Physics
Permittivity
Permittivity is a measure of how much a material can store electrical energy in an electric field. It is a fundamental property of matter and is represented by the symbol ε. Permittivity determines the capacitance of a capacitor and influences the speed of electromagnetic waves in a medium. It is a key parameter in the study of electrostatics and electromagnetism.
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12 Key excerpts on "Permittivity"
eBook - ePubBinary Polar Liquids
Structural and Dynamic Characterization Using Spectroscopic Methods
- Suresh C. Mehrotra, Ashok Kumbharkhane, Ajay Chaudhari(Authors)
- 2017(Publication Date)
- Elsevier(Publisher)
1.3. Dielectric Theories
The dielectric constant or Permittivity of a material is a measure of the extent to which the electric charge distribution in the material can be distorted or polarized by using an electric field.3 ,4The individual charges do not travel continuously for a relatively large distance, as is the case in electrical conduction by transport. There is nevertheless a flow of charge in the polarization process, for example, by the rotation of polar molecules, which tend to line up in the direction of the field.The dielectric relaxation theories can be broadly divided into two parts: (1) theories of static Permittivity and (2) theories of dynamic Permittivity. When polar dielectric materials having a permanent dipole moment are placed in a steady electric field so that all types of polarization get equilibrium with it, the Permittivity of the material under these condition is called as theories of static Permittivity (ε0 ). When dielectric materials are placed in a varying electromagnetic field with frequency f, the Permittivity of the material changes with the change in the frequency of the applied field. This is so because with increasing frequency molecular dipoles cannot orient in phase with the applied field. Thus the Permittivity of the material falls off with increasing frequency of the applied field. The frequency-dependent Permittivity of a material is called as dynamic Permittivity. The basic theories of static and dynamic Permittivity are briefly described in the following sections.1.3.1. Theories of Static Permittivity
Suppose that a charge q e.s.u. per unit area is applied to a parallel plate having area A and separated by a distance d. If vacuum is present between the two plates, then electric field intensity E is given by
- Endrik Krugel(Author)
- 2007(Publication Date)
- CRC Press(Publisher)
1 The dielectric permeability We begin by acquainting ourselves with the polarization of matter. The fun-damental quantity describing how an interstellar grain responds to an electro-magnetic wave is the dielectric permeability which relates the polarization of matter to the applied field. We recall the basic equations of electrodynamics and outline how plane waves travel in an infinite non-conducting (dielectric) medium and in a plasma. We summarize the properties of harmonic oscilla-tors, including the absorption, scattering and emission of light by individual dipoles. Approximating a solid body by an ensemble of such dipoles (identical harmonic oscillators), we learn how its dielectric permeability changes with frequency. This study is carried out for • a dielectric medium where the electron clouds oscillate about the atomic nuclei • and a metal where the electrons are free. 1.1 How the electromagnetic field acts on dust At the root of all phenomena of classical electrodynamics, such as the inter-action of light with interstellar dust, are Maxwell’s formulae. They can be written in different ways and the symbols, their names and meaning are not universal, far from it. Before we exploit Maxwell’s equations, we therefore first define the quantities which describe the electromagnetic field. 1.1.1 Electric field and magnetic induction A charge q traveling with velocity v in a fixed electric field E and a fixed magnetic field of flux density B experiences a force F = q E + 1 c v × B , (1.1) called the Lorentz force; the cross × denotes the vector product. B is also called magnetic induction. Equation (1.1) shows what happens mechanically to a charge in an electromagnetic field and we use it to define E and B . 1 2 The dielectric permeability The force F has an electric part, q E , which pulls a positive charge in the direction of E , and a magnetic component, ( q/c ) v × B , working perpendicular to v and B .
