Hello, everybody. Welcome back to our Electrodynamics and its Applications. It will be the ninth lecture. I'm with my teaching assistant, Melodie Glasser, to my right side, and my name is Professor Seungbum Hong at Kaist. So, today, we will start with the concept of the dielectric constant. So, in the past, when scientists were thinking of the insulators or dielectrics, they thought insulators will not respond to electric field because they observe, even if they apply electric field or voltage, there is no current. So, they thought there's no motion at all. However, as you can see, this dielectric constant came across the mind of Faraday when he did an experiment where he put dielectric glass in between two parallel-plates of conductors. What happened was the capacitance of a parallel-plate capacitor increased when an insulator is put between the plates, and that was very puzzling. Why? Because if there is no influence on the insulators, then there should be no change in capacitance, so something must have happened. So, the final insight they got was the peculiar properties of matter under influence of the electric field is that dielectric materials will have an instant displacement inside and react to the electric field so as to overall decrease the electric field across the gap in the parallel-plates. As you can see, capacitance, as we learned, is the parameter describing how much charge we can put on the capacitors under given voltage. Right? That capacitance is a strong function of the permeativity of the vacuum or air and the geometric factors like area of the plate and the distance between the plates. So, now, we're going to answer the question of why is there any electrical effect if the insulators do not conduct electricity. So, we will walk through this question with you and use a lot of theorems that we learned before, like Gauss Theorem, and help you understand what's happening. All right? So, for that, we're going to use this schematic picture. As you can see, you have two plates up and down and you have dielectrics in between and the arrows represent the electric field. So, the density of arrows will represent the magnitude of electric field. So, you can see the density of errors is higher in between the air, inside the air rather than the dielectric. So, we will begin with the experimental fact or observation that capacitance is increased when we put dielectric inside. All right? So, we put negative charge on the top electrode plate, as you can see, and positive charge on the bottom electrode. If C increases by putting a dielectric in between, it means the voltage is smaller for the same amount of charge on the plates. Because Q equals CV and Q is fixed, if C increases, then V has to decrease. Right? So, we must conclude that inside a capacitor, the electric field is reduced even though Q remain unchanged. So, we are going to use Gauss Law to understand that but before doing that, I'm going to ask my teaching assistant, Melodie, what Gauss Law is? So, Melodie, what is Gauss law? I think Gauss's law is the integral of E dA is equal to Q or however many Qs there are over Epsilon_0. Exactly. So, this is the Gauss law. So, the flux of electric field outside a given Tau surface is equal to the charge inside divided by permeativity. So, you can see the flux of the electric field is directly related to the enclosed charge and we're going to consider the Gaussian surface, S, which is depicted here as red dotted lines, right? If I make this Gauss surface, you instantly see, inside a conductor, there will be no electric field, inside dielectric, we might have some electric field, and outside this one, we have no flux because the electric field is parallel to the plane. So, the net charge inside the surface must be lower than it would be without an insulator. Why? Because by that way, we can have decrease in the electric field. There must be positive charges on the surface of the dielectric. In that way, we're decreasing the net charge and since the field is reduced but it's not zero, the positive charge is smaller than the negative charge in the conductor. So, in this example, you can see six minus charges for only three plus charges. So, this qualitatively tell you what kind of conditions we need to meet the experimental observation. Now, we're going to jump into some new situation so that you can refresh your mind and understand what's happening when you put a conductor instead of dielectric and see what happens. Okay? Suppose that we have a capacitor with a plate spacing, D, and we put between the place a neutral conductor whose thickness is B. So, if I replace the dielectric by a conductor, what happens is you will have no electric field inside the conductor. In order to make that happen, you have to displace positive charge on top and negative charge to bottom so that the superposition of electric field created by this dipole moment will annihilate the electric field imposed by external circuit. All right? If that happens, you can already see because there is no voltage drop across this conductor, you will have significant decrease in the overall voltage. That voltage decreases described by this equation, where voltage is equal to Sigma over Epsilon_0 times d minus b, where d is the distance between the parallel-plate conductors, b, the thickness of the conductor we insert. Right? So, we can immediately see by putting an conductor inside, you have the same effect. But again, having the same effect doesn't guarantee you have the same structure. However, having the same effect or same property gives you some hint, the feature of the structure. So, if you can see, C is increased by a factor which depends upon b over d, the proportion of volume which is occupied by the conductor. So, in the past, scientists thought, "Oh, because this has similar property or response, maybe dielectric can be modeled using metals inside the material." So, the early model of the dielectric is shown here. So, people thought most of the part, the matrix, has a fixed charge so they don't move at all, and only part of them has metal inclusion so they can move freely but they're isolated. In this way, they