, a spaceship or a rocket and this spaceship moves with the velocity



49

00:04:06,990 --> 00:04:09,570

of 80 percent of the velocity of light.



50

00:04:09,840 --> 00:04:12,900

So this is C here and now.



51

00:04:13,050 --> 00:04:18,260

The rocket shoots some projectile, which also has some velocity.



52

00:04:18,900 --> 00:04:24,900

And if you're sitting in the cockpit of this of this rocket, you will see that this projector moves



53

00:04:24,900 --> 00:04:30,570

with the velocity that is also, let's say, 80 percent of the velocity of light.



54

00:04:32,060 --> 00:04:38,510

Now, the question is, how fast is this projectile moving from the outside perspective?



55

00:04:39,740 --> 00:04:44,900

So since you're sitting in the cockpit, you are moving and you see that the projectile is moving with



56

00:04:44,900 --> 00:04:46,090

this velocity here.



57

00:04:46,580 --> 00:04:52,370

However, classically, when you're on the outside, you would say that these two velocities add up.



58

00:04:52,380 --> 00:04:58,370

So you would say this projectile moves with the velocity that is 160 percent of the speed of light.



59

00:04:59,540 --> 00:05:03,470

However, there is a very important law of special relativity.



60

00:05:03,860 --> 00:05:09,190

This tells us that no velocity can ever be larger than the speed of light.



61

00:05:09,830 --> 00:05:12,920

So it's not possible that these two velocities add up.



62

00:05:14,260 --> 00:05:21,040

And this the only solution to this problem is that we have effects that are called space contraction



63

00:05:21,040 --> 00:05:22,110

and time dilation.



64

00:05:22,690 --> 00:05:27,850

So it means that at high velocities, especially at velocities that are close to the speed of light,



65

00:05:28,410 --> 00:05:32,770

the space changes and deforms depending on the velocity.



66

00:05:33,460 --> 00:05:39,550

So if we calculate the sum of these two velocities, it gives us something like 69 percent of the speed



67

00:05:39,550 --> 00:05:43,180

of light, which you cannot understand from classical laws of physics.



68

00:05:44,710 --> 00:05:52,060

And for a long time, it was really, really a big problem that people only had the Maxwell equations



69

00:05:52,330 --> 00:05:56,580

and this Loren's force and they could not really explain how they fit together.



70

00:05:57,580 --> 00:06:05,440

And it was just at the beginning of the 20th century when Einstein developed the special relativity



71

00:06:05,440 --> 00:06:09,670

theory, when finally there was a solution for this problem.



72

00:06:10,210 --> 00:06:16,920

And the solution is that you actually need to introduce a new velocity.



73

00:06:16,930 --> 00:06:18,750

This is the so-called for velocity.



74

00:06:19,090 --> 00:06:22,800

And here you have the electromagnetic fields tensor.



75

00:06:23,020 --> 00:06:29,820

And this is the exact relation of the force and the velocity and these electromagnetic fields.



76

00:06:30,430 --> 00:06:36,550

And this is here the so-called Lowrance factor, which accounts, roughly speaking, to the subspace



77

00:06:36,550 --> 00:06:39,220

contraction and time dilation phenomena.



78

00:06:40,060 --> 00:06:46,270

So this may sound really difficult and I hope I did not totally confuse you.



79

00:06:47,320 --> 00:06:53,920

The only thing I want to tell you here is that the Lauryn's force looks really simple, but from a theoretical



80

00:06:53,920 --> 00:06:55,540

side, it's very complicated.



81

00:06:55,540 --> 00:07:02,110

And it was for a long time not known how the Lawrence Falls can be derived and how it can be related



82

00:07:02,470 --> 00:07:03,910

to the Maxwell's equations.



83

00:07:04,770 --> 00:07:11,120

So why do you need to remember is we need to Maxwell's equations and we need a fifth floor, which is



84

00:07:11,120 --> 00:07:16,160

the Lawrence force, which tells us how the charges and currents are related to the fields.



85

00:07:16,180 --> 00:07:21,130

So this is by this force and that's the Lawrence force.



86

00:07:21,130 --> 00:07:27,730

Cannot be cannot simply be derived, but is a consequence of special relativity theory.