So in this section, we want to calculate for the first time the electrostatic potential, and there
we go back to an example that we have already solved, for example, a sphere that is charged.
So we have some rotational symmetry in our charged distribution.
So in the previous section, we have derived this song equation where we have used this Maxwell equation
that is only true in electrostatic.
And it allowed us to introduce such an electrostatic potential.
And with the help of the other Maxwell equation for the electric field, we ended up with this equation
here, which is second order in the coordinate because this is yodeler plus operator.
Now, when one of the previous lectures we have already derived to Cullum's law for the electric field.
So what we had considered here is, for example, a charged sphere which is charged by the charge.
Q Or we could also labeled.
Q If we if we as we have done here and we what we basically did is we have considered this integral
formulation of the Maxwell equation, where this is the left side and this is the right side and the
right side just simplifies to the inclosed charge in this volume.
B now since we have this rotational symmetry, we know that our electric field is always pointing along
the radial direction, which is why we couldn't write it down as such a scalar function times this vector
of art.
And so this means that this E is always parallel to this integration, uh, vector here D, which is
perpendicular to the surface of our I've always feary that we consider.
And so it's very easy to calculate this integral.
This is just the surface of our sphere.
So it's just four pi r square.
And so straightforwardly we got the electric field which is proportional to one of our square.
And then here in this vector formulation, we just have to take this position back to each of our into
account.
So this was how we calculated the electric field.
Now we if we have the electric field, we can also calculate the electrostatic potential by using not
this equation because yeah, if we would solve this puzzle equation, we would have to start again from
zero.
So instead, if we already have Yovani and it's very easy, we just use this equation.
So we just integrate over E and from from some.
Yeah, from some point of reference are zero, which is not really important.
But we integrate up to the position vector R.
So hopefully you remember some integration laws, so if you have one over a square dependent's and you
integrate it, you get one of our with a minus sign and then there is another minus sign here, which
is why we end up for the electrostatic potential with one over four Pi Epsilon zero times to charge
over R.
And then of course we get this other term, which you could consider as an integration constant, for
example.
And if you remember this analogy to the classical world, we have to remind ourselves that only differences
matter in this quantity because it is something like a potential energy.
So what we can do is just we can take this integration constant and set it to zero.
Or we could also say we shift our whole electrostatic potential by some term so that this term here
cancels out.
And so this is what what what it makes sense to consider the electrostatic potential in this case.
But you could also write down that the electrostatic potential is dysfunction, plus an impossible constant.
So now we know the functional behavior of this electrostatic potential and it looks quite similar to
to the electric field.
But instead of a one of a R-squared dependence, we have here at one of our dependence.