Energy in Dielectric System

11 Key excerpts on "Energy in Dielectric System"

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  eBook - ePub

  Conceptual Electromagnetics

  • Branislav M. Notaroš(Author)
  • 2017(Publication Date)
  • CRC Press

    (Publisher)

  (2.6) ]

  w e = 1 2 ε E 2 = 1 2 E D = D 2 2 ε   ( electric   energy   density ;   unit : J / m 3 ) .

  (2.20)

  The total electric energy of the system is now obtained as We = ∫v we dv, where v denotes the volume of the system dielectric.

  CONCEPTUAL QUESTION 2.39

  Change of capacitor energy due to a change of dielectric.

  The dielectric in a spherical capacitor [Figure 2.9(a) ] is oil. The capacitor is connected to a voltage source. The source is then disconnected and the oil is drained from the capacitor. The energy of the capacitor in the final electrostatic state is

  (A)  larger than (B)  the same as (C)  smaller than before the source was disconnected.

  CONCEPTUAL QUESTION 2.40

  Dielectric drain under different circumstances.

  The oil dielectric in a spherical capacitor is completely drained while the voltage source is still connected to the capacitor. As a result, the energy of the capacitor

  (A)  increases. (B)  decreases. (C)  remains the same.

  CONCEPTUAL QUESTION 2.41

  Energies of isolated metallic spheres of different sizes.

  Consider two isolated metallic spheres with the same charges and different radii in air, and compare their energies (note that an isolated sphere can be regarded as the inner electrode of a spherical capacitor whose outer electrode has an infinite radius). The larger energy is that of

  (A)  the larger sphere. (B)  the smaller sphere. (C)  The energies are the same. (D)  Need more information.

  CONCEPTUAL QUESTION 2.42

  Change of field intensity/flux density and energy.

  Two capacitors contain the same amounts of electric energy. If the electric field intensity (E) at every point in the first capacitor becomes twice as large, while the electric flux density (D

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  Halliday and Resnick's Principles of Physics

  • David Halliday, Robert Resnick, Jearl Walker(Authors)
  • 2020(Publication Date)
  • Wiley

    (Publisher)

  This energy can be associated with the capacitor’s electric field E → . By extension we can associate stored energy with any electric field. In vacuum, the energy density u, or potential energy per unit volume, within an electric field of magnitude E is given by u = 1 2 ε 0 E 2 . (25-25) Capacitance with a Dielectric If the space between the plates of a capacitor is completely filled with a dielectric material, the capacitance C is increased by a factor κ, called the dielectric constant, which is characteristic of the material. In a region that is completely filled by a dielectric, all electrostatic equations con- taining ε 0 must be modified by replacing ε 0 with κε 0 . The effects of adding a dielectric can be understood physi- cally in terms of the action of an electric field on the permanent or induced electric dipoles in the dielectric slab. The result is the for- mation of induced charges on the surfaces of the dielectric, which results in a weakening of the field within the dielectric for a given amount of free charge on the plates. Gauss’ Law with a Dielectric When a dielectric is present, Gauss’ law may be generalized to ε 0 ∮ κ E → · dA → = q. (25-36) Here q is the free charge; any induced surface charge is accounted for by including the dielectric constant κ inside the integral. 641 PROBLEMS Figure 25-18 Problem 1. 2C 4C C 6C A B D b d Figure 25-19 Problems 3 and 4. C + – S Figure 25-20 Problem 5. (a) 0.40 μF and (b) 1.2 μF, each combination capable of withstand- ing 1000 V? 7 In Fig. 25-21, the battery has potential difference V = 9.0 V, C 2 = 3.0 μF, C 4 = 4.0 μF, and all the capacitors are initially uncharged. When switch S is closed, a total charge of 15 μC passes through point a and a total charge of 6.0 μC passes through point b. What are (a) C 1 and (b) C 3 ? 8 Figure 25-22 shows a parallel- plate capacitor with a plate area A = 3.90 cm 2 and separation d = 5.56 mm.

