Welcome back to electrodynamics and its applications, lecture 10. So today, I'm with my teaching assistant, Melodie Glasser, and my name is professor Seungbum Hong. So, in this lecture, we're going to first start with molecular dipoles. We're going to discuss why it is that materials are dielectric. So, let's recap what we learned before. So, what is dielectric? What do we mean by dielectric? In the past, people thought dielectric materials do not respond to external stimuli like electric field. However, they found something very interesting. When they put the glass in between two electrodes, the capacitance increased. So, I'm going to ask my teaching assistant Melodie. So, why does that happen? What kind of behavior do dielectric materials show when they are exposed to electric field? So, we went over it last lecture, but basically what happens is in a dielectric is, there's an induced polarization by the electric fields. So we actually have charges, and then there's an overlap here, and there's negative charges here, and there's a small induced electric field in the dielectric as well. Exactly. As Melodie just pointed out, there are limited motions of charges inside dielectric materials. So, in fact, it creates an impact or influence on the overall properties of the material. So, when an electric field is applied to a dielectric, it induces a dipole moment in the atoms. If the electric field E, induces an average dipole moment per unit volume, then kappa, the dielectric constant, is given by kappa minus one is equal to P over epsilon naught E. We have to discuss the mechanism by which polarization arises when there is an electric field inside a material. So, let's begin with the simplest possible example, the polarization of gases, ideal gases. Before doing that, let's discuss shortly about the polar molecules like water vapor, and non-polar molecules like oxygen gases. As you can see from this schematic picture, water molecule has hydrogen and oxygen, where they have different electron affinity. So, oxygen is mostly minus charge, and hydrogen is plus charge. They are forming an angle of bond, that is the center of mass of plus and center of mass of minuses, they are not coinciding. Therefore, they're creating additional dipole moment. Right? In case of oxygen gas, you don't have that kind of dipole moment, but if you apply electric field, the electron cloud around the nucleus will be polarized, to contribute to polarization. So, you have to think about two contributions for water vapor case, but only one polarization for oxygen gas. Okay. So, let's start with electronic polarization, which both water and oxygen gas have. So, we'll first depolarization of nonpolar molecules, like monatomic gas, or helium, or oxygen gas, and see what electronic polarization is. This is the displacement of the electron distribution, and that produces induced dipole moment, which is proportional to electric field. If you remember our prior lecture on plasma oscillations, where we created plasma, where we have neutral atoms, positive atoms, as well as positive ions, as well as electrons. You remember probably we were able to move the electron cloud back and forth like a spring mass system, and that solution can be also applied to this atomic model. So, the center of the charge of the electrons obeys the equation of motion when an atom is placed in an oscillating electric field as in the case of plasma oscillation. If we borrow the equation of motion from Newtonian mechanics, we know the first term is about kinetic energy or kinetic motion term, MA. F equals MA. The second term is about restoring force, KX, where K is the spring constant. We're replacing K by the mass of electron cloud or electron in the resonance frequency of the spring mass system. So, let's talk about the dipole moment of an atom as a function of electric field. So, if the electric field varies with the frequency omega, the solution for the equation of motion becomes X equal X naught, E to the power I omega T. That's a complex math number mass, and we know this is harmonic oscillator, and the coefficient can be found by solving the equation, which is qE over M times omega naught squared, minus omega squared, times E to the I omega T. At this point, I'm going to ask you as well as my teaching assistant, what happens if omega approaches the resonance frequency omega naught? So, Melodie, what happens to this spring mass system when we oscillate the system close to the resonance frequency? Then it has the most energy, when you're oscillating it close to the resonance frequency. Yes. So, you can see as omega approaches omega naught, this terms approaches zero, and therefore that whole term goes to infinity. Meaning, the amplitude is maximized, right? That's when we can have absorption of the energy. So, here we are interested only in the case of a constant field. So, let's think about more simpler case, rather than oscillating case. We're going to put omega equal zero to the solution, and when we do that, X becomes qE, over M omega naught square. Right? From this, we see that the dipole moment P of a single atom, is P equals Q times X, because dipole moment is the multiplication of the charge times the displacement. If we input the equation above into this, you find it's qE square times E over M omega naught squared. That is equal to alpha, a new term, which is called polarizability times permittivity of