Now, the idea of this whole lecture is to construct the Maxwell's equations based on symmetry arguments,
and what we want to do is we want to take the electric field and magnetic fields and consider only first
order derivatives.
So we have limited time derivatives.
We have to divergences and we have two rotations for them.
On the previous slide.
We already had to charge density and current density.
And now we want to classify all of these different objects here in terms of their behavior on the time
inversion and space inversion.
So we have this we have these Maxwell's equations and these as well.
As we have learned, these are four equations.
Now we have two of these operations.
So we can either get two plus, plus, minus, minus, plus, minus and minus plus.
So we have four different categories and we have equation with the left and on the right side, I mean,
it's clear the left side must be equal to the right side and therefore the left side must transform
in the same way as the right side.
So this means in a single Maxwell equation, we can only put objects that behave equally under time
inversion and space inversion.
So what we will do is we will go through all of these possible terms here that are, as I told you,
first order and derivatives and then classify them into the four different categories.
Or we will see is that this gives us the Maxwell's equations.
But yeah, besides the prefectures, the prefectures, we cannot really develop or understand from this
symmetry arguments, but we can understand why certain objects are in the same Maxwell equation.
So let's go ahead.
We already had the charge density, which is kind of trivial.
It's just a scalar and it's symmetric on the time inversion and space inversion.
So it means it does not change.
Now, the current density is essentially the velocity, so it's a. symmetric under both of these operations.
We have this as well.
The electric field is asymmetric on the time inversion, so it doesn't really care about if you reverse
time or not and it's a. symmetric under space inversion.
So I think that's that's quite clear.
It's just like a typical typical vector, so if you look in the mirror, it will revert.
But if you change the time and does not really revert because it's just a vector, and now you might
think that this is also true for the magnetic fields.
And this is typically a misconception because the magnetic field and the electric field are typically
both considered as being conventional vectors.
But it turns out that a magnetic field is a bit different.
I mean, you can, of course, ride it as a vector.
And we do this throughout this whole course.
Here you have a vector with the X, Y and Z component.
But it's behavior on that is transformation's is very different to a typical vector, which is also
why we call it a pseudo vector, or sometimes it's also called an actual vector.
So the reason why the magnetic field is anti symmetric under time version is the following consideration.
So let's take an electron starting from here and move it inside an area where we have a magnetic field.
So in the beginning, we have no force at all.
The electron will move with its velocity, but then we will have here the Lawrence Force, because this
is a vector product of the velocity which goes into this direction.
And the magnetic field, which you can see is located.
Yeah, not towards us, but towards the other direction.
So since we have a negative charge, this means we have a force acting in this direction.
So the electron will move on such a circular trajectory through the magnetic field and then at some
point it will hit this point where the magnetic field will go back to zero.
So the electron will move in this direction.
Now, imagine what happens if we reverse the time.
Well, of course, as we have learned, the velocity will change as we change its sine.
So you can, for example, film this trajectory here, this process, and then rewind this whole movie.
Then the electron will go from here to here.
And now you have to think in which direction must the magnetic fields be oriented so that we can move
along this trajectory?
Well, of course, the magnetic field must be reversed because if it is oriented like it's shown here,
then the electron would move like this.
But instead, it moves like this when we rewind.
So this means the orientation of the magnetic field has to flip the sign when we apply time version.
Therefore, it's a. symmetric under time inversion and it's instead symmetric on this space inversion.
OK, now the rest is pretty trivial, I would say, because we now know how the electric field behaves,
we know how the magnetic field behaves, and these derivatives are essentially just these two fields
divided by tea or multiplied or divided by the space because it is a derivative versus vector space.
So we just have to look up how the space and the time transformed as we had on the previous slide.
And then we can fill out this table, as I have done here for you.
So, for example, here, because we have the time here, we get here minus sign while we had here plus
sine and here, the minus sign stays the same for the magnetic field.
It's similar instead of the minus we get now plus and he had a plus stays the same.
And then here, for example, for the electric field, the and the behavior and the time.
Inversion remains the same similarly here.
But the behavior and the space inversion flips to sign.
So now instead of the minus side, we have a plus sine.
So if you want to understand it yourself, it's probably better to post a video here and have a careful
look.
But I think once you have understood how the time transforms into a space transforms, which is really
trivial because this is a time inversion, space inversion.
And when you have understood how electric field and magnetic fields transform, this is actually difficult
here.
You have to understand this whole cartoon.
If you understand that all of these aspects, you can fill out the whole table.