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  Steady Electric Fields and Currents

  Elementary Electromagnetic Theory

  • B. H. Chirgwin, C. Plumpton, C. W. Kilmister(Authors)
  • 2013(Publication Date)
  • Pergamon

    (Publisher)

  However, for the steady conditions of an electrostatic field of moderate intensity, K may be taken to be independent of the field quantities. When an insulator is situated in a field which presumably penetrates into the material of the insulator we call the substance of the insulator a dielectric. We now modify our theory to take account of the presence of dielectrics; our choice of modification is suggested by the results described above and by our rough ideas of the structure of matter as outlined in Chapter 1. When a dielectric is situated in a field the positive and negative charges of 111 112 ELEMENTARY ELECTROMAGNETIC THEORY which the dielectric is constructed experience forces in opposite directions along the lines of force of the field. The characteristic of an insulator (in contrast to a conductor) is that in response to such forces these charges are displaced, or strained, from their normal equilibrium positions and take up new equilibrium positions. (In a conductor the charges move until any excess charge is situated on the surface of the conductor.) In these new positions of equilibrium a pair of positive and negative charges which originally neutra-lized each other are separated by a short distance, and so constitute a close approximation to a dipole. Thus, in addition to any simple charge which may be situated on or inside the dielectric, every element of volume becomes polarized. Before developing the analysis in the general case we consider the effect in the special uniform case of a parallel plate condenser whose plates are separ-ated by a distance /. We consider two situations: (1) The plates carry charges of density ±σ in the absence of a dielectric. Then D σ = ε 0 Ε, V = Et = σΐ/ε 0 , C = σ/Κ = ε 0 /ί, where V is the potential difference between the plates and C is the capacity of unit area of the (empty) condenser.

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  Integration of Ferroelectric and Piezoelectric Thin Films

  Concepts and Applications for Microsystems

  • Emmanuel Defaÿ(Author)
  • 2013(Publication Date)
  • Wiley-ISTE

    (Publisher)

  Chapter 5

  Dielectric Formalism 1

       

  5.1. Introduction

  In this chapter we introduce well-established notions about the electric variables of an insulating material. These variables are clear enough for people who know about the physics of solids, but experience shows that the concepts are more difficult to understand in the microsystems community. It therefore seems important to us to dwell at length, just like we did for mechanical energetics, on the electric part in order to introduce the indispensable notions:

  – the average and local electric field; – electric displacement and polarization; – the relation between field strength and polarization; – the polarization catastrophe phenomenon that appears in perovskites; – dielectric relaxation and the different contributions to polarization characteristic of perovskites; and finally – electric energy density which contributes to internal energy.

  5.2. The dielectric effect seen by Faraday

  Electric charges cross a conductor when a potential difference is applied. In the case of a perfect insulator, charges cannot cross. The dielectric effect observed for the first time by Faraday corresponds to the mutual influence of opposite charges contained in the two electrodes of a plane capacitor. The most telling experiment of this dielectric effect is illustrated in Figure 5.1 . This shows the decrease in potential difference observed between the electrodes of a pre-charged isolated plane capacitor (see Figure 5.1a ) when a dielectric material is introduced in the space between the electrodes (see Figure 5.1b

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  Theory of Electric Polarization

  Dielectrics in Static Fields

  • Bozzano G Luisa(Author)
  • 2012(Publication Date)
  • Elsevier Science

    (Publisher)

  6 In this way temperatures of 0.3°K have been reached. §14. The energy of a dielectric in an external field It is often useful to have an expression for that part of the free energy of a system consisting of a dielectric and a charge distribution, that refers to the dielectric. This can be obtained by comparing the amounts of work necessary to assemble a charge distribution in the presence of a dielectric and without it. The difference between these amounts of work will be equal to the non-mechanical part of the work required to bring the dielectric into the field of the charge distribution. Thus, this difference can be considered to be the change in free energy of the dielectric upon application of the field. 104 C. J. F. BOTTCHER According to eqn. (3.42), the work required to assemble a charge distri-bution is given by: W = ~ 111 Ε · D dv. When there is no dielectric, the amount of work necessary to assemble the same charge distribution is given by (3.39): *.-sJJ]V*.««* The subscript 0 is added to indicate the absence of a dielectric, and E 0 will be called the external field. The difference AW = W — W 0 between these two amounts of work will be the change in the free energy of the dielectric when the field is applied in an isothermal reversible process (cf. section 13): A W = l l S i E D d v -L t t l E ° E ° a v -, 3 -6 0 ) 00 00 This expression can be simplified by noting that the charge distribution producing the field and polarizing the dielectric will be the same in the presence and in the absence of the dielectric, i.e.: div D = div E 0 = 4πρ, (3.61) or: div (D -£ 0 ) = 0. (3.62) If we now consider a vector field J/(D — E 0 ) where ψ is the potential of Ε or E 0 , we may write for the divergence of this field, according to (A 1.25) and (3.62): div f/(D - E 0 ) = (D — E 0 ) · grad ψ.