eBook - PDFColor Imaging
Fundamentals and Applications
- Erik Reinhard, Erum Arif Khan, Ahmet Oguz Akyuz, Garrett Johnson(Authors)
- 2008(Publication Date)
- A K Peters/CRC Press(Publisher)
Finally, for some materials, ( semiconductors ) the conductivity increases with increasing tempera-ture. See also Section 3.3.3. For most materials, the magnetic permeability μ will be close to 1. Some materials, however, have a permeability significantly different from 1, and these are then called magnetic. The speed v at which light travels through a medium is related to the material constants as follows: v = 1 √ ε μ . (2.28) We will derive this result in Section 2.2.4. The symbol μ 0 is reserved for the permeability of vacuum. The value of μ 0 is related to both Permittivity ε 0 and the speed of light c in vacuum as follows: c = 1 √ ε 0 μ 0 . (2.29) Values for all three constants are given in Table 2.2. The Permittivity and perme-ability of materials is normally given relative to those of vacuum: ε = ε 0 ε r ; (2.30a) μ = μ 0 μ r . (2.30b) Values of ε r and μ r are given for several materials in Tables 2.3 and 2.4. Normally, the three material constants, σ , ε , and μ , are independent of the field strengths. However, this is not always the case. For some materials these values also depend on past values of E or B . In this book, we will not consider such effects of hysteresis . Similarly, unless indicated otherwise, the material con-stants are considered to be isotropic , which means that their values do not change Constant Value Unit c 3 · 10 8 m s − 1 ε 0 1 36 π · 10 − 9 C 2 s 2 kg − 1 m − 3 = Fm − 1 μ 0 4 π · 10 − 7 kg m C − 2 = Hm − 1 Table 2.2. The speed of light c , Permittivity ε 0 , and permeability μ 0 (all in vacuum). 28 2. Physics of Light Material ε r Material ε r Air 1 . 0006 Paper 2 − 4 Alcohol 25 Polystyrene 2 . 56 Earth (dry) 7 Porcelain 6 Earth (wet) 30 Quartz 3 . 8 Glass 4 − 10 Snow 3 . 3 Ice 4 . 2 Water (distilled) 81 Nylon 4 Water (sea) 70 Table 2.3. Dielectric constants ε r for several materials [532].
eBook - PDFInfrared and Millimeter Waves V8
Electromagnetic Waves in Matter, Part I
- Kenneth J. Button(Author)
- 1983(Publication Date)
- Academic Press(Publisher)
It is more usual, therefore, to write P = (g-OJ?, (7) where ε^ is a high-frequency limiting Permittivity. This is unfortunately a rather amorphous and poorly defined quantity because it is not possible, in fact, to find any measurable frequency at which dispersion does not exist. It is rather to be considered as a parameter: the Permittivity in frequency regions so high that the particular dispersion mechanism under considera-tion will no longer have any effect. It is also helpful to normalize the polarization by writing 1. PROPERTIES OF DIELECTRIC MATERIALS 3 where e s is the static relative Permittivity. Then P n will go from one at ω = 0 to zero at ω = oo. At extra-high frequencies and beyond (v > 30 GHz) it is not possible, with the currently available techniques, to measure field parameters such as the vectorial electric field strength and its phase. Rather, one measures energy flow; and in this situation one is led naturally to introduce a refractive index n which is the ratio of phase velocity in free space (i.e., c) to that in the medium. The space dependence of the field is then given by E = E 0 exp(— iconx/c), (9) but the measurable quantity, the intensity, is given by I = ie 0 cEl (10) The propagating medium will usually be lossy, so one has a progressive attenuation of the field given by Lambert's law E = E Q exp(— $ax), (11) which also can be written as 7 = / 0 exp(-ox). (12) Here a is the power absorption coefficient (usually measured in nepers per centimeter). Equations (9) and (11) can be combined to give E = E 0 exp(— ίωηχ/c), (13) where the complex refractive index n is defined by n = n — i(a/4nv), (14) in which the wave number v = v/c = co/2nc (15) is introduced. The two formalisms, in terms of either ε or /?, are readily connected by means of Maxwell's celebrated relationship ε = η 2 . (16) Identifying real and imaginary components in Eq.
