dielectric constant

143

Click here to load reader

Upload: -

Post on 22-Oct-2014

179 views

Category:

Documents


13 download

TRANSCRIPT

Page 1: Dielectric Constant

Relative permittivity From Wikipedia, the free encyclopediaJump to: navigation, search

Relative static permittivities of some materials at room temperature under 1 kHz [1] (corresponds to light with wavelength of 300 km)

Material εr

Vacuum 1 (by definition)

Air1.00058986 ± 0.00000050(at STP, for 0.9 MHz),[2]

PTFE/Teflon 2.1Polyethylene 2.25

Polyimide 3.4Polypropylene 2.2–2.36

Polystyrene 2.4–2.7Carbon disulfide 2.6

Paper 3.85Electroactive polymers 2–12

Silicon dioxide 3.9 [3]

Concrete 4.5Pyrex (Glass) 4.7 (3.7–10)

Rubber 7Diamond 5.5–10

Salt 3–15Graphite 10–15Silicon 11.68

Ammonia26, 22, 20, 17

(−80, −40, 0, 20 °C)Methanol 30

Ethylene Glycol 37Furfural 42.0

Glycerol41.2, 47, 42.5(0, 20, 25 °C)

Water88, 80.1, 55.3, 34.5(0, 20, 100, 200 °C)for visible light: 1.77

Hydrofluoric acid 83.6 (0 °C)Formamide 84.0 (20 °C)

Sulfuric acid84–100

(20–25 °C)

Hydrogen peroxide128 aq–60

(−30–25 °C)

Hydrocyanic acid158.0–2.3(0–21 °C)

Titanium dioxide 86–173Strontium titanate 310

Barium strontium titanate 500

Page 2: Dielectric Constant

Relative static permittivities of some materials at room temperature under 1 kHz [1] (corresponds to light with wavelength of 300 km)

Material εr

Barium titanate1250–10,000(20–120 °C)

Lead zirconate titanate 500–6000Conjugated polymers 1.8-6 up to 100,000[4]

Calcium copper titanate >250,000[5]

Temperature dependence of the relative static permittivity of water

The relative permittivity of a material under given conditions reflects the extent to which it concentrates electrostatic lines of flux. In technical terms, it is the ratio of the amount of electrical energy stored in a material by an applied voltage, relative to that stored in a vacuum. Likewise, it is also the ratio of the capacitance of a capacitor using that material as a dielectric, compared to a similar capacitor that has a vacuum as its dielectric.

Contents [hide] 

1 Terminology 2 Measurement 3 Practical relevance 4 Chemical applications 5 Complex permittivity 6 Lossy medium 7 Metals 8 See also

9 References

[edit] Terminology

The relative permittivity of a material for a frequency of zero is known as its static relative permittivity or as its dielectric constant. Other terms used for the zero frequency relative permittivity include relative dielectric constant and static dielectric constant. While they remain very common, these terms are ambiguous and have been deprecated by some standards organizations.[6][7] The reason for the potential ambiguity is twofold. First, some older authors used "dielectric constant" or

Page 3: Dielectric Constant

"absolute dielectric constant" for the absolute permittivity ε rather than the relative permittivity.[8] Second, while in most modern usage "dielectric constant" refers to a relative permittivity,[7][9] it may be either the static or the frequency-dependent relative permittivity, depending on context.

Relative permittivity is typically denoted as εr(ω) (sometimes κ or K) and is defined as

where ε(ω) is the complex frequency-dependent absolute permittivity of the material, and ε0 is the vacuum permittivity.

Relative permittivity is a dimensionless number that is in general complex. The imaginary portion of the permittivity corresponds to a phase shift of the polarization P relative to E and leads to the attenuation of electromagnetic waves passing through the medium. By definition, the linear relative permittivity of vacuum is equal to 1,[9] that is ε = ε0, although there are theoretical nonlinear quantum effects in vacuum that exist at high field strengths.[10]

The relative permittivity of a medium is related to its electric susceptibility, χe, as εr(ω) = 1 + χe.

[edit] Measurement

The relative static permittivity, εr, can be measured for static electric fields as follows: first the capacitance of a test capacitor, C0, is measured with vacuum between its plates. Then, using the same capacitor and distance between its plates the capacitance Cx with a dielectric between the plates is measured. The relative dielectric constant can be then calculated as

For time-variant electromagnetic fields, this quantity becomes frequency-dependent and in general is called relative permittivity.

[edit] Practical relevance

The dielectric constant is an essential piece of information when designing capacitors, and in other circumstances where a material might be expected to introduce capacitance into a circuit. If a material with a high dielectric constant is placed in an electric field, the magnitude of that field will be measurably reduced within the volume of the dielectric. This fact is commonly used to increase the capacitance of a particular capacitor design. The layers beneath etched conductors in printed circuit boards (PCBs) also act as dielectrics.

Page 4: Dielectric Constant

Dielectrics are used in RF transmission lines. In a coaxial cable, polyethylene can be used between the center conductor and outside shield. It can also be placed inside waveguides to form filters. Optical fibers are examples of dielectric waveguides. They consist of dielectric materials that are purposely doped with impurities so as to control the precise value of εr within the cross-section. This controls the refractive index of the material and therefore also the optical modes of transmission. However, in these cases it is technically the relative permittivity that matters, as they are not operated in the electrostatic limit.

[edit] Chemical applications

The relative static permittivity of a solvent is a relative measure of its polarity. For example, water (very polar) has a dielectric constant of 80.10 at 20 °C while n-hexane (very non-polar) has a dielectric constant of 1.89 at 20 °C.[11] This information is of great value when designing separation, sample preparation and chromatography techniques in analytical chemistry.

The correlation should, however, be treated with caution. For instance, dichloromethane has a value of εr of 9.08 (20 °C) and is rather poorly soluble in water (13 g/L or 9.8 mL/L at 20 °C); at the same time, tetrahydrofurane has its εr = 7.52 at 22 °C, but is completely miscible with the water.

[edit] Complex permittivity

Similar as for absolute permittivity, relative permittivity can be decomposed into real and imaginary parts:[12]

.

[edit] Lossy medium

Again, similar as for absolute permittivity, relative permittivity for lossy materials can be formulated as:

in terms of a "dielectric conductivity" σ (units S/m, siemens per meter), which "sums over all the dissipative effects of the material; it may represent an actual [electrical] conductivity caused by migrating charge carriers and it may also refer to an energy loss associated with the dispersion of ε' [the real-valued permittivity]" (,[12] p. 8). Expanding the angular frequency ω = 2πc/λ and the electric constant ε0 = 1/(µ0c2), it reduces to:

Page 5: Dielectric Constant

where λ is the wavelength, c is the speed of light in vacuum and κ = µ0c/2π ≈ 60.0 S−1 is a newly-introduced constant (units reciprocal of siemens, such that σλκ = εr" remains unitless).

[edit] Metals

Although permittivity is typically associated with dielectric materials, we may still speak of an effective permittivity of a metal, with real relative permittivity equal to one (,[13] eq.(4.6), p. 121). In the low-frequency region (which extends from radiofrequencies to the far infrared region), the plasma frequency of the electron gas is much greater than the electromagnetic propagation frequency, so the complex permittivity ε of a metal is practically a purely imaginary number, expressed in terms of the imaginary unit and a real-valued electrical conductivity (,[13] eq.(4.8)-(4.9), p. 122).

Page 6: Dielectric Constant

Electrostatics From Wikipedia, the free encyclopedia  (Redirected from Electrostatic)Jump to: navigation, search For a less technical introduction, see Static electricity.

Electromagnetism

Electricity

Magnetism

Electrostatics[show]

Magnetostatics [show]

Electrodynamics [show]

Electrical Network [show]

Covariant formulation [show]

Scientists[show]

v

t

e

Paper shavings attracted by a charged CD

Electrostatics is the branch of physics that deals with the phenomena and properties of stationary or slow-moving (without acceleration) electric charges.

Since classical antiquity, it was known that some materials such as amber attract lightweight particles after rubbing. The Greek word for amber, ήλεκτρον electron,

Page 7: Dielectric Constant

was the source of the word 'electricity'. Electrostatic phenomena arise from the forces that electric charges exert on each other. Such forces are described by Coulomb's law. Even though electrostatically induced forces seem to be rather weak, the electrostatic force between e.g. an electron and a proton, that together make up a hydrogen atom, is about 40 orders of magnitude stronger than the gravitational force acting between them.

Electrostatic phenomena include many examples as simple as the attraction of the plastic wrap to your hand after you remove it from a package, to the apparently spontaneous explosion of grain silos, to damage of electronic components during manufacturing, to the operation of photocopiers. Electrostatics involves the buildup of charge on the surface of objects due to contact with other surfaces. Although charge exchange happens whenever any two surfaces contact and separate, the effects of charge exchange are usually only noticed when at least one of the surfaces has a high resistance to electrical flow. This is because the charges that transfer to or from the highly resistive surface are more or less trapped there for a long enough time for their effects to be observed. These charges then remain on the object until they either bleed off to ground or are quickly neutralized by a discharge: e.g., the familiar phenomenon of a static 'shock' is caused by the neutralization of charge built up in the body from contact with nonconductive surfaces.

Contents

 [hide]  1 Fundamental concepts

o 1.1 Coulomb's law o 1.2 Electric field o 1.3 Gauss's law o 1.4 Poisson's equation o 1.5 Laplace's equation

2 Electrostatic approximation o 2.1 Electrostatic potential

3 Electrostatic energy 4 Triboelectric series 5 Electrostatic generators 6 Charge neutralization 7 Charge induction 8 'Static' electricity

o 8.1 Static electricity and chemical industry 8.1.1 Applicable standards

9 Electrostatic induction in commercial applications 10 See also 11 References 12 Further reading

13 External links

[edit] Fundamental concepts

Page 8: Dielectric Constant

[edit] Coulomb's law

The fundamental equation of electrostatics is Coulomb's law, which describes the force between two point charges. The magnitude of the electrostatic force between two point electric charges and is directly proportional to the product of the magnitudes of each charge and inversely proportional to the surface area of a sphere whose radius is equal to the distance between the charges:

where ε0 is a constant called the vacuum permittivity or permittivity of free space, a defined value:

in  A2s4 kg-1m−3 or C2N−1m−2 or F m−1.

[edit] Electric field

The electric field (in units of volts per meter) at a point is defined as the force (in newtons) per unit charge (in coulombs) on a charge at that point:

Or we can say a charged object in an electric field feels a force F=qE

From this definition and Coulomb's law, it follows that the magnitude of the electric field E created by a single point charge Q is:

The electric field produced by a distribution of charges given by the volume charge

density is obtained by a triple integral of a vector function:

The value of the electric field gives the force that a charged particle would feel if it entered the electric field. Electric field lines gives the direction of a positive charge in the electric field.

Page 9: Dielectric Constant

[edit] Gauss's law

Gauss' law states that "the total electric flux through a closed surface is proportional to the total electric charge enclosed within the surface".

Mathematically, Gauss's law takes the form of an integral equation:

Alternatively, in differential form, the equation becomes

where is the divergence operator.

[edit] Poisson's equation

The definition of electrostatic potential, combined with the differential form of Gauss's law (above), provides a relationship between the potential Φ and the charge density ρ:

This relationship is a form of Poisson's equation.

[edit] Laplace's equation

In the absence of unpaired electric charge, the equation becomes

which is Laplace's equation.

[edit] Electrostatic approximation

The validity of the electrostatic approximation rests on the assumption that the electric field is irrotational:

From Faraday's law, this assumption implies the absence or near-absence of time-varying magnetic fields:

Page 10: Dielectric Constant

In other words, electrostatics does not require the absence of magnetic fields or electric currents. Rather, if magnetic fields or electric currents do exist, they must not change with time, or in the worst-case, they must change with time only very slowly. In some problems, both electrostatics and magnetostatics may be required for accurate predictions, but the coupling between the two can still be ignored.

[edit] Electrostatic potential

The electrostatic field (lines with arrows) of a nearby positive charge (+) causes the mobile charges in conductive objects to separate due to electrostatic induction. Negative charges (blue) are attracted and move to the surface of the object facing the external charge. Positive charges (red) are repelled and move to the surface facing away. These induced surface charges are exactly the right size and shape so their opposing electric field cancels the electric field of the external charge throughout the interior of the metal. Therefore the electrostatic field everywhere inside a conductive object is zero, and the electrostatic potential is constant.

Because the electric field is irrotational, it is possible to express the electric field as the gradient of a scalar function, called the electrostatic potential (also known as the voltage). An electric field, , points from regions of high potential, Φ, to regions of low potential, expressed mathematically as

The electrostatic potential at a point can be defined as the amount of work per unit charge required to move a charge from infinity to the given point.

[edit] Electrostatic energy

Page 11: Dielectric Constant

Energy due to a charge distribution is obtained by a triple integral:

in which V represents the volume of charge distribution.

[edit] Triboelectric series

Main article: Triboelectric effect

The triboelectric effect is a type of contact electrification in which certain materials become electrically charged when they are brought into contact with a different material and then separated. One of the materials acquires a positive charge, and the other acquires an equal negative charge. The polarity and strength of the charges produced differ according to the materials, surface roughness, temperature, strain, and other properties. Amber, for example, can acquire an electric charge by friction with a material like wool. This property, first recorded by Thales of Miletus, was the first electrical phenomenon investigated by man. Other examples of materials that can acquire a significant charge when rubbed together include glass rubbed with silk, and hard rubber rubbed with fur.

[edit] Electrostatic generators

Main article: Electrostatic generator

The presence of surface charge imbalance means that the objects will exhibit attractive or repulsive forces. This surface charge imbalance, which yields static electricity, can be generated by touching two differing surfaces together and then separating them due to the phenomena of contact electrification and the triboelectric effect. Rubbing two nonconductive objects generates a great amount of static electricity. This is not just the result of friction; two nonconductive surfaces can become charged by just being placed one on top of the other. Since most surfaces have a rough texture, it takes longer to achieve charging through contact than through rubbing. Rubbing objects together increases amount of adhesive contact between the two surfaces. Usually insulators, e.g., substances that do not conduct electricity, are good at both generating, and holding, a surface charge. Some examples of these substances are rubber, plastic, glass, and pith. Conductive objects only rarely generate charge imbalance except, for example, when a metal surface is impacted by solid or liquid nonconductors. The charge that is transferred during contact electrification is stored on the surface of each object. Static electric generators, devices which produce very high voltage at very low current and used for classroom physics demonstrations, rely on this effect.

Note that the presence of electric current does not detract from the electrostatic forces nor from the sparking, from the corona discharge, or other phenomena. Both phenomena can exist simultaneously in the same system.

See also: Friction machines, Wimshurst machines, and Van de Graaf generators.

Page 12: Dielectric Constant

[edit] Charge neutralization

Natural electrostatic phenomena are most familiar as an occasional annoyance in seasons of low humidity, but can be destructive and harmful in some situations (e.g. electronics manufacturing). When working in direct contact with integrated circuit electronics (especially delicate MOSFETs), or in the presence of flammable gas, care must be taken to avoid accumulating and suddenly discharging a static charge (see electrostatic discharge).

[edit] Charge induction

Main article: Electrostatic induction

Charge induction occurs when a negatively charged object repels electrons from the surface of a second object. This creates a region in the second object that is more positively charged. An attractive force is then exerted between the objects. For example, when a balloon is rubbed, the balloon will stick to the wall as an attractive force is exerted by two oppositely charged surfaces (the surface of the wall gains an electric charge due to charge induction, as the free electrons at the surface of the wall are repelled by the negative balloon, creating a positive wall surface, which is subsequently attracted to the surface of the balloon). You can explore the effect with a simulation of the balloon and static electricity.

[edit] 'Static' electricity

Main article: Static electricity

Lightning over Oradea in Romania

Page 13: Dielectric Constant

Before the year 1832, when Michael Faraday published the results of his experiment on the identity of electricities, physicists thought "static electricity" was somehow different from other electrical charges. Michael Faraday proved that the electricity induced from the magnet, voltaic electricity produced by a battery, and static electricity are all the same.

Static electricity is usually caused when certain materials are rubbed against each other, like wool on plastic or the soles of shoes on carpet. The process causes electrons to be pulled from the surface of one material and relocated on the surface of the other material.

A static shock occurs when the surface of the second material, negatively charged with electrons, touches a positively-charged conductor, or vice-versa.

Static electricity is commonly used in xerography, air filters, and some automotive paints. Static electricity is a build up of electric charges on two objects that have become separated from each other. Small electrical components can easily be damaged by static electricity. Component manufacturers use a number of antistatic devices to avoid this.

[edit] Static electricity and chemical industry

When different materials are brought together and then separated, an accumulation of electric charge can occur which leaves one material positively charged while the other becomes negatively charged. The mild shock that you receive when touching a grounded object after walking on carpet is an example of excess electrical charge accumulating in your body from frictional charging between your shoes and the carpet. The resulting charge build-up upon your body can generate a strong electrical discharge. Although experimenting with static electricity may be fun, similar sparks create severe hazards in those industries dealing with flammable substances, where a small electrical spark may ignite explosive mixtures with devastating consequences.

A similar charging mechanism can occur within low conductivity fluids flowing through pipelines—a process called flow electrification. Fluids which have low electrical conductivity (below 50 picosiemens per meter, where picosiemens per meter is a measure of electrical conductivity), are called accumulators. Fluids having conductivities above 50 pS/m are called non-accumulators. In non-accumulators, charges recombine as fast as they are separated and hence electrostatic charge generation is not significant. In the petrochemical industry, 50 pS/m is the recommended minimum value of electrical conductivity for adequate removal of charge from a fluid.

An important concept for insulating fluids is the static relaxation time. This is similar to the time constant (tau) within an RC circuit. For insulating materials, it is the ratio of the static dielectric constant divided by the electrical conductivity of the material. For hydrocarbon fluids, this is sometimes approximated by dividing the number 18 by the electrical conductivity of the fluid. Thus a fluid that has an electrical conductivity of 1 pS/cm (100 pS/m) will have an estimated relaxation time of about 18 seconds. The excess charge within a fluid will be almost completely dissipated after 4 to 5 times the relaxation time, or 90 seconds for the fluid in the above example.

