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University Physics Volume 2

8.5 Molecular Model of a Dielectric

University Physics Volume 28.5 Molecular Model of a Dielectric
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  1. Preface
  2. Unit 1. Thermodynamics
    1. 1 Temperature and Heat
      1. Introduction
      2. 1.1 Temperature and Thermal Equilibrium
      3. 1.2 Thermometers and Temperature Scales
      4. 1.3 Thermal Expansion
      5. 1.4 Heat Transfer, Specific Heat, and Calorimetry
      6. 1.5 Phase Changes
      7. 1.6 Mechanisms of Heat Transfer
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    2. 2 The Kinetic Theory of Gases
      1. Introduction
      2. 2.1 Molecular Model of an Ideal Gas
      3. 2.2 Pressure, Temperature, and RMS Speed
      4. 2.3 Heat Capacity and Equipartition of Energy
      5. 2.4 Distribution of Molecular Speeds
      6. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    3. 3 The First Law of Thermodynamics
      1. Introduction
      2. 3.1 Thermodynamic Systems
      3. 3.2 Work, Heat, and Internal Energy
      4. 3.3 First Law of Thermodynamics
      5. 3.4 Thermodynamic Processes
      6. 3.5 Heat Capacities of an Ideal Gas
      7. 3.6 Adiabatic Processes for an Ideal Gas
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    4. 4 The Second Law of Thermodynamics
      1. Introduction
      2. 4.1 Reversible and Irreversible Processes
      3. 4.2 Heat Engines
      4. 4.3 Refrigerators and Heat Pumps
      5. 4.4 Statements of the Second Law of Thermodynamics
      6. 4.5 The Carnot Cycle
      7. 4.6 Entropy
      8. 4.7 Entropy on a Microscopic Scale
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
  3. Unit 2. Electricity and Magnetism
    1. 5 Electric Charges and Fields
      1. Introduction
      2. 5.1 Electric Charge
      3. 5.2 Conductors, Insulators, and Charging by Induction
      4. 5.3 Coulomb's Law
      5. 5.4 Electric Field
      6. 5.5 Calculating Electric Fields of Charge Distributions
      7. 5.6 Electric Field Lines
      8. 5.7 Electric Dipoles
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
    2. 6 Gauss's Law
      1. Introduction
      2. 6.1 Electric Flux
      3. 6.2 Explaining Gauss’s Law
      4. 6.3 Applying Gauss’s Law
      5. 6.4 Conductors in Electrostatic Equilibrium
      6. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    3. 7 Electric Potential
      1. Introduction
      2. 7.1 Electric Potential Energy
      3. 7.2 Electric Potential and Potential Difference
      4. 7.3 Calculations of Electric Potential
      5. 7.4 Determining Field from Potential
      6. 7.5 Equipotential Surfaces and Conductors
      7. 7.6 Applications of Electrostatics
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    4. 8 Capacitance
      1. Introduction
      2. 8.1 Capacitors and Capacitance
      3. 8.2 Capacitors in Series and in Parallel
      4. 8.3 Energy Stored in a Capacitor
      5. 8.4 Capacitor with a Dielectric
      6. 8.5 Molecular Model of a Dielectric
      7. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    5. 9 Current and Resistance
      1. Introduction
      2. 9.1 Electrical Current
      3. 9.2 Model of Conduction in Metals
      4. 9.3 Resistivity and Resistance
      5. 9.4 Ohm's Law
      6. 9.5 Electrical Energy and Power
      7. 9.6 Superconductors
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    6. 10 Direct-Current Circuits
      1. Introduction
      2. 10.1 Electromotive Force
      3. 10.2 Resistors in Series and Parallel
      4. 10.3 Kirchhoff's Rules
      5. 10.4 Electrical Measuring Instruments
      6. 10.5 RC Circuits
      7. 10.6 Household Wiring and Electrical Safety
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    7. 11 Magnetic Forces and Fields
      1. Introduction
      2. 11.1 Magnetism and Its Historical Discoveries
      3. 11.2 Magnetic Fields and Lines
      4. 11.3 Motion of a Charged Particle in a Magnetic Field
      5. 11.4 Magnetic Force on a Current-Carrying Conductor
      6. 11.5 Force and Torque on a Current Loop
      7. 11.6 The Hall Effect
      8. 11.7 Applications of Magnetic Forces and Fields
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    8. 12 Sources of Magnetic Fields
      1. Introduction
      2. 12.1 The Biot-Savart Law
      3. 12.2 Magnetic Field Due to a Thin Straight Wire
      4. 12.3 Magnetic Force between Two Parallel Currents
      5. 12.4 Magnetic Field of a Current Loop
      6. 12.5 Ampère’s Law
      7. 12.6 Solenoids and Toroids
      8. 12.7 Magnetism in Matter
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    9. 13 Electromagnetic Induction
      1. Introduction
      2. 13.1 Faraday’s Law
      3. 13.2 Lenz's Law
      4. 13.3 Motional Emf
      5. 13.4 Induced Electric Fields
      6. 13.5 Eddy Currents
      7. 13.6 Electric Generators and Back Emf
      8. 13.7 Applications of Electromagnetic Induction
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    10. 14 Inductance
      1. Introduction
      2. 14.1 Mutual Inductance
      3. 14.2 Self-Inductance and Inductors
      4. 14.3 Energy in a Magnetic Field
      5. 14.4 RL Circuits
      6. 14.5 Oscillations in an LC Circuit
      7. 14.6 RLC Series Circuits
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    11. 15 Alternating-Current Circuits
      1. Introduction
      2. 15.1 AC Sources
      3. 15.2 Simple AC Circuits
      4. 15.3 RLC Series Circuits with AC
      5. 15.4 Power in an AC Circuit
      6. 15.5 Resonance in an AC Circuit
      7. 15.6 Transformers
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    12. 16 Electromagnetic Waves
      1. Introduction
      2. 16.1 Maxwell’s Equations and Electromagnetic Waves
      3. 16.2 Plane Electromagnetic Waves
      4. 16.3 Energy Carried by Electromagnetic Waves
      5. 16.4 Momentum and Radiation Pressure
      6. 16.5 The Electromagnetic Spectrum
      7. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
  4. A | Units
  5. B | Conversion Factors
  6. C | Fundamental Constants
  7. D | Astronomical Data
  8. E | Mathematical Formulas
  9. F | Chemistry
  10. G | The Greek Alphabet
  11. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
    12. Chapter 12
    13. Chapter 13
    14. Chapter 14
    15. Chapter 15
    16. Chapter 16
  12. Index