vacuum, times electric field, okay? So, here alpha is called the polarizability of the atom, and has dimensions of L cubic, which is dimension to the power of three, and is a measure of how easy it is to induce a moment in atom with electric field. So, let's take a look at this equation again. I'm going to ask my teaching assistant again. So, what does it mean to have large alpha? It means that the susceptibility is higher. Exactly. The susceptibility, which is the measure of how well the system respond to external electric field, and it will have more and more effect if you have a larger and larger polarizability. All right. So, dielectric constant as a function of density and omega nought can be now seen from the equation we solved. Our simple theory says, alpha is equal to q e squared over epsilon nought m omega nought squared, and we can rewrite that in terms of four pie e squared over m omega nought squared, where denominator is about the charge quantity and the denominator is about the mass and the resonant frequency of the system. So, if there are N, capital N atoms in a unit volume and N is the density of the system, the polarization P is given by, P times N times small p, which is the dipole moment, which is equal to N times alpha times epsilon naught times e. At this point, let me ask Melodie, when can we say the polarization is a simple multiplication of the density times the dipole moment. In which situation? I don't remember. Okay. So, to help you remember this, you see the polarization is some of the dipole moment divided by the volume. If you want to make some of the dipole moment just by simple multiplication, the direction of the dipole moment should be the same. So only in that case you can use this simple formula. Okay? Thank you. So, therefore, the kappa minus one is equal to p over epsilon nought e, and if I rearrange that, then you have N alpha here. So, you see the dielectric constant kappa is directly related to the density of atoms and the polarizability of atoms. So let me ask, if you have denser gas system do you have higher dielectric constant or lower dielectric constant? Higher. Exactly. You have higher dielectric constant. That is also related to the density as well as the mass and resonance frequency of the system. So now we can predict that dielectric constant kappa of different gases should depend on the density of the gas as we just discussed, and on the frequency omega nought of its optical absorption. So, the light is the electromagnetic wave where electric field is oscillating as a function of time and space and therefore, can oscillate the whole system, and if it happens to be the case that the frequency of your light matches the resonance frequency of your system, then the light will be absorbed. Okay. Now, let's cover the classical ideas to predict the dielectric constant. So, our formula is only a very rough approximation, as you can see. Because we have taken a model which ignores the complications of quantum mechanics. For example, we have assume that an atom has only one resonance frequency when in reality it has many. Right? However, the classical ideas, which we will show you in a moment, give us a reasonable estimate. That's surprising, right? So, let's try hydrogen. So how many protons do we have in hydrogen? I think we only have one. Yes, we will have one, and how many electrons do we have? Also one. Also one. So that's the simplest atom in the periodic table. So, suppose the energy needed to ionize the hydrogen atoms is equal to the energy of atomic oscillator whose natural frequency is omega nought. So, somehow, the external perturbation of electric field has the same frequency as the resonance frequency of this spring-mass oscillator of hydrogen system, then it will absorb the energy and that absorption will result in ionization. The electrons can escape from the system and hydrogen is left with only one proton. So, it's H plus. We know from quantum mechanics that that energy is roughly equal to one over two, m e to the power of four over haba squared. As you know, energy in quantum mechanics, is haba omega nought. It's related to the frequency, right? So, if I use that equation, omega nought, which is the resonance frequency of the system, will be one over two times m e to the power of four over haba to the power three. Therefore, we can calculate the electronic polarizability of hydrogen alpha becomes 16 pie times haba square over m e square, the whole thing to the power three. Therefore, as you can see, if we do the math, arithmetic, we get 1.00020, which is very close to the experimental value of 1.00026, right? So this is surprising, right? With only some of the electrostatics that we have learned and very simple quantum mechanical knowledge, we can predict the permittivity without even doing the experiment. The next thing is about polar molecules. So, we started from non-polar gas, right? For polar molecules, as I mentioned, in addition to the electronic polarizability, what else do we have to contribute to the polarizability? So let me ask Melodie. We have an already existing dipole moment within the molecule itself. Exactly, and why is that the case? Well, in the case of water, it's how the bonding is distributed in the anode. So, if you look at water, you have a negative and then it's bonded to two, sorry, two protons and I think they are paired electrons. Exactly. Here, right? So that makes the negative