Page 14: Dielectric Constant

Charge generation increases at higher fluid velocities and larger pipe diameters, becoming quite significant in pipes 8 inches (200 mm) or larger. Static charge generation in these systems is best controlled by limiting fluid velocity. The British standard BS PD CLC/TR 50404:2003 (formerly BS-5958-Part 2) Code of Practice for Control of Undesirable Static Electricity prescribes velocity limits. Because of its large impact on dielectric constant, the recommended velocity for hydrocarbon fluids containing water should be limited to 1 m/s.

Bonding and earthing are the usual ways by which charge buildup can be prevented. For fluids with electrical conductivity below 10 pS/m, bonding and earthing are not adequate for charge dissipation, and anti-static additives may be required.

[edit] Applicable standards

1.BS PD CLC/TR 50404:2003 Code of Practice for Control of Undesirable Static Electricity

2.NFPA 77 (2007) Recommended Practice on Static Electricity

3.API RP 2003 (1998) Protection Against Ignitions Arising Out of Static, Lightning, and Stray Currents

[edit] Electrostatic induction in commercial applications

The principle of electrostatic induction has been harnessed to beneficial effect in industry for many years, beginning with the introduction of electrostatic industrial painting systems for the economical and even application of enamel and polyurethane paints to consumer goods, including automobiles, bicycles, and other products.

Page 15: Dielectric Constant

Flux From Wikipedia, the free encyclopediaJump to: navigation, search This article is about the concept of flux in science and mathematics. For other uses of the word, see Flux (disambiguation).

In the various subfields of physics, there exist two common usages of the term flux, both with rigorous mathematical frameworks.

In the study of transport phenomena (heat transfer, mass transfer and fluid dynamics), flux is defined as flow per unit area, where flow is the movement of some quantity per unit time.[1] Flux, in this definition, is a vector.

In the fields of electromagnetism and mathematics, flux is usually the integral of a vector quantity, flux density, over a finite surface. It is an integral operator that acts on a vector field similarly to the gradient, divergence and curl operators found in vector analysis. The result of this integration is a scalar quantity called flux.[2] The magnetic flux is thus the integral of the magnetic vector field B over a surface, and the electric flux is defined similarly. Using this definition, the flux of the Poynting vector over a specified surface is the rate at which electromagnetic energy flows through that surface. Confusingly, the Poynting vector is sometimes called the power flux, which is an example of the first usage of flux, above.[3] It has units of watts per square metre (W/m2).

One could argue, based on the work of James Clerk Maxwell,[4] that the transport definition precedes the more recent way the term is used in electromagnetism. The specific quote from Maxwell is "In the case of fluxes, we have to take the integral, over a surface, of the flux through every element of the surface. The result of this operation is called the surface integral of the flux. It represents the quantity which passes through the surface."

In addition to these few common mathematical definitions, there are many more looser, but equally valid, usages to describe observations from other fields such as biology, the arts, history, and humanities.

Contents

 [hide]  1 Transport phenomena

o 1.1 Origin of the term o 1.2 Flux definition and theorems o 1.3 Chemical diffusion o 1.4 Quantum mechanics

2 Electromagnetism o 2.1 Flux definition and theorems

Page 16: Dielectric Constant

o 2.2 Maxwell's equations o 2.3 Poynting vector

3 Biology 4 See also 5 Notes

6 Further reading

[edit] Transport phenomena

[edit] Origin of the term

Look up flux in Wiktionary, the free dictionary.

The word flux comes from Latin: fluxus means "flow", and fluere is "to flow".[5] As fluxion, this term was introduced into differential calculus by Isaac Newton.

[edit] Flux definition and theorems

Flux is surface bombardment rate. There are many fluxes used in the study of transport phenomena. Each type of flux has its own distinct unit of measurement along with distinct physical constants. Seven of the most common forms of flux from the transport literature are defined as:

1. Momentum flux , the rate of transfer of momentum across a unit area (N·s·m−2·s−1). (Newton's law of viscosity,)

2. Heat flux , the rate of heat flow across a unit area (J·m−2·s−1). (Fourier's law of conduction)[6] (This definition of heat flux fits Maxwell's original definition.)[4]

3. Diffusion flux , the rate of movement of molecules across a unit area (mol·m−2·s−1). (Fick's law of diffusion)

4. Volumetric flux , the rate of volume flow across a unit area (m3·m−2·s−1). (Darcy's law of groundwater flow)

5. Mass flux , the rate of mass flow across a unit area (kg·m−2·s−1). (Either an alternate form of Fick's law that includes the molecular mass, or an alternate form of Darcy's law that includes the density.)

6. Radiative flux , the amount of energy moving in the form of photons at a certain distance from the source per steradian per second (J·m−2·s−1). Used in astronomy to determine the magnitude and spectral class of a star. Also acts as a generalization of heat flux, which is equal to the radiative flux when restricted to the infrared spectrum.

7. Energy flux , the rate of transfer of energy through a unit area (J·m−2·s−1). The radiative flux and heat flux are specific cases of energy flux.

8. Particle flux, the rate of transfer of particles through a unit area ([number of particles] m−2·s−1)

Flux is heathy.

Page 17: Dielectric Constant

[edit] Chemical diffusion

Chemical molar flux of a component A in an isothermal, isobaric system is defined in above-mentioned Fick's first law as:

where:

is the diffusion coefficient (m2/s) of component A diffusing through component B,

is the concentration (mol/m3) of species A.[7]

This flux has units of mol·m−2·s−1, and fits Maxwell's original definition of flux.[4]

Note: ("nabla") denotes the del operator.

For dilute gases, kinetic molecular theory relates the diffusion coefficient D to the particle density n = N/V, the molecular mass M, the collision cross section , and the absolute temperature T by

where the second factor is the mean free path and the square root (with Boltzmann's constant k) is the mean velocity of the particles.

In turbulent flows, the transport by eddy motion can be expressed as a grossly increased diffusion coefficient.

[edit] Quantum mechanicsMain article: Probability current

In quantum mechanics, particles of mass m in the state have a probability density defined as

So the probability of finding a particle in a unit of volume, say , is

Then the number of particles passing through a perpendicular unit of area per unit time is

Page 18: Dielectric Constant

This is sometimes referred to as the flux density.[8]

[edit] Electromagnetism

[edit] Flux definition and theorems

An example of the first definition of flux is the magnitude of a river's current, that is, the amount of water that flows through a cross-section of the river each second. The amount of sunlight that lands on a patch of ground each second is also a kind of flux.

To better understand the concept of flux in Electromagnetism, imagine a butterfly net. The amount of air moving through the net at any given instant in time is the flux. If the wind speed is high, then the flux through the net is large. If the net is made bigger, then the flux would be larger even though the wind speed is the same. For the most air to move through the net, the opening of the net must be facing the direction the wind is blowing. If the net opening is parallel to the wind, then no wind will be moving through the net. Perhaps the best way to think of flux abstractly is "How much stuff goes through your thing", where the stuff is a field and the thing is the virtual surface.

The flux visualized. The rings show the surface boundaries. The red arrows stand for the flow of charges, fluid particles, subatomic particles, photons, etc. The number of arrows that pass through each ring is the flux.

As a mathematical concept, flux is represented by the surface integral of a vector field,

where:

Page 19: Dielectric Constant

E is a vector field of Electric Force, dA is the vector area of the surface S, directed as the surface normal,

  is the resulting flux.

The surface has to be orientable, i.e. two sides can be distinguished: the surface does not fold back onto itself. Also, the surface has to be actually oriented, i.e. we use a convention as to flowing which way is counted positive; flowing backward is then counted negative.

The surface normal is directed accordingly, usually by the right-hand rule.

Conversely, one can consider the flux the more fundamental quantity and call the vector field the flux density.

Often a vector field is drawn by curves (field lines) following the "flow"; the magnitude of the vector field is then the line density, and the flux through a surface is the number of lines. Lines originate from areas of positive divergence (sources) and end at areas of negative divergence (sinks).

See also the image at right: the number of red arrows passing through a unit area is the flux density, the curve encircling the red arrows denotes the boundary of the surface, and the orientation of the arrows with respect to the surface denotes the sign of the inner product of the vector field with the surface normals.

If the surface encloses a 3D region, usually the surface is oriented such that the influx is counted positive; the opposite is the outflux.

The divergence theorem states that the net outflux through a closed surface, in other words the net outflux from a 3D region, is found by adding the local net outflow from each point in the region (which is expressed by the divergence).

If the surface is not closed, it has an oriented curve as boundary. Stokes' theorem states that the flux of the curl of a vector field is the line integral of the vector field over this boundary. This path integral is also called circulation, especially in fluid dynamics. Thus the curl is the circulation density.

We can apply the flux and these theorems to many disciplines in which we see currents, forces, etc., applied through areas.

[edit] Maxwell's equations

The flux of electric and magnetic field lines is frequently discussed in electrostatics. This is because Maxwell's equations in integral form involve integrals like above for electric and magnetic fields.

For instance, Gauss's law states that the flux of the electric field out of a closed surface is proportional to the electric charge enclosed in the surface (regardless of how that charge is distributed). The constant of proportionality is the reciprocal of the permittivity of free space.

Page 20: Dielectric Constant

Its integral form is:

where:

is the electric field, is the area of a differential square on the surface A with an outward

facing surface normal defining its direction, is the charge enclosed by the surface, is the permittivity of free space

is the integral over the surface A.

Either or is called the electric flux.

If one considers the flux of the electric field vector, E, for a tube near a point charge in the field the charge but not containing it with sides formed by lines tangent to the field, the flux for the sides is zero and there is an equal and opposite flux at both ends of the tube. This is a consequence of Gauss's Law applied to an inverse square field. The flux for any cross-sectional surface of the tube will be the same. The total flux for any surface surrounding a charge q is q/ε0.[9]

In free space the electric displacement vector D = ε0 E so for any bounding surface the flux of D = q, the charge within it. Here the expression "flux of" indicates a mathematical operation and, as can be seen, the result is not necessarily a "flow".

Faraday's law of induction in integral form is:

where:

is an infinitesimal element (differential) of the closed curve C (i.e. a vector with magnitude equal to the length of the infinitesimal line element, and direction given by the tangent to the curve C, with the sign determined by the integration direction).

The magnetic field is denoted by . Its flux is called the magnetic flux. The time-rate of change of the magnetic flux through a loop of wire is minus the electromotive force created in that wire. The direction is such that if current is allowed to pass through the wire, the electromotive force will cause a current which "opposes" the change in magnetic field by itself producing a magnetic field opposite to the change. This is the basis for inductors and many electric generators.

Page 21: Dielectric Constant

[edit] Poynting vector

The flux of the Poynting vector through a surface is the electromagnetic power, or energy per unit time, passing through that surface. This is commonly used in analysis of electromagnetic radiation, but has application to other electromagnetic systems as well.

[edit] Biology

In general, flux in biology relates to movement of a substance between compartments. There are several cases where the concept of flux is important.

The movement of molecules across a membrane: in this case, flux is defined by the rate of diffusion or transport of a substance across a permeable membrane. Except in the case of active transport, net flux is directly proportional to the concentration difference across the membrane, the surface area of the membrane, and the membrane permeability constant.

In ecology, flux is often considered at the ecosystem level - for instance, accurate determination of carbon fluxes using techniques like eddy covariance (at a regional and global level) is essential for modeling the causes and consequences of global warming.

Metabolic flux refers to the rate of flow of metabolites along a metabolic pathway, or even through a single enzyme. A calculation may also be made of carbon (or other elements, e.g. nitrogen) flux. It is dependent on a number of factors, including: enzyme concentration; the concentration of precursor, product, and intermediate metabolites; post-translational modification of enzymes; and the presence of metabolic activators or repressors. Metabolic control analysis and flux balance analysis provide frameworks for understanding metabolic fluxes and their constraints.

Page 22: Dielectric Constant

Frequency From Wikipedia, the free encyclopediaJump to: navigation, search For other uses, see Frequency (disambiguation).

Three cyclically flashing lights, from lowest frequency (top) to highest frequency (bottom). f is the frequency in hertz (Hz), meaning the number of cycles per second. T is the period in seconds (s), meaning the number of seconds per cycle. T and f are reciprocals.

Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency. The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency. For example, if a newborn baby's heart beats at a frequency of 120 times a minute, its period (the interval between beats) is half a second.

Contents

 [hide]  1 Definitions and units 2 Measurement

o 2.1 By counting o 2.2 By stroboscope o 2.3 By frequency counter o 2.4 Heterodyne methods

3 Frequency of waves 4 Examples

o 4.1 Physics of light o 4.2 Physics of sound o 4.3 Line current

5 Period versus frequency 6 Other types of frequency 7 Frequency ranges 8 See also 9 References

Page 23: Dielectric Constant

10 Further reading

11 External links

[edit] Definitions and units

For cyclical processes, such as rotation, oscillations, or waves, frequency is defined as a number of cycles per unit time. In physics and engineering disciplines, such as optics, acoustics, and radio, frequency is usually denoted by a Latin letter f or by a Greek letter ν (nu) .

In SI units, the unit of frequency is the hertz (Hz), named after the German physicist Heinrich Hertz: 1 Hz means that an event repeats once per second. A previous name for this unit was cycles per second.

A traditional unit of measure used with rotating mechanical devices is revolutions per minute, abbreviated RPM. 60 RPM equals one hertz.[1]

The period, usually denoted by T, is the length of time taken by one cycle, and is the reciprocal of the frequency f:

The SI unit for period is the second.

[edit] Measurement

Sinusoidal waves of various frequencies; the bottom waves have higher frequencies than those above. The horizontal axis represents time.

[edit] By counting

Calculating the frequency of a repeating event is accomplished by counting the number of times that event occurs within a specific time period, then dividing the count by the length of the time period. For example, if 71 events occur within 15 seconds the frequency is:

Page 24: Dielectric Constant

If the number of counts is not very large, it is more accurate to measure the time interval for a predetermined number of occurrences, rather than the number of occurrences within a specified time.[2] The latter method introduces a random error into the count of between zero and one count, so on average half a count. This is called gating error and causes an average error in the calculated frequency of Δf = 1/(2 Tm), or a fractional error of Δf / f = 1/(2 f Tm) where Tm is the timing interval and f is the measured frequency. This error decreases with frequency, so it is a problem at low frequencies where the number of counts N is small.

[edit] By stroboscope

An older method of measuring the frequency of rotating or vibrating objects is to use a stroboscope. This is an intense repetitively flashing light (strobe light) whose frequency can be adjusted with a calibrated timing circuit. The strobe light is pointed at the rotating object and the frequency adjusted up and down. When the frequency of the strobe equals the frequency of the rotating or vibrating object, the object completes one cycle of oscillation and returns to its original position between the flashes of light, so when illuminated by the strobe the object appears stationary. Then the frequency can be read from the calibrated readout on the stroboscope. A downside of this method is that an object rotating at an integer multiple of the strobing frequency will also appear stationary.

[edit] By frequency counter

Higher frequencies are usually measured with a frequency counter. This is an electronic instrument which measures the frequency of an applied repetitive electronic signal and displays the result in hertz on a digital display. It uses digital logic to count the number of cycles during a time interval established by a precision quartz time base. Cyclic processes that are not electrical in nature, such as the rotation rate of a shaft, mechanical vibrations, or sound waves, can be converted to a repetitive electronic signal by transducers and the signal applied to a frequency counter. Frequency counters can currently cover the range up to about 100 GHz. This represents the limit of direct counting methods; frequencies above this must be measured by indirect methods.

[edit] Heterodyne methods

Above the range of frequency counters, frequencies of electromagnetic signals are often measured indirectly by means of heterodyning (frequency conversion). A reference signal of a known frequency near the unknown frequency is mixed with the unknown frequency in a nonlinear mixing device such as a diode. This creates a heterodyne or "beat" signal at the difference between the two frequencies. If the two signals are close together in frequency the heterodyne is low enough to be measured by a frequency counter. Of course, this process just measures the unknown frequency by its offset from the reference frequency, which must be determined by some other method. To reach higher frequencies, several stages of heterodyning can be used. Current research is extending this method to infrared and light frequencies (optical heterodyne detection).

Page 25: Dielectric Constant

[edit] Frequency of waves

For periodic waves, frequency has an inverse relationship to the concept of wavelength; simply, frequency is inversely proportional to wavelength λ (lambda). The frequency f is equal to the phase velocity v of the wave divided by the wavelength λ of the wave:

In the special case of electromagnetic waves moving through a vacuum, then v = c, where c is the speed of light in a vacuum, and this expression becomes:

When waves from a monochrome source travel from one medium to another, their frequency remains exactly the same — only their wavelength and speed change.

[edit] Examples

[edit] Physics of light

Complete spectrum of electromagnetic radiation with the visible portion highlightedMain articles: Light and Electromagnetic radiation

Visible light is an electromagnetic wave, consisting of oscillating electric and magnetic fields traveling through space. The frequency of the wave determines its color: 4×1014 Hz is red light, 8×1014 Hz is violet light, and between these (in the range 4-8×1014 Hz) are all the other colors of the rainbow. An electromagnetic wave can have a frequency less than 4×1014 Hz, but it will be invisible to the human eye; such waves are called infrared (IR) radiation. At even lower frequency, the wave is called a microwave, and at still lower frequencies it is called a radio wave. Likewise, an electromagnetic wave can have a frequency higher than 8×1014 Hz, but it will be

Page 26: Dielectric Constant

invisible to the human eye; such waves are called ultraviolet (UV) radiation. Even higher-frequency waves are called X-rays, and higher still are gamma rays.

All of these waves, from the lowest-frequency radio waves to the highest-frequency gamma rays, are fundamentally the same, and they are all called electromagnetic radiation. They all travel through a vacuum at the speed of light.

Another property of an electromagnetic wave is its wavelength. The wavelength is inversely proportional to the frequency, so an electromagnetic wave with a higher frequency has a shorter wavelength, and vice-versa.

[edit] Physics of soundMain article: Sound

Sound is made up of changes in air pressure in the form of waves. Frequency is the property of sound that most determines pitch.[3] The frequencies an ear can hear are limited to a specific range of frequencies.

Mechanical vibrations perceived as sound travel through all forms of matter: gases, liquids, solids, and plasmas. The matter that supports the sound is called the medium. Sound cannot travel through a vacuum.

