So let us now come to our very first important example, which is the electric dipole.
So in the previous lecture, we have established the equation on how we can calculate the electrostatic
potential for multiple point charges.
And it turns out that we can just write it down as to some of the individual electrostatic potentials
of the individual charges.
So now that we want to consider an electric dipole, we first have to understand what this dipole means
and what it looks like.
So a dipole describes two electric charges with opposite signs that are separated by some distance away.
And so here I have defined this vector AI and this vector always points from the negative charge to
the positive charge.
And so this allows us also to introduce here to dipole moment, which is given by the individual in
charge, and then we multiply by this factor.
So this means dipole moment P is also a vector and it is a vector because, of course, it's very important
how this dipole is oriented.
So next, we want to consider this electrostatic potential for this dipole, so we, of course, get
to terms from the sun because we have two charges and both of these charges are equal in magnitude.
So we can hear pull out this cue.
And now I have to find that our zero positions here at this position of the positive charge.
So we get the first turn, which is plus one divided by hour.
And then we have here the negative charge.
So we get a minus sign and then we get instead of, ah, we get our plasma because the position here
is at minus eight.
And here we have the other minus sign.
So this is what our electrostatic potential looks like for this dipole.
So we did not really have to calculate it here.
We could just use our result from the single charge and just add up to two electrostatic potentials.
Now, what we can do next is we can do a Taylor expansion, so it's not not a big deal if you don't
perfectly remember what a Taylor expansion is and how it works, maybe you only know it from one dimension.
So in multiple dimensions, in our case, in three dimensions, a Taylor expansion looks like this.
So you have here it is derivatives, which in three dimensions is, of course, the Nahlah operator.
Then you have here as one of our faculty and then we have to function that we want to evolve.
So we have here that one divided by our Arpels eight.
So this term here can be expressed like this.
So what we can do now is that we can say that we want to look at the electric field and at the electrostatic
potential far away from this dipole.
So, for example, if you consider some very small molecule, which is has two two atoms, one of them
is positively charged, the other one is negatively charged.
So this gives us an electric dipole.
And then we want to find out what the electrostatic potential and the electric field look like far away
from this molecule.
So this means that this position R is much, much larger than this distance away.
And what this means is we don't have to take here this whole row, so we don't have to take into account
infinite numbers of terms, but we just have to take into account the first two terms.
So this is an equal zero and a new one.
So this means our electrostatic potential in this case is approximately one of her, which is this one
here.
Then we get another one of our which is this term here, four and equals zero.
So this gives us one.
This gives us one and then just one over.
And then we get another term, which gives us again here one.
This one gives us eight times Nabala and this one gives us one of our.
So that's this term here.
So you see that these two terms cancel out and we are left with just this pretty factor and then this
term here.
Now, this one is still quite difficult, we have used a number of operator acting on one of our and
you can as an exercise, calculate what this looks like.
Here you can also already see the solution.
It's the position vector of our divided by our to the power of three.
So please post a video and do the exercise and calculate this yourself.
So this is what you should have come up with.
This is not the operator acting on one of our to make it really, really simple.
I write it down in the coordinates so we have X squared plus Y square policy square and then add to
the power of minus one over two, because this position vector is the square root of X Square skyscrapers,
Z Square.
Now you can do this for every individual coordinate because you know there's not an operator.
It's a vector of partial derivatives with respect to X, Y and Z.
So let's start out with calculating the derivative with respect to X..
So this means we got this outer derivative.
We have to pull this one in front and decrease the exponent by one.
So we get minus one half times this one and the exponent is now minus three 1/2.
And then we have to also consider the inner derivative with which four X is two X and the other terms
give zero.
So we have here to X.
And now of course, if you do it for the other coordinates, you will get the same thing.
But for the inner derivative, you will get to Y and 2C.
So this means we can pull this one out front.
This one is just one of our to the power of three.
And here this one is just a position vector.
So we get minus position nektar divided by our to the power of three.
So this means our electrostatic potential far away from the dipole is given by this prefecture times
to charge, and then here we have eight times our scale, our product, and then divide it by half of
the power of three.
And of course, also we can use here just dipole moment P and write it down like this.
So this is all a result for the electrostatic potential.
Now, what does the electric field look like?
So, of course, we know that we can calculate the electric field from the electrostatic potential and
it's very easy, we just have to calculate the gradient.
So this is what it looks like in general.
And now we have to, of course, calculate this term here, which is not so simple.
So once again, it's a nice exercise, quite similar to the previous exercise.
So if you want to solve it, please post a video and do it yourself.
And here is the solution, so we calculate this gradient of P times are divided by R to the power of
three, and most often it's very simple or most simple to just write it down in terms of coordinates.
So we write it down as times X plus D, Y and Z terms.
And what we get is we first do it for the the first component, which is the partial derivative with
respect to X..
So what we get is that we differentiate here with respect to X, so we get P X divided by R to the power
of three.
But then we also have to remember that this R here is also X dependent.
So we have to use the product rule and get another term.
So that's our first term P X provided by our to the power of three and then the other term would be
P times X.
Divided by the derivative of one of our to the power of three.
And yeah, this we can also do for the other coordinates, which gives us this in total, and then of
course, we have to still calculate the derivative of one divided by our to the power of three.
And this is just as crazy gradient acting on this one here, and this is very, very similar to this
exercise on the previous slide.
Now we just get here as a as a prefect, a minus three divided by two.
And here we get it as an exponent minus five, divided by two because we have to decrease this one by
one.
Then we also have to account for the inner derivative, which is again, to X, to Y, to Z.
So this means this gradient here is equal to minus three.
Divided by the opposition, Victor must be better positioned to divide it by ah to the power of five.
So now we put all of these things together and we have our electric field, which is given by this prefecture,
and then we get to terms here due to the product rule.
The first term is the dipole moment divided by our to the power of three.
And then we have you this other term where we have to project the dipole moment on the position vector
and then we get this dependence of R divided by R to the power of five so we can write it down like
this.
And on the next slide I will show you what the electrostatic field and also the electric field look
like.
So we have now figured out what the electrostatic field and electric field look like far away from the
dipole.
And so what we have actually done here is we have used the approximation that is only valid for our
as much larger to aim for.
This means we have to look at our dipole and have to sort of say zoom out.
So this is when we zoom in and when we zoom out these to come closer to each other.
And if we zoom out very, very far, we cannot really resolve the distance anymore between them.
But please remember, there will always be this distance A.
So another important thing have to realize is that if we want to find out how this electric field behaves
far from the dipole, we have to consider this fraction here.
And you can see that in these two terms we have here, our square.
And he also our teams are which is also a square.
And then here we have our to the par five.
So this means essentially the electric field decreases with the power of one divided by our to the power
of three.
So here you can see again, yeah, what the electric field looks like for such a dipole, and if you
remember correctly, we have already figured out this electric field in one of the very first lecture's.
So this is just the addition of the two individual electric fields and on the previous slides, we have
simplified this or generalised this dipole to the so-called mathematical dipole, where we have considered
the approximation that the position vector is much larger than the distance between these two charges.
And another thing that I want to mention here is that we can find cuts along these electric field lines
where the electrostatic potential is constant.
And, for example, if he cuts right between these two charges, we will find that in on this red line
here, the electrostatic potential is zero in all cases because the individual electrostatic potentials
of these two charges cancel out because they have opposite sides.
And so this line where the electrostatic potential to zero will be important later.
So maybe you remember this later.