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The first lecture of this section, I just want to go ahead and show you the Maxwell's equations that



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Maxwell formulated in the 18 60s.



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So these are the four equations in differential form.



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So we have here their divergence of the electric field and the divergence of the magnetic fields.



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So these are the sources of these two fields.



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And for the electric field, we get that it's given by one over Epsilon zero, which is a constant called



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to vacuum permeability.



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And then we multiply here by the charged density, which is a quantity that depends on the position.



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So it's actually row of R and that tells us the charges are the sources of the electric field.



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The second equation is that the divergence of the magnetic field is zero.



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So when we compare the electric field and the magnetic field, this means the magnetic field does not



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have any sources.



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So it means there are not such things as charges for the magnetic field.



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So there are no magnetic monopoles.



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Magnets always come in pairs.



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So you always have the North and South Pole, but no monopoles.



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The other two equations relate the rotation of the electric field to the time derivative of the magnetic



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field and vice versa.



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So we have year rotation of the electric field is equal to minus one times the time derivative of the



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magnetic field.



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So it means that when we have a change in the magnetic field over time, this generates electric fields.



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And also, it goes the other way around when we have time changes in the electric field.



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This gives rise to magnetic fields.



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But here we have a factor, which is the product of this vacuum perpetuity and the vacuum permeability.



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And then we also have the other term that we have found already in the first section of the Holocaust



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where we found that charges.



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So basically moving, moving charges that give rise to currents generate magnetic fields.



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So remember these rotational symmetric field lines around a wire.



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So these are the form Maxwell equations and these are the most important equations of this whole course.



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And we were out the following lectures and sections.



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We will discuss these equations in much detail and we will solve special cases, for example, where



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the fields do not depend on time, but only on the position.



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And also, I want to show you how we can find the Coulomb law and the amperes law and the law of electromagnetic



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induction in these Maxwell's equations.



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Therefore, we also have to formulate these differential versions of the Maxwell equations as integral



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versions.



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So this is one of the upcoming lectures.



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But before I want to start with an optional lecture where I want to show you how we can motivate these



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four equation based on symmetry arguments.



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So this is a very theoretical lecture.



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And if you are not that interested in the topic, that is totally fine.



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You can go ahead and skip it.



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But I think it's worth giving it a shot and try to understand because it's really rewarding to understand



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where to Maxwell's equations come from.