Learning Objectives

By the end of this section, you will be able to:
  • Explain the polarization of a dielectric in a uniform electrical field
  • Describe the effect of a polarized dielectric on the electrical field between capacitor plates
  • Explain dielectric breakdown

We can understand the effect of a dielectric on capacitance by looking at its behavior at the molecular level. As we have seen in earlier chapters, in general, all molecules can be classified as either polar or nonpolar. There is a net separation of positive and negative charges in an isolated polar molecule, whereas there is no charge separation in an isolated nonpolar molecule (Figure 8.19). In other words, polar molecules have permanent electric-dipole moments and nonpolar molecules do not. For example, a molecule of water is polar, and a molecule of oxygen is nonpolar. Nonpolar molecules can become polar in the presence of an external electrical field, which is called induced polarization.

The figures show a large scale view of an atom. Figure a shows an unpolarized atom, with protons and neutrons in the center and a circular electron cloud surrounding the nucleus. Figure b shows a polarized atom and positive and negative external charges. The atom is oblong in shape with the electron cloud being pulled towards the positive external charge, and the nucleus being pulled towards the negative external charge. Figure c shows another oblong polarized atom.
Figure 8.19 The concept of polarization: In an unpolarized atom or molecule, a negatively charged electron cloud is evenly distributed around positively charged centers, whereas a polarized atom or molecule has an excess of negative charge at one side so that the other side has an excess of positive charge. However, the entire system remains electrically neutral. The charge polarization may be caused by an external electrical field. Some molecules and atoms are permanently polarized (electric dipoles) even in the absence of an external electrical field (polar molecules and atoms).

