Great job, you have now finished this section on Maranello Static's, so let us come to the last lecture
of the section, which is a summary, and also I will compare here the results to our results in Electrostatic.
So here you remember our board from the last section where we had this is our summary.
And now in Magneto Static's, we actually went along a very similar path.
And you probably have already noticed in the previous lectures that many of the results in Magneto's
Static's are very closely related to the results in Electrostatic.
So here, instead of starting with these Maxwell equations for the electric field, of course we have
started with the Maxwell equations for the magnetic fields.
And here we have that.
The magnetic field does not have any sources or things.
So the divergence of B is zero and instead the rotation of the magnetic field is generated by currents.
So that's a bit different here compared to the electric field.
So the rotation and divergence have swapped their roles.
So now, because of these Maxwell's equations, it allows us to define a vector potential.
So this is for the same reason as for electrostatic.
So this is because when we take this expression here, calculated divergence of the rotation of any
vector, this will always be zero.
So this means it must always be possible to write the magnetic field in terms of a rotation of some
function.
And this function is a vector that depends on R and we call it the vector potential.
Now, if we take this and put it in two of the other Maxwell's equation, we get this equation here
that looks like this song equation.
And in fact, it is for the equation, but for every individual component, because here A and G of
actors and here fi and draw are scalars.
Please also keep in mind that we have considered here a special gaige for the vector potential, which
means that the divergence of A is zero.
And this is always possible to realize.
Now, since these equations look so similar, of course, also the solutions for A and B are very similar
compared to Phi and E.
So here, just instead of roll, we have J and hear the same thing for the magnetic field compared to
the electric field.
Then also, we have considered the magnetic dipole, so physically speaking or based on the physical
system, this magnetic dipole moment is quite different compared to the electric dipole moment.
In the electric case, we had two charges that are separated by a vector.
So disconnecting, connecting vector here defines the orientation of the electric dipole moment.
In the magnetic Kate case.
We have circular currents, for example, of current loops.
So what this means is that electrons go around to circle here and this leads to the emergence of a magnetic
moment perpendicular to this plane.
So if it is really a plane here, then this expression is true.
It's the the yeah.
The currents multiplied by the area of this loop.
And then the orientation is given by the perpendicular normal vector.
Now, since this result is already very similar to this one here, we can already understand that our
solution for the magnetic fields looks very similar.
In fact, it's the same relation.
It's just that we have to exchange the electric dipole moment with the magnetic dipole moment and then
for the force and to talk.
It's also the same thing as in the electric case.
So this means if we take such a magnetic dipole moment and put it into a homogeneous magnetic field,
there will not be any net force because this gradient of constant function is zero.
But there will be a talk that leads to the reorientation of this vector and or the spectrem parallel
to the homogeneous and constant magnetic fields.
But if you think about it on the microscopic level, you have to consider the individual electrons that
move around the circle and you have to consider the Lawrence forces.
Then in the last lecture, we have discussed the energy of the magnetic field and like in these cases
here, it's again very similar to electrostatic.
So we just have to integrate over this energy density, which we have derived from the continuity equation.
But this time we only consider the terms.
Yeah, that depend on the magnetic field and not on the electric field.
And if we do this, we get an energy that is given by the current times to vector potential instead
of having here the charged density times, the scale of potential.
So you can see that it really corresponds to these two pieces here that are exchanged.