Let us now discuss our eight Maxwell equations in detail, so we started out with the differential formulations
of the Maxwell equations, and I already told you that these four equations correspond to the statements
that charges are the sources of the electric field while the magnetic field has no sources.
So there are no magnetic monopoles.
And then for the other two equations, we had that time dependent magnetic fields generate electric
fields and vice versa, but also that currents can generate magnetic fields.
The previous lecture, we have also derived integral formulations and so their interpretation can be
a bit different.
But of course, it's it's still the same equation just from a different point of view.
So the physical consequences will be quite similar of these equations.
So let us start with the first two equation equations.
These are the Gauss laws.
So here we have the differential formulation and the integrated formulation.
And it means that charges are the sources of the electric field, which you can see, for example,
here when we have two charges, both of them are sources or sinks of the electric field.
And we have these lines of the electric field that connect to two sources.
Then in the integrated formulation we have that the electric flux through a closed surface is determined
by a deep and close charge.
So when we have a volume here, the electric field or the electric flux is determined by the charge
inside of the volume.
Now, we can also formulate Koslov on magnetic fields, and it is kind of similar, but the right hand
side of these equations is zero in both cases.
So this means a magnetic field has no source or sinks, so monopoles do not exist.
So there is not something like QM, like a magnetic charge just does not exist.
We know, for example, when we have such a magnet, it always consists out of the north and the South
Pole.
So it is more of a vectorial quantity and not such a divergence in the sense of a monopole.
And these magnetic fields, field lines, for example, they go through the magnets, as you can see
here.
Now, in the integrated version, it means that the magnetic flux vanishes when we integrate over every
closed surface.
So that's very interesting, I think, because you can take any magnetic system and you can take any
volume, and if you just integrate the magnetic flux, it will always vanish.
Now, in the very first section of this course, we have talked about early theories of electrodynamics
and we have learned that people already knew about induction and the electromagnetic induction law.
And this is the interpretation of the third Maxwell equation.
So this is here the differential formulation and the integrated formulation and means that time dependent
changes in the magnetic field give rise to electric fields and then the integrated formulation.
It means that an electric field is generated by changes of the magnetic flux.
So this means not only does the magnetic field have to change, but also the area of which we can integrate,
all of which we have to integrate could also change.
So this is something we have discussed earlier in terms of the electromagnetic induction, where we
take a close to wire and bring it into a magnetic field or take it out of the magnetic field.
And this leads to the generation of an electric field and therefore to an induction voltage that can
be used, for example, to charge devices or to detect cars going over a red light, for example.
And here you can see that this is the flux.
So we have we have to integrate the magnetic field over an area.
And if the magnetic field is constant in the area, then we can just ride the flux as equal to A times
B and then this year is our induction voltage is equal to minus five minus the time derivative of fire.
And this line integral here where we integrate over the surrounding of this area.
Now, the last Maxwell equation is this one in the differential formulation.
And here we have the integral formulation, and this is an extended or generalised version of unpause
law.
So we have encountered Amperes Law already in our first section about early theories of electrodynamics
because in 18, 25, thirsted and Ampere.
Notice that when you take a wire and apply the current, then a compass needle will always orient perpendicular
to this wire.
And this is because this is this current.
So basically a moving charge will generate a magnetic field.
So this was what I understood came up with.
You can see it is kind of similar to this equation.
But we have also learned that when we use this identity here, which is true for every vector.
So it's also true for B that the divergence of the rotation is always zero.
This means that the divergence of J is zero if we take this equation here.
However, we also know that the charge must follow a continuity equation where we have on the right
hand side some time derivative term of some charge density.
So this term was missing in this equation.
T it turns out in the Maxwell equation we have this term.
So we have here a time derivative of music, zero, Epsilon Zero and the electric field.
So in this amperes law, the continuity equation is finally fulfilled.
So we know that this was a nice guess to describe this experiment, but it was not sufficient to really
fulfill the whole theory.
So this is the differential formulation that electric fields give rise or changes in the electric field
or currents give rise to magnetic fields.
And also we have here the the integrated version that this is also true if we integrate over the surrounding
of.
An area.