direction going this way in the positive direction going this way. The other way, right? Yeah. So you have different gravitational centers for plus and minus ions. So, very good. So, let's consider a polymeric molecule such as a water molecule, which carries permanent dipole moment p nought, and with no electric field, as you can see from this schematic diagram, the individual dipoles point in random directions so the net polarization will be zero. That is understandable because if they are aligned, they create huge electric field and they will raise the free energy of the system. Now, if I apply an electric field to the system, then the story change, right? There is an extra dipole moment induced because of the forces on the electrons. We already learned about that. So the electron cloud will be shifted, which is called electronic polarizability. Then the electric field tends to line up the individual dipoles to molecular dipoles to produce the overall net polarization p. We're going to study this in more detail. So, the net dipole moment under electric field will be covered in this slide. So, if all the dipoles in a gas were to line up, there would be a very large P as you can imagine. Which doesn't happen because at ordinary temperature and electric field, the collisions of the molecules in the thermal motion keep them from lining up very much. In other words, entropy term dominates. So, you have more chaotic situation or random orientation. The resulting P can be computed by the methods of statistical mechanics, and we're going to use Boltzmann distribution. The energy of a dipole U, which is related to the enthalpy of the system, is in this case as you see, electric field is pointing from bottom to top, and you have a slanted dipole moment that is at an angle of theta with the electric field, can be written like Q five of one, minus Q five of two, which is the electrostatic potential of each charge, each component of the dipole moment, which can be rewritten as qd dot del phi. So, we saw a similar equation before. In lecture six, we saw the electrostatic potential of a dipole moment. Let me remind you what you can see from this equation. So here, we have del phi, which is the gradient of electrostatic potential. So, what is the gradient of the electrostatic potential? In the electric field. Exactly. So, Melody just told you the right answer. The gradient of electrostatic potential is related to the electric field in a way that the downhill slope is electric field. So, it's minus electric field, and QD, this is dipole moment. So, in other terms, you can use it minus P naught.e. So, the. product between the dipole moment, and the external electric field, will be the major factors affecting your potential energy. If you rewrite that in a scalar form, then it is minus p naught e, times cosine theta. So, if you want to lower the energy of a dipole, do we have to have theta equals zero? Or, theta equals 180 degrees? Melody, which one lowers the energy? When it's zero. Yes. Zero means, your dipole moment is aligned with the electric field when they are both thumbs up. In that case they feel happy, and that's when they have minimum. If you're anti-parallel, then it's very unhappy. In a state of equilibrium, the relative numbers of molecules with the potential energy U according to Boltzmann distribution, is proportional to exponential to the power of minus U over KT. So, now we know U, and now we know distribution, so we're going to plug them in to our equation. So, let's take a look at this picture. So, imagine now, we're going to gather all the dipoles in the world, and preserving their direction and magnitude, but putting the starting point to the origin of a sphere. Once you do that, you can imagine you will have all arrows pointing everywhere. But if you apply electric field, you can imagine you've got more population around the northern hemisphere than the southern hemisphere. If I apply electric field downward, then the situation will reverse. We're going to solve this problem quantitatively using the known quantities that we have just discussed. So, let's make N of theta, be the number of molecules at theta per unit solid angle. So, you can see, when you have angle relationship between the electric field and the dipole moment to be theta, then the number of those dipole moment and of theta, is equal to n naught times E to the power of P naught E, cosine theta over KT, because this is Boltzmann distribution. For normal temperature in electric field, the exponent is small. So, we can approximate by expanding the exponential to this form. This is binomial expansion to the first order. If I put that into the equation to evaluate the total number of dipoles per unit volume. So, this is the density of our dipole moment, or density of our molecules, then we have to cover from zero to four pi. So, at this point, I'm going to ask my teaching assistant Melody. What radian is? And how we define radian? So, radians are a measurement of the angle with a in the circle. So, if you have a circle, and then you look at the angular distance here, then your radian is actually going to be the ratio of this outer section, to the radius. Yes, exactly. So, that's the definition of radian. So, now we're going to expand this knowledge to solid angle. So, how do we define a solid angle? So, solid angle, is a ratio between the area of this strip on the surface of the sphere, to