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  Electron Phonon Interactions: A Novel Semiclassified Approach

  A Novel Semiclassical Approach

  • Albert Rose(Author)
  • 1989(Publication Date)
  • World Scientific

    (Publisher)

  The act of imparting energy is defined by the operation of suddenly displacing the electron in the medium. The energy left behind in the medium where the electron was is the energy imparted. This operational definition is, of course, designed to match the manner in which a moving electron leaves behind a trail of energy in the medium. In addition to the formal definition of /? as the fraction of available coulomb energy imparted to the medium, two other expressions for f$ were obtained for use in particular cases, namely, P = , [6] K L and Electrical energy [7] Lattice deformation For well-separated resonances in a solid as, for example, the ionic resonances at a few hundredth of a volt and the electronic resonances at a few volts, the dielectric constants K H and K L in Eq. [6] refer to values above and below the resonance in question and on the flat parts of the dielectric constant versus frequency curves (see Fig. 2). RCA Review • Vol. 32 • September 1971 467 7 7 •ad) v polarizi '(K H d) C H d) when 111CU1U / r) 2 , W nsfer o v a ] mec 'r) 2 , isfei ne meaium. l T/T) 2 , which ( lb energy imparl Dbtained for use £ ^L — # # # L Kr Electrical energy I 0 = Total energy J Lattice deformation were ooiamea lor use : K L &L — %H K L Eq. [7] is more general. If one thinks of any elastic deformation of a solid, there will be associated with this deformation an elastic energy and, in general, an electrical field energy. The total energy of the deformation will be the sum of the elastic energy and the accom-panying electric field energy as, for example, in the case of a deforma-tion in a piezoelectric solid. This is the total energy that appears in the denominator of Eq. [7], The electrical energy that appears in the numerator denotes the energy that electrons can gain by relaxing to the new conditions of the deformed medium.

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  Fundamentals of Physics, Extended

  • David Halliday, Robert Resnick, Jearl Walker(Authors)
  • 2018(Publication Date)
  • Wiley

    (Publisher)

  If the slab were allowed to slide between the plates with no restraint and if there were no friction, the slab would oscillate back and forth between the plates with a (constant) mechani- cal energy of 893 pJ, and this system energy would transfer back and forth between kinetic energy of the moving slab and potential energy stored in the electric field. Sample Problem 25.05 Work and energy when a dielectric is inserted into a capacitor A parallel-plate capacitor whose capacitance C is 13.5 pF is charged by a battery to a potential difference V = 12.5 V between its plates. The charging battery is now discon- nected, and a porcelain slab (κ = 6.50) is slipped between the plates. (a) What is the potential energy of the capacitor before the slab is inserted? KEY IDEA We can relate the potential energy U i of the capacitor to the capacitance C and either the potential V (with Eq. 25-22) or the charge q (with Eq. 25-21): U i = 1 2 CV 2 = q 2 2C . Calculation: Because we are given the initial potential V (= 12.5 V), we use Eq. 25-22 to find the initial stored energy: U i = 1 2 CV 2 = 1 2 (13.5 × 10 −12 F)(12.5 V) 2 = 1.055 × 10 −9 J = 1055 pJ ≈ 1100 pJ. (Answer) (b) What is the potential energy of the capacitor–slab device after the slab is inserted? Additional examples, video, and practice available at WileyPLUS Dielectrics: An Atomic View What happens, in atomic and molecular terms, when we put a dielectric in an electric field? There are two possibilities, depending on the type of molecule: 1. Polar dielectrics. The molecules of some dielectrics, like water, have perma- nent electric dipole moments. In such materials (called polar dielectrics), the 734 CHAPTER 25 CAPACITANCE (a) E 0 = 0 The initial electric field inside this nonpolar dielectric slab is zero. + – + – + – + – + – + – + – + – + – + – + – + – E 0 (b) + + + + + + + + – – – – – – – – The applied field aligns the atomic dipole moments.