eBook - PDF- Léon Brillouin, H. S. W. Massey(Authors)
- 2013(Publication Date)
- Academic Press(Publisher)
C H A P T E R I V P R O P A G A T I O N O F E L E C T R O M A G N E T I C W A V E S IN MATERIAL MEDIA'' 1. Definitions: Role of a Dielectric Coefficient Depending on Density and Temperature The discussion of Chapter III indicates a variety of circumstances in which the group velocity plays an important role. These results were obtained on a special example, but their significance appears to be very general. It is therefore appropriate to state the problem in general terms, without using a special model, and to see how much can be proved in this way. We shall see that aU the most important results can be obtained, provided the absorption coefficient is small enough to be neglected. Let us first recall the fundamental equations of electromagnetism in vacuum: calhng D and Ε the displacement and electric field, Β and JFÍ the induction field and magnetic intensity, ρ and / the charge density and electric current density, Maxwell's equations are written as (1) c n n H = á n J + ^ , d i v ß ^ O OD (2) curlE = — , div D = 4 π ρ (3) D = ε ,Ε, Β = μ ,Η SQ is the dielectric coefficient (or Permittivity) and μQ the magnetic permeability of free space. These two coefficients have magnitudes * Chapters IV and V were first published by L. Brillouin: Congres International d'Electricité, Paris, 1932, Vol. 2, pp. 739-788. Gauthier-Villars, Paris, 1933. 85 86 IV. WAVES IN MATERIAL MEDIA which depend on the chosen system of units, and are connected by the relation (4) εομ^ε^ = 1 c being the velocity of light in vacuum. Since the coefficients BQ and are constants, it is possible to define an energy density é given by (5) In general, these equations are written in the same form for a material medium, with the coefficients ε and μ being characteristic of the medium.
- Rohit P. Prasankumar, Antoinette J. Taylor(Authors)
- 2016(Publication Date)
- CRC Press(Publisher)
However, an earlier connection between the atomic polarizability and the electric Permittivity was developed by Mossotti in 1850, and independently by Clausius in 1879. This relationship, known as the Clausius–Mossotti equation, established that for any given material, the dielectric constant ε should be proportional to the density of atoms ε ε α -+ = ∑ 1 2 1 3 0 ε N j j , (1.27) where N j is the concentration α j is the atomic polarizability Equation 1.27 thus provides an important connection between macroscopic and microscopic theories. The atomic polarizability would be calculated by a microscopic theory, which could then be connected to a dielectric function, which would describe the response of the ions to the local field acting on them. The dielectric constant (also called the electric Permittivity) ε and the magnetic perme-ability μ are called the optical constants. Although in some cases it may be suitable to treat them as constant, they are, however, functions of frequency and, in general, may also depend on the wave vector, that is, ε = ε ( ω , k ), and μ = μ ( ω , k ). Here we denote angular frequency (cycles per unit time) as ω , which has units of rad/s, and the wave vector (wavelengths in a length of 2 π ) as k , which has units of 1/m. The relationship between the angular frequency and frequency is ω = 2 π f . It is typical to refer to ω as “frequency,” as it is encountered more often than f . As they have been derived here the optical constants are linear response functions that obey causality. Thus we may write them as complex functions ε = ε 1 + i ε 2 and μ = μ 1 + i μ 2 , where the subscripts 1 and 2 denote the real and imaginary parts, respectively. The real part of the complex optical constants describes the amplitude “response,” and imaginary portions describe the damping or loss, due to external electric or magnetic fields.