the square of the radius. So, if I'm covering the whole surface of the sphere, then my solid angle will be four pi. That's why we're running from zero to four pi, to cover all of the distributions that we're discussing right now. So, knowing that, we're going to see what is then the portion of the dipole moment in a infinitesimal change of the solid angle d omega. So, you can first think about the area ratio of the whole surface area, versus the ratio of the area of the strip. The area of this strip is rd theta, which is the arc length of the strip, times the periphery of the circle. The periphery of the circle is just when you think about the radius of the circle here is, r sine theta, then is two pi r sine theta. So, that's why we will replace d omega, with the parameters in terms of theta and r. So, once we do that, we will see the tool number n per unit volume, is four pi n naught. In ot her words, the n naught, the coefficient here, is n over four pi. Now, let's see, what happens if we do not use the approximation for the exponential function? So, this will be your homework question. But of course, we will give you hint which you can use to solve this problem. So, we are going to calculate the polarization of polar molecules under electric field based on the equations which we just derived. We know now, the n naught which was the coefficient in front of this distribution, was N, capital N over four pi. Now, we see that there will be more molecules oriented along the field where cosine theta equals one, in other words, theta equals zero degrees, because as you can see, if I put theta equals zero, you have increase in this parenthesis. Whereas, if you put cosine theta equals minus one, then you have decrease. So, you can instantly see r, n is larger when theta is close to zero degrees, then it is close to 180 degrees. Now, we're going to calculate P, and in order to do that, vector sum of all molecular moments in a unit volume. However, we already saw that in in-plane direction, you have random distribution. So, you can imagine, in-plane component will be zero. Only out of plane component will matter. So, we will calculate only along the actual component, which is out of plane, which is just summation of P naught cosine theta. This is i, which means each component. So, let's calculate the average polarization under electric field for polar molecules. We can evaluate the sum by integrating over the angular distribution. So, here's the thing, we already know the density, the number of dipoles that has the same angle theta. So, if I sum them up is just the number of dipoles at that moment times P_naught cosine theta. If I do integration from zero to four pi, then I will get the average polarization. So, we can go step-by-step over this equation where you see from zero to four pi n of theta times P_naught cosine theta times the solid angle d omega, and if I do it step-by-step and insert the numbers that we have already got for example, for N theta, then we integrated it and we got this number. So, polarization is equal to the density of atoms N, capital N times the dipole moment squared times electric field over three KT. So, let's do some discussion here. So, from this equation if I increase electric field. Melody, what will happen to polarization? Polarization will increase. Polarization will increase. So, let's think about this, if I apply more or stronger electric field they will tend to align more. So, the polarization will increase, that's understandable. Now, if I increase temperature, what happens to my polarization? It decreases. It decreases. So, if I heat the system up, they want to randomize more, they don't want to align with the electric field. So, I can make a joke out of this. So, if somebody, the leaders want to align the values of the society applying big pressure or electric field they can do it, but if the society has high temperature where they have more freedom of expression, where they tend to think more freely, then it's very hard to align. If you lower the temperature, it's easier to align. Okay. So, here's we're going to discuss Langevin Function, and from this discussion, I hope you are able to solve the homework problem that we just discussed. Langevin function is a function for magnetic moments, but as you can see, magnetic moments behave similarly to electric dipole moment. So, if you just replace the parameters in this equation you can solve the problem for electric dipole moment as well. So, let's take a look at this. For paramagnetic substance with magnetic moments M in an applied field H at temperature T, then if we make the assumption that there is no preferred alignment with the substance, we can assume that the number of moments N of theta again, the same between angles theta and theta plus D theta with respect to the external magnetic field H is proportional the solid angle two pi sine theta d theta. Then the probability density function where you can see the net magnetic moment over the maximum moment is equal to this function. Where you see the denominator is a whole number of dipole moments, and the nominator is the dipole moment that is within the strip of the same solid angle. If you do this calculation using Boltzmann equation as we did, where the distribution function follows the Boltzmann function. Then without doing the approximation, you can do the mathematics for exponential function by doing a partial