The audible frequency range for humans is typically given as being between about 20 Hz and 20,000 Hz (20 kHz). High frequencies often become more difficult to hear with age. Other species have different hearing ranges. For example, some dog breeds can perceive vibrations up to 60,000 Hz.[4]

[edit] Line current

In Europe, Africa, Australia, Southern South America, most of Asia, and Russia, the frequency of the alternating current in household electrical outlets is 50 Hz (close to the tone G), whereas in North America and Northern South America, the frequency of the alternating current in household electrical outlets is 60 Hz (between the tones B ♭and B; that is, a minor third above the European frequency). The frequency of the 'hum' in an audio recording can show where the recording was made, in countries using a European, or an American, grid frequency.

[edit] Period versus frequency

As a matter of convenience, longer and slower waves, such as ocean surface waves, tend to be described by wave period rather than frequency. Short and fast waves, like audio and radio, are usually described by their frequency instead of period. These commonly used conversions are listed below:

Frequency1 mHz (10−3)

1 Hz (100)

1 kHz (103)

1 MHz (106)

1 GHz (109)

1 THz (1012)

Period (time) 1 ks (103) 1 s (100) 1 ms (10−3) 1 µs (10−6) 1 ns (10−9) 1 ps (10−12)

Page 27: Dielectric Constant

[edit] Other types of frequency

Angular frequency ω is defined as the rate of change of angular displacement, θ, (during rotation), or the rate of change of the phase of a sinusoidal waveform (e.g. in oscillations and waves), or as the rate of change of the argument to the sine function:

Angular frequency is commonly measured in radians per second (rad/s) but, for discrete-time signals, can also be expressed as radians per sample time, which is a dimensionless quantity.

Spatial frequency is analogous to temporal frequency, but the time axis is replaced by one or more spatial displacement axes. E.g.:

Wavenumber, k, sometimes means the spatial frequency analogue of angular temporal frequency. In case of more than one spatial dimension, wavenumber is a vector quantity.

[edit] Frequency ranges

The frequency range of a system is the range over which it is considered to provide a useful level of signal with acceptable distortion characteristics. A listing of the upper and lower limits of frequency limits for a system is not useful without a criterion for what the range represents.

Many systems are characterized by the range of frequencies to which they respond. Musical instruments produce different ranges of notes within the hearing range. The electromagnetic spectrum can be divided into many different ranges such as visible light, infrared or ultraviolet radiation, radio waves, X-rays and so on, and each of these ranges can in turn be divided into smaller ranges. A radio communications signal must occupy a range of frequencies carrying most of its energy, called its bandwidth. Allocation of radio frequency ranges to different uses is a major function of radio spectrum allocation.

Page 28: Dielectric Constant

Complex number From Wikipedia, the free encyclopediaJump to: navigation, search

A complex number can be visually represented as a pair of numbers (a,b) forming a vector on a diagram called an Argand diagram, representing the complex plane. Re is the real axis, Im is the imaginary axis, and i is the imaginary unit, satisfying i2 = −1.

A complex number is a number which can be put in the form a + bi, in which a and b are real numbers and i is called the imaginary unit, where i2 = −1.[1] In this expression, a is called the real part and b the imaginary part of the complex number. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part. The complex number can be identified with the point (a, b). A complex number whose real part is zero is said to be purely imaginary, whereas a complex number whose imaginary part is zero is a real number. In this way the complex numbers contain the ordinary real numbers while extending them in order to solve problems that cannot be solved with only real numbers.

Complex numbers are used in many scientific fields, including engineering, electromagnetism, quantum physics, applied mathematics, and chaos theory. Italian mathematician Gerolamo Cardano is the first known to have introduced complex numbers. He called them "fictitious", during his attempts to find solutions to cubic equations in the 16th century.[2]

Contents

 [hide]  1 Overview

o 1.1 Definition o 1.2 Complex plane o 1.3 History in brief

2 Elementary operations

Page 29: Dielectric Constant

o 2.1 Conjugation o 2.2 Addition and subtraction o 2.3 Multiplication and division o 2.4 Square root

3 Polar form o 3.1 Absolute value and argument o 3.2 Multiplication, division and exponentiation in polar form

4 Properties o 4.1 Field structure o 4.2 Solutions of polynomial equations o 4.3 Algebraic characterization o 4.4 Characterization as a topological field

5 Formal construction o 5.1 Formal development o 5.2 Matrix representation of complex numbers

6 Complex analysis o 6.1 Complex exponential and related functions o 6.2 Holomorphic functions

7 Applications o 7.1 Control theory o 7.2 Improper integrals o 7.3 Fluid dynamics o 7.4 Dynamic equations o 7.5 Electromagnetism and electrical engineering o 7.6 Signal analysis o 7.7 Quantum mechanics o 7.8 Relativity o 7.9 Geometry

7.9.1 Fractals 7.9.2 Triangles

o 7.10 Algebraic number theory o 7.11 Analytic number theory o 7.12 Quality Adjusted Life Years

8 History 9 Generalizations and related notions 10 See also 11 Notes 12 References

o 12.1 Mathematical references o 12.2 Historical references

13 Further reading

14 External links

[edit] Overview

Complex numbers allow for solutions to certain equations that have no real solution: the equation

Page 30: Dielectric Constant

has no real solution, since the square of a real number is 0 or positive. Complex numbers provide a solution to this problem. The idea is to extend the real numbers

with the imaginary unit i where , so that solutions to equations like the preceding one can be found. In this case the solutions are −1 ± 3i. In fact not only quadratic equations, but all polynomial equations in a single variable can be solved using complex numbers.

[edit] Definition

An illustration of the complex plane. The real part of a complex number z = x + iy is x, and its imaginary part is y.

A complex number is a number that can be expressed in the form

where a and b are real numbers and i is the imaginary unit, satisfying i2 = −1. For example, −3.5 + 2i is a complex number. It is common to write a for a + 0i and bi for 0 + bi. Moreover, when the imaginary part is negative, it is common to write a − bi with b > 0 instead of a + (−b)i, for example 3 − 4i instead of 3 + (−4)i.

The set of all complex numbers is denoted by or .

The real number a of the complex number z = a + bi is called the real part of z, and the real number b is often called the imaginary part. By this convention the imaginary part is a real number – not including the imaginary unit: hence b, not bi, is the imaginary part.[3][4] The real part is denoted by Re(z) or ℜ(z), and the imaginary part b is denoted by Im(z) or ℑ(z). For example,

Some authors write a+ib instead of a+bi (scalar multiplication between b and i is commutative). In some disciplines, in particular electromagnetism and electrical

Page 31: Dielectric Constant

engineering, j is used instead of i, since i is frequently used for electric current. In these cases complex numbers are written as a + bj or a + jb.

A real number a can usually be regarded as a complex number with an imaginary part of zero, that is to say, a + 0i. However the sets are defined differently and have slightly different operations defined, for instance comparison operations are not defined for complex numbers. A pure imaginary number is a complex number whose real part is zero, that is to say, of the form 0 + bi.

[edit] Complex planeMain article: Complex plane

Figure 1: A complex number plotted as a point (red) and position vector (blue) on an Argand diagram; is the rectangular expression of the point.

A complex number can be viewed as a point or position vector in a two-dimensional Cartesian coordinate system called the complex plane or Argand diagram (see Pedoe 1988 and Solomentsev 2001), named after Jean-Robert Argand. The numbers are conventionally plotted using the real part as the horizontal component, and imaginary part as vertical (see Figure 1). These two values used to identify a given complex number are therefore called its Cartesian, rectangular, or algebraic form.

The defining characteristic of a position vector is that it has magnitude and direction. These are emphasised in a complex number's polar form and it turns out notably that the operations of addition and multiplication take on a very natural geometric character when complex numbers are viewed as position vectors: addition corresponds to vector addition while multiplication corresponds to multiplying their magnitudes and adding their arguments (i.e. the angles they make with the x axis). Viewed in this way the multiplication of a complex number by i corresponds to rotating a complex number counterclockwise through 90° about the origin:

.

[edit] History in briefMain section: History

The solution in radicals (without trigonometric functions) of a general cubic equation contains the square roots of negative numbers when all three roots are real numbers, a situation that cannot be rectified by factoring aided by the rational root test if the cubic is irreducible (the so-called casus irreducibilis). This conundrum led Italian mathematician Gerolamo Cardano to conceive of complex numbers in around 1545, though his understanding was rudimentary.

Page 32: Dielectric Constant

Work on the problem of general polynomials ultimately led to the fundamental theorem of algebra, which shows that with complex numbers, a solution exists to every polynomial equation of degree one or higher. Complex numbers thus form an algebraically closed field, where any polynomial equation has a root.

Many mathematicians contributed to the full development of complex numbers. The rules for addition, subtraction, multiplication, and division of complex numbers were developed by the Italian mathematician Rafael Bombelli.[5] A more abstract formalism for the complex numbers was further developed by the Irish mathematician William Rowan Hamilton, who extended this abstraction to the theory of quaternions.

[edit] Elementary operations

[edit] Conjugation

Geometric representation of and its conjugate in the complex plane

The complex conjugate of the complex number z = x + yi is defined to be x − yi. It is denoted or . Geometrically, is the "reflection" of z about the real axis. In particular, conjugating twice gives the original complex number: .

The real and imaginary parts of a complex number can be extracted using the conjugate:

Moreover, a complex number is real if and only if it equals its conjugate.

Conjugation distributes over the standard arithmetic operations:

Page 33: Dielectric Constant

The reciprocal of a nonzero complex number z = x + yi is given by

This formula can be used to compute the multiplicative inverse of a complex number if it is given in rectangular coordinates. Inversive geometry, a branch of geometry studying more general reflections than ones about a line, can also be expressed in terms of complex numbers.

[edit] Addition and subtraction

Addition of two complex numbers can be done geometrically by constructing a parallelogram.

Complex numbers are added by adding the real and imaginary parts of the summands. That is to say:

Similarly, subtraction is defined by

Using the visualization of complex numbers in the complex plane, the addition has the following geometric interpretation: the sum of two complex numbers A and B, interpreted as points of the complex plane, is the point X obtained by building a parallelogram three of whose vertices are O, A and B. Equivalently, X is the point such that the triangles with vertices O, A, B, and X, B, A, are congruent.

[edit] Multiplication and division

The multiplication of two complex numbers is defined by the following formula:

Page 34: Dielectric Constant

In particular, the square of the imaginary unit is −1:

The preceding definition of multiplication of general complex numbers follows naturally from this fundamental property of the imaginary unit. Indeed, if i is treated as a number so that di means d times i, the above multiplication rule is identical to the usual rule for multiplying two sums of two terms.

(distributive law) (commutative law of addition—the order of the

summands can be changed)

(commutative law of multiplication—the order of the multiplicands can be changed)

(fundamental property of the imaginary unit).

The division of two complex numbers is defined in terms of complex multiplication, which is described above, and real division:

Division can be defined in this way because of the following observation:

As shown earlier, is the complex conjugate of the denominator . The real part c and the imaginary part d of the denominator must not both be zero for division to be defined.

[edit] Square rootSee also: Square roots of negative and complex numbers

The square roots of a + bi (with b ≠ 0) are , where

and

Page 35: Dielectric Constant

where sgn is the signum function. This can be seen by squaring to obtain

a + bi.[6][7] Here is called the modulus of a + bi, and the square root with non-negative real part is called the principal square root.

[edit] Polar form

Main article: Polar coordinate system

Figure 2: The argument φ and modulus r locate a point on an Argand diagram;

or are polar expressions of the point.

[edit] Absolute value and argument

Another way of encoding points in the complex plane other than using the x- and y-coordinates is to use the distance of a point P to O, the point whose coordinates are (0, 0) (the origin), and the angle of the line through P and O. This idea leads to the polar form of complex numbers.

The absolute value (or modulus or magnitude) of a complex number z = x + yi is

If z is a real number (i.e., y = 0), then r = |x|. In general, by Pythagoras' theorem, r is the distance of the point P representing the complex number z to the origin.

The argument or phase of z is the angle of the radius OP with the positive real axis,

and is written as . As with the modulus, the argument can be found from the rectangular form :[8]

Page 36: Dielectric Constant

The value of φ must always be expressed in radians. It can change by any multiple of 2π and still give the same angle. Hence, the arg function is sometimes considered as

multivalued. Normally, as given above, the principal value in the interval is

chosen. Values in the range are obtained by adding if the value is negative. The polar angle for the complex number 0 is undefined, but arbitrary choice of the angle 0 is common.

The value of φ equals the result of atan2: .

Together, r and φ give another way of representing complex numbers, the polar form, as the combination of modulus and argument fully specify the position of a point on the plane. Recovering the original rectangular co-ordinates from the polar form is done by the formula called trigonometric form

Using Euler's formula this can be written as

Using the cis function, this is sometimes abbreviated to

In angle notation, often used in electronics to represent a phasor with amplitude r and phase φ it is written as[9]

Page 37: Dielectric Constant

[edit] Multiplication, division and exponentiation in polar form

Multiplication of 2+i (blue triangle) and 3+i (red triangle). The red triangle is rotated to match the vertex of the blue one and stretched by √5, the length of the hypotenuse of the blue triangle.

The relevance of representing complex numbers in polar form stems from the fact that the formulas for multiplication, division and exponentiation are simpler than the ones using Cartesian coordinates. Given two complex numbers z1 = r1(cos φ1 + isin φ1) and z2 =r2(cos φ2 + isin φ2) the formula for multiplication is

In other words, the absolute values are multiplied and the arguments are added to yield the polar form of the product. For example, multiplying by i corresponds to a quarter-rotation counter-clockwise, which gives back i 2 = −1. The picture at the right illustrates the multiplication of

Since the real and imaginary part of 5+5i are equal, the argument of that number is 45 degrees, or π/4 (in radian). On the other hand, it is also the sum of the angles at the origin of the red and blue triangle are arctan(1/3) and arctan(1/2), respectively. Thus, the formula

holds. As the arctan function can be approximated highly efficiently, formulas like this—known as Machin-like formulas—are used for high-precision approximations of π.

Similarly, division is given by

Page 38: Dielectric Constant

This also implies de Moivre's formula for exponentiation of complex numbers with integer exponents:

The n-th roots of z are given by

for any integer k satisfying 0 ≤ k ≤ n − 1. Here is the usual (positive) nth root of the positive real number r. While the nth root of a positive real number r is chosen to be the positive real number c satisfying cn = x there is no natural way of distinguishing one particular complex nth root of a complex number. Therefore, the nth root of z is considered as a multivalued function (in z), as opposed to a usual function f, for which f(z) is a uniquely defined number. Formulas such as

(which holds for positive real numbers), do in general not hold for complex numbers.

[edit] Properties

[edit] Field structure

The set C of complex numbers is a field. Briefly, this means that the following facts hold: first, any two complex numbers can be added and multiplied to yield another complex number. Second, for any complex number z, its negative −z is also a complex number; and third, every nonzero complex number has a reciprocal complex number. Moreover, these operations satisfy a number of laws, for example the law of commutativity of addition and multiplication for any two complex numbers z1 and z2:

These two laws and the other requirements on a field can be proven by the formulas given above, using the fact that the real numbers themselves form a field.

Unlike the reals, C is not an ordered field, that is to say, it is not possible to define a relation z1 < z2 that is compatible with the addition and multiplication. In fact, in any ordered field, the square of any element is necessarily positive, so i2 = −1 precludes the existence of an ordering on C.

When the underlying field for a mathematical topic or construct is the field of complex numbers, the thing's name is usually modified to reflect that fact. For example: complex analysis, complex matrix, complex polynomial, and complex Lie algebra.

Page 39: Dielectric Constant

[edit] Solutions of polynomial equations

Given any complex numbers (called coefficients) a0, ..., an, the equation

has at least one complex solution z, provided that at least one of the higher coefficients, a1, ..., an, is nonzero. This is the statement of the fundamental theorem of algebra. Because of this fact, C is called an algebraically closed field. This property does not hold for the field of rational numbers Q (the polynomial x2 − 2 does not have a rational root, since √2 is not a rational number) nor the real numbers R (the polynomial x2 + a does not have a real root for a > 0, since the square of x is positive for any real number x).

There are various proofs of this theorem, either by analytic methods such as Liouville's theorem, or topological ones such as the winding number, or a proof combining Galois theory and the fact that any real polynomial of odd degree has at least one root.

Because of this fact, theorems that hold "for any algebraically closed field", apply to C. For example, any complex matrix has at least one (complex) eigenvalue.

[edit] Algebraic characterization

The field C has the following three properties: first, it has characteristic 0. This means that 1 + 1 + ... + 1 ≠ 0 for any number of summands (all of which equal one). Second, its transcendence degree over Q, the prime field of C is the cardinality of the continuum. Third, it is algebraically closed (see above). It can be shown that any field having these properties is isomorphic (as a field) to C. For example, the algebraic closure of Qp also satisfies these three properties, so these two fields are isomorphic. Also, C is isomorphic to the field of complex Puiseux series. However, specifying an isomorphism requires the axiom of choice. Another consequence of this algebraic characterization is that C contains many proper subfields which are isomorphic to C.

[edit] Characterization as a topological field

The preceding characterization of C describes the algebraic aspects of C, only. That is to say, the properties of nearness and continuity, which matter in areas such as analysis and topology, are not dealt with. The following description of C as a topological field (that is, a field that is equipped with a topology, which allows the notion of convergence) does take into account the topological properties. C contains a subset P (namely the set of positive real numbers) of nonzero elements satisfying the following three conditions:

P is closed under addition, multiplication and taking inverses. If x and y are distinct elements of P, then either x − y or y − x is in P. If S is any nonempty subset of P, then S + P = x + P for some x in C.

Page 40: Dielectric Constant

Moreover, C has a nontrivial involutive automorphism (namely the complex conjugation), such that xx∗ is in P for any nonzero x in C.

Any field F with these properties can be endowed with a topology by taking the sets B(x, p) = {y | p − (y − x)(y − x)∗ ∈ P} as a base, where x ranges over the field and p ranges over P. With this topology F is isomorphic as a topological field to C.

The only connected locally compact topological fields are R and C. This gives another characterization of C as a topological field, since C can be distinguished from R because the nonzero complex numbers are connected, while the nonzero real numbers are not.

[edit] Formal construction

[edit] Formal development

Above, complex numbers have been defined by introducing i, the imaginary unit, as a symbol. More rigorously, the set C of complex numbers can be defined as the set R2 of ordered pairs (a, b) of real numbers. In this notation, the above formulas for addition and multiplication read

It is then just a matter of notation to express (a, b) as a + bi.

