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Let us get started with our news section on dependent problems, and once again, I want to show you



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a slide that I've shown you already many times throughout this course.



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It's all about the Maxwell's equations again.



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So you can see them as you know them already, we have the divergence of ENP and the rotation of E and



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B, and we have here terms that depend on the charge and the current and the time derivatives of the



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magnetic fields and the electric fields.



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And so far, we have discussed several special cases, for example, we have discussed the electromagnetic



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waves so light, for example, where we have considered a vacuum, which means that the charges are



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zero and that the currents are zero there after we have discussed electrostatic and magnetic static's,



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which means that there are no time dependent magnetic and electric fields.



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And in this section, for the very first time, we will discuss the Maxwell equations as they are written



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down here without any approximations and without any special cases.



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So let's get started.



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In this section, we want to rewrite these Maxwell's equations, and our goal is to write them in a



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way that we can somehow transfer our previous knowledge from electrostatic and magnetic statics to this



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general case here where we have time dependent fields.



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And, yeah, also charges and currents, of course.



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So we start with this Maxwell equation, which is the divergence of the magnetic field is zero.



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So that's something that, you know, the magnetic field does not have monopoles.



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And this is, of course, exactly the same as in magnetic aesthetics.



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So nothing really has changed.



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And once again, since we have this divergence here of magnetic field, this means because of the zero,



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this means that the magnetic fields must be expressible in terms of a rotation of some other field,



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because, as you know, the divergence of a rotation of any vector is always zero, no matter what this



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vector looks like.



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Now, the other Maxwell equation for the electric field is that the rotation of the electric field is



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this time derivative of the magnetic field and the minus sign here.



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So this is different compared to electromagnetics, because maybe you remember this in electromagnetics,



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we did not have this term here, so we had that the rotation of E is zero.



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And this allowed us to write down the electric field in terms of a gradient of some scalar function,



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which was the electrostatic potential.



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Now, since we have this term here.



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This is not possible anymore.



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However, we can do a special trick or we can just rewrite this equation here.



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And we do this by using this expression for the magnetic field and put it in here.



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So we have our rotation of the electric field and then we have the other term brought to the other side.



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So we have plus the time derivative of the rotation of the vector potentially.



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Now we have these two derivatives here.



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This one is with respect to time.



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And this one, if you would write it down in terms of the components, this would be partial derivatives



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with respect to X, Y and Z.



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And so since the time and the spatial coordinates are independent variables, at least in classical



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theory, we can switch around and swap these derivatives so we can write that the rotation of E plus



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the rotation of the time derivative of A is zero.



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And now we have something very similar to this equation.



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We have now a new vector field here, which is E plus the time, the time derivative of A..



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So this means this whole expression can be expressed as a gradient of a scalar function, just like



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an electrostatic.



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And now we can we can write it down like this.



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So we have an expression for our electric field.



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And this is here minus the time derivative of a minus the gradient of this electrostatic potential.



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And if you write this vector potential and this scalar potential here as a four component vector, this



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is sometimes also called the full potential.



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But it's not so important here to we will not use this.



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So we will just use this equation here because it allows us to determine e if we have determined A and



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this scalar potential.



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So this means we have now transformed these first two Maxwell equations.



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But so far it's not really much easier here.



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We had B and E and now we have A and fire and here also.



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Now let's go ahead and transform the other equations.



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So here there is this Maxwell equation that the divergence of E basically the sources and sinks of the



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electric field are given by charges.



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So this is here to charge density.



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And we have here that the rotation of B is generated by currents, but also by the time derivative of



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the electric field.



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So we will get started with the left hand side to you where we have this Maxwell equation and we take



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his expression for E from the previous slide and put it into this Maxwell equation.



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And now we can again switch around these two derivatives.



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So we have the time derivative of the divergence of a and here we have the lipless operator, because



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the divergence of the gradient of a scalar function is equal to this lipless operator, which is basically



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the sum of the second order of derivatives, which back to the core in its.



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Now, we will come back later to this equation here, but let's first transform this right inside so



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here we can re express B and E from the left hand side, we get this double rotation of A, and here



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we get, you know, the time derivative from here.



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And then we get this expression here another time derivative and the gradient, which is from from this



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one here.



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Now, another thing is here, I have substituted these two constants which are equal to one over the



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velocity of light squared.



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Now, this looks kind of OK, I guess, but what we can also do is we can transform this double the



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double rotation here because it may look a bit complicated.



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And also we will later see why this is useful.



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So we will use here and identity that this rotation of the rotation of A is equal to this minus LaBella's



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operator of A plus the gradient of the divergence of a.



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So we write it down like this, and also here I have calculated these brackets and brought it to the



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other side, so we have now four terms.



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We have second order time derivative plus operator, which is the second order based derivative.



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We have the gradient of the divergence and we have here the gradient of the time, derivative of our



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scalar potential.



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And all of this is equal to these current densities.



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Now we introduce a new operator.



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This is not really changing anything, it just makes it look a bit more simple, a bit more beautiful.



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And this new operator is basically given by the second order time derivative and the second order spatial



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derivative.



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So this goes along the same language as the four potential from the previous slide where we write down



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the vector potential in the scale of potential in one expression and the four component vector.



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Here we write the second order time derivative and the second order based derivative in terms of one



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operator.



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So now this equation looks more simple and this is what we end up with.



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And now the cool thing is we can do a very similar thing to the left hand side to this maximal equation



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here.



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And first, I want to show you the solution and then you can see that this really works out.



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So we have here to time derivative of the divergence.



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So this is this term here.



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Then we have the right hand side.



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This does not change.



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And then we have here this is the plant's operator acting on fire, which is basically this one here.



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But we do not have to second order time derivative, so we have to include it.



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And yeah, this is this term here.



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And then we have this one coming from the plant operator.



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So these two expressions are the same.



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And now if you compare these two equations, they have exactly the same mathematical shape.



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So we have here this square operator.



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Then we have some complicated looking expression.



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And on the right hand side, we have just our charge density and our current density.



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So what we have done is we have transformed our Maxwell's equations in the most general case without



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any approximations or special cases.



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We have transformed them to form new equations.



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And so far it's not really that much easier.



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But you will soon see why this made sense and why this is helpful.



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And this we will discuss in the next lecture, which is about the Lawrence H.