Welcome back after this long and difficult lecture about the -- potentials in this lecture.
We will discuss the halcyon dipole.
And this is a dipole that was named after the German physicist Gustav hats, so you already know to
dipoles the electric and the magnetic dipole.
We have discussed these two examples in the previous sections about electrostatic and magneto static's.
And it turned out that the equations for the magnetic field of the magnetic dipole and the electric
field of the electric dipole look very similar.
And in fact, if you are far, far away from the dipole, they will have the same functional dependence.
The only difference is that here you have the electric dipole moment Q10 say, and here you have the
magnetic dipole moment, which in case of a plane, our current loop is the current times, the area
of the surface times, the normal vector here.
So far away, these fields look exactly the same.
What we do now is we consider time dependent dipoles.
So here you see that we will have a kind of different electric and magnetic fields for this example,
and that they will even be time dependent.
So you can see here the electric field in the color.
And the darker this blue color is, the larger is the electric field.
So you see here, it's large.
Here it's very low.
And this will change over time.
And these wave fronts here, they will travel along the radial direction over time.
And we want to calculate these fields, for this example of the Hartshorn dipole.
First of all, we have to understand what exactly is a hertz in dipole.
So roughly speaking, it is a tiny, infinitesimally small segment of an antenna.
So you have a wire here in black where a current oscillates, so you have a time dependent current,
and this is why our here is infinitesimally small.
In fact, it is condensed to a single point.
So the current is zero everywhere except for the center position here.
This corresponds to when you have a wire and you zoom out very, very much.
And then in this case, you can write down the current density with this Delta function that we also
had introduced in the previous lecture and this Delta function condensers exactly this property that
the current is zero everywhere, except for one particular position, which in our case is zero because
we want to place our wire in the center.
So now you may say that this is very unrealistic, that we have just an infinitesimally small wire,
but it turns out that this is the most simple example that we can solve with our -- potentials.
And still, it is very useful because when we want to build a real antenna, which has a finite length,
we can just take many, many infinitesimally small antennas which are hurting dipoles and placed them
along each other and build our real antenna.
And then we just have to add up the solutions for these electric and magnetic fields.
So the result that we will get, they are really useful and they are not just some mathematically arbitrary
concept for practice here.
OK, so we have now our current density oscillating in the wire so it doesn't have to be a harmonic
oscillation can be any time dependent in general.
And this current density is giving given by the change in the dipole moment because of this equation
here and then this Delta function, as we have discussed, to get also the space dependent.
So you can see we have separated our space dependence and our time dependence.
So it is time dependent.
Now and the next step we used to feature of this Delta function, so we integrate over P dot times Delta,
which means we integrate over the current density.
So if we integrate over the current density, of course we get the real current.
And if we get if we integrate over P dot times Delta, we get P at the position R equals zero.
So this means we get P dot is equal to the current.
OK, so now let's use our current density, which is this relation that we have established on the previous
slide, and let's calculate the vector potential and the electrostatic potential from this current density.
This is all we have to do.
So now this is this is all we take for from our model.
And now we have to do the math.
So we write down our equation for this -- potential for the vector potential.
So this is basically the equation from electrostatic.
But as you remember, we have here the current density not at the position are Komati, but we have
these new arguments here and especially we have this -- time because the information about the
current density at the position of this Vector R here has to travel a certain distance to our vector
R that we want to consider here for our vector potential.
And so this will take some time because at maximum the information can travel with the velocity of light.
So now let's take our current density here and put it in so we have now a P dot that only depends on
time and we have the Delta function, which is our space dependent.
But in fact, it's a very simple space dependent because as I explained already, this Delta function
is zero everywhere except at the position of our equals zero.
So this means our integral will.
We will get this whole function here.
So P dots divided by this distance here, but we have to set one of these hours to zero.
So this means what we get is this equation here we have P dots and we have we have got rid of the integral
and we just get P dots at this time argument.
So we still have this retardation divided by R, which is the absolute value of our position vector.
So that was kind of easy.
We just had to have some knowledge about the Delta distribution.
So please, if you do not really feel so well with this Delta distribution, it's not not such a big
problem and this is really the only time we will use it in this course.
So please just accept that the feature of this Delta distribution is that you get rid of the integral
and then you take this into ground here and use that this are here to zero.
OK, so this is our vector potential now, and we also need, of course, the electrostatic potential
because after all, we want to calculate the electric field and the magnetic field and for their calculation,
we need both.
We need to we need to fight and a.
So we could now calculate fire also with the -- potential, but we can make it also a bit easier
and used as Lowrance gauge condition.
So this is something we have used previously before we have even derived the -- potentials.
We have said we want to work in this Lawrence Cage.
So this relation here must always be true.
So we know now we have a so we can calculate this one here and then we can calculate this one and get
our.
So our time derivative of fire is equal to see square.
Times minus divergence of eight.
So this is what I have written down here, si, square minus and then the divergence of a Soheir divergence
basically of P dots and the time argument divided by R.
So now we need to use the product rule and also the chain rule.
So we have here a product of two functions.
We have P dots and we have R or better.
Just say one of our and you have to apply this novel operator also to pitot because its argument at
the time is also our dependent.
So you really have to consider it for the derivation.
So what we get is two terms due to the product rule we have to get.
So we have two terms.
The first one is one of our times the divergence of this p dot here and the other one is just P dot,
uh, times the times, the gradient of one of our.
So this one here is actually the gradient.
So the dot has to be here, OK.
So now we can calculate this and we get for the divergence of P dots, we get P double dots which is
the outer derivative, and then we need to consider know the derivative, which is the gradient of this
argument here to gradient of T minus are divided by seat.
So this one gives zero here, but the gradient of miners are divided by C gives a finite value.
And then here on the right hand side, we just have to calculate the gradient of one of our and we already
did this in the previous lecture.
So this is just the position vector are divided by R to the power of three.
So if you have skipped the previous lecture, this would be a good point to practice, so please post
a video and calculate the gradient of one over and you will see when you do it in Cartesian coordinates,
you will get are divided by our to the power of three.
So now we just need to calculate this one here.
So we basically have to calculate the gradient of art, which is this are back to divided by our.
So this is here what we get we get the prefecture than we have here.
Second order time derivative of the, um, of the electric dipole moment with the scale of product with
our and then we get to the first all the time the relative scale of product with our and then we have
here some are dependents in the denominator.
And also please remember that these PS here are time dependent support and that time dependent.
And the time argument is not simply T but it is T minus are divided by C due to this retardation.
OK, so now we can just simplify these terms.
This is really easy.
So what we get so we can pull this are divided by out of the brackets and then we just get here one
of our and here we get one of our square.
And then this is our final result that we get.
So again, please remember this time argument.
OK, so now we know the time derivative of our electrostatic potential, but what we really want to
have is the electrostatic potential.
So what we have to do is we have to integrate over time to get rid of this derivative here and now.
That's really easy because the only time dependent functions are P.
So what we get is we get P dot instead of double dots.
If we integrate and we get P instead of P Dot when we integrate.
So this is here our final solution for that electrostatic potential.
And you can see here, I have said in this time argument to really make sure that we think about the
retardation.
OK, that's really good.
We have now a to potential and our electrostatic potential, and in the next step we can calculate the
electric field and the magnetic fields based on these equations.