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  Fundamentals of Physics

  • David Halliday, Robert Resnick, Jearl Walker(Authors)
  • 2021(Publication Date)
  • Wiley

    (Publisher)

  776 CHAPTER 25 CAPACITANCE Thus, the magnitude of the electric field produced by a point charge inside a dielectric is given by this modified form of Eq. 23.6.1: E = 1 _____ 4πκε 0 q __ r 2 . (25.5.3) Also, the expression for the electric field just outside an isolated conductor immersed in a dielectric (see Eq. 23.3.1) becomes E = σ ___ κε 0 . (25.5.4) Because κ is always greater than unity, both these equations show that for a fixed distribution of charges, the effect of a dielectric is to weaken the electric field that would otherwise be present. Dielectrics: An Atomic View What happens, in atomic and molecular terms, when we put a dielectric in an electric field? There are two possibilities, depending on the type of molecule: 1. Polar dielectrics. The molecules of some dielectrics, like water, have perma- nent electric dipole moments. In such materials (called polar dielectrics), the electric dipoles tend to line up with an external electric field as in Fig. 25.5.3. Sample Problem 25.5.1 Work and energy when a dielectric is inserted into a capacitor A parallel-plate capacitor whose capacitance C is 13.5 pF is charged by a battery to a potential difference V = 12.5 V between its plates. The charging battery is now discon- nected, and a porcelain slab (κ = 6.50) is slipped between the plates. (a) What is the potential energy of the capacitor before the slab is inserted? KEY IDEA We can relate the potential energy U i of the capacitor to the capacitance C and either the potential V (with Eq. 25.4.2) or the charge q (with Eq. 25.4.1): U i = 1 _ 2 CV 2 = q 2 ___ 2C . Calculation: Because we are given the initial potential V (= 12.5 V), we use Eq. 25.4.2 to find the initial stored energy: U i = 1 _ 2 CV 2 = 1 _ 2 (13.5 × 10 −12 F)(12.5 V) 2 = 1.055 × 10 −9 J = 1055 pJ ≈ 1100 pJ.

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  Engineering Electromagnetics

  Pergamon Unified Engineering Series

  • David T. Thomas, Thomas F. Irvine, James P. Hartnett, William F. Hughes(Authors)
  • 2013(Publication Date)
  • Pergamon

    (Publisher)

  The basic happening in dielectric materials is polarization, or separation of charge centers of orbital electrons and nuclei, of positive and negative ions with-out physically breaking their bonds and freeing them. EXERCISES 4-1. The plane z = 0 is the boundary between dielectric and free space. Near z = 0 the polarization, P, varies as shown in Fig. E.4-1. Estimate p P everywhere, including surface polarization charge, p SP , on the dielectric surface. 4-2. A solid dielectric cylinder of length L and radius a is uniformly polarized with Fig. E.4-1. Polarization near a boundary. 138 Dielectric Materials axially directed polarization, P. Determine the electric field along the axis, both within and outside the cylinder. 4-3. Show from the definition of P that dP/dt has dimensions of a current density, J P . Show }p is consistent with the expression, where p, v are charge density and velocity, respectively. 4-4. An electric field of 10,000 V/m is applied to a helium gas. For atom density n = 5 x 10 25 atoms/m 3 and an average electron cloud shift of 10 18 m find the polarization, P, polarization charge, p P , and permittivity of the helium. 4-5. A coaxial capacitor is partially filled with dielectric as shown. The inner and outer conductor radii are a, c respectively. The dielectric fills the coaxial to radius b. Find the fields. How does the presence of dielectric affect possible breakdown of the air? 4-6. A long coaxial capacitor is immersed in water (e=8l€ 0 ) (see Fig. E.4-6). If a voltage V 0 is applied, how far will the liquid rise in the capacitor? 4-7. The plasma frequency is the natural oscillatory frequency of an ionized gas. Fig.E.4-5. Dielectric loaded coaxial capacitor. Assuming 100% ionization, what is the plasma frequency of a hydrogen gas (density 10-3 mmHg)? HINT: The force on an electron in an electric field is eE, which causes motion predicted by Newton's Law, 4-8.

