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Great job, you have now finished this section on Maranello Static's, so let us come to the last lecture



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of the section, which is a summary, and also I will compare here the results to our results in Electrostatic.



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So here you remember our board from the last section where we had this is our summary.



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And now in Magneto Static's, we actually went along a very similar path.



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And you probably have already noticed in the previous lectures that many of the results in Magneto's



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Static's are very closely related to the results in Electrostatic.



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So here, instead of starting with these Maxwell equations for the electric field, of course we have



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started with the Maxwell equations for the magnetic fields.



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And here we have that.



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The magnetic field does not have any sources or things.



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So the divergence of B is zero and instead the rotation of the magnetic field is generated by currents.



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So that's a bit different here compared to the electric field.



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So the rotation and divergence have swapped their roles.



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So now, because of these Maxwell's equations, it allows us to define a vector potential.



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So this is for the same reason as for electrostatic.



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So this is because when we take this expression here, calculated divergence of the rotation of any



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vector, this will always be zero.



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So this means it must always be possible to write the magnetic field in terms of a rotation of some



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function.



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And this function is a vector that depends on R and we call it the vector potential.



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Now, if we take this and put it in two of the other Maxwell's equation, we get this equation here



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that looks like this song equation.



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And in fact, it is for the equation, but for every individual component, because here A and G of



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actors and here fi and draw are scalars.



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Please also keep in mind that we have considered here a special gaige for the vector potential, which



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means that the divergence of A is zero.



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And this is always possible to realize.



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Now, since these equations look so similar, of course, also the solutions for A and B are very similar



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compared to Phi and E.



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So here, just instead of roll, we have J and hear the same thing for the magnetic field compared to



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the electric field.



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Then also, we have considered the magnetic dipole, so physically speaking or based on the physical



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system, this magnetic dipole moment is quite different compared to the electric dipole moment.



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In the electric case, we had two charges that are separated by a vector.



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So disconnecting, connecting vector here defines the orientation of the electric dipole moment.



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In the magnetic Kate case.



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We have circular currents, for example, of current loops.



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So what this means is that electrons go around to circle here and this leads to the emergence of a magnetic



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moment perpendicular to this plane.



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So if it is really a plane here, then this expression is true.



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It's the the yeah.



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The currents multiplied by the area of this loop.



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And then the orientation is given by the perpendicular normal vector.



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Now, since this result is already very similar to this one here, we can already understand that our



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solution for the magnetic fields looks very similar.



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In fact, it's the same relation.



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It's just that we have to exchange the electric dipole moment with the magnetic dipole moment and then



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for the force and to talk.



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It's also the same thing as in the electric case.



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So this means if we take such a magnetic dipole moment and put it into a homogeneous magnetic field,



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there will not be any net force because this gradient of constant function is zero.



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But there will be a talk that leads to the reorientation of this vector and or the spectrem parallel



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to the homogeneous and constant magnetic fields.



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But if you think about it on the microscopic level, you have to consider the individual electrons that



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move around the circle and you have to consider the Lawrence forces.



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Then in the last lecture, we have discussed the energy of the magnetic field and like in these cases



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here, it's again very similar to electrostatic.



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So we just have to integrate over this energy density, which we have derived from the continuity equation.



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But this time we only consider the terms.



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Yeah, that depend on the magnetic field and not on the electric field.



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And if we do this, we get an energy that is given by the current times to vector potential instead



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of having here the charged density times, the scale of potential.



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So you can see that it really corresponds to these two pieces here that are exchanged.