In the previous two exercises, you already get a first impression about boundary conditions.
So, for example, we had to determine our integration constants in such a way that the electrostatic
potential would be continuous at all positions.
And so, in effect, an electrostatic the boundary conditions often play a very important role.
And in this lecture, I want to explain to you a bit more why this is the case.
And also in this lecture, we will learn about the concept of Mireia charges, which sometimes make
our life a bit more easy.
So we start again from our personal equation, which is the second order differential equations.
So remember, this lipless operator is the sum of the second or the partial derivative with respect
to the coordinates.
And on the right hand side, we had here the charge density, which is in general different from zero.
And also we have already determined a special solution for this differential equation, which looks
like this for the charged distribution.
So if you remember, we started from a point charge or from a sphere, and then we considered multiple
spheres.
We have added them up and then we made these spheres smaller and smaller and ended up with this integral
here.
Now, if you really are an expert of differential equations and if you know the mathematics quite well,
you will know that this is here and in homogeneous differential equation, because we have here this
idea that these terms, including the derivative and on the right hand side, we have a term that does
not include derivatives and it is different from zero.
So such an inner homogeneous differential equation always has a general solution.
And this general solution is to some of the special solution is one we have already here, plus a solution
to the homogeneous differential equation.
And the homogeneous differential equation just means that you take all the terms that include the differentials.
But you said the other terms to zero.
So this means the lower class operator of this homogeneous yeah.
Solution, which is our electrostatic potential, must be zero.
And also it's very important that this solution must fulfill the boundary conditions.
So previously we always had that electrostatic potential for Arrigo's to infinity must be a constant
number.
And in our case I have often chosen it to be zero, to make it look a lot better, to make it more easy.
And as you can see here, when the class in acting on the homogeneous solution must be zero, then a
possible solution would be that this fire of H must also be zero.
So this is really a valid solution.
So this is why our previous examples had the correct solutions.
But in general, they exist even more functions that solve this homogeneous differential equations.
And you always have to consider the boundaries to make it fit.
The problem?
So typical boundaries are, for example, charged wires or metal surfaces, and for such examples,
you will always find that the surfaces of metals or the charged virus will have constant electrostatic
potentials.
So this is because the electrons that live on them will reorient or will behave in such a way that they
are really nicely ordered and so that the electrostatic potential is equal.
Otherwise, you would have your voltage drops or voltages between different points of a wire.
And that's, of course, you cannot really happen.
So, for example, here we have, um, yeah, different surfaces of metal, for example, here we have
cut a hole in the metal and then we have here some wire going towards us and another wire also coming
towards us.
And so on the surface of this wire and on this wire and here on the surface of the metal, there will
always be a constant electrostatic potential.
But the potentials can be different between the different wires and the metal.
So in general, we would have here find one, which is a constant here.
We would have to a. we would have five, three.
And then in between these different regions, we would have electric fields, would have our electric
field lines.
So in this case here, I want to remind you of some of the electric dipole that we have discussed previously,
so the electric field of an electric dipole could be considered the sum of the electric fields of two
charges.
For example, here we have a negative charge and we have this radial distribution of the electric field
lines.
So you can see them here in the background.
And to you, we would have the same electric field lines, but with an opposite orientation.
So the opposite sine.
And now if you add these two lines up here, these two profiles, then we would get such a profile here,
which corresponds to this electric dipole.
Now, a very, very different example, which has a very similar solution, would be that we take this
individual charge here and put it beneath a metal.
And so, as you know, inside of the metal, we have electrons that can move and that can react to their
environment.
And so what happens here is we have here there's this line where we cut through this electric field
profile where the electrostatic potential zero is because here we have an electrostatic potential that
is radial symmetric and here as well.
And right between these two charges, both of these electrostatic potentials cancel out into electrostatic
potential is zero on this whole line.
And so if we have here just charged beneath a metal, we also know that there must be a boundary condition
that enforces that the electrostatic potential is constant.
So we could just take as a constant zero.
Doesn't really matter here.
Just must be constant.
And so what this means is that our electrostatic potential on this line for the single charge beneath
the metal looks exactly like the profile here between these two charges of a dipole.
And this automatically means that the electric field of this charge here must look also like the electric
field of a dipole.
So that's pretty remarkable.
You can see here, if we would have just a single charge, we would have this radial symmetry.
And if we just bring a metal in the vicinity of this charge, the boundary conditions strongly affect
the electric field and they change the orientation of the field lines.
But we see here that these field lines look exactly like the field lines of this dipole.
So what happens here is that the electrons react in such a way that they form something like a mirror
charge.
So you could say the electrons which are negatively charged, they are not homogeneous in this metal
anymore, but they leave something like a gap.
And this gap has then the positive charge, of course.
So just in our imagination, we could just continue this electric field lines to this imaginary charge
here.
And so we have something like people called this a charge.
So that's not really not really a charge in the classical sense that you really, really have there
as a sphere.
But it's more like a gap or something mythic missing in the electron charge of the reality of the matter.
And so here there are really two very important things.
The first one is if you compare this example here to this example, you will see that the boundary conditions
of this metal strongly affect the electric field lines and also the electrostatic potential.
And then also you can see that metals behave quite interestingly because their electrons really are
affected by the charges in the vicinity.
And so they form something like a dipole.
So this means they want to shield the charge as that is in the vicinity of the metal because they exhibit
something like a mirror charge, which is the same magnitude, but with an opposite sign.