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Welcome back to this lecture about -- potentials, which is part of the section time dependent



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problems.



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So in the previous two lectures, we have transformed these General Maxwell's equations without any



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approximations to these different types of textual equations.



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And then we have used the Lawrence Gage to transform them even more.



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So here we have four simple looking equations.



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And the special thing is we have now here the square operator acting on A and the square operator acting



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on fire.



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And these two equations are uncoupled.



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They only depend on the charged density and on the current density.



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Now let's take these four equations and write them down in the extended version.



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We are right down this square operator as the second order time derivative and this lipless operator.



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So here you can see the equations.



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It's exactly the same as on the previous slide.



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If you look at the first two equations and if you would not consider this term here, we have something



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that looks very similar to electrostatic.



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And also for the other two equations, if it would not have this term here, we would get the equations



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of Magneto Static's.



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So the truth is here we also have an additional term, but that's a bit unproblematic.



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Unproblematic, I could say, because if you first solve the little static problem, then you already



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know what a looks like.



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So you know what this term looks like.



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So you really have decouples here.



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But can you do static's and electrostatic if you would not have these two terms here?



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Now, since these equations are so similar and related to each other, we can maybe use the results



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from Electron Mikoto Static's to arrive at the solutions for the general case.



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So here are the solutions again that we have derived extensively in these two sections here on the electromagnetic



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static's.



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And here you see again that they look kind of similar.



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So we have a prefect and we have here an integral over the charge density divided by the distance of



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these two vectors.



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And here we have the the current density instead.



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And this is, of course, a scalar because it's the electrostatic potential and this one is the three



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component vector because it's our vector potential.



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And yeah, when we have these two fields or these two potentials, we can calculate B and E now since



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we have these two additional terms, our solution in the general case looks kind of similar.



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In fact, it looks exactly the same.



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But the only difference is that we have a different time argument.



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So here we have this charged density and the current density and the time argument is not any more t



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the same as here, but we have some retardation, so the time lags behind.



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And this is because the electrostatic potential and the vector potential are determined by charges and



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currents at every possible position in the three dimensional space.



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And so the information from position, from this position are to this position are needed to travel



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a certain distance.



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And this will take a certain time.



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And this is why this information lags behind.



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So we have a retardation and this is why these solutions are also called the -- potentials, not



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the solution.



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Looks really simple, but it's not trivial to see that they are indeed the solution.



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In fact, it's very difficult to really arrive at.



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And in the next section I want to prove and I want to show you that these two terms here are indeed



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the solutions to these four Maxwell equations.