Let’s first consider a dielectric composed of polar molecules. In the absence of any external electrical field, the electric dipoles are oriented randomly, as illustrated in Figure 8.20(a). However, if the dielectric is placed in an external electrical field E0E0, the polar molecules align with the external field, as shown in part (b) of the figure. Opposite charges on adjacent dipoles within the volume of dielectric neutralize each other, so there is no net charge within the dielectric (see the dashed circles in part (b)). However, this is not the case very close to the upper and lower surfaces that border the dielectric (the region enclosed by the dashed rectangles in part (b)), where the alignment does produce a net charge. Since the external electrical field merely aligns the dipoles, the dielectric as a whole is neutral, and the surface charges induced on its opposite faces are equal and opposite. These induced surface charges +Qi+Qi and QiQi produce an additional electrical field EiEi (an induced electrical field), which opposes the external field E0E0, as illustrated in part (c).

Figure a shows a dielectric material with oblong shaped molecules within it. Each molecule has a plus sign on one side and a minus sign on the other. They are all randomly arranged. Figure b shows the same molecules now perfectly aligned in rows and columns, with the negative end of each molecule on the upper side. An external electric field E0 points downwards. A dashed line encompasses the negative signs of all the molecules in the topmost row. Similarly, a dashed line encompasses the positive signs of all the molecules in the bottommost row. Figure c shows negative signs at the top of the dielectric, labeled minus Qi, and positive signs at the bottom, labeled plus Qi. The induced field Ei within the dielectric, points upwards.
Figure 8.20 A dielectric with polar molecules: (a) In the absence of an external electrical field; (b) in the presence of an external electrical field E0E0. The dashed lines indicate the regions immediately adjacent to the capacitor plates. (c) The induced electrical field EiEi inside the dielectric produced by the induced surface charge QiQi of the dielectric. Note that, in reality, the individual molecules are not perfectly aligned with an external field because of thermal fluctuations; however, the average alignment is along the field lines as shown.

The same effect is produced when the molecules of a dielectric are nonpolar. In this case, a nonpolar molecule acquires an induced electric-dipole moment because the external field E0E0 causes a separation between its positive and negative charges. The induced dipoles of the nonpolar molecules align with E0E0 in the same way as the permanent dipoles of the polar molecules are aligned (shown in part (b)). Hence, the electrical field within the dielectric is weakened regardless of whether its molecules are polar or nonpolar.

Therefore, when the region between the parallel plates of a charged capacitor, such as that shown in Figure 8.21(a), is filled with a dielectric, within the dielectric there is an electrical field E0E0 due to the free charge Q0Q0 on the capacitor plates and an electrical field EiEi due to the induced charge QiQi on the surfaces of the dielectric. Their vector sum gives the net electrical field EE within the dielectric between the capacitor plates (shown in part (b) of the figure):

E=E0+Ei.E=E0+Ei.
(8.13)

This net field can be considered to be the field produced by an effective charge Q0QiQ0Qi on the capacitor.

Figure a shows an empty charged capacitor. Arrows representing electric field E0 go from the positive plate to the negative one. Figure b shows a dielectric-filled charged capacitor. Arrows representing electric field E go from the positive plate to the negative one. The dielectric has negative charges accumulated near the surface of the positive plate and positive charges accumulated near the surface of the negative plate.
Figure 8.21 Electrical field: (a) In an empty capacitor, electrical field E0E0. (b) In a dielectric-filled capacitor, electrical field EE.

In most dielectrics, the net electrical field EE is proportional to the field E0E0 produced by the free charge. In terms of these two electrical fields, the dielectric constant κκ of the material is defined as

κ=E0E.κ=E0E.
(8.14)

Since E0E0 and EiEi point in opposite directions, the magnitude E is smaller than the magnitude E0E0 and therefore κ>1.κ>1. Combining Equation 8.14 with Equation 8.13, and rearranging the terms, yields the following expression for the induced electrical field in a dielectric:

Ei=(1κ1)E0.Ei=(1κ1)E0.
(8.15)

