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Welcome back after this long and difficult lecture about the -- potentials in this lecture.



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We will discuss the halcyon dipole.



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And this is a dipole that was named after the German physicist Gustav hats, so you already know to



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dipoles the electric and the magnetic dipole.



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We have discussed these two examples in the previous sections about electrostatic and magneto static's.



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And it turned out that the equations for the magnetic field of the magnetic dipole and the electric



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field of the electric dipole look very similar.



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And in fact, if you are far, far away from the dipole, they will have the same functional dependence.



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The only difference is that here you have the electric dipole moment Q10 say, and here you have the



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magnetic dipole moment, which in case of a plane, our current loop is the current times, the area



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of the surface times, the normal vector here.



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So far away, these fields look exactly the same.



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What we do now is we consider time dependent dipoles.



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So here you see that we will have a kind of different electric and magnetic fields for this example,



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and that they will even be time dependent.



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So you can see here the electric field in the color.



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And the darker this blue color is, the larger is the electric field.



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So you see here, it's large.



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Here it's very low.



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And this will change over time.



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And these wave fronts here, they will travel along the radial direction over time.



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And we want to calculate these fields, for this example of the Hartshorn dipole.



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First of all, we have to understand what exactly is a hertz in dipole.



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So roughly speaking, it is a tiny, infinitesimally small segment of an antenna.



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So you have a wire here in black where a current oscillates, so you have a time dependent current,



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and this is why our here is infinitesimally small.



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In fact, it is condensed to a single point.



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So the current is zero everywhere except for the center position here.



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This corresponds to when you have a wire and you zoom out very, very much.



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And then in this case, you can write down the current density with this Delta function that we also



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had introduced in the previous lecture and this Delta function condensers exactly this property that



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the current is zero everywhere, except for one particular position, which in our case is zero because



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we want to place our wire in the center.



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So now you may say that this is very unrealistic, that we have just an infinitesimally small wire,



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but it turns out that this is the most simple example that we can solve with our -- potentials.



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And still, it is very useful because when we want to build a real antenna, which has a finite length,



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we can just take many, many infinitesimally small antennas which are hurting dipoles and placed them



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along each other and build our real antenna.



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And then we just have to add up the solutions for these electric and magnetic fields.



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So the result that we will get, they are really useful and they are not just some mathematically arbitrary



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concept for practice here.



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OK, so we have now our current density oscillating in the wire so it doesn't have to be a harmonic



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oscillation can be any time dependent in general.



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And this current density is giving given by the change in the dipole moment because of this equation



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here and then this Delta function, as we have discussed, to get also the space dependent.



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So you can see we have separated our space dependence and our time dependence.



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So it is time dependent.



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Now and the next step we used to feature of this Delta function, so we integrate over P dot times Delta,



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which means we integrate over the current density.



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So if we integrate over the current density, of course we get the real current.



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And if we get if we integrate over P dot times Delta, we get P at the position R equals zero.



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So this means we get P dot is equal to the current.



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OK, so now let's use our current density, which is this relation that we have established on the previous



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slide, and let's calculate the vector potential and the electrostatic potential from this current density.



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This is all we have to do.



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So now this is this is all we take for from our model.



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And now we have to do the math.



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So we write down our equation for this -- potential for the vector potential.



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So this is basically the equation from electrostatic.



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But as you remember, we have here the current density not at the position are Komati, but we have



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these new arguments here and especially we have this -- time because the information about the



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current density at the position of this Vector R here has to travel a certain distance to our vector



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R that we want to consider here for our vector potential.



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And so this will take some time because at maximum the information can travel with the velocity of light.



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So now let's take our current density here and put it in so we have now a P dot that only depends on



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time and we have the Delta function, which is our space dependent.



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But in fact, it's a very simple space dependent because as I explained already, this Delta function



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is zero everywhere except at the position of our equals zero.



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So this means our integral will.



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We will get this whole function here.



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So P dots divided by this distance here, but we have to set one of these hours to zero.



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So this means what we get is this equation here we have P dots and we have we have got rid of the integral



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and we just get P dots at this time argument.



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So we still have this retardation divided by R, which is the absolute value of our position vector.



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So that was kind of easy.



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We just had to have some knowledge about the Delta distribution.



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So please, if you do not really feel so well with this Delta distribution, it's not not such a big



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problem and this is really the only time we will use it in this course.



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So please just accept that the feature of this Delta distribution is that you get rid of the integral



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and then you take this into ground here and use that this are here to zero.



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OK, so this is our vector potential now, and we also need, of course, the electrostatic potential



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because after all, we want to calculate the electric field and the magnetic field and for their calculation,



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we need both.



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We need to we need to fight and a.



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So we could now calculate fire also with the -- potential, but we can make it also a bit easier



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and used as Lowrance gauge condition.



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So this is something we have used previously before we have even derived the -- potentials.



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We have said we want to work in this Lawrence Cage.



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So this relation here must always be true.



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So we know now we have a so we can calculate this one here and then we can calculate this one and get



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our.



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So our time derivative of fire is equal to see square.



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Times minus divergence of eight.



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So this is what I have written down here, si, square minus and then the divergence of a Soheir divergence



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basically of P dots and the time argument divided by R.



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So now we need to use the product rule and also the chain rule.



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So we have here a product of two functions.



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We have P dots and we have R or better.



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Just say one of our and you have to apply this novel operator also to pitot because its argument at



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the time is also our dependent.



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So you really have to consider it for the derivation.



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So what we get is two terms due to the product rule we have to get.



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So we have two terms.



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The first one is one of our times the divergence of this p dot here and the other one is just P dot,



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uh, times the times, the gradient of one of our.



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So this one here is actually the gradient.



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So the dot has to be here, OK.



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So now we can calculate this and we get for the divergence of P dots, we get P double dots which is



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the outer derivative, and then we need to consider know the derivative, which is the gradient of this



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argument here to gradient of T minus are divided by seat.



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So this one gives zero here, but the gradient of miners are divided by C gives a finite value.



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And then here on the right hand side, we just have to calculate the gradient of one of our and we already



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did this in the previous lecture.



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So this is just the position vector are divided by R to the power of three.



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So if you have skipped the previous lecture, this would be a good point to practice, so please post



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a video and calculate the gradient of one over and you will see when you do it in Cartesian coordinates,



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you will get are divided by our to the power of three.



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So now we just need to calculate this one here.



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So we basically have to calculate the gradient of art, which is this are back to divided by our.



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So this is here what we get we get the prefecture than we have here.



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Second order time derivative of the, um, of the electric dipole moment with the scale of product with



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our and then we get to the first all the time the relative scale of product with our and then we have



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here some are dependents in the denominator.



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And also please remember that these PS here are time dependent support and that time dependent.



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And the time argument is not simply T but it is T minus are divided by C due to this retardation.



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OK, so now we can just simplify these terms.



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This is really easy.



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So what we get so we can pull this are divided by out of the brackets and then we just get here one



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of our and here we get one of our square.



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And then this is our final result that we get.



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So again, please remember this time argument.



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OK, so now we know the time derivative of our electrostatic potential, but what we really want to



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have is the electrostatic potential.



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So what we have to do is we have to integrate over time to get rid of this derivative here and now.



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That's really easy because the only time dependent functions are P.



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So what we get is we get P dot instead of double dots.



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If we integrate and we get P instead of P Dot when we integrate.



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So this is here our final solution for that electrostatic potential.



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And you can see here, I have said in this time argument to really make sure that we think about the



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retardation.



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OK, that's really good.



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We have now a to potential and our electrostatic potential, and in the next step we can calculate the



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electric field and the magnetic fields based on these equations.