Chapter 7: Polarization

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1 Chapter 7: Polarization Joaquín Bernal Méndez Group 4 1 Index Introduction Polarization Vector The Electric Displacement Vector Constitutive Laws: Linear Dielectrics Energy in Dielectric Systems Forces on Dielectrics 2

2 Introduction Conductors: contain a great amount of free charge Dielectrics: all charges are attached to specific atoms or molecules Examples: wood, plastic, stone... Then How does a dielectric substance respond to an external electrostatic field? Charges attached to molecules or atoms undergo microscopic displacements 3 Induced Dipoles An atom has a positively charged core (the nucleus) and a negatively charged electron cloud surrounding it The nucleus is pushed in the direction of the field and the electrons the opposite way: The atom gets polarized Induced dipole moment: Polarizability If the electric field is too strong this relationship can become nonlinear and the atom can even be ionized 4

3 Alignment of Polar Molecules Some molecules have permanent dipole moments that are not due to the action of an external electric field Example: water molecule Polar Molecules This polar molecules tends to rotate to line up its dipole moment parallel to the external electric field 5 Polarization Dielectrics with neutral atoms or nonpolar molecules: A dipole moment parallel to the field is induced in each atom or molecule by the applied electric field Dielectrics with polar molecules: The external electric field exerts a torque on each molecule that tends to line it up along the field direction This will not be a complete alignment due to the effect of random thermal motion In both cases we obtain a polarized dielectric: a lot of little dipoles aligned with the external field 6

4 Index Introduction Polarization Vector The Electric Displacement Vector Constitutive Laws: Linear Dielectrics Energy in Dielectric Systems Forces on Dielectrics 7 Polarization Vector We are going to study the field due to a piece of polarized material We will forget for a moment about the cause of the polarization Each molecule has a dipole moment: From a macroscopic point of view we define the polarization vector: Dipole moment per unit volume 8

5 Electric Field Due to a Polarized Material Let's suppose that we know the polarization vector. Can we calculate the the electric field created by the polarized material? Idea: the total field can be obtained as a superposition of the fields of all the tiny dipoles inside the material Potential due to a dipole at the origin: If the dipole is located at an arbitrary point 9 Electric Field Due to a Polarized Material Dipole moment due to a volume element : Potential created by this volume element: Integrating over the volume of the polarized material: 10

6 Electric Field Due to a Polarized Material This potential can be expressed in a different way By using: We can write down the integrand as: 11 Electric Field Due to a Polarized Material And we arrive to: By applying the Divergence Theorem: : Volume of the polarized material : Surface boundary of the polarized material 12

7 Polarization Charges Potential created by a volume and a surface charge densities: By analogy we van define: Surface density of polarization charges Volume density of polarization charges 13 Polarization Charges We can calculate the electric field produced by the polarized material by finding the polarization charges and calculating the field that they produce We get to a problem of electrostatics (chapter 3) We must know the polarization vector to apply this technique Questions about polarization charges: Are they actual charges or just a mathematical tool? If they are true charges, How does polarization lead to such accumulation of charge in a neutral material? 14

8 Physical Meaning of the Polarization Charges Uniformly polarized material: The head of a dipole cancels the tail of its neighbor But at the ends are two layers of charges left over: 15 Physical Meaning of the Polarization Charges Piece of material with nonuniform polarization There is not complete compensation between adjacent positive and negative layers net bound charge within the material Polarization charges are real accumulations of charge 16

9 Total Polarization Charge The total charge can be calculated by summing the surface and volume polarization charges: Divergence Theorem There is no total polarization charge in a polarized material (unless free charge has been deposited) 17 Index Introduction Polarization Vector The Electric Displacement Vector Constitutive Laws: Linear Dielectrics Energy in Dielectric Systems Forces on Dielectrics 18

10 The Electric Displacement Vector We have already calculated the field crated by a polarized material: polarization charges The total electric field is that produced by both the polarization charges and the free charges. This field obeys the Gauss's Law: with: Electric displacement vector 19 The Electric Displacement Vector Gauss's Law can be written in terms of the electric displacement vector: Differential form Integral form This is an auxiliary field: it can not be measured Units: C/m2 (same as Its scalar sources are only the free charges Boundary condition: ) 20

11 Vector Sources of the Electric Displacement Vector A vector field is determined by its divergence (scalar sources) and its curl (vector sources) From the definition: (Electrostatics) We get to: The curl of the polarization vector is the vector source of the electric displacement vector 21 Usefulness of the Electric Displacement Vector The parallel between and is subtle: The electric displacement vector can NOT be obtained in the same way as the electric field but forgetting about the polarization charges However for highly symmetric situations we usually have: and then the electric displacement vector can be calculated in terms of the free charge from the Gauss's Law: 22