When the magnitude of an external electrical field becomes too large, the molecules of dielectric material start to become ionized. A molecule or an atom is ionized when one or more electrons are removed from it and become free electrons, no longer bound to the molecular or atomic structure. When this happens, the material can conduct, thereby allowing charge to move through the dielectric from one capacitor plate to the other. This phenomenon is called dielectric breakdown. (Figure 8.1 shows typical random-path patterns of electrical discharge during dielectric breakdown.) The critical value, EcEc, of the electrical field at which the molecules of an insulator become ionized is called the dielectric strength of the material. The dielectric strength imposes a limit on the voltage that can be applied for a given plate separation in a capacitor. For example, the dielectric strength of air is Ec=3.0MV/mEc=3.0MV/m, so for an air-filled capacitor with a plate separation of d=1.00mm,d=1.00mm, the limit on the potential difference that can be safely applied across its plates without causing dielectric breakdown is V=Ecd=(3.0×106V/m)(1.00×10−3m)=3.0kVV=Ecd=(3.0×106V/m)(1.00×10−3m)=3.0kV.

However, this limit becomes 60.0 kV when the same capacitor is filled with Teflon™, whose dielectric strength is about 60.0MV/m60.0MV/m. Because of this limit imposed by the dielectric strength, the amount of charge that an air-filled capacitor can store is only Q0=κairC0(3.0kV)Q0=κairC0(3.0kV) and the charge stored on the same Teflon™-filled capacitor can be as much as

Q=κteflonC0(60.0kV)=κteflonQ0κair(3.0kV)(60.0kV)=20κteflonκairQ0=202.11.00059Q042Q0,Q=κteflonC0(60.0kV)=κteflonQ0κair(3.0kV)(60.0kV)=20κteflonκairQ0=202.11.00059Q042Q0,

which is about 42 times greater than a charge stored on an air-filled capacitor. Typical values of dielectric constants and dielectric strengths for various materials are given in Table 8.1. Notice that the dielectric constant κκ is exactly 1.0 for a vacuum (the empty space serves as a reference condition) and very close to 1.0 for air under normal conditions (normal pressure at room temperature). These two values are so close that, in fact, the properties of an air-filled capacitor are essentially the same as those of an empty capacitor.

Material Dielectric constant κκ Dielectric strength Ec[×106V/m]Ec[×106V/m]
Vacuum 1
Dry air (1 atm) 1.00059 3.0
Teflon™ 2.1 60 to 173
Paraffin 2.3 11
Silicon oil 2.5 10 to 15
Polystyrene 2.56 19.7
Nylon 3.4 14
Paper 3.7 16
Fused quartz 3.78 8
Glass 4 to 6 9.8 to 13.8
Concrete 4.5
Bakelite 4.9 24
Diamond 5.5 2,000
Pyrex glass 5.6 14
Mica 6.0 118
Neoprene rubber 6.7 15.7 to 26.7
Water 80
Sulfuric acid 84 to 100
Titanium dioxide 86 to 173
Strontium titanate 310 8
Barium titanate 1,200 to 10,000
Calcium copper titanate > 250,000
Table 8.1 Representative Values of Dielectric Constants and Dielectric Strengths of Various Materials at Room Temperature

Not all substances listed in the table are good insulators, despite their high dielectric constants. Water, for example, consists of polar molecules and has a large dielectric constant of about 80. In a water molecule, electrons are more likely found around the oxygen nucleus than around the hydrogen nuclei. This makes the oxygen end of the molecule slightly negative and leaves the hydrogens end slightly positive, which makes the molecule easy to align along an external electrical field, and thus water has a large dielectric constant. However, the polar nature of water molecules also makes water a good solvent for many substances, which produces undesirable effects, because any concentration of free ions in water conducts electricity.

Example 8.11

Electrical Field and Induced Surface Charge Suppose that the distance between the plates of the capacitor in Example 8.10 is 2.0 mm and the area of each plate is 4.5×10−3m24.5×10−3m2. Determine: (a) the electrical field between the plates before and after the Teflon™ is inserted, and (b) the surface charge induced on the Teflon™ surfaces.

Strategy In part (a), we know that the voltage across the empty capacitor is V0=40VV0=40V, so to find the electrical fields we use the relation V=EdV=Ed and Equation 8.14. In part (b), knowing the magnitude of the electrical field, we use the expression for the magnitude of electrical field near a charged plate E=σ/ε0E=σ/ε0, where σσ is a uniform surface charge density caused by the surface charge. We use the value of free charge Q0=8.0×10−10CQ0=8.0×10−10C obtained in Example 8.10.

