Welcome back to this lecture about -- potentials, which is part of the section time dependent
problems.
So in the previous two lectures, we have transformed these General Maxwell's equations without any
approximations to these different types of textual equations.
And then we have used the Lawrence Gage to transform them even more.
So here we have four simple looking equations.
And the special thing is we have now here the square operator acting on A and the square operator acting
on fire.
And these two equations are uncoupled.
They only depend on the charged density and on the current density.
Now let's take these four equations and write them down in the extended version.
We are right down this square operator as the second order time derivative and this lipless operator.
So here you can see the equations.
It's exactly the same as on the previous slide.
If you look at the first two equations and if you would not consider this term here, we have something
that looks very similar to electrostatic.
And also for the other two equations, if it would not have this term here, we would get the equations
of Magneto Static's.
So the truth is here we also have an additional term, but that's a bit unproblematic.
Unproblematic, I could say, because if you first solve the little static problem, then you already
know what a looks like.
So you know what this term looks like.
So you really have decouples here.
But can you do static's and electrostatic if you would not have these two terms here?
Now, since these equations are so similar and related to each other, we can maybe use the results
from Electron Mikoto Static's to arrive at the solutions for the general case.
So here are the solutions again that we have derived extensively in these two sections here on the electromagnetic
static's.
And here you see again that they look kind of similar.
So we have a prefect and we have here an integral over the charge density divided by the distance of
these two vectors.
And here we have the the current density instead.
And this is, of course, a scalar because it's the electrostatic potential and this one is the three
component vector because it's our vector potential.
And yeah, when we have these two fields or these two potentials, we can calculate B and E now since
we have these two additional terms, our solution in the general case looks kind of similar.
In fact, it looks exactly the same.
But the only difference is that we have a different time argument.
So here we have this charged density and the current density and the time argument is not any more t
the same as here, but we have some retardation, so the time lags behind.
And this is because the electrostatic potential and the vector potential are determined by charges and
currents at every possible position in the three dimensional space.
And so the information from position, from this position are to this position are needed to travel
a certain distance.
And this will take a certain time.
And this is why this information lags behind.
So we have a retardation and this is why these solutions are also called the -- potentials, not
the solution.
Looks really simple, but it's not trivial to see that they are indeed the solution.
In fact, it's very difficult to really arrive at.
And in the next section I want to prove and I want to show you that these two terms here are indeed
the solutions to these four Maxwell equations.