So in this lecture, we will introduce to Victor potential, which is somewhat similar to the electrostatic
potential, and also we will derive an equation for the vector potential, which is a bit similar to
the song equation from Electrostatic.
So first, let us consider a quick recap of the electrostatic potential in electrostatic.
So there we have considered this Maxwell equation where the rotation of the electric field is zero.
And this led us to the possibility to introduce this electrostatic potential, because when we have
the rotation of a gradient, this is always zero.
So since we have this Maxwell equation here, this means that we will always have the possibility to
express this electric field in terms of a gradient of some function.
And also we could immediately understand that this electrostatic potential fire has certain properties.
For example, we could add any possible constant without changing the physics.
So this means that the electrostatic potential cannot really be a measurable quantity.
But only differences in the electrostatic potential can be measured and the gradient can be measured
because this is the electric field.
Now, in Agneta Static's, we have the Maxwell equation that the divergence of the magnetic field is
zero and not the rotation.
So this again allows us to introduce a new potential, which we call the vector potential.
And it has the probability that it's now are or the magnetic field has not a property, that it's the
rotation of this vector potential, because this time we have the divergence of a rotation of a vector.
And this identity is always zero for every vector.
So it's the same logic, just like in a different order.
So we know that the divergence of the magnetic field is zero.
This means we must have the possibility to express the magnetic field in terms of a rotation of some
other vector field.
Now, since this magnetic field is expressed as the rotation of a this Eigg has no special properties,
one of them is that the vector potentially can be shifted by any gradient of a function so we can introduce
a new vector potential and it's the old vector potential plus the gradient of some function.
And this function can even depend on the position vector.
And while this means we have two different vector potentials, they will both give rise to the same
magnetic field, which you can see in this quick calculation, because here we will get the idea, the
rotation of A and then the other term, which is related to this one here.
So we have to rotation of the gradient of chi.
And so the rotation of a gradient of a circular function is always zero.
So this means this B is equal to the old B.
So again, this means that this vector potential, just like the electrostatic potential, is not really
a measurable quantity.
It's more that the rotation of this quantity can be measured because this is the magnetic field.
Now we have again, like an electrostatic the possibility to tackle our problems in a different way,
so instead of directly calculating it or B, we can first calculate fine or A and then from these properties
can calculate B and by the gradient or by the rotation, respectively.
Now, what what other advantage does it have to introduce such a vector potential?
So we have introduced this vector potential and we can now use the other next equation.
So here we have the rotation of the magnetic field is basically determined by the currents or by the
current density of our.
So this means if we take this relation here and put it in, we have the rotation of the rotation of
A is equal to zero times J of our.
Now we can use an identity, so that's really something you should practice yourself and please be warned,
it's kind of difficult to show.
So you should go ahead and consider all of the three components individually.
Best would be you take the left hand side, write down what it is, you take the right hand side.
So first this term, then this term and write down what it is.
And then you will see that both of these sides are equal.
So you will see that the rotation of the rotation of any vector must not be the vector.
Potential can be really any vector that is are dependent is equal to the gradient of the divergence
of the same vector minus the lab class operator acting on the vector and know something kind of difficult
will happen.
Or we will have to.
Yeah.
Do something difficult or something unintuitive.
So I have already told you that this vector potentially can be shifted by any gradient of a function.
So this means we can take our old vector potential and add gradient of some function which will give
us a new vector potential.
And we are allowed to do this because this does not change to physics, because this does not change
the magnetic field.
B And in particular, this means we always have the chance to choose a gauge.
So it shows a gradient of chi in the way that the divergence of eight is zero.
So maybe you can think of it like this.
You first of all, calculate some vector potential A that fulfils this relation so that its rotation
is the magnetic field.
But then you figure out, OK, it's divergence is not zero.
So what do you do then is you choose a function chi whose gradient is exactly the minus sign of this
divergence of a and so your new AI has a divergence of zero.
So that's something that's, as I mentioned, is a bit unintuitive.
It's just a very clever trick.
So it's not something to be worried about because probably the first physicist who has ever done it
took many days or even weeks or month to figure this out.
It's very it's really a very clever trick that we are using now.
So we take our freedom to gauge our vector potential, to gauge it in such a way that the divergence
of A is zero.
So we are allowed to do this.
But why should we do this?
The answer is we do this because then this term here becomes zero.
So this means this rotation of the rotation of A is then just minus the plus operator acting on a so
we can write, mindful of how separate of A is equal to this one, or we write it like this.
And now if you remember the same equation from electrostatic, this equation looks kind of similar.
So here we had the lipless operator acting on five is equal to minus a constant times row of R.
So it's basically the same thing, just a different constant and it's a vector equation.
So here we have three components of this equation and here we have a scalar differential equation.
So what we can do now is we can solve the equation for every individual component and the solution will
look just like the solution for the equation.
So we can take all of our results from electrostatic and carry it over to Manado Static's.
So that's really the reason why we have considered this difficult looking trick here, where we have
chosen a gauge such that the divergence of eight is zero.
So here all the magic happens because this allows us to simplify our.
Yeah, our differential equation for a in such a way that it looks like the differential equation for
the electrostatic potential, and this allows us to carry over all of the old results.
So one more comment before I continue with the next lecture is that it's not necessary to do this trick.
We do not have to use a gauge so that the divergence of A is zero.
But if we don't do this, we will get this additional term here that also would appear here in this
equation.
So if we do not choose such a very clever gauge, then our differential equation for a looks more difficult
and then it would be more difficult to solve it because then we would have to start again from scratch
and we could not carry over our old results from electrostatic.
And this is why I've considered it here so that we now know the solutions from our previous calculations.