12 Example Parallel-plate capacitor filled with a dielectric slab Plane symmetry By applying Gauss's Law: 23 Example Infinite straight line with a uniform line charge λ surrounded by a dielectric cylinder can be expressed in terms of the free charges: If we knew we could calculate by using: BUT USUALLY: we need to know the functional form of this relationship (constitutive equation) 24

13 Index Introduction Polarization Vector The Electric Displacement Vector Constitutive Laws: Linear Dielectrics Energy in Dielectric Systems Forces on Dielectrics 25 Constitutive Laws A dielectric is usually polarized due to an external electric field For many substances the polarization is proportional to the field: : electric susceptibility (dimensionless) In vacuum: is the total electric field (due to free and polarization charges), not the externally applied electric field Substances verifying this constitutive equation are referred to as linear dielectrics 26

14 Linear Dielectrics Homogeneous: its susceptibility is independent of position Isotropic: its susceptibility is a scalar magnitude (in instead of a tensor) Linear: polarization is proportional to the field This is true as long as the field is not too strong There exit substances not obeying this law: Ferroelectric materials: the polarization depends on the history of the particular chunk of material Electrets: materials which are able to hold a permanent electric polarization in absence of an external field 27 Linear Dielectrics Relationship between the electric displacement and the electric field: We define: Permittivity of the material: Relative permittivity: Therefore: ;(F/m) and: 28

15 Dielectric Constants for Some Common Substances Material Air Glass 4-10 Paper 2-4 Wood Porcelain 6-8 Rubber Ethyl Alcohol 28.4 sodium chloride 6.1 Sea water 72 Distilled water Example Parallel-plate capacitor filled with a linear dielectric We have already obtained: The capacitance is increased by a factor of 30

16 Example Parallel-plate capacitor: polarization charges In general, for linear dielectrics: 31 Capacitor filled with Insulating Material For a given free charge, the potential difference is smaller when the capacitor is filled with a dielectric material: Because the electric field between the plates is partially shielded by the polarization charges and hence its magnitude is smaller than in the vacuum case For a given difference of potential, the accumulated free charge is larger when the capacitor is filled with a dielectric material: Because an extra amount of free charge is needed to counteract the effect of the polarization charges in order to attain the same electric field between the plates 32

17 Example Parallel plate capacitor partially filled with a dielectric Applying Gauss's Law: Capacitance of two capacitors connected in series 33 Example Parallel-plate capacitor partially filled with a dielectric In both regions must be verified that: Capacitance of two capacitors connected in parallel 34

18 Example Conducting sphere carrying a charge q surrounded by a dielectric sphere Symmetry: Gauss's Law: 35 Example Conducting sphere carrying a charge q surrounded by a dielectric sphere: polarization charges Inside the material: (linear dielectric) Exercise: check that the total polarization charge is zero 36

19 Index Introduction Polarization Vector The Electric Displacement Vector Constitutive Laws: Linear Dielectrics Energy in Dielectric Systems Forces on Dielectrics 37 Energy in Dielectric Systems We already know: This equation give us the work that it takes to bring all the charges from infinity to their final positions When dealing with dielectric systems it is more convenient to use this formula: This equation give us the work that it takes to bring the free charges from infinity to their final positions Both formulas are correct, but they represent different things 38

20 Energy Stored in a Parallel-Plate Capacitor For a parallel-plate capacitor filled with a dielectric: We have calculated: Therefore: 39 Forces on Dielectrics The force exerted on the dielectric material in a direction can be calculated by using the principle of virtual work: Example: dielectric slab partially inserted between the plates of a parallel-plate capacitor: 40

21 Summary (I) Polarization is the response of dielectric materials to external electric fields: The dipole moments of the molecules of the dielectric tends to line up in the direction of the electric field. The polarization vector describes the polarization of the material from a macroscopic point of view. Polarization charges account for the electric field created by the polarized material. The electric displacement vector is an auxiliary vector field whose scalar sources are the free charges. 41 Summary (II) In highly symmetric distributions the electric displacement vector can be calculated as a function of the free charges by using Gauss's Law. To calculate the total electric field we also need a constitutive equation of the medium, which gives us the relationship between the polarization vector and the electric field. For linear media the polarization vector is proportional to the electric field. To calculate the energy of dielectric systems we have introduced an alternative formula of the energy that does not include the work required to bring the polarization charges from infinity. 42

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