Physikalisches Praktikum für Fortgeschrittene 1 Dielectric Constant Assistent: Herr Dr. M. Ghatkesar by Therese Challand theres.

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1 Physikalisches Praktikum für Fortgeschrittene 1 Dielectric Constant Assistent: Herr Dr. M. Ghatkesar by Therese Challand theres.challand stud.unibas.ch

2 Contents 1 Theory Introduction Elementary Treatment of Electrostatics with Ponderable Media 1.3 Molecular Polarizability and Electric Susceptibility Models for the Molecular Polarizability Experiment 8.1 Aim of the Experiment Experimental Set-Up Measurement Data Analysis Error Evaluation Problems 10 4 Conclusion 11 5 Sources 11 1 Theory 1.1 Introduction An electric field in a dielectric medium is interacting with the charges present in it inducing its polarization. There are mainly two different mechanisms throughout which polarisation can be induced: Deformation Polarisation: in every molecule the charge distribution is deformed by the electric field producing a dipole moment aligned to the electric field. Orientation Polarization: this is only present in the case of molecule with a permanent dipole moment. The molecular dipoles tend to align to the external electric field. Following Langevin-Debeye theory it is possible to describe how the total molecular polarizability depends on deformation polarizability, molecular permanent dipole moment and temperature: γ mol = γ i + 1 T p 0 3k B = A + B T 1

3 What is measured in the experiment is not the molecular polarizability but the dielectric constant. Theory allows to link the property of a single molecule (molecular polarizability) to a property of the media (dielectric constant): γ mol = 3 ɛ r 1 4πN ɛ r+ MKS: γ mol = 3ɛ 0 N ɛ r 1 ɛ r+ This equation is called the Clausius-Mosotti equation. 1. Elementary Treatment of Electrostatics with Ponderable Media We know about electrostatic potentials and fields caused by the presence of charges or conductors. In this case we don t need to make a distinction between microscopic and macroscopic field in that medium, because air is tenuous so that we can neglect its dielectric properties. We consider now the case of ponderable media present. In ponderable media electric responses must be taken into account. Now we prove if the Maxwell equations for ponderable media still are correct: Generally, to obtain the Maxwell equations for macroscopic phenomena we need to average over macroscopically small but microscopically large regions, e.g. volumes V. So we take the homogeneous equation E micro = 0 and make an averaging and see that E macro = E = 0 also holds for the averaged macroscopic, electric field E. So the field is still derivable from a potential Φ( x) in electrostatics. If an electric field is applied to a medium made of many atoms or molecules, the charge bound of the molecules will respond to the applied field and will execute perturbed motions. So the molecular charge density will be distorted. The multipole moments of each molecule will be different from what they were without field. When there is no applied field the multipole moments are all zero in simple substances, at least when averaged over many molecules. The dominant molecular multipole with the applied fields is the dipole. So there is produced in the medium an electric polarization P, this is the dipole moment per unit volume, given by P ( x) = i N i < p i > where p i is the dipole moment of the ith molecule. N i is the average number per unit volume of the ith molecule at the point x. The charge density will be ρ(x) = i N i e i + ρ excess if the molecules have a net charge e i and additionally there is macroscopic excess or free charge. Usually the average molecular charge i N i e i is zero, then the

4 charge density will only be the excess or free charge. If we look at the medium in a macroscopic way, we can build up the potential or field by linear superposition of the contributions from each volume element V at the variable point x. So the charge of V is ρ( x ) V and the dipole moment of V is P ( x ) V. If there are no higher macroscopic multipole moment densities, the potential Φ( x, x ) caused by the configuration of moments in V can [ be seen as: ] Φ( x, x ) = 1 ρ( x ) V + P ( x ) ( x x ) 4πɛ 0 V x x x x 3 provided x is outside V. Now we treat V as infinitesimal and put it equal to d 3 x to integrate over all spaces: Φ( x) = 1 4πɛ 0 d 3 x [ ρ( x ) + P ( x ) 1 ( x x x x )] An integration by parts transforms the potential into: Φ( x) = 1 [ 4πɛ 0 d 3 x 1 x x ρ( x ) P ( x ) ] This is just the expression for the potential caused by the charge distribution ρ P. With E = φ (1st Maxwell equation) it follows: E [ ] = 1 ɛ 0 ρ P The presence of the divergence of P in the effective charge density can be understood qualitatively. If the polarisation is nonuniform there can be a net increase or decrease of charge within any small volume as indicated in Figure 1. With the definition of the electric displacement D, D = ɛ 0E + P the equation E [ ] = 1 ɛ 0 ρ P becomes clear: D = ρ. So a constitutive relation connecting D and Eis necessary before a solution for the electrostatic potential or fields can be obtained. We assume that the response of the system to an applied field is linear. As a simplification we suppose that the medium is isotropic. Then the induced polarization P is parallel to E with a coefficient of proportionality that is independent of direction: P = ɛ 0 χ ee. The constant χ e is called the electric susceptibility of the medium. The displacement D is therefore proportional to E, D = ɛe, with ɛ = ɛ 0 (1 + χ e ) called the electric permitivity. We call ɛ r = ɛ ɛ 0 = 1 + χ e the relativ electric permitivity. When the dielectric is isotropic and uniform the divergence equation D = ρ can be written as E = ρ/ɛ. All problems with the dielectric are the same in presence of ponderable media exept the electric field. It s reduced by the factor 1/ɛ r = ɛ 0 ɛ. The 3