Though this low-level construction does accurately describe the structure of the complex numbers, the following equivalent definition reveals the algebraic nature of C more immediately. This characterization relies on the notion of fields and polynomials. A field is a set endowed with an addition, subtraction, multiplication and division operations which behave as is familiar from, say, rational numbers. For example, the distributive law

must hold for any three elements x, y and z of a field. The set R of real numbers does form a field. A polynomial p(X) with real coefficients is an expression of the form

where the a0, ..., an are real numbers. The usual addition and multiplication of polynomials endows the set R[X] of all such polynomials with a ring structure. This ring is called polynomial ring. The quotient ring R[X]/(X2+1) can be shown to be a field. This extension field contains two square roots of −1, namely (the cosets of) X and −X, respectively. (The cosets of) 1 and X form a basis of R[X]/(X2+1) as a real vector space, which means that each element of the extension field can be uniquely written as a linear combination in these two elements. Equivalently, elements of the extension field can be written as ordered pairs (a,b) of real numbers. Moreover, the above formulas for addition etc. correspond to the ones yielded by this abstract

Page 41: Dielectric Constant

algebraic approach – the two definitions of the field C are said to be isomorphic (as fields). Together with the above-mentioned fact that C is algebraically closed, this also shows that C is an algebraic closure of R.

[edit] Matrix representation of complex numbers

Complex numbers a+ib can also be represented by 2×2 matrices that have the following form:

Here the entries a and b are real numbers. The sum and product of two such matrices is again of this form, and the sum and product of complex numbers corresponds to the sum and product of such matrices. The geometric description of the multiplication of complex numbers can also be phrased in terms of rotation matrices by using this correspondence between complex numbers and such matrices. Moreover, the square of the absolute value of a complex number expressed as a matrix is equal to the determinant of that matrix:

The conjugate corresponds to the transpose of the matrix.

Though this representation of complex numbers with matricies is the most common,

many other representations arise from matrices other than that square to the negative of the identity matrix. See the article on 2 × 2 real matrices for other representations of complex numbers.

[edit] Complex analysis

Page 42: Dielectric Constant

Color wheel graph of sin(1/z). Black parts inside refer to numbers having large absolute values.Main article: Complex analysis

The study of functions of a complex variable is known as complex analysis and has enormous practical use in applied mathematics as well as in other branches of mathematics. Often, the most natural proofs for statements in real analysis or even number theory employ techniques from complex analysis (see prime number theorem for an example). Unlike real functions which are commonly represented as two-dimensional graphs, complex functions have four-dimensional graphs and may usefully be illustrated by color coding a three-dimensional graph to suggest four dimensions, or by animating the complex function's dynamic transformation of the complex plane.

[edit] Complex exponential and related functions

The notions of convergent series and continuous functions in (real) analysis have natural analogs in complex analysis. A sequence of complex numbers is said to converge if and only if its real and imaginary parts do. This is equivalent to the (ε, δ)-definition of limits, where the absolute value of real numbers is replaced by the one of complex numbers. From a more abstract point of view, C, endowed with the metric

is a complete metric space, which notably includes the triangle inequality

for any two complex numbers z1 and z2.

Like in real analysis, this notion of convergence is used to construct a number of elementary functions: the exponential function exp(z), also written ez, is defined as the infinite series

and the series defining the real trigonometric functions sine and cosine, as well as hyperbolic functions such as sinh also carry over to complex arguments without change. Euler's identity states:

for any real number φ, in particular

Page 43: Dielectric Constant

Unlike in the situation of real numbers, there is an infinitude of complex solutions z of the equation

for any complex number w ≠ 0. It can be shown that any such solution z—called complex logarithm of a—satisfies

where arg is the argument defined above, and ln the (real) natural logarithm. As arg is a multivalued function, unique only up to a multiple of 2π, log is also multivalued. The principal value of log is often taken by restricting the imaginary part to the interval (−π,π].

Complex exponentiation zω is defined as

Consequently, they are in general multi-valued. For ω = 1 / n, for some natural number n, this recovers the non-unicity of n-th roots mentioned above.

Complex numbers, unlike real numbers, do not in general satisfy the unmodified power and logarithm identities, particularly when naïvely treated as single-valued functions; see failure of power and logarithm identities. For example they do not satisfy

Both sides of the equation are multivalued by the definition of complex exponentiation given here, and the values on the left are a subset of those on the right.

[edit] Holomorphic functions

A function f : C → C is called holomorphic if it satisfies the Cauchy-Riemann equations. For example, any R -linear map C → C can be written in the form

with complex coefficients a and b. This map is holomorphic if and only if b = 0. The second summand is real-differentiable, but does not satisfy the Cauchy-Riemann equations.

Complex analysis shows some features not apparent in real analysis. For example, any two holomorphic functions f and g that agree on an arbitrarily small open subset of C necessarily agree everywhere. Meromorphic functions, functions that can locally be written as f(z)/(z − z0)n with a holomorphic function f(z), still share some of the features of holomorphic functions. Other functions have essential singularities, such as sin(1/z) at z = 0.

Page 44: Dielectric Constant

[edit] Applications

Some applications of complex numbers are:

[edit] Control theory

In control theory, systems are often transformed from the time domain to the frequency domain using the Laplace transform. The system's poles and zeros are then analyzed in the complex plane. The root locus, Nyquist plot, and Nichols plot techniques all make use of the complex plane.

In the root locus method, it is especially important whether the poles and zeros are in the left or right half planes, i.e. have real part greater than or less than zero. If a system has poles that are

in the right half plane, it will be unstable, all in the left half plane, it will be stable, on the imaginary axis, it will have marginal stability.

If a system has zeros in the right half plane, it is a nonminimum phase system.

[edit] Improper integrals

In applied fields, complex numbers are often used to compute certain real-valued improper integrals, by means of complex-valued functions. Several methods exist to do this; see methods of contour integration.

[edit] Fluid dynamics

In fluid dynamics, complex functions are used to describe potential flow in two dimensions.

[edit] Dynamic equations

In differential equations, it is common to first find all complex roots r of the characteristic equation of a linear differential equation or equation system and then attempt to solve the system in terms of base functions of the form f(t) = ert. Likewise, in difference equations, the complex roots r of the characteristic equation of the difference equation system are used, to attempt to solve the system in terms of base functions of the form f(t) = r t.

[edit] Electromagnetism and electrical engineeringMain article: Alternating current

In electrical engineering, the Fourier transform is used to analyze varying voltages and currents. The treatment of resistors, capacitors, and inductors can then be unified by introducing imaginary, frequency-dependent resistances for the latter two and combining all three in a single complex number called the impedance. This approach is called phasor calculus.

Page 45: Dielectric Constant

In electrical engineering, the imaginary unit is denoted by j, to avoid confusion with I which is generally in use to denote electric current.

Since the voltage in an AC circuit is oscillating, it can be represented as

To obtain the measurable quantity, the real part is taken:

See for example.[10]

[edit] Signal analysis

Complex numbers are used in signal analysis and other fields for a convenient description for periodically varying signals. For given real functions representing actual physical quantities, often in terms of sines and cosines, corresponding complex functions are considered of which the real parts are the original quantities. For a sine wave of a given frequency, the absolute value |z| of the corresponding z is the amplitude and the argument arg(z) the phase.

If Fourier analysis is employed to write a given real-valued signal as a sum of periodic functions, these periodic functions are often written as complex valued functions of the form

where ω represents the angular frequency and the complex number z encodes the phase and amplitude as explained above.

This use is also extended into digital signal processing and digital image processing, which utilize digital versions of Fourier analysis (and wavelet analysis) to transmit, compress, restore, and otherwise process digital audio signals, still images, and video signals.

Another example, relevant to the two side bands of amplitude modulation of AM radio, is:

Page 46: Dielectric Constant

[edit] Quantum mechanics

The complex number field is intrinsic to the mathematical formulations of quantum mechanics, where complex Hilbert spaces provide the context for one such formulation that is convenient and perhaps most standard. The original foundation formulas of quantum mechanics – the Schrödinger equation and Heisenberg's matrix mechanics – make use of complex numbers.

[edit] Relativity

In special and general relativity, some formulas for the metric on spacetime become simpler if one takes the time variable to be imaginary. (This is no longer standard in classical relativity, but is used in an essential way in quantum field theory.) Complex numbers are essential to spinors, which are a generalization of the tensors used in relativity.

[edit] Geometry

[edit] Fractals

Certain fractals are plotted in the complex plane, e.g. the Mandelbrot set and Julia sets.

[edit] Triangles

Every triangle has a unique Steiner inellipse—an ellipse inside the triangle and tangent to the midpoints of the three sides of the triangle. The foci of a triangle's Steiner inellipse can be found as follows, according to Marden's theorem:[11][12] Denote the triangle's vertices in the complex plane as a=xA+yAi, b=xB+yBi, and c=xC+yCi. Write

the cubic equation , take its derivative, and equate the (quadratic) derivative to zero. Marden's Theorem says that the solutions of this equation are the complex numbers denoting the locations of the two foci of the Steiner inellipse.

[edit] Algebraic number theory

Construction of a regular polygon using straightedge and compass.

Page 47: Dielectric Constant

As mentioned above, any nonconstant polynomial equation (in complex coefficients) has a solution in C. A fortiori, the same is true if the equation has rational coefficients. The roots of such equations are called algebraic numbers – they are a principal object of study in algebraic number theory. Compared to Q, the algebraic closure of Q, which also contains all algebraic numbers, C has the advantage of being easily understandable in geometric terms. In this way, algebraic methods can be used to study geometric questions and vice versa. With algebraic methods, more specifically applying the machinery of field theory to the number field containing roots of unity, it can be shown that it is not possible to construct a regular nonagon using only compass and straightedge – a purely geometric problem.

Another example are Pythagorean triples (a, b, c), that is to say integers satisfying

(which implies that the triangle having sidelengths a, b, and c is a right triangle). They can be studied by considering Gaussian integers, that is, numbers of the form x + iy, where x and y are integers.

[edit] Analytic number theory

Analytic number theory studies numbers, often integers or rationals, by taking advantage of the fact that they can be regarded as complex numbers, in which analytic methods can be used. This is done by encoding number-theoretic information in complex-valued functions. For example, the Riemann zeta-function ζ(s) is related to the distribution of prime numbers.

[edit] Quality Adjusted Life Years

Complex numbers are used in the calculation of quality-adjusted life years (QALYs).[13]

[edit] History

The earliest fleeting reference to square roots of negative numbers can perhaps be said to occur in the work of the Greek mathematician Heron of Alexandria in the 1st century AD, where in his Stereometrica he considers, apparently in error, the volume

of an impossible frustum of a pyramid to arrive at the term in his calculations, although negative quantities were not conceived of in Hellenistic mathematics and Heron merely replaced it by its positive.[14]

The impetus to study complex numbers proper first arose in the 16th century when algebraic solutions for the roots of cubic and quartic polynomials were discovered by Italian mathematicians (see Niccolo Fontana Tartaglia, Gerolamo Cardano). It was soon realized that these formulas, even if one was only interested in real solutions, sometimes required the manipulation of square roots of negative numbers. As an example, Tartaglia's cubic formula gives the solution to the equation x3 − x = 0 as

Page 48: Dielectric Constant

The three cube roots of −1, two of which are complex

At first glance this looks like nonsense. However formal calculations with complex

numbers show that the equation z3 = i has solutions –i, and .

Substituting these in turn for in Tartaglia's cubic formula and simplifying, one gets 0, 1 and −1 as the solutions of x3 – x = 0. Of course this particular equation can be solved at sight but it does illustrate that when general formulas are used to solve cubic equations with real roots then, as later mathematicians showed rigorously, the use of complex numbers is unavoidable. Rafael Bombelli was the first to explicitly address these seemingly paradoxical solutions of cubic equations and developed the rules for complex arithmetic trying to resolve these issues.

The term "imaginary" for these quantities was coined by René Descartes in 1637, although he was at pains to stress their imaginary nature[15]

[...] quelquefois seulement imaginaires c’est-à-dire que l’on peut toujours en imaginer autant que j'ai dit en chaque équation, mais qu’il n’y a quelquefois aucune quantité qui corresponde à celle qu’on imagine. ([...] sometimes only imaginary, that is one can imagine as many as I said in each equation, but sometimes there exists no quantity that matches that which we imagine.)

A further source of confusion was that the equation

seemed to be capriciously inconsistent with the algebraic identity , which is valid for non-negative real numbers a and b, and which was also used in complex number calculations with one of a, b positive and the other negative. The

incorrect use of this identity (and the related identity ) in the case when both a and b are negative even bedeviled Euler. This difficulty eventually led to the

convention of using the special symbol i in place of to guard against this mistake[citation needed]. Even so Euler considered it natural to introduce students to complex numbers much earlier than we do today. In his elementary algebra text book, Elements of Algebra, he introduces these numbers almost at once and then uses them in a natural way throughout.

Page 49: Dielectric Constant

In the 18th century complex numbers gained wider use, as it was noticed that formal manipulation of complex expressions could be used to simplify calculations involving trigonometric functions. For instance, in 1730 Abraham de Moivre noted that the complicated identities relating trigonometric functions of an integer multiple of an angle to powers of trigonometric functions of that angle could be simply re-expressed by the following well-known formula which bears his name, de Moivre's formula:

In 1748 Leonhard Euler went further and obtained Euler's formula of complex analysis:

by formally manipulating complex power series and observed that this formula could be used to reduce any trigonometric identity to much simpler exponential identities.

The idea of a complex number as a point in the complex plane (above) was first described by Caspar Wessel in 1799, although it had been anticipated as early as 1685 in Wallis's De Algebra tractatus.

Wessel's memoir appeared in the Proceedings of the Copenhagen Academy but went largely unnoticed. In 1806 Jean-Robert Argand independently issued a pamphlet on complex numbers and provided a rigorous proof of the fundamental theorem of algebra. Gauss had earlier published an essentially topological proof of the theorem in 1797 but expressed his doubts at the time about "the true metaphysics of the square root of −1". It was not until 1831 that he overcame these doubts and published his treatise on complex numbers as points in the plane, largely establishing modern notation and terminology. The English mathematician G. H. Hardy remarked that Gauss was the first mathematician to use complex numbers in 'a really confident and scientific way' although mathematicians such as Niels Henrik Abel and Carl Gustav Jacob Jacobi were necessarily using them routinely before Gauss published his 1831 treatise.[16] Augustin Louis Cauchy and Bernhard Riemann together brought the fundamental ideas of complex analysis to a high state of completion, commencing around 1825 in Cauchy's case.

The common terms used in the theory are chiefly due to the founders. Argand called

the direction factor, and the modulus; Cauchy (1828) called the reduced form (l'expression réduite) and apparently

introduced the term argument; Gauss used i for , introduced the term complex number for a + bi, and called a2 + b2 the norm. The expression direction coefficient, often used for , is due to Hankel (1867), and absolute value, for modulus, is due to Weierstrass.

Later classical writers on the general theory include Richard Dedekind, Otto Hölder, Felix Klein, Henri Poincaré, Hermann Schwarz, Karl Weierstrass and many others.

[edit] Generalizations and related notions

Page 50: Dielectric Constant

The process of extending the field R of reals to C is known as Cayley-Dickson construction. It can be carried further to higher dimensions, yielding the quaternions H and octonions O which (as a real vector space) are of dimension 4 and 8, respectively. However, with increasing dimension, the algebraic properties familiar from real and complex numbers vanish: the quaternions are only a skew field, i.e. x·y ≠ y·x for two quaternions, the multiplication of octonions fails (in addition to not being commutative) to be associative: (x·y)·z ≠ x·(y·z). However, all of these are normed division algebras over R. By Hurwitz's theorem they are the only ones. The next step in the Cayley-Dickson construction, the sedenions fail to have this structure.

The Cayley-Dickson construction is closely related to the regular representation of C, thought of as an R-algebra (an R-vector space with a multiplication), with respect to the basis 1, i. This means the following: the R-linear map

for some fixed complex number w can be represented by a 2×2 matrix (once a basis has been chosen). With respect to the basis 1, i, this matrix is

i.e., the one mentioned in the section on matrix representation of complex numbers above. While this is a linear representation of C in the 2 × 2 real matrices, it is not the only one. Any matrix

has the property that its square is the negative of the identity matrix: J2 = −I. Then

is also isomorphic to the field C, and gives an alternative complex structure on R2. This is generalized by the notion of a linear complex structure.

Hypercomplex numbers also generalize R, C, H, and O. For example this notion contains the split-complex numbers, which are elements of the ring R[x]/(x2 − 1) (as opposed to R[x]/(x2 + 1)). In this ring, the equation a2 = 1 has four solutions.

The field R is the completion of Q, the field of rational numbers, with respect to the usual absolute value metric. Other choices of metrics on Q lead to the fields Qp of p - adic numbers (for any prime number p), which are thereby analogous to R. There are no other nontrivial ways of completing Q than R and Qp, by Ostrowski's theorem. The

algebraic closure of Qp still carry a norm, but (unlike C) are not complete with

respect to it. The completion of turns out to be algebraically closed. This field is called p-adic complex numbers by analogy.

Page 51: Dielectric Constant

The fields R and Qp and their finite field extensions, including C, are local fields.

Page 52: Dielectric Constant

Permittivity From Wikipedia, the free encyclopediaJump to: navigation, search

A dielectric medium showing orientation of charged particles creating polarization effects. Such a medium can have a higher ratio of electric flux to charge (permittivity) than empty space

In electromagnetism, absolute permittivity is the measure of the resistance that is encountered when forming an electric field in a medium. In other words, permittivity is a measure of how an electric field affects, and is affected by, a dielectric medium. The permittivity of a medium describes how much electric field (more correctly, flux) is 'generated' per unit charge in that medium. Less electric flux exists in a medium with a high permittivity (per unit charge) because of polarization effects. Permittivity is directly related to electric susceptibility, which is a measure of how easily a dielectric polarizes in response to an electric field. Thus, permittivity relates to a material's ability to transmit (or "permit") an electric field.

In SI units, permittivity ε is measured in farads per meter (F/m); electric susceptibility χ is dimensionless. They are related to each other through

where εr is the relative permittivity of the material, and = 8.85… × 10−12 F/m is the vacuum permittivity.

