All right, so we are almost finished with our course, let's continue this section on the electrodynamics
and matter.
So in the previous two lectures, we have derived the polarization and the magnetization, and we have
shown that you can express these polarization charges as the divergence of deeper polarization and that
you can express these circular currents that correspond to magnetic moments in terms of the rotation
of the magnetization.
And furthermore, we have also shown that you can split up these charge densities into the current densities,
into several contributions.
And now the whole important idea here is that we separate our external charges and Curran's these are
the ones that we also had previously in vacuum from the charges and the currents that occur in the material.
Since we have these nice results, we can now go ahead and take our Maxwell's equations.
So far we have only discussed them in vacuum, but they are also true in.
However, then they become really difficult to solve because we thought we would then have to use here
all of the charged densities.
So the idea was Eros, but also the polarization charges, which are rather difficult to calculate.
If you have some piece of matter and you want to find out what all of the polarization charges look
like.
So this would probably be some very complicated expression here.
And the same also is true for this fourth Maxwell equation, where we would have to consider now all
of the different contributions to.
So even these polarisation currents in the material that are very hard to write down and to take into
account is what we will do.
Instead, we will transform the first and to fourth maximal equation and we will use the results from
the previous slide.
So here we have these two Maxwell's equations that we want to transform and here we have the result
for the charged density.
So we have separated these into these two terms and we can write down the road ropy in terms of this
divergence of the polarisation.
So now let's take it and put it in here so we get one term.
That is Rosero when we take Epsilon Zero to the other side and we get the other term, that is the divergence
of the polarization.
Now, you can see we have here the divergence of E and the divergence of P, so we can just write down
Rosero is the divergence of the sum of these two terms here.
And so we have an equation that looks very similar to this equation.
And we can introduce a new field, which is the that is called the displacement vector.
And so instead of writing down the divergence of E is basically row, we write down the divergence of
the is zero zero.
And that is the very important difference we have now, a different field, which is the old field E
plus the polarisation.
And here we have a different charge.
Here we only consider role zero, which is the same as we did before.
So now when we have a piece of matter with charges that arise due to charge separation to polarisation,
for example, we do not consider these charges here.
We only consider the charges, for example, if we have some defect, some atem, some electron, some
wire interacting with our material, but not the charges in the material itself.
That's the whole trick.
Then we can do a similar transformation for the other Maxwell equation.
So here we need the current density on this.
We had separated into three terms and one of them becomes the time derivative of the polarization,
and the other one is the rotation of the magnetization, the one we put all of these three terms in
here.
We must not forget that we also have this term as well.
So there's that settlement here.
So we have, in fact, four terms here on the right hand side.
And now you can see we have two terms that are a rotation of some vector and we have two terms that
are the time derivative of some vector.
So let's write it down like this.
We have the rotation of this sum here and the time derivative of the sun.
And once again, this gives us an equation that looks mathematically just like this equation.
And this allows us to introduce a new field, which is this magnetizing field H.
And here we have again the field from the left hand side, which is this displacement field so we can
write the rotation of the magnetizing field is equal to the dots plus J zero.
So we have, uh, an equation that looks very similar to the maximal equation, but with different fields.
And the nice thing like on the left hand side is now that here we do not consider all of the currents
that are difficult to calculate.
We only consider the J zeros, which excludes the currents that are in the piece of matter due to a
chart separation, also the circulating currents, for example.
So this means we have now transformed our Maxwell's equations that are true in a vacuum, but also in
matter, of course.
But there they would look very difficult.
We have transformed them to these Maxwell equations in matter via the introduction of these two new
fields that are based on the old fields and the polarisation and democratization respectively.
Of course, we can go ahead and do the same transformation also for the integral formulation of these
Maxwell equations.
So you really have to one to one correspondence where you take the Epsilon zero and the one over zero
and put them in the new fields here and here.
And then you're right here instead of the total row, only the row zero and instead of the total J only
the J zero.
And the other terms, for example, the polarization charge which turns 10 into polarization, they
are also put in this new field here.
So this means we have ended up with this integral version of these Maxwell's equations and matter and
the differential version of the BEXELL Equations and Mather and these make our lives much more easy.
And furthermore, what this means is we can now go ahead and take all of the nice results that we have
derived throughout this course and apply them also to matter.
So, for example, the magnetic dipole or the electric dipole or even here Hatsune Dipole, they are
not only true in vacuum, but they are true in matter.
But instead of just calculating E, we now need to also have a dependence on off of these on E age,
on B, and this is typically done phenomenologically and then most materials, we simply assume a linear
dependence with some proportionality constant that is taken from the experiment, for example.