OK, that was really difficult, but we have done it, we have calculated the electric field and the
magnetic field for our Hatzius Dipole, so let's go ahead and discuss the results.
So what we had considered is a time dependent current in an infinitesimally small wire.
So basically we have here a tiny segment of an antenna and our results where this be field with the
two terms and the electric field with these six terms here.
And in both cases, we still have our -- time argument, which comes, of course, from the --
potentials.
So first of all, we can do something very simple.
We can check if this result is equal to our results from electrostatic, where we have considered the,
you know, the static dipole.
And this we can do by considering considering a time independent dipole moment.
So this means that pitot a zero and also P is zero.
So this means we get a magnetic field of zero, which is, of course, clear because we have yeah,
we have static electric charges and we just got an electric field with these two terms.
And when we compare it to our previous results from Electrostatic, we see it's exactly the same result.
That's really satisfying.
I think because we had this difficult calculation.
We have first had this difficult lecture on proving that the -- potentials are right.
Then we had the difficult lecture on calculating the electric field and it turns out that's in the limit
of time, independent type of moments.
We get the same result as previously, which means we did a good job.
And that is correct.
But now, of course, our aim was not to calculate time independent problems, because then we could
have done it in a much more simple way.
Instead, we want to calculate time dependent on, for example, harmonically oscillating dipoles.
And so what we can assume here is some some cosine dependence on the time or as you would typically
do it, you write it down in terms of an exponential function with an imaginary argument.
So you have your electric dipole moment at the time, zero, and then it oscillates and thus you do
by multiplying such an E to the eye on McAtee, where Armacost the frequency of the oscillation.
Now you can go ahead and calculate p dot and people dot dot is just minus eight times, oh my God,
times this whole function and then for people that you do it one more time, so you get plus I square
Iomega which plus square Omega's Square, which is of course minus Omega Square Times Pete.
And now we can go ahead and take these and put them into B and E.
So when we start with B what we get is the following we gets here is P in both cases and then of course
here the different prefectures and you can see we have all we got of a C, and this is typically something
that you introduce as a as the K vector, which is which was for example, for an electromagnetic wave,
the so-called wave factor.
Now what we must also consider is that we have in both cases here our P, which was P zero times E to
the Omega T, but it's also really important to remember that this T argument is, in fact, not simply
our T, but it's also has to take into account the retardation.
So we have to write instead of t t minus are divided by C.
So we get here I omega T minus, I only got T plus I Omega are divided by sea and this is our K.R. minus
Omega T and this is something that you already know from our electromagnetic waves.
So in some sense we get to hear again an electromagnetic wave, but it has some difficult space dependence
and some difficult.
Now, one other thing that you can do is you can introduce the wavelength of this of this wave.
And this is just, you know, two pi, one of akei or vice versa as it's written down here.
And what I have done here is I have ordered these terms with respect to their power.
In a SO later, we can discuss the near field and the far field.
So here we get as our sold for B that we have some unit vector along R and then the cross products of
these terms and the vectorial orientation is along P0.
All right, so we have the results for B, let's do the same thing for E!
So here what we can can do or what we must do is we must take P dot and P double dots and put it in.
So in all cases we get no P and we get these three factors here.
So this is what we get here.
You get the term four for these two, you get a term for these two.
And here this one is for these two and you can see the prefectures minus I omegle divided by C and minus
Omega squared, divided by C squared, because here we have C Square, here we have C, here we have
nothing.
All right, so now we once again take this expression for.
So we get P0 in all cases.
So yeah, this one also has to be p0 in fact, and then we pull out this exponential function and also
have to consider the -- argument.
So this is what we what we get, we get.
Yeah, OK, we get the same thing as here, so nothing has really changed, but the only thing that
I have changed is this term here, because this one looks exactly like this double, double, double
vector product where we used to be a C minus C, a B rule.
So we can write this one down as this one.
So to make it just a bit more look more simple and a bit more beautiful, but it's actually not really
that important, to be honest.
OK, now we are almost done, the only thing that I want to do is I want to order these terms with respect
to their powers and are just in the same way as we did for the magnetic field.
So this means we have to replace all of these actors are with the unit, Victor E. R. and have to pull
the legs are out of this vector.
So what we get is we get basically the same thing, but we have pulled out the the RCA everywhere and
have canceled them with our to the power of five.
So now we got here on the R to the power of three because here we have these two hours, here we have
these two hours and then the similar thing we can do for the other terms here.
So we finally have arrived at the results for a harmonically oscillating electric dipole.
And we have also ordered our terms with respect to the powers of R.
So what this means is in the very close vicinity to our antenna, we have terms that are dominating
which are these ones here.
So it's just a single term.
This is because for for, you know, in the vicinity we have an R that is very small.
So this means that one of our to the power of free will be very large compared to these other two terms
and also to these two terms.
And then far away we have a small R, so this means here to terms with the power, um, our to the minus
one, they dominate with respect to all of the other terms.
So this means here we will mainly get these two terms.
So what does it mean.
So in the near field, alien blue, what we get is that the electric field dominates because the electric
field is dominated by this term.
And this is much larger than this term of the magnetic field, because here we have our to the power
of two instead of three.
And it also means that the electric field in the near field is rather similar to our dipole from electrostatic
so that you can see here.
So along the Z axis, we get a field profile that is pointing along the radio direction.
So in this case, Homsi and especially close to the x y plane, we get a field that is pointing along
this Edyta direction, which is here along this one.
So in the near field, it looks rather similar to what we already know, and this is why you could also
call it a very slowly oscillating dipole in the near field.
Now, in the far field, it looks very different because here we have these two terms to dominate.
And you can also or you can immediately see that in the far field, the electric field is perpendicular
to the electric field.
And also you can see that the electric field, as large as near the x y plane is because of this triple
vector product here.
And in the X Y plane, once again, like here, the electric field is pointing along the etha to the
direction.
The magnetic field is also largest near the XY plane, but it's pointing along the fire direction,
as is indicated here.
So this means in the far field, the majority of the energy is transmitted near or close to the X Y
plane, and it turns out that the data point in Vector s, which is essentially a prefect at times to
vector product of E and B, is pointing along the radial direction because the product of Edyta and
if you feel if I is E r, which is to radial vector.
So this means all of the energy is transferred along the radial direction close to the x y plane.