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All right, so we are almost finished with our course, let's continue this section on the electrodynamics



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and matter.



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So in the previous two lectures, we have derived the polarization and the magnetization, and we have



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shown that you can express these polarization charges as the divergence of deeper polarization and that



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you can express these circular currents that correspond to magnetic moments in terms of the rotation



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of the magnetization.



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And furthermore, we have also shown that you can split up these charge densities into the current densities,



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into several contributions.



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And now the whole important idea here is that we separate our external charges and Curran's these are



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the ones that we also had previously in vacuum from the charges and the currents that occur in the material.



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Since we have these nice results, we can now go ahead and take our Maxwell's equations.



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So far we have only discussed them in vacuum, but they are also true in.



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However, then they become really difficult to solve because we thought we would then have to use here



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all of the charged densities.



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So the idea was Eros, but also the polarization charges, which are rather difficult to calculate.



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If you have some piece of matter and you want to find out what all of the polarization charges look



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like.



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So this would probably be some very complicated expression here.



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And the same also is true for this fourth Maxwell equation, where we would have to consider now all



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of the different contributions to.



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So even these polarisation currents in the material that are very hard to write down and to take into



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account is what we will do.



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Instead, we will transform the first and to fourth maximal equation and we will use the results from



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the previous slide.



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So here we have these two Maxwell's equations that we want to transform and here we have the result



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for the charged density.



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So we have separated these into these two terms and we can write down the road ropy in terms of this



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divergence of the polarisation.



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So now let's take it and put it in here so we get one term.



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That is Rosero when we take Epsilon Zero to the other side and we get the other term, that is the divergence



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of the polarization.



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Now, you can see we have here the divergence of E and the divergence of P, so we can just write down



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Rosero is the divergence of the sum of these two terms here.



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And so we have an equation that looks very similar to this equation.



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And we can introduce a new field, which is the that is called the displacement vector.



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And so instead of writing down the divergence of E is basically row, we write down the divergence of



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the is zero zero.



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And that is the very important difference we have now, a different field, which is the old field E



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plus the polarisation.



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And here we have a different charge.



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Here we only consider role zero, which is the same as we did before.



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So now when we have a piece of matter with charges that arise due to charge separation to polarisation,



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for example, we do not consider these charges here.



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We only consider the charges, for example, if we have some defect, some atem, some electron, some



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wire interacting with our material, but not the charges in the material itself.



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That's the whole trick.



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Then we can do a similar transformation for the other Maxwell equation.



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So here we need the current density on this.



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We had separated into three terms and one of them becomes the time derivative of the polarization,



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and the other one is the rotation of the magnetization, the one we put all of these three terms in



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here.



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We must not forget that we also have this term as well.



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So there's that settlement here.



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So we have, in fact, four terms here on the right hand side.



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And now you can see we have two terms that are a rotation of some vector and we have two terms that



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are the time derivative of some vector.



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So let's write it down like this.



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We have the rotation of this sum here and the time derivative of the sun.



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And once again, this gives us an equation that looks mathematically just like this equation.



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And this allows us to introduce a new field, which is this magnetizing field H.



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And here we have again the field from the left hand side, which is this displacement field so we can



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write the rotation of the magnetizing field is equal to the dots plus J zero.



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So we have, uh, an equation that looks very similar to the maximal equation, but with different fields.



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And the nice thing like on the left hand side is now that here we do not consider all of the currents



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that are difficult to calculate.



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We only consider the J zeros, which excludes the currents that are in the piece of matter due to a



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chart separation, also the circulating currents, for example.



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So this means we have now transformed our Maxwell's equations that are true in a vacuum, but also in



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matter, of course.



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But there they would look very difficult.



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We have transformed them to these Maxwell equations in matter via the introduction of these two new



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fields that are based on the old fields and the polarisation and democratization respectively.



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Of course, we can go ahead and do the same transformation also for the integral formulation of these



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Maxwell equations.



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So you really have to one to one correspondence where you take the Epsilon zero and the one over zero



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and put them in the new fields here and here.



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And then you're right here instead of the total row, only the row zero and instead of the total J only



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the J zero.



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And the other terms, for example, the polarization charge which turns 10 into polarization, they



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are also put in this new field here.



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So this means we have ended up with this integral version of these Maxwell's equations and matter and



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the differential version of the BEXELL Equations and Mather and these make our lives much more easy.



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And furthermore, what this means is we can now go ahead and take all of the nice results that we have



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derived throughout this course and apply them also to matter.



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So, for example, the magnetic dipole or the electric dipole or even here Hatsune Dipole, they are



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not only true in vacuum, but they are true in matter.



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But instead of just calculating E, we now need to also have a dependence on off of these on E age,



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on B, and this is typically done phenomenologically and then most materials, we simply assume a linear



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dependence with some proportionality constant that is taken from the experiment, for example.