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  Handbook of Electrostatic Processes

  • Jen-Shih Chang, Arnold J. Kelly, Joseph M. Crowley(Authors)
  • 2018(Publication Date)
  • CRC Press

    (Publisher)

  4 Electrical Phenomena of Dielectric Materials R. Tobaz6on Centre National de la Recherche Scientifique Grenoble , France I. INTRODUCTION If we exclude metals, all remaining materials are dielectrics, whatever the state of the matter in question (solid, liquid, gas), and a permittivity e can be ascribed to any substance, with vacuum as the reference dielectric. Dielectrics can be employed either as passive devices (capacitors, ca­ bles) or in active devices (electrets, electrostatic motors), and they are required to function in our near or far environment (air, seawater, soil, space). Generally, materials are subjected to a voltage (dc, ac, impulse), and, in exceptional cases, to an electromagnetic field produced by, for example, an intense laser beam. The spatiotemporal distribution of the field inside the matter not only is imposed by the geometry of the elec­ trodes (whether to insure a uniform or a nonuniform field) and the shape of the voltage wave but also depends on space charges: charge carriers can be generated or blocked at interfaces or interphases, when different dielectric substances come into contact with each other. Among environmental constraints, we may consider the actions of pres­ sure, temperature, radiation, chemical attack, etc. Time is often a funda­ mental parameter in the study of dielectrics, e.g.: A perfect insulator would be a medium through which no conduction current could flow. In fact, the transition from “capacitive” behavior to “resistive” behavior depends on the conduction relaxation time 51 52 TOBAZEON tc = ep (insofar as a resistivity p can be ascribed to the material). Thus a dielectric may behave in a completely different manner under dc, ac, or impulse voltages.

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  General Physics Electromagnetism Optics

  • Pierluigi Zotto, Sergio Lo Russo, Paolo Sartori(Authors)
  • 2022(Publication Date)
  • Società Editrice Esculapio

    (Publisher)

  The former case can be managed, while in the latter one an infinite energy is needed to build-up a source, since a great num- ber of same sign infinitesimal charges are constrained to be at null distance. Further information about electrostatic energy will be presented in paragraphs 5.12 and 14.10. Chapter 2 Electrostatic Field and Electrostatic Potential 23  E P == q 4 + 2 ( ) 16πε 0 a 2 ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥  u x + q 4 + 2 ( ) 16πε 0 a 2 ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥  u y so it is directed along the direction θ = π 4 with respect to axis x and it has magnitude E P = E x 2 + E y 2 = q 2 2 + 1 ( ) 8πε 0 a 2 . The electrostatic potential in P is obtained as the scalar sum of the electrostatic poten- tials associated with each field independently of other fields. Reminding the expression of the potential associated with a coulomb field if V ∞ = 0, the electrostatic potential is V P = V 1 + V 2 + V 3 = q 4πε 0 a + q 4πε 0 a 2 + q 4πε 0 a = q 4πε 0 a 4 + 2 2 . The electrostatic energy of the system is the sum of the electrostatic energy of all the pairs of charges, U = U 12 + U 13 + U 23 = q 1 q 2 4πε 0 a + q 1 q 3 4πε 0 a 2 + q 2 q 3 4πε 0 a = q 2 4πε 0 a 4 + 2 2 . 2.9 Electric Dipole The simplest charge system is an electric dipole, which consists of two equal and oppo- site point-like charges, q and –q, separated by a fixed distance d. a) Electrostatic Potential The electrostatic potential of a dipole in a point P of the space is given by the algebraic sum of the electro- static potentials, with respect to infinity, due to the two charges V = 1 4πε 0 q r 1 − 1 4πε 0 q r 2 = q 4πε 0 r 2 − r 1 r 1 r 2 , which, by multiplying and dividing by r 2 + r 1 , can be rewritten as V = q 4πε 0 r 2 − r 1 r 1 r 2 r 2 + r 1 r 2 + r 1 = q 4πε 0 r 2 2 − r 1 2 r 1 r 2 r 2 + r 1 ( ) . Consider a vector  d , whose magnitude is equal to the distance between the charges and whose direction is given by the line connecting them oriented from the negative one to the positive one.

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