Contents

 [hide]  1 Explanation 2 Vacuum permittivity 3 Relative permittivity 4 Dispersion and causality

o 4.1 Complex permittivity o 4.2 Classification of materials o 4.3 Lossy medium o 4.4 Quantum-mechanical interpretation

Page 53: Dielectric Constant

5 Measurement 6 See also 7 References 8 Further reading

9 External links

[edit] Explanation

In electromagnetism, the electric displacement field D represents how an electric field E influences the organization of electrical charges in a given medium, including charge migration and electric dipole reorientation. Its relation to permittivity in the very simple case of linear, homogeneous, isotropic materials with "instantaneous" response to changes in electric field is

where the permittivity ε is a scalar. If the medium is anisotropic, the permittivity is a second rank tensor.

In general, permittivity is not a constant, as it can vary with the position in the medium, the frequency of the field applied, humidity, temperature, and other parameters. In a nonlinear medium, the permittivity can depend on the strength of the electric field. Permittivity as a function of frequency can take on real or complex values.

In SI units, permittivity is measured in farads per meter (F/m or A2·s4·kg−1·m−3). The displacement field D is measured in units of coulombs per square meter (C/m2), while the electric field E is measured in volts per meter (V/m). D and E describe the interaction between charged objects. D is related to the charge densities associated with this interaction, while E is related to the forces and potential differences.

[edit] Vacuum permittivity

Main article: vacuum permittivity

The vacuum permittivity ε0 (also called permittivity of free space or the electric constant) is the ratio D/E in free space. It also appears in the Coulomb force constant 1/4πε0.

Its value is[1]

where

c0 is the speed of light in free space,[2]

Page 54: Dielectric Constant

µ0 is the vacuum permeability.

Constants c0 and μ0 are defined in SI units to have exact numerical values, shifting responsibility of experiment to the determination of the meter and the ampere.[3] (The approximation in the second value of ε0 above stems from π being an irrational number.)

[edit] Relative permittivity

Main article: relative permittivity

The linear permittivity of a homogeneous material is usually given relative to that of free space, as a relative permittivity εr (also called dielectric constant, although this sometimes only refers to the static, zero-frequency relative permittivity). In an anisotropic material, the relative permittivity may be a tensor, causing birefringence. The actual permittivity is then calculated by multiplying the relative permittivity by ε0:

where

χ (frequently written χe) is the electric susceptibility of the material.

The susceptibility is defined as the constant of proportionality (which may be a tensor) relating an electric field E to the induced dielectric polarization density P such that

where is the electric permittivity of free space.

The susceptibility of a medium is related to its relative permittivity by

So in the case of a vacuum,

The susceptibility is also related to the polarizability of individual particles in the medium by the Clausius-Mossotti relation.

The electric displacement D is related to the polarization density P by

The permittivity ε and permeability µ of a medium together determine the phase velocity v = c/n of electromagnetic radiation through that medium:

Page 55: Dielectric Constant

[edit] Dispersion and causality

In general, a material cannot polarize instantaneously in response to an applied field, and so the more general formulation as a function of time is

That is, the polarization is a convolution of the electric field at previous times with

time-dependent susceptibility given by . The upper limit of this integral can be

extended to infinity as well if one defines for . An instantaneous response corresponds to Dirac delta function susceptibility

.

It is more convenient in a linear system to take the Fourier transform and write this relationship as a function of frequency. Because of the convolution theorem, the integral becomes a simple product,

This frequency dependence of the susceptibility leads to frequency dependence of the permittivity. The shape of the susceptibility with respect to frequency characterizes the dispersion properties of the material.

Moreover, the fact that the polarization can only depend on the electric field at

previous times (i.e. for ), a consequence of causality, imposes

Kramers–Kronig constraints on the susceptibility .

Page 56: Dielectric Constant

[edit] Complex permittivity

A dielectric permittivity spectrum over a wide range of frequencies. ε′ and ε″ denote the real and the imaginary part of the permittivity, respectively. Various processes are labeled on the image: ionic and dipolar relaxation, and atomic and electronic resonances at higher energies.[4]

As opposed to the response of a vacuum, the response of normal materials to external fields generally depends on the frequency of the field. This frequency dependence reflects the fact that a material's polarization does not respond instantaneously to an applied field. The response must always be causal (arising after the applied field) which can be represented by a phase difference. For this reason permittivity is often treated as a complex function (since complex numbers allow specification of

magnitude and phase) of the (angular) frequency of the applied field ω, . The definition of permittivity therefore becomes

where

D0 and E0 are the amplitudes of the displacement and electrical fields, respectively,i is the imaginary unit, i 2 = −1.

The response of a medium to static electric fields is described by the low-frequency limit of permittivity, also called the static permittivity εs (also εDC ):

Page 57: Dielectric Constant

At the high-frequency limit, the complex permittivity is commonly referred to as ε∞. At the plasma frequency and above, dielectrics behave as ideal metals, with electron gas behavior. The static permittivity is a good approximation for alternating fields of low frequencies, and as the frequency increases a measurable phase difference δ emerges between D and E. The frequency at which the phase shift becomes noticeable depends on temperature and the details of the medium. For moderate fields strength (E0), D and E remain proportional, and

Since the response of materials to alternating fields is characterized by a complex permittivity, it is natural to separate its real and imaginary parts, which is done by convention in the following way:

where

ε" is the imaginary part of the permittivity, which is related to the dissipation (or loss) of energy within the medium.ε' is the real part of the permittivity, which is related to the stored energy within the medium.

It is important to realize that the choice of sign for time-dependence, , dictates the sign convention for the imaginary part of permittivity. The signs used here correspond to those commonly used in physics, whereas for the engineering convention one should reverse all imaginary quantities.

The complex permittivity is usually a complicated function of frequency ω, since it is a superimposed description of dispersion phenomena occurring at multiple frequencies. The dielectric function ε(ω) must have poles only for frequencies with positive imaginary parts, and therefore satisfies the Kramers–Kronig relations. However, in the narrow frequency ranges that are often studied in practice, the permittivity can be approximated as frequency-independent or by model functions.

At a given frequency, the imaginary part of leads to absorption loss if it is positive (in the above sign convention) and gain if it is negative. More generally, the imaginary parts of the eigenvalues of the anisotropic dielectric tensor should be considered.

In the case of solids, the complex dielectric function is intimately connected to band structure. The primary quantity that characterizes the electronic structure of any crystalline material is the probability of photon absorption, which is directly related to

Page 58: Dielectric Constant

the imaginary part of the optical dielectric function ε(ω). The optical dielectric function is given by the fundamental expression:[5]

In this expression, Wcv(E) represents the product of the Brillouin zone-averaged transition probability at the energy E with the joint density of states,[6][7] Jcv(E); is a broadening function, representing the role of scattering in smearing out the energy levels.[8] In general, the broadening is intermediate between Lorentzian and Gaussian;[9][10] for an alloy it is somewhat closer to Gaussian because of strong scattering from statistical fluctuations in the local composition on a nanometer scale.

[edit] Classification of materials

Materials can be classified according to their permittivity and conductivity, σ. Materials with a large amount of loss inhibit the propagation of electromagnetic waves. In this case, generally when σ/(ωε') >> 1, we consider the material to be a good conductor. Dielectrics are associated with lossless or low-loss materials, where σ/(ωε') << 1. Those that do not fall under either limit are considered to be general media. A perfect dielectric is a material that has no conductivity, thus exhibiting only a displacement current. Therefore it stores and returns electrical energy as if it were an ideal capacitor.

[edit] Lossy medium

In the case of lossy medium, i.e. when the conduction current is not negligible, the total current density flowing is:

where

σ is the conductivity of the medium;ε' is the real part of the permittivity.

is the complex permittivity

The size of the displacement current is dependent on the frequency ω of the applied field E; there is no displacement current in a constant field.

In this formalism, the complex permittivity is defined as[11]:

In general, the absorption of electromagnetic energy by dielectrics is covered by a few different mechanisms that influence the shape of the permittivity as a function of frequency:

Page 59: Dielectric Constant

First, are the relaxation effects associated with permanent and induced molecular dipoles. At low frequencies the field changes slowly enough to allow dipoles to reach equilibrium before the field has measurably changed. For frequencies at which dipole orientations cannot follow the applied field because of the viscosity of the medium, absorption of the field's energy leads to energy dissipation. The mechanism of dipoles relaxing is called dielectric relaxation and for ideal dipoles is described by classic Debye relaxation.

Second are the resonance effects, which arise from the rotations or vibrations of atoms, ions, or electrons. These processes are observed in the neighborhood of their characteristic absorption frequencies.

The above effects often combine to cause non-linear effects within capacitors. For example, dielectric absorption refers to the inability of a capacitor that has been charged for a long time to completely discharge when briefly discharged. Although an ideal capacitor would remain at zero volts after being discharged, real capacitors will develop a small voltage, a phenomenon that is also called soakage or battery action. For some dielectrics, such as many polymer films, the resulting voltage may be less than 1-2% of the original voltage. However, it can be as much as 15 - 25% in the case of electrolytic capacitors or supercapacitors.

[edit] Quantum-mechanical interpretation

In terms of quantum mechanics, permittivity is explained by atomic and molecular interactions.

At low frequencies, molecules in polar dielectrics are polarized by an applied electric field, which induces periodic rotations. For example, at the microwave frequency, the microwave field causes the periodic rotation of water molecules, sufficient to break hydrogen bonds. The field does work against the bonds and the energy is absorbed by the material as heat. This is why microwave ovens work very well for materials containing water. There are two maxima of the imaginary component (the absorptive index) of water, one at the microwave frequency, and the other at far ultraviolet (UV) frequency. Both of these resonances are at higher frequencies than the operating frequency of microwave ovens.

At moderate frequencies, the energy is too high to cause rotation, yet too low to affect electrons directly, and is absorbed in the form of resonant molecular vibrations. In water, this is where the absorptive index starts to drop sharply, and the minimum of the imaginary permittivity is at the frequency of blue light (optical regime).

At high frequencies (such as UV and above), molecules cannot relax, and the energy is purely absorbed by atoms, exciting electron energy levels. Thus, these frequencies are classified as ionizing radiation.

While carrying out a complete ab initio (that is, first-principles) modelling is now computationally possible, it has not been widely applied yet. Thus, a phenomenological model is accepted as being an adequate method of capturing experimental behaviors. The Debye model and the Lorentz model use a 1st-order and

Page 60: Dielectric Constant

2nd-order (respectively) lumped system parameter linear representation (such as an RC and an LRC resonant circuit).

[edit] Measurement

Main article: dielectric spectroscopy

The dielectric constant of a material can be found by a variety of static electrical measurements. The complex permittivity is evaluated over a wide range of frequencies by using different variants of dielectric spectroscopy, covering nearly 21 orders of magnitude from 10−6 to 1015 Hz. Also, by using cryostats and ovens, the dielectric properties of a medium can be characterized over an array of temperatures. In order to study systems for such diverse excitation fields, a number of measurement setups are used, each adequate for a special frequency range.

Various microwave measurement techniques are outlined in Chen et al..[12] Typical errors for the Hakki-Coleman method employing a puck of material between conducting planes are about 0.3%.[13]

Low-frequency time domain measurements (10−6-103 Hz) Low-frequency frequency domain measurements (10−5-106 Hz) Reflective coaxial methods (106-1010 Hz) Transmission coaxial method (108-1011 Hz) Quasi-optical methods (109-1010 Hz) Terahertz time-domain spectroscopy (1011-1013 Hz) Fourier-transform methods (1011-1015 Hz)

At infrared and optical frequencies, a common technique is ellipsometry. Dual polarisation interferometry is also used to measure the complex refractive index for very thin films at optical frequencies.

[edit] See also

Density functional theory Electric-field screening Green-Kubo relations Green's function (many-body theory) Linear response function Rotational Brownian motion Electromagnetic permeability

[edit] References

1. ̂ electric constant2. ̂ Current practice of standards organizations such as NIST and BIPM

is to use c0, rather than c, to denote the speed of light in vacuum according to ISO 31. In the original Recommendation of 1983, the symbol c was used for this purpose. See NIST Special Publication 330 , Appendix 2, p. 45 .

Page 61: Dielectric Constant

3. ̂ Latest (2006) values of the constants (NIST)4. ̂ Dielectric Spectroscopy5. ̂ Peter Y. Yu, Manuel Cardona (2001). Fundamentals of

Semiconductors: Physics and Materials Properties. Berlin: Springer. p. 261. ISBN 3-540-25470-6.

6. ̂ José García Solé, Jose Solé, Luisa Bausa, (2001). An introduction to the optical spectroscopy of inorganic solids. Wiley. Appendix A1, pp, 263. ISBN 0-470-86885-6.

7. ̂ John H. Moore, Nicholas D. Spencer (2001). Encyclopedia of chemical physics and physical chemistry. Taylor and Francis. p. 105. ISBN 0-7503-0798-6.

Page 62: Dielectric Constant

acuum permittivity From Wikipedia, the free encyclopediaJump to: navigation, search This article is about the electric constant. For the analogous magnetic constant, see vacuum permeability. For the ordinal in mathematics ε0, see epsilon naught.

The physical constant ε0, commonly called the vacuum permittivity, permittivity of free space or electric constant is an ideal, (baseline) physical constant, which is the value of the absolute (not relative) dielectric permittivity of classical vacuum. Its value is:[1]

ε0 ≈ 8.854187817620... × 10−12 Farads per metre (F·m−1).

The ellipsis ‘...’ does not represent an experimental inaccuracy (the value is exact) but the error introduced by truncation of a non-terminating decimal value.

This constant relates the units for electric charge to mechanical quantities such as length and force.[2] For example, the force between two separated electric charges (in the vacuum of classical electromagnetism) is given by Coulomb's law:

where q1 and q2 are the charges, and r is the distance between them. Likewise, ε0 appears in Maxwell's equations, which describe the properties of electric and magnetic fields and electromagnetic radiation, and relate them to their sources.

Contents

 [hide]  1 Value

o 1.1 Redefinition of the SI units 2 Terminology 3 Historical origin of the parameter ε 0

o 3.1 Rationalization of units o 3.2 Determination of a value for ε 0

4 Permittivity of real media 5 See also

6 Notes

[edit] Value

The value of ε0 is defined by the formula[3]

Page 63: Dielectric Constant

where c0 is the defined value for the speed of light in classical vacuum in SI units,[4] and μ0 is the parameter that international Standards Organizations call the "magnetic constant" (commonly called vacuum permeability). Since μ0 has the defined value 4π × 10−7 H m−1,[5] and c0 has the defined value 299792458 m·s−1,[6] it follows that ε0 has a defined value given approximately by

ε0 ≈ 8.854187817620... × 10−12 F·m−1 (or A2·s4·kg−1·m−3 in SI base units, or C2·N−1·m−2 or C·V−1·m−1 using other SI coherent units).[7][8]

The ellipsis (...) does not indicate experimental uncertainty, but the arbitrary termination of a nonrecurring decimal. The historical origins of the electric constant ε0, and its value, are explained in more detail below.

[edit] Redefinition of the SI unitsMain article: New SI definitions

Under the proposals to redefine the ampere as a fixed number of elementary charges per second,[9] the electric constant would no longer have an exact fixed value. The value of the electron charge would become a defined number, not measured, making μ0 a measured quantity. Consequently, ε0 also would not be exact. As before, it would be defined by the equation ε0= 1/(μ0c0

2), but now with a measurement error related to the error related to that in μ0, the magnetic constant. This measurement error can be related to that in the fine structure constant α:

with e the exact elementary charge, h the exact Planck constant, and c0 the exact speed of light in vacuum. Here use is made of the relation for the fine structure constant:

The relative uncertainty in the value of ε0 therefore would be the same as that for the fine structure constant, currently 6.8×10−10.[7]

[edit] Terminology

Historically, the parameter ε0 has been known by many different names. The terms "vacuum permittivity" or its variants, such as "permittivity in/of vacuum",[10][11] "permittivity of empty space",[12] or "permittivity of free space"[13] are widespread. Standards Organizations worldwide now use "electric constant" as a uniform term for this quantity,[7] and official standards documents have adopted the term (although they continue to list the older terms as synonyms).[14][15]

Page 64: Dielectric Constant

Another historical synonym was "dielectric constant of vacuum", as "dielectric constant" was sometimes used in the past for the absolute permittivity.[16][17] However, in modern usage "dielectric constant" typically refers exclusively to a relative permittivity ε/ε0 and even this usage is considered "obsolete" by some standards bodies in favor of relative static permittivity.[15][18] Hence, the term "dielectric constant of vacuum" for the electric constant ε0 is considered obsolete by most modern authors, although occasional examples of continuing usage can be found.

As for notation, the constant can be denoted by either or , using either of the common glyphs for the letter epsilon.

[edit] Historical origin of the parameter ε0

As indicated above, the parameter ε0 is a measurement-system constant. Its presence in the equations now used to define electromagnetic quantities is the result of the so-called "rationalization" process described below. But the method of allocating a value to it is a consequence of the result that Maxwell's equations predict that, in free space, electromagnetic waves move with the speed of light. Understanding why ε0 has the value it does requires a brief understanding of the history of how electromagnetic measurement systems developed.

[edit] Rationalization of units

The experiments of Coulomb and others showed that the force F between two equal point-like "amounts" of electricity, situated a distance r apart in free space, should be given by a formula that has the form

where Q is a quantity that represents the amount of electricity present at each of the two points, and ke is Coulomb's constant. If one is starting with no constraints, then the value of ke may be chosen arbitrarily.[19] For each different choice of ke there is a different "interpretation" of Q: to avoid confusion, each different "interpretation" has to be allocated a distinctive name and symbol.

In one of the systems of equations and units agreed in the late 19th century, called the "centimetre-gram-second electrostatic system of units" (the cgs esu system), the constant ke was taken equal to 1, and a quantity now called "gaussian electric charge" qs was defined by the resulting equation

The unit of gaussian charge, the statcoulomb, is such that two units, a distance of 1 centimetre apart, repel each other with a force equal to the cgs unit of force, the dyne. Thus the unit of gaussian charge can also be written 1 dyne1/2 cm. "Gaussian electric charge" is not the same mathematical quantity as modern (rmks) electric charge and is not measured in coulombs.