could model all the behavior of dielectrics, and in fact, that was a very good approximation. Indeed, in the future, when I was a Project Leader in Samsung, we used this early model to make a new type of structure for memory devices. So, we put insulative films and we included metal nanoparticles inside so we could increase the dielectric constant without changing the composition. So, sometimes, the wrong model can be used to have an innovation in other fields. As you can see, the phenomenon, the dielectric constant increase is explained by the effect of charges which would be induced in each sphere and it was assumed that each of the atoms of the material was a perfect conductor but insulated from the others, and the dielectric constant, Kappa, would depend on the proportion of space, which was occupied by the conducting spheres. So, to some extent, this was successful model, but some of the important features that we will discuss in the later slides, they couldn't explain using this model, okay? So, in this slide, we are going to look into more details of polarization. But before starting with polarization, let me ask my teaching assistant, Melodie, what polarization is. Polarization is the sum of the dipole moment over volume, right? Yes, so it is the sum of the dipole moments over the volume they occupy. So, you can imagine the dipole moment that we learned in the earlier lecture where you have q times d, the d being the displacement vector, is really the direction of polarization. So, the only thing that is essential to understanding of dielectric is that there are many little dipoles induced in the material, and that many little dipoles will sum up to make a huge polarization. So, we will start with atomic nucleus in electron cloud. As an atom has a positive charge on the nucleus which is surrounded by negative electrons in an electric field, the center of gravity of the negative charge will be displaced and will no longer coincide with the positive charge of the nucleus as shown in this picture. So if we look from a distance, such a neutral configuration is equivalent to a first approximation to a little dipole. So, it seems reasonable that if the field is not two enormous, the amount of induced dipole moment will be proportional to the field. So it's like minus charge and plus charge connected to a linear spring. We will assume that's the case in our materials. We will assume that in each atom there are charge q separated by a distance delta, this is displacement vector, so that q delta is the dipole moment per atom. So, polarization will be the dipole moment per unit volume which is equal to N times q times delta. N is the number of atoms per unit volume, q is the charge, and delta is the displacement. So as Melody has explained to us, polarization is summation of all dipoles over the volume that they are occupying. So, you can understand why P is equal to N times q times delta. Of course, here's an assumption. When you apply electric field, all the dipoles will align in the same direction. In that case, you can use this. So in general, p will vary from place to place in the dielectric, and we will suppose that there exists a mechanism by which the dipole moment is induced which is proportional to the electric field. Now, let's move on to a new concept that is written here, polarization charge. They are bound charge. As I mentioned before, in the past scientists believed in insulators charges are not moving. They were right to some extent, but they didn't think about the fact that they can move instantly and stop. So they are bound, but they can make a motion that is instantaneous and doesn't result in overall flow. So, let us see that what this model gives for the theory of a condenser with a dielectric. Let's consider sheet of material in which there is a certain amount of dipole moment per unit volume as depicted in this picture. So, we're going to think about uniformly charge slap instead of spheres that we considered before, and we're going to displace them by a bit. So imagine you have a slab like this with plus charges, the same amount of plus charges as in this slab. We have same amount of minus charges. The charge density, they are the same, and then I'm going to overlap them. So, if I perfectly overlap them, then there will be no charge density. But if I decide to display them by a little bit delta, then I start to see excess charge on top and bottom as you can see. So, for the top surface as depicted here as well, you will have positive charge, bottom you have a negative charge, and inside you will have no charge. The net charge is zero. So then you may ask, will there be on the average any charge density produced by this? Meaning, will there be any charge density inside the material not the outside? You will not have anything if the polarization is uniform. However, what if p is not uniform? You will have it and I will show you why that's the case in the later slides. So, suppose that p is uniform for simplicity and see what happens at the surfaces. So, we will have a surface charge density of charge which we will call the surface polarization charge. So, we're going to prove that sigma polarization is n times q_e times delta. So, think about this: first, we have a slab on the surface where the volume of this surface portion is equal to delta times a, whereas delta is the thickness, a is the area of the slab. If I want to count the number of atoms in the slab, then we multiply V by N. N is the atomic density times, if I multiply that by q_e, then we have total charge in this slab. This is equal to the surface charge density sigma polarization times the area. So if I rearrange this, you may find sigma polarization is equal to N.V.q_e over A. V over A as you can see here is delta. This is delta. So therefore, I just prove that sigma polarization is N times q_e times delta. From the former slide, we know that sigma polarization is equal to P because P equals Nq delta. So, now we understand the surface charge density when you have uniform p with displacement parallel to the polarization, then the surface charge density is just polarization. So, the surface charge