Now, what can we do with these equations that describe electrostatic?
It turns out that we can derive a new equation, which is called the Poisson Equation, named after
the famous French mathematician and physicist.
So we start from our Maxwell equation that deportation of the electric field is zero.
So this equation is only true if we consider electrostatic because otherwise we would have here the
time dependent magnetic fields.
But we assume that there is no time dependence in this field.
So the derivative is zero, which is why we have this equation here.
So as it turns out, due to this Maxwell equation, it allows us to make an additional statement about
the electric field e it means that we must always be able to write this electric field in terms of a
gradient of some scalar function.
So it means because the rotation of E is equal to zero, it must always be possible to write E as negative
gradient of this function and this function.
Five.
Are we named the electrostatic potential?
So yeah, if you want to find out what the electrostatic potential is, if you already know the electric
field, you just have to integrate to get rid of this novel right here, which is essentially just a
vector of partial derivatives with respect to the coordinates.
So what you get is that your your firepower is equal to this integral here.
And then you have here this other boundary, which could just be considered some constant.
So as you as you know, from integration, there will always be some integration constant.
And also it turns out that this constant doesn't really matter, physically speaking, because there's
potential here is something like a potential energy.
And as you know from classical mechanics, you can you can engage the zero value of your potential as
you want.
And it does not matter because only differences in the potential energy and also in the electrostatic
potential matter.
So, as I mentioned, it's quite similar to classical mechanics in classical mechanics.
We have to force FFR and we have our potential potential energy in these two quantities are related
by this gradient.
And so if you want to find out what the what the potential looks like, you just have to integrate over
the force.
So, for example, you could have such a landscape here and you want to find out or, you know, the
forces to to push this ball here from R zero to our then you can calculate the potential like this and
of course, vice versa.
If you know the potential, you can quite easily calculate the force via this equation.
And in our case, it's very similar.
So instead of the force we have now, the electric field and instead of the potential, we have this
electrostatic potential.
Now, so far, it does not really have as much to have to potential, but we also have another Maxwell
equation, which is not concerned with the rotation of E, but with the divergence of E.
So this is this Maxwell equation is generally true.
It's not only true in electrostatic, but also if we have time dependent magnetic fields, for example,
it just tells us that the sources and things of the electric fields are the charges.
So here had divergence of E is given by the space dependent charge density ratio of R.
And now if we take this equation and put here the electrostatic potential in this gradient instead of
E, we end up with such an equation.
So we get here another gradient, another Nabala operator.
So we end up with this large class operator, which is the sum of the second partial derivatives with
respect to the coordinates, as you hopefully remember.
So we know now that the lower plus operator acting on this electrostatic potential is equal to this
constant times.
The charge density ratio of our sources means now if we have a lower charged density, Rolph, are we
can directly calculate this this electrostatic potential and then from the electrostatic potential,
we can calculate the electric field.
So in some cases, this is not really helpful because it's much easier to directly calculate the electric
field, but in other cases it is much simpler to calculate first the electrostatic potential.
And the reason is that this electrostatic potential is a scalar function, whereas the electric field
is a vector.
So it has different components.
So you already know that.
Yeah, there's I would say less information in this electrostatic potential, which is why it's sometimes
easier to calculate this this scalar field first.