Solution

  1. The electrical field E0E0 between the plates of an empty capacitor is
    E0=V0d=40V2.0×10−3m=2.0×104V/m.E0=V0d=40V2.0×10−3m=2.0×104V/m.

    The electrical field E with the Teflon™ in place is
    E=1κE0=12.12.0×104V/m=9.5×103V/m.E=1κE0=12.12.0×104V/m=9.5×103V/m.
  2. The effective charge on the capacitor is the difference between the free charge Q0Q0 and the induced charge QiQi. The electrical field in the Teflon™ is caused by this effective charge. Thus
    E=1ε0σ=1ε0Q0QiA.E=1ε0σ=1ε0Q0QiA.

    We invert this equation to obtain QiQi, which yields
    Qi=Q0ε0AE=8.0×10−10C(8.85×10−12C2N·m2)(4.5×10−3m2)(9.5×103Vm)=4.2×10−10C=0.42nC.Qi=Q0ε0AE=8.0×10−10C(8.85×10−12C2N·m2)(4.5×10−3m2)(9.5×103Vm)=4.2×10−10C=0.42nC.

Example 8.12

Inserting a Dielectric into a Capacitor Connected to a Battery When a battery of voltage V0V0 is connected across an empty capacitor of capacitance C0C0, the charge on its plates is Q0Q0, and the electrical field between its plates is E0E0. A dielectric of dielectric constant κκ is inserted between the plates while the battery remains in place, as shown in Figure 8.22. (a) Find the capacitance C, the voltage V across the capacitor, and the electrical field E between the plates after the dielectric is inserted. (b) Obtain an expression for the free charge Q on the plates of the filled capacitor and the induced charge QiQi on the dielectric surface in terms of the original plate charge Q0Q0.

Figure a shows a capacitor connected to a battery. The capacitor has voltage V0 across it. The positive and negative plates of the capacitor have charge plus Q0 and minus Q0 respectively. Figure b shows the same capacitor with a dielectric inserted in it. The charge on the positive and negative plates is now plus Q and minus Q respectively. Negative charges are shown accumulated near the inner surface of the positive plate. These are labeled minus Qi. Positive charges are shown accumulated near the inner surface of the negative plate. These are labeled plus Qi.
Figure 8.22 A dielectric is inserted into the charged capacitor while the capacitor remains connected to the battery.

Strategy We identify the known values: V0V0, C0C0,E0E0, κκ, and Q0Q0. Our task is to express the unknown values in terms of these known values.

Solution (a) The capacitance of the filled capacitor is C=κC0C=κC0. Since the battery is always connected to the capacitor plates, the potential difference between them does not change; hence, V=V0V=V0. Because of that, the electrical field in the filled capacitor is the same as the field in the empty capacitor, so we can obtain directly that

E=Vd=V0d=E0.E=Vd=V0d=E0.

(b) For the filled capacitor, the free charge on the plates is

Q=CV=(κC0)V0=κ(C0V0)=κQ0.Q=CV=(κC0)V0=κ(C0V0)=κQ0.

The electrical field E in the filled capacitor is due to the effective charge QQiQQi (Figure 8.22(b)). Since E=E0E=E0, we have

QQiε0A=Q0ε0A.QQiε0A=Q0ε0A.

Solving this equation for QiQi, we obtain for the induced charge

Qi=QQ0=κQ0Q0=(κ1)Q0.Qi=QQ0=κQ0Q0=(κ1)Q0.

Significance Notice that for materials with dielectric constants larger than 2 (see Table 8.1), the induced charge on the surface of dielectric is larger than the charge on the plates of a vacuum capacitor. The opposite is true for gasses like air whose dielectric constant is smaller than 2.

Check Your Understanding 8.8

Continuing with Example 8.12, show that when the battery is connected across the plates the energy stored in dielectric-filled capacitor is U=κU0U=κU0 (larger than the energy U0U0 of an empty capacitor kept at the same voltage). Compare this result with the result U=U0/κU=U0/κ found previously for an isolated, charged capacitor.

Check Your Understanding 8.9

Repeat the calculations of Example 8.10 for the case in which the battery remains connected while the dielectric is placed in the capacitor.

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