integration and then you will get this function which will result in hyperbolic cotangent alpha minus one over A where you see the A function is MU naught H over KT, and MU naught H is something similar to external field times the P_naught, the dipole moment. So, if you replace those parameters with the parameters we discussed, then you will have the same function. The graphical representation of Langevin function is like this. As you can see, in the electric field case, if you increase the electric field, you have linearly increasing portion and then it goes way off from this linear portion and saturates to the asymptotic values here. So, the portion that we just solve together was this starting point. So, that's the same thing, if I apply huge magnetic field to magnetic dipole moment they tend to align but if the temperature increases they tend to be randomized. Okay. So, as you can see from this graph, the slope of the initial part of the graph is exactly the same as the one that we derived, the polarization is equal to N P_naught square times E over three KT. So, you can see the polarization is proportional to the field E. So, there will be normal dielectric behavior, we also expect that p depends inversely on T because at higher temperature, there is more disalignment by collisions. So that's, one over T dependence and which is called Curie's law. So, Curie's law is permittivity, is the inverse permittivity or susceptibility is proportional to one over T. Okay. Let me ask Melody, what susceptibility was? The susceptibility was the influence that a particular system would get from an electric field. Exactly. So, let me explain what Melody just explained to us in equations. So, P polarization is equal to susceptibility times permittivity of vacuum times electric field. If you compare this equation with this equation, what you see is the susceptibility is within this side. In other words, susceptibility is proportional to one over T, and that is exactly what happens for paraelectric materials, it falls off exponentially as a function of temperature from the Curie temperature, meaning at Curie temperature susceptibility will diverge to infinity. So, from this simple Langevin Function we can get many messages or knowledge from this simple equation. Now, why do we have P_naught square dependence here? Why do we have P_naught square? So, let's think about it. In a given electric field, the aligning force depends on P_naught we know that. Right? The mean moment that is produced by the lining up is again proportional to P_naught, so that's why we have P_naught square. So, this is a very important factor in determining the polarization of the system. So, let's think about a little bit more complicated situation where you have a steam, in other words, water molecules instead of hydrogen atom. In the case of water molecule, we already discussed it. In addition to electronic polarizability, we have the molecular dipole. Based on our discussion on the alignment of molecular dipole moment, we know the kappa minus one which is dielectric constant minus one is equal to polarization one over epsilon_naught E is equal to N times P_naught squared or three epsilon_naught KT. Since we don't know what P_naught is for water molecule we cannot compute P directly but we can use kappa minus one using the equation above to see how our equation predicts the behavior of the water molecule as a function of temperature. So, dielectric constant has been measured at several different pressures and temperature chosen such that the number of molecules in a unit volume remains fixed, and doing so you can see here the real experimental values which lies in a very good match with the calculation based on this equation. So again, we can see how this theoretical approach and experimental approach meet together. So, let's discuss about the permittivity or dielectric constant as a function of the frequency of electric field as well. So, there is another characteristics of kappa its variation with the frequency of applied electric field in addition to electric field itself the magnitude and the temperature. That due to the moment of inertia of the molecules, it takes a certain amount of time for the heavy molecules to turn towards the direction of field you can imagine. So, if we apply frequency in high microwave region or above, the polar contribution to the dielectric constants begins to fall away because the molecules cannot follow, and as you can see from this graph, you see the space charge region as well as dipolar region will die out, we call that relaxation. When relaxation happens, you have decrease of permittivity or dielectric constant. For atomic vibrations or electronic polarizability is like spring mass systems, so you will have resonance. So, that's why you have resonance and after resonance you have smaller contribution from that component and that's why it's going down and down and down as a function of frequency. The highest component that is surviving is only electronic polarizability. So, at the frequency approaching the light you only have to think about electronic polarizability and that's why the refractive index is related to the permittivity due to the electronic polarizability. So, in contrast to this, as you can see, the electronic polarizability still remains the same up to optical frequencies as we just discussed, because of the smaller inertia in the electrons. Okay?