5 reduction comes from the polarisation of the atoms that produce fields in opposition to that of the given charge. So the capacitance of a capacitor is increased by the factor ɛ r = ɛ ɛ 0 if there is a dielectric in the capacitor with relative electric permittivity ɛ r when there are no fringing fields. If there are in the capacitor different media juxtaposed, we must consider the question of boundary conditions on D and E at the interfaces between media. The boundary conditions are derived from the full set of Maxwell equations. 1.3 Molecular Polarizability and Electric Susceptibility We consider the relation between molecular properties and the macroscopically defined parameter, the electric susceptibility χ e. We will just talk about simple classical models of the molecular properties, although a proper treatment would involve quantum-mechanical considerations. But the simple properties of dielectrics are amenable to classical analysis. Before we talk about how the detailed properties of the molecules are related to the susceptibility, we must make a distinction between the fields acting on the molecules in the medium and the applied field. The susceptibility is defined through P = ɛ 0 χ ee, where E is the macroscopic electric field. In rarefied media there is little difference between the macroscopic field and that field acting on any molecule or group of molecules. In dense media with closely packed molecules the polarization of neighbouring molecules give rise to an internal field at any molecule E i in addition to the average macroscopic field E, so that the total field at the molecule is E + E i. Ei can be written as: Ei = E near E p. E near is the actual contribution of the molecules close to the given molecule and E p is the contribution from those molecules treated in an average continuum approximation described by the polarization P. Let s say: close to the molecule in question we must take care to recognize the specific atomic configuration and locations of the nearby molecules. So in a specific volume V we subtract out the smoothed macroscopic equivalent of the nearby molecular contributions ( E p ) and replace it with the correctly evaluated contribution ( E near ). This difference is the extra internal field E i. To calculate E p we use the result for the integral of the electric field inside a Volume V of radius R containing a charge distribution. The volume V is chosen to be a sphere of radius R containing many molecules. So the total dipole moment inside is p = 4πR3 P provided that P is constant throughout 3 the volume since the volume is small enough. So the average electric field 4

6 inside the sphere is (as we expect E p to be): E p = 3 4πR 3 r<r Ed 3 P x = 3ɛ 0. The internal field can therefore be written as: E i = 1 3ɛ 0 P + Enear We want determine the field due to the molecules near by. Lorentz showed that for atoms in a simple cubic lattice E near vanishes at any lattice side. This argument depends on the symmetry of the problem. I won t type the prove of this argument here. If Enear = 0 for a highly symmetric situation, it seems plausible that E near = 0 for a completely random situation, too. So, amorphous substances are expected to have no internal field due to nearby molecules. The polarization vector P was defined above as P = N p with p i the average dipole moment of the molecules. This dipole moment is approximatively proportional to the electric field acting on the molecule. To show this dependence on electric field we define the molecular polarizability γ mol as the ratio of the average molecular dipole moment to ɛ 0 times the applied field at the molecule. Taking into account the internal field E i = 1 3ɛ 0 P + Enear, it follows: p mol = ɛ 0 γ mol ( E + E i ) In principle, γ mol is a function of the electric field, but for a wide range of field strengths γ mol is a constant that characterizes the response of the molecules to an applied field. Taking p mol = ɛ 0 γ mol ( E + E i ), P = N p and E i = 1 3ɛ 0 P + Enear, we get: P = Nγmol(ɛ 0 E P ), assuming that E near = 0. Solving this for P in terms of E and using the fact that P = ɛ 0 χ e E defines the electric susceptibility χe of a substance we find: χ e = Nγ mol Nγ mol as the relation between electric susceptibility χ e, a macroscopic parameter, and molecular polarizability γ mol, a microscopic parameter. Since the dielectric constant is ɛ ɛ 0 = 1 + χ e, it can be expressed in terms of γ mol, or : the the molecular polarizability can be expressed in terms of the dielectric constant: γ mol = 3 ( (ɛ/ɛ 0) 1 ). N (ɛ/ɛ 0 )+ This equation is called the Clausius-Mosotti equation. Clausius and Mosotti estabilshed independently that for any given substance the density should be proportional to ( (ɛ/ɛ 0) 1 (ɛ/ɛ 0 ). For diluted substances such as gases, )+ the relation holds best; for liquids and solids, the equation is approximatively valid if the dielectric constant is very large. At optical frequencies, it 5

