So far, I've told you that the Maxwell's equation are the four main equations of electrodynamics and
that we only need these four Maxwell's equations to explain all of the phenomena in this course.
But this is not the exact truth, because we need a fifth law of electrodynamics and this is a law that
you probably know already about.
This is the Lauryn's force.
So why do we need the Lawrence force?
So here you can see again the full Maxwell equations and these relate the charges and the currents.
So, for example, Roe and J with the fields, E and B, however, how are these charges and currents
interacting with the fields?
It's not really clear, not really intuitive, but of course you would say these charges and currents
interact via forces with these fields.
But these terms here in these equations, they don't tell us anything about forces.
The only force that we know and that we must know is the Lawrence force.
And this is given by the discharge queue times e so this is something we have considered already.
Plus, there is another term which occurs when there is a moving charge.
Then we have queue times the velocity of this charge and the vector product with the magnetic field.
So this explains as how the fields act on the currents and charges and this also makes them physically
relevant.
Or you could even say the Lawrence forces define what these fields even mean.
So far they were just abstract vectors.
So let's start from this Lawrence force, and I want to explain to you again how this Loren's force
acts on electrons.
So this is something I have already shown you and one of the optional lectures.
But I want to tell to you again.
So consider we have an electron here and it moves towards a magnetic field.
And let's forget about the electric field so far, because this is really trivial and we have discussed
this already.
Now, if the electron is outside of the magnetic field, then the force is zero.
However, once it enters the region where the magnetic field is present, then there is a force that
is given by the vector product of the velocity and the magnetic field.
So the velocity is pointing into this direction here and the magnetic field is pointing into the plain
of your screen.
So the cross product is pointing along this direction.
If you assume such a negative charge here, because for an electron the charge is negative and there's
the reason why this electron is forced onto such a spherical trajectory here through this magnetic field.
Then at one point it exits the magnetic field and will then not feel any force anymore.
And it will move along this direction here.
Now, what's really important to realize here is that you cannot really drive this force in terms of
classical physics, it looks quite simple and you probably know it since school, but it is actually
really difficult to derive this equation and you need special relativity.
So I do not want to explain this here.
And I do not want to go into detail.
I just want to make sure that you understand that here, even though it looks quite simple, it's really
difficult.
So the starting point of special relativity or the typical starting point of special relativity is a
thought experiment.
You probably heard of this already.
So you can consider, for example, a spaceship or a rocket and this spaceship moves with the velocity
of 80 percent of the velocity of light.
So this is C here and now.
The rocket shoots some projectile, which also has some velocity.
And if you're sitting in the cockpit of this of this rocket, you will see that this projector moves
with the velocity that is also, let's say, 80 percent of the velocity of light.
Now, the question is, how fast is this projectile moving from the outside perspective?
So since you're sitting in the cockpit, you are moving and you see that the projectile is moving with
this velocity here.
However, classically, when you're on the outside, you would say that these two velocities add up.
So you would say this projectile moves with the velocity that is 160 percent of the speed of light.
However, there is a very important law of special relativity.
This tells us that no velocity can ever be larger than the speed of light.
So it's not possible that these two velocities add up.
And this the only solution to this problem is that we have effects that are called space contraction
and time dilation.
So it means that at high velocities, especially at velocities that are close to the speed of light,
the space changes and deforms depending on the velocity.
So if we calculate the sum of these two velocities, it gives us something like 69 percent of the speed
of light, which you cannot understand from classical laws of physics.
And for a long time, it was really, really a big problem that people only had the Maxwell equations
and this Loren's force and they could not really explain how they fit together.
And it was just at the beginning of the 20th century when Einstein developed the special relativity
theory, when finally there was a solution for this problem.
And the solution is that you actually need to introduce a new velocity.
This is the so-called for velocity.
And here you have the electromagnetic fields tensor.
And this is the exact relation of the force and the velocity and these electromagnetic fields.
And this is here the so-called Lowrance factor, which accounts, roughly speaking, to the subspace
contraction and time dilation phenomena.
So this may sound really difficult and I hope I did not totally confuse you.
The only thing I want to tell you here is that the Lauryn's force looks really simple, but from a theoretical
side, it's very complicated.
And it was for a long time not known how the Lawrence Falls can be derived and how it can be related
to the Maxwell's equations.
So why do you need to remember is we need to Maxwell's equations and we need a fifth floor, which is
the Lawrence force, which tells us how the charges and currents are related to the fields.
So this is by this force and that's the Lawrence force.
Cannot be cannot simply be derived, but is a consequence of special relativity theory.