Page 65: Dielectric Constant

The idea subsequently developed that it would be better, in situations of spherical geometry, to include a factor 4π in equations like Coulomb's law, and write it in the form:

This idea is called "rationalization". The quantities q's and ke' are not the same as those in the older convention. Putting ke'=1 generates a unit of electricity of different size, but it still has the same dimensions as the cgs esu system.

The next step was to treat the quantity representing "amount of electricity" as a fundamental quantity in its own right, denoted by the symbol q, and to write Coulomb's Law in its modern form:

The system of equations thus generated is known as the rationalized metre-kilogram-second (rmks) equation system, or "metre-kilogram-second-ampere (mksa)" equation system. This is the system used to define the SI units.[20] The new quantity q is given the name "rmks electric charge", or (nowadays) just "electric charge". Clearly, the quantity qs used in the old cgs esu system is related to the new quantity q by

[edit] Determination of a value for ε0

One now adds the requirement that one wants force to be measured in newtons, distance in metres, and charge to be measured in the engineers' practical unit, the coulomb, which is defined as the charge accumulated when a current of 1 ampere flows for one second. This shows that the parameter ε0 should be allocated the unit C2·N−1·m−2 (or equivalent units - in practice "farads per metre").

In order to establish the numerical value of ε0, one makes use of the fact that if one uses the rationalized forms of Coulomb's law and Ampère's force law (and other ideas) to develop Maxwell's equations, then the relationship stated above is found to exist between ε0, μ0 and c0. In principle, one has a choice of deciding whether to make the coulomb or the ampere the fundamental unit of electricity and magnetism. The decision was taken internationally to use the ampere. This means that the value of ε0 is determined by the values of c0 and μ0, as stated above. For a brief explanation of how the value of μ0 is decided, see the article about μ0.

[edit] Permittivity of real media

By convention, the electric constant ε0 appears in the relationship that defines the electric displacement field D in terms of the electric field E and classical electrical polarization density P of the medium. In general, this relationship has the form:

.

Page 66: Dielectric Constant

For a linear dielectric, P is assumed to be proportional to E, but a delayed response is permitted, and a spatially non-local response, so one has:[21]

In the event that nonlocality and delay of response are not important, the result is:

where ε is the permittivity and εr the relative static permittivity. In the vacuum of classical electromagnetism, the polarization P = 0, so εr = 1 and ε = ε0.

Page 67: Dielectric Constant

Electric susceptibility From Wikipedia, the free encyclopediaJump to: navigation, search

This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. (November 2010)

In electromagnetism, the electric susceptibility (latin: susceptibilis “receptiveness”) is a dimensionless proportionality constant that indicates the degree of polarization of a dielectric material in response to an applied electric field. The greater the electric susceptibility, the greater the ability of a material to polarize in response to the field, and thereby reduce the total electric field inside the material (and store energy). It is in this way that the electric susceptibility influences the electric permittivity of the material and thus influences many other phenomena in that medium, from the capacitance of capacitors to the speed of light.[1][2]

Contents [hide] 

1 Definition of Volume Susceptibility 2 Molecular Polarizability 3 Dispersion and causality 4 See also

5 References

[edit] Definition of Volume Susceptibility

Electric susceptibility is defined as the constant of proportionality (which may be a tensor) relating an electric field E to the induced dielectric polarization density P such that:

Where:

is the Polarization Density is the Electric Permittivity of Free Space is the Electric Susceptibility is the Electric Field

The susceptibility is also related to the polarizability of individual particles in the medium by the Clausius-Mossotti relation. The susceptibility is related to its relative permittivity by:

Page 68: Dielectric Constant

So in the case of a vacuum:

At the same time, the electric displacement D is related to the polarization density P by:

[edit] Molecular Polarizability

A similar parameter exists to relate the magnitude of the induced dipole moment p of an individual molecule to the local electric field E that induced the dipole. This parameter is the molecular polarizability and the dipole moment resulting from the local electric field is given by:

This introduces a complication however, as locally the field can differ significantly from the overall applied field. We have:

where P is the polarization per unit volume, and N is the number of molecules per unit volume contributing to the polarization. Thus, if the local electric field is parallel to the ambient electric field, we have:

Thus only if the local field equals the ambient field can we write:

[edit] Dispersion and causality

In general, a material cannot polarize instantaneously in response to an applied field, and so the more general formulation as a function of time is

That is, the polarization is a convolution of the electric field at previous times with

time-dependent susceptibility given by . The upper limit of this integral can

be extended to infinity as well if one defines for . An instantaneous response corresponds to Dirac delta function susceptibility

.

Page 69: Dielectric Constant

It is more convenient in a linear system to take the Fourier transform and write this relationship as a function of frequency. Due to the convolution theorem, the integral becomes a simple product,

This frequency dependence of the susceptibility leads to frequency dependence of the permittivity. The shape of the susceptibility with respect to frequency characterizes the dispersion properties of the material.

Moreover, the fact that the polarization can only depend on the electric field at

previous times (i.e. for ), a consequence of causality, imposes

Kramers–Kronig constraints on the susceptibility .

Page 70: Dielectric Constant

Capacitance From Wikipedia, the free encyclopediaJump to: navigation, search

Electromagnetism

Electricity

Magnetism

Electrostatics [show]

Magnetostatics [show]

Electrodynamics [show]

Electrical Network [hide]

Electrical conduction

Electrical resistance

Capacitance

Inductance

Impedance

Resonant cavities

Waveguides

Covariant formulation [show]

Scientists[show]

v

t

e

In electromagnetism and electronics, capacitance is the ability of a capacitor to store charge in an electric field. Capacitance is also a measure of the amount of electric potential energy stored (or separated) for a given electric potential. A common form of energy storage device is a parallel-plate capacitor. In a parallel plate capacitor, capacitance is directly proportional to the surface area of the conductor plates and inversely proportional to the separation distance between the plates. If the charges on

Page 71: Dielectric Constant

the plates are +q and −q, and V gives the voltage between the plates, then the capacitance is given by

The SI unit of capacitance is the farad; 1 farad is 1 coulomb per volt.

The energy (measured in joules) stored in a capacitor is equal to the work done to charge it. Consider a capacitor of capacitance C, holding a charge +q on one plate and −q on the other. Moving a small element of charge dq from one plate to the other against the potential difference V = q/C requires the work dW:

where W is the work measured in joules, q is the charge measured in coulombs and C is the capacitance, measured in farads.

The energy stored in a capacitor is found by integrating this equation. Starting with an uncharged capacitance (q = 0) and moving charge from one plate to the other until the plates have charge +Q and −Q requires the work W:

Contents

 [hide]  1 Capacitors

o 1.1 Voltage dependent capacitors o 1.2 Frequency dependent capacitors

2 Capacitance matrix 3 Self-capacitance 4 Elastance 5 Stray capacitance 6 Capacitance of simple systems 7 See also 8 References

9 Further reading

[edit] Capacitors

Main article: Capacitor

The capacitance of the majority of capacitors used in electronic circuits is several orders of magnitude smaller than the farad. The most common subunits of capacitance

Page 72: Dielectric Constant

in use today are the millifarad (mF), microfarad (µF), nanofarad (nF), picofarad (pF), and femtofarad (fF).

Capacitance can be calculated if the geometry of the conductors and the dielectric properties of the insulator between the conductors are known. For example, the capacitance of a parallel-plate capacitor constructed of two parallel plates both of area A separated by a distance d is approximately equal to the following:

where

C is the capacitance;A is the area of overlap of the two plates;εr is the relative static permittivity (sometimes called the dielectric constant) of the material between the plates (for a vacuum, εr = 1);ε0 is the electric constant (ε0 ≈ 8.854×10−12 F m–1); andd is the separation between the plates.

Capacitance is proportional to the area of overlap and inversely proportional to the separation between conducting sheets. The closer the sheets are to each other, the greater the capacitance. The equation is a good approximation if d is small compared to the other dimensions of the plates so the field in the capacitor over most of its area is uniform, and the so-called fringing field around the periphery provides a small contribution. In CGS units the equation has the form:[1]

where C in this case has the units of length. Combining the SI equation for capacitance with the above equation for the energy stored in a capacitance, for a flat-plate capacitor the energy stored is:

.

where W is the energy, in joules; C is the capacitance, in farads; and V is the voltage, in volts.

[edit] Voltage dependent capacitors

The dielectric constant for a number of very useful dielectrics changes as a function of the applied electrical field, for example ferroelectric materials, so the capacitance for these devices is more complex. For example, in charging such a capacitor the differential increase in voltage with charge is governed by:

Page 73: Dielectric Constant

where the voltage dependence of capacitance, C(V), stems from the field, which in a large area parallel plate device is given by ε = V/d. This field polarizes the dielectric, which polarization, in the case of a ferroelectric, is a nonlinear S-shaped function of field, which, in the case of a large area parallel plate device, translates into a capacitance that is a nonlinear function of the voltage causing the field.[2][3]

Corresponding to the voltage-dependent capacitance, to charge the capacitor to voltage V an integral relation is found:

which agrees with Q = CV only when C is voltage independent.

By the same token, the energy stored in the capacitor now is given by

Integrating:

   

where interchange of the order of integration is used.

The nonlinear capacitance of a microscope probe scanned along a ferroelectric surface is used to study the domain structure of ferroelectric materials.[4]

Another example of voltage dependent capacitance occurs in semiconductor devices such as semiconductor diodes, where the voltage dependence stems not from a change in dielectric constant but in a voltage dependence of the spacing between the charges on the two sides of the capacitor.[5] This effect is intentionally exploited in diode-like devices known as varicaps.

[edit] Frequency dependent capacitors

If a capacitor is driven with a time-varying voltage that changes rapidly enough, then the polarization of the dielectric cannot follow the signal. As an example of the origin of this mechanism, the internal microscopic dipoles contributing to the dielectric constant cannot move instantly, and so as frequency of an applied alternating voltage increases, the dipole response is limited and the dielectric constant diminishes. A changing dielectric constant with frequency is referred to as dielectric dispersion, and is governed by dielectric relaxation processes, such as Debye relaxation. Under transient conditions, the displacement field can be expressed as (see electric susceptibility):

Page 74: Dielectric Constant

indicating the lag in response by the time dependence of εr, calculated in principle from an underlying microscopic analysis, for example, of the dipole behavior in the dielectric. See, for example, linear response function.[6][7] The integral extends over the entire past history up to the present time. A Fourier transform in time then results in:

where εr(ω) is now a complex function, with an imaginary part related to absorption of energy from the field by the medium. See permittivity. The capacitance, being proportional to the dielectric constant, also exhibits this frequency behavior. Fourier transforming Gauss's law with this form for displacement field:

where j is the imaginary unit, V(ω) is the voltage component at angular frequency ω, G(ω) is the real part of the current, called the conductance, and C(ω) determines the imaginary part of the current and is the capacitance. Z(ω) is the complex impedance.

When a parallel-plate capacitor is filled with a dielectric, the measurement of dielectric properties of the medium is based upon the relation:

where a single prime denotes the real part and a double prime the imaginary part, Z(ω) is the complex impedance with the dielectric present, C(ω) is the so-called complex capacitance with the dielectric present, and C0 is the capacitance without the dielectric.[8][9] (Measurement "without the dielectric" in principle means measurement in free space, an unattainable goal inasmuch as even the quantum vacuum is predicted to exhibit nonideal behavior, such as dichroism. For practical purposes, when measurement errors are taken into account, often a measurement in terrestrial vacuum, or simply a calculation of C0, is sufficiently accurate.[10])

Using this measurement method, the dielectric constant may exhibit a resonance at certain frequencies corresponding to characteristic response frequencies (excitation energies) of contributors to the dielectric constant. These resonances are the basis for a number of experimental techniques for detecting defects. The conductance method measures absorption as a function of frequency.[11] Alternatively, the time response of the capacitance can be used directly, as in deep-level transient spectroscopy.[12]

Page 75: Dielectric Constant

Another example of frequency dependent capacitance occurs with MOS capacitors, where the slow generation of minority carriers means that at high frequencies the capacitance measures only the majority carrier response, while at low frequencies both types of carrier respond.[13][14]

At optical frequencies, in semiconductors the dielectric constant exhibits structure related to the band structure of the solid. Sophisticated modulation spectroscopy measurement methods based upon modulating the crystal structure by pressure or by other stresses and observing the related changes in absorption or reflection of light have advanced our knowledge of these materials.[15]

[edit] Capacitance matrix

The discussion above is limited to the case of two conducting plates, although of arbitrary size and shape. The definition C=Q/V still holds for a single plate given a charge, in which case the field lines produced by that charge terminate as if the plate were at the center of an oppositely charged sphere at infinity.

does not apply when there are more than two charged plates, or when the net charge on the two plates is non-zero. To handle this case, Maxwell introduced his coefficients of potential. If three plates are given charges , then the voltage of plate 1 is given by

,

and similarly for the other voltages. Hermann von Helmholtz and Sir William Thomson showed that the coefficients of potential are symmetric, so that , etc. Thus the system can be described by a collection of coefficients known as the elastance matrix or reciprocal capacitance matrix, which is defined as:

From this, the mutual capacitance between two objects can be defined[16] by

solving for the total charge Q and using .

Since no actual device holds perfectly equal and opposite charges on each of the two "plates", it is the mutual capacitance that is reported on capacitors.

The collection of coefficients is known as the capacitance matrix,[17][18] and is the inverse of the elastance matrix.

Page 76: Dielectric Constant

[edit] Self-capacitance

In electrical circuits, the term capacitance is usually a shorthand for the mutual capacitance between two adjacent conductors, such as the two plates of a capacitor. However, for an isolated conductor there also exists a property called self-capacitance, which is the amount of electrical charge that must be added to an isolated conductor to raise its electrical potential by one unit (i.e. one volt, in most measurement systems).[19] The reference point for this potential is a theoretical hollow conducting sphere, of infinite radius, centered on the conductor. Using this method, the self-capacitance of a conducting sphere of radius R is given by:[20]

Example values of self-capacitance are:

for the top "plate" of a van de Graaff generator, typically a sphere 20 cm in radius: 20 pF

the planet Earth: about 710 µF[21]

The capacitative component of a coil, which reduces its impedance at high frequencies and can lead to resonance and self-oscillation, is also called self-capacitance[22] as well as stray or parasitic capacitance.

[edit] Elastance

The reciprocal of capacitance is called elastance. The unit of elastance is the daraf, but is not recognised by SI.

[edit] Stray capacitance

Any two adjacent conductors can be considered a capacitor, although the capacitance will be small unless the conductors are close together for long distances or over a large area. This (often unwanted) effect is termed "stray capacitance". Stray capacitance can allow signals to leak between otherwise isolated circuits (an effect called crosstalk), and it can be a limiting factor for proper functioning of circuits at high frequency.

Stray capacitance is often encountered in amplifier circuits in the form of "feedthrough" capacitance that interconnects the input and output nodes (both defined relative to a common ground). It is often convenient for analytical purposes to replace this capacitance with a combination of one input-to-ground capacitance and one output-to-ground capacitance; the original configuration — including the input-to-output capacitance — is often referred to as a pi-configuration. Miller's theorem can be used to effect this replacement: it states that, if the gain ratio of two nodes is 1/K, then an impedance of Z connecting the two nodes can be replaced with a Z/(1-k) impedance between the first node and ground and a KZ/(K-1) impedance between the second node and ground. Since impedance varies inversely with capacitance, the internode capacitance, C, will be seen to have been replaced by a capacitance of KC

Page 77: Dielectric Constant

from input to ground and a capacitance of (K-1)C/K from output to ground. When the input-to-output gain is very large, the equivalent input-to-ground impedance is very small while the output-to-ground impedance is essentially equal to the original (input-to-output) impedance.

[edit] Capacitance of simple systems

Calculating the capacitance of a system amounts to solving the Laplace equation ∇2φ=0 with a constant potential φ on the surface of the conductors. This is trivial in cases with high symmetry. There is no solution in terms of elementary functions in more complicated cases.

For quasi-two-dimensional situations analytic functions may be used to map different geometries to each other. See also Schwarz-Christoffel mapping.

Capacitance of simple systems

Type CapacitanceComme

nt

Parallel-plate

capacitor

A: Aread: Distanceε: Permittivity

Coaxial cable

a1: Inner radiusa2: Outer radius: Length

Pair of parallel wires[23]

a: Wire radiusd: Distance, d > 2a: Length

of pair

Wire parallel to

wall[23]

a: Wire radiusd: Distance, d > a: Wire

lengthTwo

parallelcoplanar strips[24]

d: Distancew1, w2: Strip widthki:

Page 79: Dielectric Constant

Electric field From Wikipedia, the free encyclopediaJump to: navigation, search

Electromagnetism

Electricity

Magnetism

Electrostatics [show]

Magnetostatics [show]

Electrodynamics [show]

Electrical Network [show]

Covariant formulation [show]

Scientists[show]

v

t

e

In physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding. The concept of an electric field was introduced by Michael Faraday.

Contents

 [hide]  1 Qualitative description 2 Quantitative definition 3 Superposition

o 3.1 Array of discrete point charges o 3.2 Continuum of charges

4 Electrostatic fields o 4.1 Uniform fields

Page 80: Dielectric Constant

o 4.2 Parallels between electrostatic and gravitational fields 5 Electrodynamic fields 6 Energy in the electric field 7 Further extensions

o 7.1 Definitive equation of vector fields o 7.2 Constitutive relation

8 See also 9 References

10 External links

[edit] Qualitative description

The electric field is a vector field with SI units of newtons per coulomb (N C−1) or, equivalently, volts per metre (V m−1). The SI base units of the electric field are kg•m•s−3•A−1. The strength or magnitude of the field at a given point is defined as the force that would be exerted on a positive test charge of 1 coulomb placed at that point; the direction of the field is given by the direction of that force. Electric fields contain electrical energy with energy density proportional to the square of the field amplitude. The electric field is to charge as gravitational acceleration is to mass and force density is to volume.

An electric field that changes with time, such as due to the motion of charged particles in the field, influences the local magnetic field. That is, the electric and magnetic fields are not completely separate phenomena; what one observer perceives as an electric field, another observer in a different frame of reference perceives as a mixture of electric and magnetic fields. For this reason, one speaks of "electromagnetism" or "electromagnetic fields". In quantum electrodynamics, disturbances in the electromagnetic fields are called photons, and the energy of photons is quantized.

