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Let's now come to the very early concepts of magnetism and let us also discuss the famous amperes law.



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So already in the sixth century B.C., Aristotle realized that load stones are natural, naturally magnetized,



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so they can attract other pieces of iron.



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So, for example, here you can see how these iron nails are attracted by this dark stone, which is



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consistent out of iron and oxide.



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So oxygen.



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So it's basically rust.



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So here you can see these iron, the three oxygen four compound and its crystalline shape.



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But in nature, it looks more like this because it is mixed with other iron compounds.



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And today we know that the reason for this attractive interaction is the magnetic field.



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So here you see you have a magnet of our magnet and it exhibits these field lines of the magnetic field.



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And when we take a magnetized piece of iron and put it into these field lines, the individual magnets



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off the iron will orient according to these field lines, and this will lead to an attractive interaction.



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So this is why these nails here are attracted.



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And already in the 12th century, there was the first famous application of this concept and this was



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the compass.



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So when you take a needle, for example, here, you could take a nail and you you make it able to to



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rotate.



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Then you will find that this nail will always orient along a certain direction on Earth.



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And in 600, Gilbert found out that the earth is in fact magnetic and that this needle will always orient



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towards the North Pole or the South Pole.



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However, it turns out today we know that these north and south poles are actually a bit different to



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the geographic north and south poles, but still it's quite a good approximation.



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Now, in 1825, our and Amper realized another very important effect, so what they did is they took



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this compass and then they also took such a wire, basically.



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So you have a current going through his wire.



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So here the wire is coming out of out of the plane.



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So it's coming.



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It's coming towards us.



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So when the electrons are flowing through this wire, what happens is the needle of the compass will



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reorient so that it's always perpendicular to the wire.



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So the reason for this is that we have this magnetic field around wire, which goes in such circles,



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so the needle will orient along the and tangled of this of this profile of these rotational symmetric



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lines of the magnetic fields.



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And if you expressed us in terms of mathematics, you can say that the rotation of the magnetic field



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is basically given by the current or by the current density.



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So this year is on APLA operator, which is the vector of the individual derivatives, and here we calculate



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the vector product with the magnetic field.



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Now another very famous observation is that when you take two of such vires and you apply the current



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along different directions, you will see that they attract each other.



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And this is because you have here does this profile of the magnetic field that is circulating into opposite



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directions and so forth between these two wires is given by the product of the two currents, and it



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is of one of our distant dependent's.



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Now, the problem with this equation here, which is amperes law, is actually that it's that it violates



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the continuity equation.



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So for every vector, this relationship is true.



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So this is not only true for magnetic fields, but it's true for every Wiktor.



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We take the rotation and calculate then the divergence.



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This will always give zero.



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So that's a good exercise for you.



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If you want to understand what is happening, post a video, take some vector, write down, for example,



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Vector Vectibix consisting of the components B one, be two three and calculate first the rotation and



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then the divergence.



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And you will see that all of the terms were cancel out and that this is zero.



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Now, the problem with this is that since we have this observation from the experiments here, people



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thought that this rotation of B is equal to some constant times the current.



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So what we can right here is that the divergence of the current is zero.



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However, we are missing the term because the continuity equation implies sort of that this is an equation



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for all conserved vector fields and current is one of these vector fields.



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This implies that the divergence of such a vector must always be equal to the time derivative of some



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density.



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So here it's a charged density and divergence of the current is equal to minus the time derivative of



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the charged density.



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This is basically because charge is a conserved quantity.



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So you cannot really lose charge or generate charge.



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It's more that when you look at a whole closed system, the charge will always remain the same.



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So when you have in a certain proportion of the system a temporal change of the charge, this means



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it must generate a current because otherwise the charge must be conserved.



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So this means we are missing here, this particular term.



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And this equation cannot be correct because it violates discharged conservation.



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So this is something we have to keep in mind, because later in the Maxwell equations, we will see



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that is Paris, LA is not really correct, but we will also introduce here another term.