density of charges equal to polarization inside the material and we're going to check this in other ways. I'm going to ask one question to my teaching assistant melody. What is the physical unit of polarization? The unit for polarization in standard units are columns which represent the charge per meters squared and then you can see we have surface charge density. Exactly. So, from the physical unit or polarization, we can also understand that the polarization is equal to the surface charge density. All right. The surface charge is positive on one surface and negative on the other as depicted before. Let's assume that our slab is the dielectric or a parallel plate capacitor. Now, we're going to categorize two types of charge: one is sigma free, which is surface charge, we can move freely anywhere on the conductor, usually Electrons and holes that we inject through the circuit will be the surface charges. The charge we put on them when we charge capacitor. sigma sub pol exists only because of sigma sub free in case of paraelectric. Of course, in case of ferroelectric, we have sigma sub pol even without sigma sub free. It goes back to zero when sigma sub free is removed by the relaxation of the polarization inside of the material. So, now we're going to revisit the picture we had before and use the Gauss law to the Gaussian surface S as we draw here. Note electric field inside metal is zero, as you can see from geometry. From geometry electric field is directed parallel to the thickness direction. So, the electric field E in the dielectric is equal to the total surface charge density divided by Epsilon naught. That was what melody just explained to us using Gauss law. So, inside this Gauss surface you have sigma free, that's the free charge, minus, minus, minus on the conductor and we have sigma pol, that is plus, plus, plus, but sigma pol has opposite sign, so we put minus here to represent sum. So, sigma free minus sigma pol is really the net charge of this Gauss volume, and divided by Epsilon naught is equal to E. E, electric field. Note that electric field E naught between a metal plate and the surface of the dielectric is higher than electric field E. As you can see, there you have more lines in the air than in the dielectric. Here we are concerned with the field inside the dielectric which, if the dialectic nearly fills the gap is the field over nearly the whole volume. So, imagine you are decreasing the gap between the dielectric and metal and you can see all of the gap will be replaced by this less denser electric field. In that case, you can see this one, sigma free minus sigma pol will be the really the charge inside. We can replace sigma pol by P from what we have just derived. Then E becomes sigma free minus P over Epsilon naught. So, that's very clear. So, here is one picture that I want to present to you to let you understand what we mean by susceptibility. So this is the picture of Roman Phalanx defending with spears. If you imagine spears are like electric field, then you can see how dielectric response to electric field from external side annihilate the electric field by creating electric field from dipole moment and have less penetration. So in other words, how strongly does electric field influence the polarization of the dielectric? So the more influence, they will defend more vigorously, and if they defend more vigorously, susceptibility will go higher. So, we will see that in the next slide. So electric susceptibility, Chi, is defined by the proportional constant between polarization and electric field. As you can see, if Chi goes higher for a given electric field polarization will go higher. So I'm going to ask my teaching assistant, Melody. What will happen to the overall electric field between capacitor if Chi goes higher? So, if Chi goes higher then the polarization will also increase. Then since over here, you can see that the polarization is negative, so if the polarization goes down the electric field goes down. So, the voltage goes down and then look, you need less voltage for the capacitor. Exactly. So, as Melody mentioned, if P goes up because we have negative signs over here, the overall denominator will decrease, as a result, electric field will decrease. We will arrange the equation using these two equations like this. As you can see, electric field is equal to sigma sub free over Epsilon naught times one over one plus Chi. With this equation, so this is the factor by which the field is reduced. With this equation we can think of some extreme cases, what if x tends to either zero or infinity? So mathematically, you can see if it goes to zero, you have the same case as you have nothing. So, it will be like transparent to electric field. If it goes to infinity, as you can see here, electric field goes to zero. So if it goes to zero, it means for a given charge capacitance will go to infinity. So Melody, can you think of any material class that mimic these extreme cases? To those metals? Then there's also ferroelectrics. Exactly, exactly. So in case of ferroelectrics, if you have switching voltage the capacitance tends to infinity because with small voltage you can change the polarization from up and down. Now, the total charge on the capacitor is sigma sub free times A, so the capacitance C is C equals Q over V. Now, we're going to replace Q by sigma sub free times the area and voltage by the equations that we just proved, and when we do that sigma sub free will be annihilated. So, we will end up having Epsilon naught A plus one plus Chi over D where Epsilon naught times one plus Chi is Kappa, which is the dielectric constant. So, Kappa Epsilon naught A over D will be the equation. So, when a parallel plate capacitor is filled with a dielectric, C is increased by the factor Kappa which is a property of the material. So Kappa equals one plus Chi. So, here's one surprise question. So, do you happen to know the Kappa of water? I actually don't. It's about 80. Really? Yes, it's about 80. It's huge. So you can see, water has 80 times higher dielectric constant than air or vacuum, and that's why it's so dipolar and electric field can change property of waters.