[edit] Quantitative definition

Electric field from a positive Q

Page 81: Dielectric Constant

Electric field from a negative Q

Electric fields are generated by charges. Suppose a stationary charge Q (the "source charge") creates an electric field E, and that another separate charge q (a "test charge") is placed in the E-field due to Q.

The electric field intensity E is defined as the force F experienced by a stationary positive unit point charge q at position r (relative to Q) in the field:[1][2]

Since the E field can vary from point to point in space, i.e. depends on r, it is a vector field. Using Coulomb's law, the E-field at a point in space due to Q is given by:

where r = |r| is the magnitude of the position vector, is the unit vector corresponding to r (pointing from Q to q), and ε0 is the electric constant.

From the definition, the direction of the electric field is the same as the direction of the force it would exert on a positively-charged particle, and opposite the direction of the force on a negatively-charged particle. Since like charges repel and opposites attract, the electric field is directed away from positive charges and towards negative charges.

According to Coulomb's law the electric field is dependent on position. The electric field due to any single charge falls off as the square of the distance from that charge, an example of an inverse-square law. Adding or moving another source charge will alter the electric field distribution. Therefore an electric field is defined with respect to a particular configuration of source charges.

[edit] Superposition

Page 82: Dielectric Constant

[edit] Array of discrete point charges

Electric fields satisfy the superposition principle. If more than one charge is present, the total electric field at any point is equal to the vector sum of the separate electric fields that each point charge would create in the absence of the others.

The total E-field due to N point charges is simply the superposition of the E-fields due to each point charge:

where ri is the position of charge qi, the corresponding unit vector.

[edit] Continuum of charges

The superposition principle holds for an infinite number of infinitesimally small elements of charges - i.e. a continuous distribution of charge. The limit of the above sum is the integral:

where ρ is the charge density (the amount of charge per unit volume), and dV is the differential volume element. This integral is a volume integral over the region of the charge distribution.

The electric field at a point is equal to the negative gradient of the electric potential there, :

Coulomb's law is actually a special case of Gauss's Law, a more fundamental description of the relationship between the distribution of electric charge in space and

Page 83: Dielectric Constant

the resulting electric field. While Columb's law (as given above) is only true for stationary point charges, Gauss's law is true for all charges either in static or in motion. Gauss's law is one of Maxwell's equations governing electromagnetism.

Gauss's law allows the E-field to be calculated in terms of a continuous distribution of charge density

where • is the ∇ divergence operator, ρ is the total charge density, including free and bound charge, in other words all the charge present in the system (per unit volume).

[edit] Electrostatic fields

Main article: Electrostatics

Electrostatic fields are E-fields which do not change with time, which happens when the charges are stationary.

Illustration of the electric field surrounding a positive (red) and a negative (blue) charge in one dimension if the right charge is changing from positive to negative

Illustration of the electric field surrounding a positive (red) and a negative (blue) charge.

The electric field at a point E(r) is equal to the negative gradient of the electric potential Φ(r), a scalar field at the same point:

Page 84: Dielectric Constant

where is the ∇ gradient. This is equivalent to the force definition above, since electric potential Φ is defined by the electric potential energy U per unit (test) positive charge:

and force is the negative of potential energy gradient:

If several spatially distributed charges generate such an electric potential, e.g. in a solid, an electric field gradient may also be defined.

[edit] Uniform fields

A uniform field is one in which the electric field is constant at every point. It can be approximated by placing two conducting plates parallel to each other and maintaining a voltage (potential difference) between them; it is only an approximation because of edge effects. Ignoring such effects, the equation for the magnitude of the electric field E is:

where Δϕ is the potential difference between the plates d is the distance separating the plates.

The negative sign arises as positive charges repel, so a positive charge will experience a force away from the positively charged plate, in the opposite direction to that in which the voltage increases.

[edit] Parallels between electrostatic and gravitational fields

Electric field from a negative Q where

Page 85: Dielectric Constant

Coulomb's law, which describes the interaction of electric charges:

is similar to Newton's law of universal gravitation:

This suggests similarities between the electric field E and the gravitational field g, so sometimes mass is called "gravitational charge".

Similarities between electrostatic and gravitational forces:

1. Both act in a vacuum.2. Both are central and conservative.3. Both obey an inverse-square law (both are inversely proportional to square of

r).4. Both propagate with finite speed c, the speed of light.5. Electric charge and relativistic mass are conserved; note, though, that rest

mass is not conserved.

Differences between electrostatic and gravitational forces:

1. Electrostatic forces are much greater than gravitational forces (by about 1036 times).

2. Gravitational forces are attractive for like charges, whereas electrostatic forces are repulsive for like charges.

3. There are no negative gravitational charges (no negative mass) while there are both positive and negative electric charges. This difference combined with previous implies that gravitational forces are always attractive, while electrostatic forces may be either attractive or repulsive.

[edit] Electrodynamic fields

Main article: Electrodynamics

Electrodynamic fields are E-fields which do change with time, when charges are in motion.

An electric field can be produced, not only by a static charge, but also by a changing magnetic field. The electric field is given by:

in which B satisfies

Page 86: Dielectric Constant

and × denotes the ∇ curl. The vector field B is the magnetic flux density and the vector A is the magnetic vector potential. Taking the curl of the electric field equation we obtain,

which is Faraday's law of induction, another one of Maxwell's equations.[3]

[edit] Energy in the electric field

Main article: Electric energy

The electrostatic field stores energy. The energy density u (energy per unit volume) is given by[4]

where ε is the permittivity of the medium in which the field exists, and E is the electric field vector.

The total energy U stored in the electric field in a given volume V is therefore

[edit] Further extensions

[edit] Definitive equation of vector fieldsMain article: Defining equation (physics)

In the presence of matter, it is helpful in electromagnetism to extend the notion of the electric field into three vector fields, rather than just one:[5]

where P is the electric polarization - the volume density of electric dipole moments, and D is the electric displacement field. Since E and P are defined separately, this equation can be used to define D. The physical interpretation of D is not as clear as E (effectively the field applied to the material) or P (induced field due to the dipoles in the material), but still serves as a convenient mathematical simplification, since Maxwell's equations can be simplified in terms of free charges and currents.

Page 87: Dielectric Constant

[edit] Constitutive relationMain article: Constitutive equation

The E and D fields are related by the permittivity of the material, ε.[6][7]

For linear, homogeneous, isotropic materials E and D are proportional and constant throughout the region, there is no position dependence: For inhomogeneous materials, there is a position dependence throughout the material:

For anisotropic materials the E and D fields are not parallel, and so E and D are related by the permittivity tensor (a 2nd order tensor field), in component form:

For non-linear media, E and D are not proportional. Materials can have varying extents of linearity, homogeneity and isotropy.

[edit] See also

Classical electromagnetism Magnetism Teltron Tube

[edit] References

1. ̂ Electromagnetism (2nd Edition), I.S. Grant, W.R. Phillips, Manchester Physics, John Wiley & Sons, 2008, ISBN 978-0-471-92712-9

2. ̂ Introduction to Electrodynamics (3rd Edition), D.J. Griffiths, Pearson Education, Dorling Kindersley, 2007, ISBN 81-7758-293-3

3. ̂ Huray, Paul G. (2009), Maxwell's Equations, Wiley-IEEE, p. 205, ISBN 0-470-54276-4, Chapter 7, p 205

4. ̂ Introduction to Electrodynamics (3rd Edition), D.J. Griffiths, Pearson Education, Dorling Kindersley, 2007, ISBN 81-7758-293-3

5. ̂ Electromagnetism (2nd Edition), I.S. Grant, W.R. Phillips, Manchester Physics, John Wiley & Sons, 2008, ISBN 978-0-471-92712-9

6. ̂ Electricity and Modern Physics (2nd Edition), G.A.G. Bennet, Edward Arnold (UK), 1974, ISBN 0-7131-2459-8

7. ̂ Electromagnetism (2nd Edition), I.S. Grant, W.R. Phillips, Manchester Physics, John Wiley & Sons, 2008, ISBN 978-0-471-92712-9

[edit] External links

Electric field in "Electricity and Magnetism", R Nave - Hyperphysics, Georgia State University

'Gauss's Law' - Chapter 24 of Frank Wolfs's lectures at University of Rochester

Page 88: Dielectric Constant

'The Electric Field' - Chapter 23 of Frank Wolfs's lectures at University of Rochester

[1] - An applet that shows the electric field of a moving point charge. Fields - a chapter from an online textbook Learning by Simulations Interactive simulation of an electric field of up to

four point charges Java simulations of electrostatics in 2-D and 3-D Electric Fields Applet - An applet that shows electric field lines as well as

potential gradients. The inverse cube law The inverse cube law for dipoles (PDF file) by Eng.

Xavier Borg Interactive Flash simulation picturing the electric field of user-defined or

preselected sets of point charges by field vectors, field lines, or equipotential lines. Author: David Chappell

Page 89: Dielectric Constant

Dielectric From Wikipedia, the free encyclopediaJump to: navigation, search

A dielectric is an electrical insulator that can be polarized by an applied electric field. When a dielectric is placed in an electric field, electric charges do not flow through the material, as in a conductor, but only slightly shift from their average equilibrium positions causing dielectric polarization. Because of dielectric polarization, positive charges are displaced toward the field and negative charges shift in the opposite direction. This creates an internal electric field which reduces the overall field within the dielectric itself.[1] If a dielectric is composed of weakly bonded molecules, those molecules not only become polarized, but also reorient so that their symmetry axis aligns to the field.[1]

Although the term "insulator" implies low electrical conduction, "dielectric" is typically used to describe materials with a high polarizability. The latter is expressed by a number called the dielectric constant. A common, yet notable example of a dielectric is the electrically insulating material between the metallic plates of a capacitor. The polarization of the dielectric by the applied electric field increases the capacitor's surface charge.[1]

The study of dielectric properties is concerned with the storage and dissipation of electric and magnetic energy in materials.[2] It is important to explain various phenomena in electronics, optics, and solid-state physics.

The term "dielectric" was coined by William Whewell (from "dia-electric") in response to a request from Michael Faraday.[3]

Contents

 [hide]  1 Electric susceptibility

o 1.1 Dispersion and causality 2 Dielectric polarization

o 2.1 Basic atomic model o 2.2 Dipolar polarization o 2.3 Ionic polarization

3 Dielectric dispersion 4 Dielectric relaxation

o 4.1 Debye relaxation o 4.2 Variants of the Debye equation

5 Applications o 5.1 Capacitors o 5.2 Dielectric resonator

6 Some practical dielectrics 7 See also

Page 90: Dielectric Constant

8 References 9 Further reading

10 External links

[edit] Electric susceptibility

Main article: permittivity

The electric susceptibility χe of a dielectric material is a measure of how easily it polarizes in response to an electric field. This, in turn, determines the electric permittivity of the material and thus influences many other phenomena in that medium, from the capacitance of capacitors to the speed of light.

It is defined as the constant of proportionality (which may be a tensor) relating an electric field E to the induced dielectric polarization density P such that

where is the electric permittivity of free space.

The susceptibility of a medium is related to its relative permittivity by

So in the case of a vacuum,

The electric displacement D is related to the polarization density P by

[edit] Dispersion and causality

In general, a material cannot polarize instantaneously in response to an applied field. The more general formulation as a function of time is

That is, the polarization is a convolution of the electric field at previous times with

time-dependent susceptibility given by . The upper limit of this integral can

be extended to infinity as well if one defines for . An instantaneous response corresponds to Dirac delta function susceptibility

.

Page 91: Dielectric Constant

It is more convenient in a linear system to take the Fourier transform and write this relationship as a function of frequency. Due to the convolution theorem, the integral becomes a simple product,

Note the simple frequency dependence of the susceptibility, or equivalently the permittivity. The shape of the susceptibility with respect to frequency characterizes the dispersion properties of the material.

Moreover, the fact that the polarization can only depend on the electric field at

previous times (i.e. for ), a consequence of causality, imposes

Kramers–Kronig constraints on the susceptibility .

[edit] Dielectric polarization

[edit] Basic atomic model

Electric field interaction with an atom under the classical dielectric model.

In the classical approach to the dielectric model, a material is made up of atoms. Each atom consists of a cloud of negative charge (Electrons) bound to and surrounding a positive point charge at its center. In the presence of an electric field the charge cloud is distorted, as shown in the top right of the figure.

This can be reduced to a simple dipole using the superposition principle. A dipole is characterized by its dipole moment, a vector quantity shown in the figure as the blue arrow labeled M. It is the relationship between the electric field and the dipole moment that gives rise to the behavior of the dielectric. (Note that the dipole moment is shown to be pointing in the same direction as the electric field. This isn't always correct, and it is a major simplification, but it is suitable for many materials.)

When the electric field is removed the atom returns to its original state. The time required to do so is the so-called relaxation time; an exponential decay.

Page 92: Dielectric Constant

This is the essence of the model in physics. The behavior of the dielectric now depends on the situation. The more complicated the situation the richer the model has to be in order to accurately describe the behavior. Important questions are:

Is the electric field constant or does it vary with time? o If the electric field does vary, at what rate?

What are the characteristics of the material? o Is the direction of the field important (isotropy)?o Is the material the same all the way through (homogeneous)?o Are there any boundaries/interfaces that have to be taken into account?

Is the system linear or do nonlinearities have to be taken into account?

The relationship between the electric field E and the dipole moment M gives rise to the behavior of the dielectric, which, for a given material, can be characterized by the function F defined by the equation:

.

When both the type of electric field and the type of material have been defined, one then chooses the simplest function F that correctly predicts the phenomena of interest. Examples of phenomena that can be so modeled include:

Refractive index Group velocity dispersion Birefringence Self-focusing Harmonic generation

[edit] Dipolar polarization

Dipolar polarization is a polarization that is either inherent to polar molecules (orientation polarization), or can be induced in any molecule in which the asymmetric distortion of the nuclei is possible (distortion polarization). Orientation polarization results from a permanent dipole, e.g. that arising from the ca. 104ο angle between the asymmetric bonds between oxygen and hydrogen atoms in the water molecule, which retains polarization in the absence of an external electric field. The assembly of these dipoles forms a macroscopic polarization.

When an external electric field is applied, the distance between charges, which is related to chemical bonding, remains constant in orientation polarization; however, the polarization itself rotates. This rotation occurs on a timescale which depends on the torque and the surrounding local viscosity of the molecules. Because the rotation is not instantaneous, dipolar polarizations lose the response to electric fields at the lowest frequency in polarizations. A molecule rotates about 1ps per radian in a fluid, thus this loss occurs at about 1011 Hz (in the microwave region). The delay of the response to the change of the electric field causes friction and heat.

When an external electric field is applied in the infrared, a molecule is bent and stretched by the field and the molecular moment changes in response. The molecular

Page 93: Dielectric Constant

vibration frequency is approximately the inverse of the time taken for the molecule to bend, and the distortion polarization disappears above the infrared.

[edit] Ionic polarization

Ionic polarization is polarization which is caused by relative displacements between positive and negative ions in ionic crystals (for example, NaCl).

If crystals or molecules do not consist of only atoms of the same kind, the distribution of charges around an atom in the crystals or molecules leans to positive or negative. As a result, when lattice vibrations or molecular vibrations induce relative displacements of the atoms, the centers of positive and negative charges might be in different locations. These center positions are affected by the symmetry of the displacements. When the centers don't correspond, polarizations arise in molecules or crystals. This polarization is called ionic polarization.

Ionic polarization causes ferroelectric transition as well as dipolar polarization. The transition, which is caused by the order of the directional orientations of permanent dipoles along a particular direction, is called order-disorder phase transition. The transition which is caused by ionic polarizations in crystals is called displacive phase transition.

[edit] Dielectric dispersion

In physics, dielectric dispersion is the dependence of the permittivity of a dielectric material on the frequency of an applied electric field. Because there is always a lag between changes in polarization and changes in an electric field, the permittivity of the dielectric is a complicated, complex-valued function of frequency of the electric field. It is very important for the application of dielectric materials and the analysis of polarization systems.

This is one instance of a general phenomenon known as material dispersion: a frequency-dependent response of a medium for wave propagation.

When the frequency becomes higher:

1. it becomes impossible for dipolar polarization to follow the electric field in the microwave region around 1010 Hz;

2. in the infrared or far-infrared region around 1013 Hz, ionic polarization and molecular distortion polarization lose the response to the electric field;

3. electronic polarization loses its response in the ultraviolet region around 1015 Hz.

In the frequency region above ultraviolet, permittivity approaches the constant ε0 in every substance, where ε0 is the permittivity of the free space. Because permittivity indicates the strength of the relation between an electric field and polarization, if a polarization process loses its response, permittivity decreases.

[edit] Dielectric relaxation

Page 94: Dielectric Constant

Dielectric relaxation is the momentary delay (or lag) in the dielectric constant of a material. This is usually caused by the delay in molecular polarization[disambiguation needed  ]

with respect to a changing electric field in a dielectric medium (e.g. inside capacitors or between two large conducting surfaces). Dielectric relaxation in changing electric fields could be considered analogous to hysteresis in changing magnetic fields (for inductors or transformers). Relaxation in general is a delay or lag in the response of a linear system, and therefore dielectric relaxation is measured relative to the expected linear steady state (equilibrium) dielectric values. The time lag between electrical field and polarization implies an irreversible degradation of free energy(G).

In physics, dielectric relaxation refers to the relaxation response of a dielectric medium to an external electric field of microwave frequencies. This relaxation is often described in terms of permittivity as a function of frequency, which can, for ideal systems, be described by the Debye equation. On the other hand, the distortion related to ionic and electronic polarization shows behavior of the resonance or oscillator type. The character of the distortion process depends on the structure, composition, and surroundings of the sample.

The number of possible wavelengths of emitted radiation due to dielectric relaxation can be equated using Hemmings' first law (named after Mark Hemmings)

where

n is the number of different possible wavelengths of emitted radiationis the number of energy levels (including ground level).

[edit] Debye relaxation

Debye relaxation is the dielectric relaxation response of an ideal, noninteracting population of dipoles to an alternating external electric field. It is usually expressed in the complex permittivity of a medium as a function of the field's frequency :

where is the permittivity at the high frequency limit, where is the static, low frequency permittivity, and is the characteristic relaxation time of the medium.