- Marc J. Madou(Author)
- 2011(Publication Date)
- CRC Press(Publisher)
E is the electrical field, and P is the polarization. The entire free electron gas is displaced over a small distance, δ x . Quantum Mechanics and the Band Theory of Solids 87 density P ,and ε 0 isthePermittivityoffreespacewith avalueof8.854pF/m.Inthecaseofalinearhomo-geneousandisotropicmaterial, χ e becomesascalar constant.Thepolarization P isfurtherlinkedtothe electricdisplacementfield D ,expressedincoulombs persquaremeter(C/m 2 ),andtothedielectriccon-stant ε(ω , k ) or Permittivity ( As / Vm ), a materials- dependentconstant,as: D E P E k E = + = + = ε χ ε ε ω 0 e ) ( ) ( , 1 0 (3.37) In the expression D E P = + ε 0 we have separated theelectricdisplacement D initsmaterials( P )and vacuumparts( ε 0 E ).ThePermittivityofamaterial isusuallygivenasarelativePermittivity ε r ( ω )(also called the dielectric constant). The Permittivity ε(ω , k ) of a medium is an intensive parameter and is calculated by multiplying the relative permittiv-ityby ε 0 ,or ε(ω) = ε 0 ε r ( ω ).ThePermittivityofairis ε air = 8.876pF/m,sotherelativePermittivityofair is ε r,air = 1.0005,andforvacuumitis1bydefinition. FromEquation3.37,itthenalsofollowsthat ε r ( ω ) = 1 + χ e .Theelectricdisplacementfield, D ,isinturn relatedtotheelectricfield E (inunitsofV/m)bythe constitutiverelation: D = ε ( ω , k ) E (3.38) UsingEquation3.36forthepolarization P ,wecan rewriteEquation3.37as: D k E E P E E = = = + ε ω ε + ε -ω γω ( , uni00A0 uni00A0 ( ) ) ne m i 2 e 0 0 2 (3.39) From Equation 3.39 we arrive at the following expressionforthecomplexdielectricfunction: ε ω ε ε ω γω ε ω ω ( , ( ) k ) ne m i ( 2 e p 2 2 = -+ uni23A1 uni23A3 uni23A2 uni23A4 uni23A6 uni23A5 = -0 0 2 0 1 1 + uni23A1 uni23A3 uni23A2 uni23A2 uni23A4 uni23A6 uni23A5 uni23A5 i ) γω (3.40) The plasma frequency ω p in Equation 3.40 is de -finedas: ω ε p 2 e ne m = 0 (3.41) where n is the density of electrons in the metal.
- Changzhi Li, Mohammad-Reza Tofighi, Dominique Schreurs, Tzyy-Sheng Jason Horng(Authors)
- 2016(Publication Date)
- Academic Press(Publisher)
Because the focus of this chapter is on the fundamentals of dielectric theory as related to RF and microwave applications in medicine, it attempts to approach the topic from a perspective that would be of interest to RF and microwave engineers. Therefore, the focus of the discussion that follows will be on the macroscopic property (bulk complex Permittivity) modeling. The underlying physics, biochemistry, and microscopic theory of dielectrics can be found elsewhere, such as in some of the above-mentioned references.2.2.1 Complex Permittivity and Loss
Since for all practical purposes, the magnetic permeability of biological materials is that of free space [3] , conductivity and Permittivity are the two constitutive parameters that need to be discussed. Recall the Maxwell’s curl equation for the magnetic field( H )in time domain∇ × H ( t ) =(2.1)σ fE ( t ) + ∂ D ( t ) / ∂ t(2.1)where E and D are the electric field intensity and electric displacement, respectively. Moreover, σ f is the conductivity associated with free charges, which could be in general free electrons in conductors, electrons and holes in semiconductors, or free ions in electrolytes and biological media. For this presentation, the two terms at the right side of the equation then can be viewed as the conduction and displacement current densities, respectively. The presence of an electric field E causes the polarization of atoms, polarization of nonpolar molecules, reorientation of dipolar moment of polar molecules (e.g., water), or polarization of charges due to the build-up of charge at the interface between two dissimilar dielectrics in heterogeneous media such as biological tissues [1 –3 ,25 ,26] . This leads to a net electric moment and thus the electric polarization P that plays the main role in establishing the relationship between D and E :D ( t ) =(2.2)ε 0E ( t ) + P ( t )(2.2)where ε 0 =8.854×10−12 (F/m) is the free space Permittivity. This equation maintains its form for the time-harmonic case (after replacing the time domain fields and polarization with their phasors), where the steady-state situation for single frequency sine waves is considered. For a linear medium (which is a realistic assumption for biological tissues and the electric field intensity less than 105 V/m [25]
