Let us get started with our news section on dependent problems, and once again, I want to show you
a slide that I've shown you already many times throughout this course.
It's all about the Maxwell's equations again.
So you can see them as you know them already, we have the divergence of ENP and the rotation of E and
B, and we have here terms that depend on the charge and the current and the time derivatives of the
magnetic fields and the electric fields.
And so far, we have discussed several special cases, for example, we have discussed the electromagnetic
waves so light, for example, where we have considered a vacuum, which means that the charges are
zero and that the currents are zero there after we have discussed electrostatic and magnetic static's,
which means that there are no time dependent magnetic and electric fields.
And in this section, for the very first time, we will discuss the Maxwell equations as they are written
down here without any approximations and without any special cases.
So let's get started.
In this section, we want to rewrite these Maxwell's equations, and our goal is to write them in a
way that we can somehow transfer our previous knowledge from electrostatic and magnetic statics to this
general case here where we have time dependent fields.
And, yeah, also charges and currents, of course.
So we start with this Maxwell equation, which is the divergence of the magnetic field is zero.
So that's something that, you know, the magnetic field does not have monopoles.
And this is, of course, exactly the same as in magnetic aesthetics.
So nothing really has changed.
And once again, since we have this divergence here of magnetic field, this means because of the zero,
this means that the magnetic fields must be expressible in terms of a rotation of some other field,
because, as you know, the divergence of a rotation of any vector is always zero, no matter what this
vector looks like.
Now, the other Maxwell equation for the electric field is that the rotation of the electric field is
this time derivative of the magnetic field and the minus sign here.
So this is different compared to electromagnetics, because maybe you remember this in electromagnetics,
we did not have this term here, so we had that the rotation of E is zero.
And this allowed us to write down the electric field in terms of a gradient of some scalar function,
which was the electrostatic potential.
Now, since we have this term here.
This is not possible anymore.
However, we can do a special trick or we can just rewrite this equation here.
And we do this by using this expression for the magnetic field and put it in here.
So we have our rotation of the electric field and then we have the other term brought to the other side.
So we have plus the time derivative of the rotation of the vector potentially.
Now we have these two derivatives here.
This one is with respect to time.
And this one, if you would write it down in terms of the components, this would be partial derivatives
with respect to X, Y and Z.
And so since the time and the spatial coordinates are independent variables, at least in classical
theory, we can switch around and swap these derivatives so we can write that the rotation of E plus
the rotation of the time derivative of A is zero.
And now we have something very similar to this equation.
We have now a new vector field here, which is E plus the time, the time derivative of A..
So this means this whole expression can be expressed as a gradient of a scalar function, just like
an electrostatic.
And now we can we can write it down like this.
So we have an expression for our electric field.
And this is here minus the time derivative of a minus the gradient of this electrostatic potential.
And if you write this vector potential and this scalar potential here as a four component vector, this
is sometimes also called the full potential.
But it's not so important here to we will not use this.
So we will just use this equation here because it allows us to determine e if we have determined A and
this scalar potential.
So this means we have now transformed these first two Maxwell equations.
But so far it's not really much easier here.
We had B and E and now we have A and fire and here also.
Now let's go ahead and transform the other equations.
So here there is this Maxwell equation that the divergence of E basically the sources and sinks of the
electric field are given by charges.
So this is here to charge density.
And we have here that the rotation of B is generated by currents, but also by the time derivative of
the electric field.
So we will get started with the left hand side to you where we have this Maxwell equation and we take
his expression for E from the previous slide and put it into this Maxwell equation.
And now we can again switch around these two derivatives.
So we have the time derivative of the divergence of a and here we have the lipless operator, because
the divergence of the gradient of a scalar function is equal to this lipless operator, which is basically
the sum of the second order of derivatives, which back to the core in its.
Now, we will come back later to this equation here, but let's first transform this right inside so
here we can re express B and E from the left hand side, we get this double rotation of A, and here
we get, you know, the time derivative from here.
And then we get this expression here another time derivative and the gradient, which is from from this
one here.
Now, another thing is here, I have substituted these two constants which are equal to one over the
velocity of light squared.
Now, this looks kind of OK, I guess, but what we can also do is we can transform this double the
double rotation here because it may look a bit complicated.
And also we will later see why this is useful.
So we will use here and identity that this rotation of the rotation of A is equal to this minus LaBella's
operator of A plus the gradient of the divergence of a.
So we write it down like this, and also here I have calculated these brackets and brought it to the
other side, so we have now four terms.
We have second order time derivative plus operator, which is the second order based derivative.
We have the gradient of the divergence and we have here the gradient of the time, derivative of our
scalar potential.
And all of this is equal to these current densities.
Now we introduce a new operator.
This is not really changing anything, it just makes it look a bit more simple, a bit more beautiful.
And this new operator is basically given by the second order time derivative and the second order spatial
derivative.
So this goes along the same language as the four potential from the previous slide where we write down
the vector potential in the scale of potential in one expression and the four component vector.
Here we write the second order time derivative and the second order based derivative in terms of one
operator.
So now this equation looks more simple and this is what we end up with.
And now the cool thing is we can do a very similar thing to the left hand side to this maximal equation
here.
And first, I want to show you the solution and then you can see that this really works out.
So we have here to time derivative of the divergence.
So this is this term here.
Then we have the right hand side.
This does not change.
And then we have here this is the plant's operator acting on fire, which is basically this one here.
But we do not have to second order time derivative, so we have to include it.
And yeah, this is this term here.
And then we have this one coming from the plant operator.
So these two expressions are the same.
And now if you compare these two equations, they have exactly the same mathematical shape.
So we have here this square operator.
Then we have some complicated looking expression.
And on the right hand side, we have just our charge density and our current density.
So what we have done is we have transformed our Maxwell's equations in the most general case without
any approximations or special cases.
We have transformed them to form new equations.
And so far it's not really that much easier.
But you will soon see why this made sense and why this is helpful.
And this we will discuss in the next lecture, which is about the Lawrence H.