7 holds that ɛ ɛ 0 = n, where n is the index of refraction. Then the equation is called Lorentz-Lorentz equation. 1.4 Models for the Molecular Polarizability The polarization of a collection of molecules can arise in two ways the applied field distorts the charge distribution and so produces an induced dipole moment in each molecule the applied field tends to line up the randomly oriented permanent dipole moments of the molecules We consider a simple model of harmonically bound charges to estimate the induced moments. Each charge e is bound under the action of a restoring force F = mω0 x where m is the mass of the charge and ω 0 is the frequency of oscillation about equilibrium. Under the action of an electric field E the charge is displaced from its equilibrium by an amount x which is given by mω0 x = ee. So the induced dipole moment is p mol = e x = e E. mω0 From this we conclude by using the equations in the part above that the polarizability γ is: γ = e. mω0 ɛ 0 If there are a set of charges e j with masses m j and oscillation frequencies ω j in each molecule then the polarizability is : γ mol = 1 ɛ 0 j m j ωj We must consider the possibility that the thermal agitation of the molecules modify the result for the induced dipole polarizability e j γ mol = 1 ɛ 0 j m j. ωj In statistical mechanics the probability distributon of particles in phase space, i. e. p or q space, is some function f(h) of the Hamiltonian. For classical systems, f(h) = e H/kT is the Boltzmann factor. For the simple problem of the harmonically bound charge with an applied field in z direction, the Hamiltonian is H = 1 m p + m ω 0 x eez where p is the momentum of the charged particle. The average value of the dipole moment in the z direction is d p mol = 3 p d 3 x(ez)f(h) d 3 p d 3 xf(h) If we introduce a displacement coordinate x = x eeˆ z/mω0 then H = 1 m p + mω 0 ( x ) e E mω0 6 e j

8 and d 3 p d 3 x (ez + e E mω 0 p mol = )f(h) d 3 p d 3 xf(h) Since H is even in z the first integral vanishes. So, independent of the form of f(h), we obtain p mol = e E, mω0 that s just what we found in the beginning of this part: p mol = e x = e E. mω0 The second type of polarizability is that caused by the partial orientation of normally random permanent dipole moments. This orientation polarization is important in polar substances such as HCl and H O. All molecules are assumed to possess a permanent dipole moment p 0, which can be oriented in any direction. In the absence of a field, thermal agitations keeps the molecules randomly oriented so that there is no net dipole moment. When a field is applied, there is a tendency to line up along the field in the configuration of lowest energy. So there will be an average dipole moment. To calculate this we note that the Hamiltonian is given by H = H 0 p 0 E, where H 0 is a function of only the internal coordinates of the molecule. Using the Boltzmann factor f(h) = e H/kT, we can write the average dipole moment as: p mol = dωp0 cosθexp( p 0 EcosΘ kt ) dωexp( p 0 EcosΘ kt ) where E is along the z axis. We note also that the component of p 0 parallel to the field is different from zero. Generally, (p 0 E/kT ) is very small compared to 1, except at low temperatures. So we can expand the exponentials and obtain: p mol = 1 p 0 E. 3 kt The orientation polarization depends inversely on the temperature. We can consider it as an effect in which the applied field must overcome the opposition of thermal agitation. In general both types of polarization, induced and orientation, are present, and the general form of the molecular polarization is: γ mol = γi + 1 p 0. 3ɛ 0 kt This shows a temperature dependence of the form (a + b ) so that the T two types of polarization can be separated in the experiment as you can see on figure. For polar molecules, such as HCl and H O, the observed permanent dipole moments are of the order of an electronic charge times 10 8 cm according with molecular dimensions. 7

9 Experiment.1 Aim of the Experiment The aim of the experiment is to measure the dielectric constant in function of the temperature for two different dielectric media: a molecule with permanent dipole: 4-chlorotoluene a non polar molecule: p-xylene. Experimental Set-Up We have a capacitor filled with one of the two substances. We measure the capacitance in the capacitor with an extern Faradmeter. The capacitor is heated by an extern heating. The temperature we measure by some bi-metal thermometer which we just put into the liquid inside the capacitor..3 Measurement First of all we measure with a ruler the dimensions of the capacitor: outer part of the capacitor is something like a box with 5 sides (open top of the box): depth: 0, 017m length: 0, 085m highth: 0, 0805m with a thickness of walls of 0, 003m. On this part of the capacitor we put one part of the potential difference. inner part is just a square piece of metal which devides the inner part in the box in two parts: length: 0, 053m depth: 0, 0045m highth: 0, 069m. On this part of the capacitor we put the other part of the potential difference. So we try to calculate C 0 from these geometrical arguments. We know that the capacitance of a parallel-plate capacitor is given by: C = Q = ɛ 0A, U d where is Q the charge, U the potential difference between the parallel plates, A the area of the plate and d the distance of the plates. Since our box acts like two parallel plugged capacitors, we can add the capacitances of both: C tot = C 1 + C + C (For serial capacitors the total capacitance would be: 1 C tot = 1 C C + 1 C ) 8