- Constantine A. Balanis(Author)
- 2023(Publication Date)
- Wiley(Publisher)
When a material is examined mac- roscopically, the presence of all the electric dipoles is accounted for by introducing an electric polarization vector P [see (2-3) and (2-10)]. Ultimately, the static Permittivity s ε [see (2-11a)] is introduced to account for the presence of P . A similar procedure is used to account for the orbit- ing and spinning of the electrons of atoms (which are represented electrically by small electric current-carrying loops) when magnetic materials are subjected to applied static magnetic fields. When the material is examined macroscopically, the presence of all the loops is accounted for by introducing the magnetic polarization (magnetization) vector M [see (2-15) and (2-21)]. In turn the static permeability s µ [see (2-22a)] is introduced to account for the presence of M. When the applied fields begin to alternate in polarity, the polarization vectors P and M, and in turn the permittivities and permeabilities, are affected and they are functions of the frequency of the alternating fields. By this action of the alternating fields, there are simultaneous changes imposed upon the static conductivity s σ [see (2-39) and (2-40)] of the material. In fact, the incremental changes in the conductivity that are attributable to the reverses in polarity of the applied fields (frequency) are responsible for the heating of materials using microwaves (for example, microwave cooking of food) [13, 18]. In the sections that follow, the variations of ε, σ , and µ as a function of frequency of the applied fields will be examined. 2.9.1 Complex Permittivity Let us assume that each atom of a material in the absence of an applied electric field (unpolarized atom) is represented by positive (representing the nucleus) and negative (representing the elec- trons) charges whose respective centroids coincide. The electrical and mechanical equivalents of a typical atom are shown in Figure 2-16a [6]. The large positive sphere of a mass M represents the
eBook - PDFPhysics of Continuous Media
Problems and Solutions in Electromagnetism, Fluid Mechanics and MHD, Second Edition
- Grigory Vekstein(Author)
- 2013(Publication Date)
- CRC Press(Publisher)
Problem 2.1.1 The electromagnetic properties of a homogeneous isotropic medium with-out spatial dispersion can be described by the “traditional” electric per-mittivity, ǫ ( ω ), and the magnetic permeability, µ ( ω ). Express ǫ ( ω ) and µ ( ω ) in terms of the limiting as k → 0 values of ǫ || ( k,ω ) and ǫ ⊥ ( k,ω ), introduced in equation (2.10). In terms of ǫ and µ , the electric current in a medium is equal to vector j = ∂ vector P ∂t + c vector ∇ × vector M, where vector P and vector M are, respectively, the polarization and the magnetization vectors given by the following relations: vector P = ( ǫ − 1) 4 π vector E, vector M = ( µ − 1) 4 πµ vector B Then, in the ( vector k,ω ) representation, using vector B = c ( vector k × vector E ) /ω from (2.2), one gets vector j = − iω vector P + ic ( vector k × vector M ) = − iω ( ǫ − 1) 4 π vector E + ic 2 ( µ − 1) 4 πµω [ vector k ( vector k · vector E ) − k 2 vector E ] This yields the conductivity tensor σ αβ = − iω ( ǫ − 1) 4 π δ αβ − ik 2 c 2 ( µ − 1) 4 πµω parenleftbigg δ αβ − k α k β k 2 parenrightbigg , and, according to relation (2.7), the dielectric permeability tensor ǫ αβ = ǫδ αβ + k 2 c 2 ( µ − 1) µω 2 parenleftbigg δ αβ − k α k β k 2 parenrightbigg (2.14) 24 Physics of Continuous Media, Second Edition It follows now from equations (2.10) and (2.14), that in the limit of k → 0, when spatial dispersion is absent, lim k → 0 ǫ bardbl ( k,ω ) = lim k → 0 ǫ ⊥ ( k,ω ) , so that ǫ ( ω ) = lim k → 0 ǫ bardbl ( k,ω ) , µ − 1 ( ω ) = 1 − ω 2 c 2 lim k → 0 ( ǫ ⊥ − ǫ bardbl ) k 2 Problem 2.1.2 Derive the dielectric permeability tensor for a “cold” plasma (a gas of im-mobile heavy ions and initially resting electrons with the number density n ), which is immersed into an external magnetic field vector B . Since the ions are immobile, they do not contribute to the electric current (their only role is to compensate the space charge of electrons).
