Let's now come to the very early concepts of magnetism and let us also discuss the famous amperes law.
So already in the sixth century B.C., Aristotle realized that load stones are natural, naturally magnetized,
so they can attract other pieces of iron.
So, for example, here you can see how these iron nails are attracted by this dark stone, which is
consistent out of iron and oxide.
So oxygen.
So it's basically rust.
So here you can see these iron, the three oxygen four compound and its crystalline shape.
But in nature, it looks more like this because it is mixed with other iron compounds.
And today we know that the reason for this attractive interaction is the magnetic field.
So here you see you have a magnet of our magnet and it exhibits these field lines of the magnetic field.
And when we take a magnetized piece of iron and put it into these field lines, the individual magnets
off the iron will orient according to these field lines, and this will lead to an attractive interaction.
So this is why these nails here are attracted.
And already in the 12th century, there was the first famous application of this concept and this was
the compass.
So when you take a needle, for example, here, you could take a nail and you you make it able to to
rotate.
Then you will find that this nail will always orient along a certain direction on Earth.
And in 600, Gilbert found out that the earth is in fact magnetic and that this needle will always orient
towards the North Pole or the South Pole.
However, it turns out today we know that these north and south poles are actually a bit different to
the geographic north and south poles, but still it's quite a good approximation.
Now, in 1825, our and Amper realized another very important effect, so what they did is they took
this compass and then they also took such a wire, basically.
So you have a current going through his wire.
So here the wire is coming out of out of the plane.
So it's coming.
It's coming towards us.
So when the electrons are flowing through this wire, what happens is the needle of the compass will
reorient so that it's always perpendicular to the wire.
So the reason for this is that we have this magnetic field around wire, which goes in such circles,
so the needle will orient along the and tangled of this of this profile of these rotational symmetric
lines of the magnetic fields.
And if you expressed us in terms of mathematics, you can say that the rotation of the magnetic field
is basically given by the current or by the current density.
So this year is on APLA operator, which is the vector of the individual derivatives, and here we calculate
the vector product with the magnetic field.
Now another very famous observation is that when you take two of such vires and you apply the current
along different directions, you will see that they attract each other.
And this is because you have here does this profile of the magnetic field that is circulating into opposite
directions and so forth between these two wires is given by the product of the two currents, and it
is of one of our distant dependent's.
Now, the problem with this equation here, which is amperes law, is actually that it's that it violates
the continuity equation.
So for every vector, this relationship is true.
So this is not only true for magnetic fields, but it's true for every Wiktor.
We take the rotation and calculate then the divergence.
This will always give zero.
So that's a good exercise for you.
If you want to understand what is happening, post a video, take some vector, write down, for example,
Vector Vectibix consisting of the components B one, be two three and calculate first the rotation and
then the divergence.
And you will see that all of the terms were cancel out and that this is zero.
Now, the problem with this is that since we have this observation from the experiments here, people
thought that this rotation of B is equal to some constant times the current.
So what we can right here is that the divergence of the current is zero.
However, we are missing the term because the continuity equation implies sort of that this is an equation
for all conserved vector fields and current is one of these vector fields.
This implies that the divergence of such a vector must always be equal to the time derivative of some
density.
So here it's a charged density and divergence of the current is equal to minus the time derivative of
the charged density.
This is basically because charge is a conserved quantity.
So you cannot really lose charge or generate charge.
It's more that when you look at a whole closed system, the charge will always remain the same.
So when you have in a certain proportion of the system a temporal change of the charge, this means
it must generate a current because otherwise the charge must be conserved.
So this means we are missing here, this particular term.
And this equation cannot be correct because it violates discharged conservation.
So this is something we have to keep in mind, because later in the Maxwell equations, we will see
that is Paris, LA is not really correct, but we will also introduce here another term.