10 So C tot = ɛ0a d = 8, As V m 0,0805m 0,085m 0,017m/ = 1, [ As m m V m m = As = C = F ] V V This value we call C 0. The capacitance of the coaxial wire is C coax = 54µF = F. Further I measured the capacitance of the capacitor filled with air included the capacitance of the coaxial wire: C cap+coax = 80µF, so the capacitance of the capacitor itself is: (80-54) µf = 6µF = C cap = C 0. When we compare the values for C 0 and C 0 we see that there s a huge difference in size. We use C 0 for the calculation..4 Data Analysis We know that ɛ r = C. C 0 The Clausius-Mossotti equation was: γ mol = 3ɛ 0 ( ɛr 1) = ( 3ɛ 0 ) ( C 1 C 0 ) = [ A s m 3 N ɛ r+ N C + V m C 0 = ] A s m A s m = V kg m s 3 A = A s4 = C s 1 kg kg and we expect from the theory γ mol to behave like a + b for the chemical T with permanent dipole. For the non polar chemical we expect to behave γ mol like a constant (e.g. a). So we plot our value of γ mol versus 1 and so we will find out which one T of the two chemicals is the one made of molecules with permanent dipole moments according to the slopes you can see in Figure. If you have a look at the data I measured twice for both substances you will see that there s not much difference in the slope of the curve for the 4- chlorotoluene and the slope of the curve for the p-xylene. I will explain what I did with the data I measured. I took the temperature in [ C] I measured, then I calculated the temperature in [K] and then I calculated 1 T [K]. Then I took the capacitance I measured, calculated it in SI units, then devided it by C 0 I measured (capacitor filled with air), then calculated γ mol by using N [m 3 ], the density of the molecules. Then I plotted 1 versus γ T [K] mol. When I do a linear fit for Y=BX+A on all four data, I get a similar slope B and similar A for all four slopes. A and B are so small, that I can t decide which substance behaves like a constant and which one behaves like a linear curve. I even controlled if the data I measured are the data I took for calculation. When I did the experiment, I really controlled if the substances I took are not yet dense, which means old. The only thing we could prove that the measurement does make sense is when we calculate B for the 4-chlorotoluene by the data given on the exercise sheet. B = p 0 3 k B, where k B = 1, J K the Boltzmann con- 9

11 stant and p 0 = 7, Cm ] the dipole moment. So B = 1, =. As you can see, B is very small and [ C m J K C m K s = C K s kg m kg moreover has the correct SI unit. In the theory part we wrote: For polar molecules, such as HCl and H O, the observed permanent dipole moments are of the order of an electronic charge times 10 8 cm according with molecular dimensions. So this indicates that all these values are very small, especially for the polar substance 4- chlorotoluene..5 Error Evaluation Since we plot the data for: γ mol = 3ɛ 0 ( ɛr 1) = ( 3ɛ 0 ) ( C 1 C 0 ) N ɛ r+ N C + C 0 but fit the data for: γ mol = γi + 1 p 0 A + B, 3ɛ 0 kt T there are only the errors the computer is calculating for A and B. No other error is important. Since the values for A and B in the linear fit are so small, the error calculated by the computer would be at least 10 times smaller and the computer decided not to give me any error values. 3 Problems I had difficulties to decide which formula in Jackson are written in MKS and which of them in cgs system. I note: In MKS, ɛ 0 often appears and when in some formula 1 appears, it s cgs, and to get this formula in MKS 4π just replace 1 by ɛ 4π 0. I m sorry for just having written down the stuff from Jackson ; the first section about Elementary Treatment of Ponderable Media, where the Maxwell equation in ponderable media are derived, I really should have skipped, since this is well known. The main problem was to determine when the system of capacitor and medium is in thermal stable state. So I waited each measurement about 3 4 hours. I think the data and the fit are all nonsense. I expected a clearly determinable constant for the p-xylene and a curve with a nicely positive slope for all measurements of the 4-chlorotoluene. 10

12 4 Conclusion 5 Sources J.D. Jackson, Classical Electrodynamics 11