This relaxation model was introduced by and named after the chemist Peter Debye (1913).[4]

[edit] Variants of the Debye equation Cole–Cole equation Cole–Davidson equation Havriliak–Negami relaxation

Page 95: Dielectric Constant

Kohlrausch–Williams–Watts function (Fourier transform of stretched exponential function)

[edit] Applications

[edit] CapacitorsMain article: capacitor

Charge separation in a parallel-plate capacitor causes an internal electric field. A dielectric (orange) reduces the field and increases the capacitance.

Commercially manufactured capacitors typically use a solid dielectric material with high permittivity as the intervening medium between the stored positive and negative charges. This material is often referred to in technical contexts as the "capacitor dielectric".[5]

The most obvious advantage to using such a dielectric material is that it prevents the conducting plates on which the charges are stored from coming into direct electrical contact. More significant, however, a high permittivity allows a greater charge to be stored at a given voltage. This can be seen by treating the case of a linear dielectric with permittivity ε and thickness d between two conducting plates with uniform charge density σε. In this case the charge density is given by

and the capacitance per unit area by

From this, it can easily be seen that a larger ε leads to greater charge stored and thus greater capacitance.

Dielectric materials used for capacitors are also chosen such that they are resistant to ionization. This allows the capacitor to operate at higher voltages before the insulating dielectric ionizes and begins to allow undesirable current.

Page 96: Dielectric Constant

[edit] Dielectric resonatorMain article: dielectric resonator

A dielectric resonator oscillator (DRO) is an electronic component that exhibits resonance for a narrow range of frequencies, generally in the microwave band. It consists of a "puck" of ceramic that has a large dielectric constant and a low dissipation factor. Such resonators are often used to provide a frequency reference in an oscillator circuit. An unshielded dielectric resonator can be used as a Dielectric Resonator Antenna (DRA).

[edit] Some practical dielectrics

Dielectric materials can be solids, liquids, or gases. In addition, a high vacuum can also be a useful, lossless dielectric even though its relative dielectric constant is only unity.

Solid dielectrics are perhaps the most commonly used dielectrics in electrical engineering, and many solids are very good insulators. Some examples include porcelain, glass, and most plastics. Air, nitrogen and sulfur hexafluoride are the three most commonly used gaseous dielectrics.

Industrial coatings such as parylene provide a dielectric barrier between the substrate and its environment.

Mineral oil is used extensively inside electrical transformers as a fluid dielectric and to assist in cooling. Dielectric fluids with higher dielectric constants, such as electrical grade castor oil, are often used in high voltage capacitors to help prevent corona discharge and increase capacitance.

Because dielectrics resist the flow of electricity, the surface of a dielectric may retain stranded excess electrical charges. This may occur accidentally when the dielectric is rubbed (the triboelectric effect). This can be useful, as in a Van de Graaff generator or electrophorus, or it can be potentially destructive as in the case of electrostatic discharge.

Specially processed dielectrics, called electrets (which should not be confused with ferroelectrics), may retain excess internal charge or "frozen in" polarization. Electrets have a semipermanent external electric field, and are the electrostatic equivalent to magnets. Electrets have numerous practical applications in the home and industry.

Some dielectrics can generate a potential difference when subjected to mechanical stress, or change physical shape if an external voltage is applied across the material. This property is called piezoelectricity. Piezoelectric materials are another class of very useful dielectrics.

Some ionic crystals and polymer dielectrics exhibit a spontaneous dipole moment which can be reversed by an externally applied electric field. This behavior is called the ferroelectric effect. These materials are analogous to the way ferromagnetic materials behave within an externally applied magnetic field. Ferroelectric materials often have very high dielectric constants, making them quite useful for capacitors.

Page 97: Dielectric Constant

Electromagnetic field From Wikipedia, the free encyclopediaJump to: navigation, search

It has been suggested that Flux density be merged into this article or section. (Discuss) Proposed since October 2011.

Electromagnetism

Electricity

Magnetism

Electrostatics [show]

Magnetostatics [show]

Electrodynamics [show]

Electrical Network [show]

Covariant formulation [show]

Scientists[show]

v

t

e

An electromagnetic field (also EMF or EM field) is a physical field produced by moving electrically charged objects. It affects the behavior of charged objects in the vicinity of the field. The electromagnetic field extends indefinitely throughout space and describes the electromagnetic interaction. It is one of the four fundamental forces of nature (the others are gravitation, the weak interaction, and the strong interaction).

The field can be viewed as the combination of an electric field and a magnetic field. The electric field is produced by stationary charges, and the magnetic field by moving charges (currents); these two are often described as the sources of the field. The way in which charges and currents interact with the electromagnetic field is described by Maxwell's equations and the Lorentz force law.

Page 98: Dielectric Constant

From a classical perspective, the electromagnetic field can be regarded as a smooth, continuous field, propagated in a wavelike manner; whereas from the perspective of quantum field theory, the field is seen as quantized, being composed of individual particles.[citation needed]

Contents

 [hide]  1 Structure of the electromagnetic field

o 1.1 Continuous structure o 1.2 Discrete structure

2 Dynamics of the electromagnetic field 3 Electromagnetic field as a feedback loop 4 Mathematical description 5 Properties of the field

o 5.1 Reciprocal behavior of electric and magnetic fields o 5.2 Light as an electromagnetic disturbance

6 Relation to and comparison with other physical fields o 6.1 Electromagnetic and gravitational fields

7 Applications o 7.1 Static E and B fields and static EM fields o 7.2 Time-varying EM fields in Maxwell’s equations

8 Health and safety 9 See also 10 References

11 External links

[edit] Structure of the electromagnetic field

The electromagnetic field may be viewed in two distinct ways: a continuous structure or a discrete structure.

[edit] Continuous structure

Classically, electric and magnetic fields are thought of as being produced by smooth motions of charged objects. For example, oscillating charges produce electric and magnetic fields that may be viewed in a 'smooth', continuous, wavelike fashion. In this case, energy is viewed as being transferred continuously through the electromagnetic field between any two locations. For instance, the metal atoms in a radio transmitter appear to transfer energy continuously. This view is useful to a certain extent (radiation of low frequency), but problems are found at high frequencies (see ultraviolet catastrophe).

[edit] Discrete structure

The electromagnetic field may be thought of in a more 'coarse' way. Experiments reveal that in some circumstances electromagnetic energy transfer is better described

Page 99: Dielectric Constant

as being carried in the form of packets called quanta (in this case, photons) with a fixed frequency. Planck's relation links the energy of a photon to its frequency through the equation:

where is Planck's constant, named in honor of Max Planck, and is the frequency of the photon . Although modern quantum optics tells us that there also is a semi-classical explanation of the photoelectric effect —the emission of electrons from metallic surfaces subjected to electromagnetic radiation— the photon was historically (although strictly unnecessarily) used to explain certain observations. It is found that increasing the intensity of the incident radiation (so long as one remains in the linear regime) increases only the number of electrons ejected, and has almost no effect on the energy distribution of their ejection. Only the frequency of the radiation is relevant to the energy of the ejected electrons.

This quantum picture of the electromagnetic field (which treats it as analogous to harmonic oscillators) has proved very successful, giving rise to quantum electrodynamics, a quantum field theory describing the interaction of electromagnetic radiation with charged matter. It also gives rise to Quantum optics, which is different from quantum electrodynamics in that the matter itself is modelled using quantum mechanics rather than Quantum field theory.

[edit] Dynamics of the electromagnetic field

In the past, electrically charged objects were thought to produce two different, unrelated types of field associated with their charge property. An electric field is produced when the charge is stationary with respect to an observer measuring the properties of the charge, and a magnetic field (as well as an electric field) is produced when the charge moves (creating an electric current) with respect to this observer. Over time, it was realized that the electric and magnetic fields are better thought of as two parts of a greater whole — the electromagnetic field.

Once this electromagnetic field has been produced from a given charge distribution, other charged objects in this field will experience a force (in a similar way that planets experience a force in the gravitational field of the Sun). If these other charges and currents are comparable in size to the sources producing the above electromagnetic field, then a new net electromagnetic field will be produced. Thus, the electromagnetic field may be viewed as a dynamic entity that causes other charges and currents to move, and which is also affected by them. These interactions are described by Maxwell's equations and the Lorentz force law. (This discussion ignores the radiation reaction force.)

[edit] Electromagnetic field as a feedback loop

The behavior of the electromagnetic field can be resolved into four different parts of a loop:

the electric and magnetic fields are generated by electric charges,

Page 100: Dielectric Constant

the electric and magnetic fields interact with each other, the electric and magnetic fields produce forces on electric charges, the electric charges move in space.

A common misunderstanding is that (a) the quanta of the fields act in the same manner as (b) the charged particles that generate the fields. In our everyday world, charged particles, such as electrons, move slowly through matter, typically on the order of a few inches (or centimeters) per second[citation needed], but fields propagate at the speed of light - approximately 300 thousand kilometers (or 186 thousand miles) a second. The mundane speed difference between charged particles and field quanta is on the order of one to a million, more or less. Maxwell's equations relate (a) the presence and movement of charged particles with (b) the generation of fields. Those fields can then affect the force on, and can then move, other slowly moving charged particles. Charged particles can move at relativistic speeds nearing field propagation speeds, but, as Einstein showed[citation needed], this requires enormous field energies, which are not present in our everyday experiences with electricity, magnetism, matter, and time.

The feedback loop can be summarized in a list, including phenomena belonging to each part of the loop:

charged particles generate electric and magnetic fields the fields interact with each other

o changing electric field acts like a current, generating 'vortex' of magnetic field

o Faraday induction : changing magnetic field induces (negative) vortex of electric field

o Lenz's law : negative feedback loop between electric and magnetic fields

fields act upon particles o Lorentz force: force due to electromagnetic field

electric force: same direction as electric field magnetic force: perpendicular both to magnetic field and to

velocity of charge particles move

o current is movement of particles particles generate more electric and magnetic fields; cycle repeats

[edit] Mathematical description

Main article: Mathematical descriptions of the electromagnetic field

There are different mathematical ways of representing the electromagnetic field. The first one views the electric and magnetic fields as three-dimensional vector fields. These vector fields each have a value defined at every point of space and time and are thus often regarded as functions of the space and time coordinates. As such, they are

often written as (electric field) and (magnetic field).

Page 101: Dielectric Constant

If only the electric field ( ) is non-zero, and is constant in time, the field is said to be an electrostatic field. Similarly, if only the magnetic field ( ) is non-zero and is constant in time, the field is said to be a magnetostatic field. However, if either the electric or magnetic field has a time-dependence, then both fields must be considered together as a coupled electromagnetic field using Maxwell's equations.[1]

With the advent of special relativity, physical laws became susceptible to the formalism of tensors. Maxwell's equations can be written in tensor form, generally viewed by physicists as a more elegant means of expressing physical laws.

The behaviour of electric and magnetic fields, whether in cases of electrostatics, magnetostatics, or electrodynamics (electromagnetic fields), is governed in a vacuum by Maxwell's equations. In the vector field formalism, these are:

(Gauss's law)(Gauss's law for magnetism)

(Faraday's law)

(Ampère-Maxwell law)

where is the charge density, which can (and often does) depend on time and position, is the permittivity of free space, is the permeability of free space, and is the current density vector, also a function of time and position. The units used above are the standard SI units. Inside a linear material, Maxwell's equations change by switching the permeability and permittivity of free space with the permeability and permittivity of the linear material in question. Inside other materials which possess more complex responses to electromagnetic fields, these terms are often represented by complex numbers, or tensors.

The Lorentz force law governs the interaction of the electromagnetic field with charged matter.

When a field travels across to different media, the properties of the field change according to the various boundary conditions. These equations are derived from Maxwell's equations. The tangential components of the electric and magnetic fields as they relate on the boundary of two media are as follows[2]:

(current-free)(charge-free)

The angle of refraction of an electric field between media is related to the permittivity

of each media:

Page 102: Dielectric Constant

The angle of refraction of a magnetic field between media is related to the

permeability of each media:

[edit] Properties of the field

[edit] Reciprocal behavior of electric and magnetic fields

The two Maxwell equations, Faraday's Law and the Ampère-Maxwell Law, illustrate a very practical feature of the electromagnetic field Faraday's Law may be stated roughly as 'a changing magnetic field creates an electric field'. This is the principle behind the electric generator.

Ampere's Law roughly states that 'a changing electric field creates a magnetic field'. Thus, this law can be applied to generate a magnetic field and run an electric motor.

[edit] Light as an electromagnetic disturbance

Maxwell's equations take the form of an electromagnetic wave in an area that is very far away from any charges or currents (free space) – that is, where and are zero. It can be shown, that, under these conditions, the electric and magnetic fields satisfy the electromagnetic wave equation [3] :

James Clerk Maxwell was the first to obtain this relationship by his completion of Maxwell's equations with the addition of a displacement current term to Ampere's Circuital law.

[edit] Relation to and comparison with other physical fields

Main article: Fundamental forcesThis section requires expansion.

Being one of the four fundamental forces of nature, it is useful to compare the electromagnetic field with the gravitational, strong and weak fields. The word 'force'

Page 103: Dielectric Constant

is sometimes replaced by 'interaction' because the fundamental forces operate by exchanging what are now known to be gauge bosons.

[edit] Electromagnetic and gravitational fields

Sources of electromagnetic fields consist of two types of charge – positive and negative. This contrasts with the sources of the gravitational field, which are masses. Masses are sometimes described as gravitational charges, the important feature of them being that there is only one type (no negative masses), or, in more colloquial terms, 'gravity is always attractive'.

The relative strengths and ranges of the four interactions and other information are tabulated below:

Theory Interaction mediatorRelative

MagnitudeBehavior Range

Chromodynamics Strong interaction gluon 1038 1 10−15 m

ElectrodynamicsElectromagnetic interaction

photon 1036 1/r2 infinite

Flavordynamics Weak interactionW and Z bosons

1025 1/r5 to 1/r7 10−16 m

Geometrodynamics Gravitation graviton 100 1/r2 infinite

[edit] Applications

This section requires expansion.

[edit] Static E and B fields and static EM fieldsMain articles: electrostatics, magnetostatics, and magnetism

When an EM field (see electromagnetic tensor) is not varying in time, it may be seen as a purely electrical field or a purely magnetic field, or a mixture of both. However the general case of a static EM field with both electric and magnetic components present, is the case that appears to most observers. Observers who see only an electric or magnetic field component of a static EM field, have the other (electric or magnetic) component suppressed, due to the special case of the immobile state of the charges that produce the EM field in that case. In such cases the other component becomes manifest in other observer frames.

A consequence of this, is that any case that seems to consist of a "pure" static electric or magnetic field, can be converted to an EM field, with both E and B components present, by simply moving the observer into a frame of reference which is moving with regard to the frame in which only the “pure” electric or magnetic field appears. That is, a pure static electric field will show the familiar magnetic field associated with a current, in any frame of reference where the charge moves. Likewise, any new motion of a charge in a region that seemed previously to contain only a magnetic field, will show that that the space now contains an electric field as well, which will be found to produces an additional Lorentz force upon the moving charge.

Page 104: Dielectric Constant

Thus, electrostatics, as well as magnetism and magnetostatics, are now seen as studies of the static EM field when a particular frame has been selected to suppress the other type of field, and since an EM field with both electric and magnetic will appear in any other frame, these "simpler" effects are merely the observer's. The "applications" of all such non-time varying (static) fields are discussed in the main articles linked in this section.

[edit] Time-varying EM fields in Maxwell’s equationsMain articles: near and far field, electromagnetic radiation, virtual particle, dielectric heating, and magnetic induction

An EM field that varies in time has two “causes” in Maxwell’s equations. One is charges and currents (so-called “sources”), and the other cause for an E or B field is a change in the other type of field (this last cause also appears in “free space” very far from currents and charges).

An electromagnetic field very far from currents and charges (sources) is called electromagnetic radiation (EMR) since it radiates from the charges and currents in the source, and has no "feedback" effect on them, and is also not affected directly by them in the present time (rather, it is indirectly produced by a sequences of changes in fields radiating out from them in the past). EMR consists of the radiations in the electromagnetic spectrum, including radio waves, microwave, infrared, visible light, ultraviolet light, X-rays, and gamma rays. The many commercial applications of these radiations are discussed in the named and linked articles.

A notable application of visible light is that this type of energy from the Sun powers all life on Earth that either makes or uses oxygen.

A changing electromagnetic field which is physically close to currents and charges (see near and far field for a definition of “close”) will have a dipole characteristic that is dominated by either a changing electric dipole, or a changing magnetic dipole. This type of dipole field near sources is called an electromagnetic near-field.

Changing electric dipole fields, as such, are used commercially as near-fields mainly as a source of dielectric heating. Otherwise, they appear parasitically around conductors which absorb EMR, and around antennas which have the purpose of generating EMR at greater distances.

Changing magnetic dipole fields (i.e., magnetic near-fields) are used commercially for many types of magnetic induction devices. These include motors and electrical transformers at low frequencies, and devices such as metal detectors and MRI scanner coils at higher frequencies. Sometimes these high-frequency magnetic fields change at radio frequencies without being far-field waves and thus radio waves; see RFID tags.

Further uses of near-field EM effects commercially, may be found in the article on virtual photons, since at the quantum level, these fields are represented by these particles. Far-field effects (EMR) in the quantum picture of radiation, are represented by ordinary photons.

Page 105: Dielectric Constant

[edit] Health and safety

The potential health effects of the very low frequency EMFs surrounding power lines and electrical devices are the subject of on-going research and a significant amount of public debate. In workplace environments, where EMF exposures can be up to 10,000 times greater than the average, the US National Institute for Occupational Safety and Health (NIOSH) has issued some cautionary advisories but stresses that the data is currently too limited to draw good conclusions.[4]

The potential effects of electromagnetic fields on human health vary widely depending on the frequency and intensity of the fields. For more information on the health effects due to specific parts of the electromagnetic spectrum, see the following articles:

Static electric fields: see Electric shock Static magnetic fields: see MRI#Safety Extremely low frequency (ELF): see Power lines#Health concerns Radio frequency (RF): see Electromagnetic radiation and health Light: see Laser safety Ultraviolet (UV): see Sunburn Gamma rays: see Gamma ray Mobile telephony: see Mobile phone radiation and health