eBook - ePubFundamentals of the Optics of Materials
Tutorial and Problem Solving
- Vladimir I. Gavrilenko, Volodymyr S. Ovechko(Authors)
- 2023(Publication Date)
- Jenny Stanford Publishing(Publisher)
After a brief overview of the electron energy structure, the classical theory of light and the optical functions of materials (dielectric Permittivity, polarization functions) are considered within both the classical and quantum mechanical theories. This provides a bridge between the classical theory of optics and the modern state-of-theart approaches in the optics of materials from first principles._________________Fundamentals of the Optics of Materials: Tutorial and Problem SolvingVladimir I. Gavrilenko and Volodymyr S. Ovechko Copyright © 2024 Jenny Stanford Publishing Pte. Ltd. ISBN 978-981-4877-93-0 (Hardcover), 978-1-003-25694-6 (eBook) www.jennystanford.com1.1 Introduction: Description of the Electromagnetic Field in Medium
Within the electromagnetic (EM) Maxwell’s theory, a signature of the medium is embodied in the relationship between the polarization vector function (P(r, t)) (or in magnetic medium in magnetization vector function, M(r, t)), on the one hand, and the electric (E(r, t)) (or magnetic, H(r, t)) field of external EM radiation, on the other [Born and Wolf (1999 ); Jackson (1998 ); Zahn (1979 );Vysotskii et al. (2011); Ovechko and Sheka (2006 ); Ovechko (2017 )]. These are known as constitutive relations (or materials equations) [Born and Wolf (1999 ); Saleh and Teich (2007 )].First we consider nondispersive, homogeneous, and isotropic medium. In nonmagnetic materials (which are mostly addressed in this book) the constitutive relation describes a relationship between P and E according to the equation(1.1)P = χ E ,where χ is the dielectric susceptibility of the medium. According to EM theory (see e.g. [Born and Wolf (1999 ); Saleh and Teich (2007 ); Zahn (1979 )]) the electric field in a medium is described in terms of the electric flux density (or the electric displacement) D according to(1.2)D = ε E = E + 4 π P .Equations (1.1) and (1.2) define the dielectric Permittivity function of the medium:(1.3)ε = 1 + 4 π χ .The dielectric Permittivity function in the static case is also called the dielectric constant. In the same way the magnetic field in a medium (B) can be written as(1.4)B = μ H ,where μ is the magnetic permeability
eBook - ePub- H. John Pain, Patricia Rankin(Authors)
- 2015(Publication Date)
- Wiley(Publisher)
Thus, for a dielectric the electrostatic energy per unit volume in an electromagnetic wave equals the magnetic energy per unit volume and the total energy is the sum. This gives the instantaneous value of the energy per unit volume and we know that, in the wave, and so that the time average value of the energy per unit volume is Now the amount of energy in an electromagnetic wave which crosses unit area in unit time is called the intensity, I, of the wave and is evidently () where v is the velocity of the wave. This gives the time averaged value of the Poynting vector and, for an electromagnetic wave in free space we have 9.6 Electromagnetic Waves in a Medium of Properties μ, ε and σ (where σ ≠ 0) From a physical point of view the electric vector in electromagnetic waves plays a much more significant role than the magnetic vector, e.g. most optical effects are associated with the electric vector. We shall therefore concentrate our discussion on the electric field behaviour. In a medium of conductivity σ = 0 we have obtained the wave equation where the right-hand term, rewritten shows that we are considering a term When σ ≠ 0 we must also consider the conduction currents which flow. These currents are given by Ohm’s Law as I = V / R, and we define the current density; that is, the current per unit area, as where σ is the conductivity 1/(R × Length) and E is the electric field. J = σ E is another form of Ohm’s Law. With both displacement and conduction currents flowing, Maxwell’s second time-varying equation reads, in vector form, (9.5) each term on the right-hand side having dimensions of current per unit area
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