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In the previous two exercises, you already get a first impression about boundary conditions.



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So, for example, we had to determine our integration constants in such a way that the electrostatic



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potential would be continuous at all positions.



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And so, in effect, an electrostatic the boundary conditions often play a very important role.



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And in this lecture, I want to explain to you a bit more why this is the case.



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And also in this lecture, we will learn about the concept of Mireia charges, which sometimes make



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our life a bit more easy.



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So we start again from our personal equation, which is the second order differential equations.



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So remember, this lipless operator is the sum of the second or the partial derivative with respect



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to the coordinates.



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And on the right hand side, we had here the charge density, which is in general different from zero.



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And also we have already determined a special solution for this differential equation, which looks



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like this for the charged distribution.



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So if you remember, we started from a point charge or from a sphere, and then we considered multiple



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spheres.



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We have added them up and then we made these spheres smaller and smaller and ended up with this integral



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here.



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Now, if you really are an expert of differential equations and if you know the mathematics quite well,



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you will know that this is here and in homogeneous differential equation, because we have here this



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idea that these terms, including the derivative and on the right hand side, we have a term that does



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not include derivatives and it is different from zero.



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So such an inner homogeneous differential equation always has a general solution.



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And this general solution is to some of the special solution is one we have already here, plus a solution



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to the homogeneous differential equation.



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And the homogeneous differential equation just means that you take all the terms that include the differentials.



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But you said the other terms to zero.



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So this means the lower class operator of this homogeneous yeah.



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Solution, which is our electrostatic potential, must be zero.



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And also it's very important that this solution must fulfill the boundary conditions.



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So previously we always had that electrostatic potential for Arrigo's to infinity must be a constant



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number.



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And in our case I have often chosen it to be zero, to make it look a lot better, to make it more easy.



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And as you can see here, when the class in acting on the homogeneous solution must be zero, then a



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possible solution would be that this fire of H must also be zero.



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So this is really a valid solution.



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So this is why our previous examples had the correct solutions.



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But in general, they exist even more functions that solve this homogeneous differential equations.



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And you always have to consider the boundaries to make it fit.



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The problem?



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So typical boundaries are, for example, charged wires or metal surfaces, and for such examples,



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you will always find that the surfaces of metals or the charged virus will have constant electrostatic



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potentials.



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So this is because the electrons that live on them will reorient or will behave in such a way that they



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are really nicely ordered and so that the electrostatic potential is equal.



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Otherwise, you would have your voltage drops or voltages between different points of a wire.



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And that's, of course, you cannot really happen.



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So, for example, here we have, um, yeah, different surfaces of metal, for example, here we have



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cut a hole in the metal and then we have here some wire going towards us and another wire also coming



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towards us.



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And so on the surface of this wire and on this wire and here on the surface of the metal, there will



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always be a constant electrostatic potential.



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But the potentials can be different between the different wires and the metal.



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So in general, we would have here find one, which is a constant here.



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We would have to a. we would have five, three.



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And then in between these different regions, we would have electric fields, would have our electric



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field lines.



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So in this case here, I want to remind you of some of the electric dipole that we have discussed previously,



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so the electric field of an electric dipole could be considered the sum of the electric fields of two



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charges.



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For example, here we have a negative charge and we have this radial distribution of the electric field



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lines.



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So you can see them here in the background.



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And to you, we would have the same electric field lines, but with an opposite orientation.



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So the opposite sine.



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And now if you add these two lines up here, these two profiles, then we would get such a profile here,



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which corresponds to this electric dipole.



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Now, a very, very different example, which has a very similar solution, would be that we take this



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individual charge here and put it beneath a metal.



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And so, as you know, inside of the metal, we have electrons that can move and that can react to their



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environment.



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And so what happens here is we have here there's this line where we cut through this electric field



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profile where the electrostatic potential zero is because here we have an electrostatic potential that



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is radial symmetric and here as well.



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And right between these two charges, both of these electrostatic potentials cancel out into electrostatic



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potential is zero on this whole line.



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And so if we have here just charged beneath a metal, we also know that there must be a boundary condition



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that enforces that the electrostatic potential is constant.



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So we could just take as a constant zero.



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Doesn't really matter here.



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Just must be constant.



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And so what this means is that our electrostatic potential on this line for the single charge beneath



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the metal looks exactly like the profile here between these two charges of a dipole.



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And this automatically means that the electric field of this charge here must look also like the electric



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field of a dipole.



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So that's pretty remarkable.



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You can see here, if we would have just a single charge, we would have this radial symmetry.



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And if we just bring a metal in the vicinity of this charge, the boundary conditions strongly affect



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the electric field and they change the orientation of the field lines.



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But we see here that these field lines look exactly like the field lines of this dipole.



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So what happens here is that the electrons react in such a way that they form something like a mirror



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charge.



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So you could say the electrons which are negatively charged, they are not homogeneous in this metal



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anymore, but they leave something like a gap.



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And this gap has then the positive charge, of course.



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So just in our imagination, we could just continue this electric field lines to this imaginary charge



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here.



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And so we have something like people called this a charge.



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So that's not really not really a charge in the classical sense that you really, really have there



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as a sphere.



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But it's more like a gap or something mythic missing in the electron charge of the reality of the matter.



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And so here there are really two very important things.



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The first one is if you compare this example here to this example, you will see that the boundary conditions



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of this metal strongly affect the electric field lines and also the electrostatic potential.



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And then also you can see that metals behave quite interestingly because their electrons really are



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affected by the charges in the vicinity.



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And so they form something like a dipole.



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So this means they want to shield the charge as that is in the vicinity of the metal because they exhibit



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something like a mirror charge, which is the same magnitude, but with an opposite sign.