Notch filter

A generic ideal band-stop filter, showing both positive and negative angular frequencies

In signal processing, a band-stop filter or band-rejection filter is a filter that passes most frequencies unaltered, but attenuates those in a specific range to very low levels. It is the opposite of a band-pass filter. A notch filter is a band-stop filter with a narrow stopband (high Q factor). Notch filters are used in live sound reproduction (Public Address systems, also known as PA systems) and in instrument amplifier (especially amplifiers or preamplifiers for acoustic instruments such as acoustic guitar, mandolin, bass instrument amplifier, etc.) to reduce or prevent feedback, while having little noticeable effect on the rest of the frequency spectrum. Other names include 'band limit filter', 'T-notch filter', 'band-elimination filter', and 'band-reject filter'.
Typically, the width of the stopband is less than 1 to 2 decades (that is, the highest frequency attenuated is less than 10 to 100 times the lowest frequency attenuated). In the audio band, a notch filter uses high and low frequencies that may be only semitones apart.
Audio example 1: Anti-hum filter

• Low Freq: 59 Hz
• High Freq: 61 Hz

This means that the filter passes all frequencies, except for the range of 59–61 Hz. This would be used to filter out the mains hum from a 60 Hz power line, though its higher harmonics could still be present. The common European version of the filter would have a 49–51 Hz range.
Audio example 2: Anti-presence filter

• Low Freq: 1 kHz
• High Freq: 4 kHz

RF example 1: Non-linearities of power amplifiers For instance, when measuring non-linearities of power amplifiers a very narrow notch filter could be very useful to avoid the carrier so maximum input power of e.g. a spectrum analyser used to detect spurious content will not be exceeded.

Magnetostatics

Magnetostatics is the study of static magnetic fields. In electrostatics, the charges are stationary, whereas here, the currents are stationary or dc(direct current). As it turns out magnetostatics is a good approximation even when the currents are not static as long as the currents do not alternate rapidly.

Applications

Magnetostatics as a special case of Maxwell's equations

Starting from Maxwell's equations, the following simplifications can be made:

• ignore any electrostatic charge
• ignore the electric field
• presume the magnetic field is constant with respect to time

Name	Partial differential form	Integral form
presumption
Gauss's law for magnetism:
presumption
Ampère's law:

The quality of this approximation may be guessed by comparing the above equations with the full version of Maxwell's equations and considering the importance of the terms that have been removed. Of particular significance is the comparison of the term against the term. If the term is substantially larger, then the smaller term may be ignored without significant loss of accuracy.

Re-introducing Faraday's law

A common technique is to solve a series of magnetostatic problems at incremental time steps and then use these solutions to approximate the term . Plugging this result into Faraday's Law finds a value for (which had previously been ignored). This method is not a true solution of Maxwell's equations but can provide a good approximation for slowly changing fields.

Solving magnetostatic problems

If all currents in a system are known (i.e. if a complete description of is available) then the magnetic field can be determined from the currents by the Biot-Savart equation:

This technique works well for problems where the medium is a vacuum or air or some similar material with a relative permeability of 1. This includes Air core inductors and Air core transformers. One advantage of this technique is that a complex coil geometry can be integrated in sections, or for a very difficult geometry numerical integration may be used. Since this equation is primarily used to solve linear problems, the complete answer will be a sum of the integral of each component section.
One pitfall in the use of the Biot-Savart equation is that it does not implicitly enforce Gauss's law for magnetism so it is possible to come up with an answer that includes magnetic monopoles. This will occur if some section of the current path has not been included in the integral (implying that electrons are being continuously created in one place and destroyed in another).
Using Biot-Savart in the presence of Ferromagnetic, Ferrimagnetic or Paramagnetic materials is difficult because the external current induces a surface current in the magnetic material which in turn must be included in the integral. The value of the surface current depends on the magnetic field which was what you were trying to calculate in the first place. For these problems, using Ampère's law (usually in integral form) is a better choice. For problems where the dominant magnetic material is a highly permeable magnetic core with relatively small air gaps, a magnetic circuit approach is useful. When the air gaps are large in comparison to the magnetic circuit length, fringing becomes significant and usually requires a finite element calculation. The finite element calculation uses a modified form of the magnetostatic equations above in order to calculate magnetic potential. The value of can be found from the magnetic potential.

Hysteresis

In a deterministic system with no dynamics or hysteresis, it is possible to predict the system's output at an instant in time, given only its input at that instant in time. In a system with hysteresis, this is not possible; there is no way to predict the output without knowing the system's current state, and there is no way to know the system's state without looking at the history of the input. This means that it is necessary to know the path that the input followed before it reached its current value.
Many physical systems naturally exhibit hysteresis. A piece of iron that is brought into a magnetic field retains some magnetization, even after the external magnetic field is removed. Once magnetized, the iron will stay magnetized indefinitely. To demagnetize the iron, it would be necessary to apply a magnetic field in the opposite direction. This is the effect that provides the element of memory in a hard disk drive.
A system may be explicitly designed to exhibit hysteresis, especially in control theory. For example, consider a thermostat that controls a furnace. The furnace is either off or on, with nothing in between. The thermostat is a system; the input is the temperature, and the output is the furnace state. If one wishes to maintain a temperature of 20 °C, then one might set the thermostat to turn the furnace on when the temperature drops below 18 °C, and turn it off when the temperature exceeds 22 °C. This thermostat has hysteresis. If the temperature is 21 °C, then it is not possible to predict whether the furnace is on or off without knowing the history of the temperature.
The word hysteresis is often used specifically to represent rate-independent state. This means that if some set of inputs X(t) produce an output Y(t), then the inputs X(αt) produce output Y(αt) for any α > 0. The magnetized iron or the thermostat have this property. Not all systems with state (or, equivalently, with memory) have this property; for example, a linear low-pass filter has state, but its state is rate-dependent.
The term is derived from ὑστέρησις, an ancient Greek word meaning "deficiency" or "lagging behind". It was coined by Sir James Alfred Ewing.

Introduction

Hysteresis phenomena occur in magnetic and ferromagnetic materials, as well as in the elastic, electric, and magnetic behavior of materials, in which a lag occurs between the application and the removal of a force or field and its subsequent effect. Electric hysteresis occurs when applying a varying electric field, and elastic hysteresis occurs in response to a varying force. The term "hysteresis" is sometimes used in other fields, such as economics or biology; where it describes a memory, or lagging effect, in which the order of previous events can influence the order of subsequent events.^{[citation needed]}
The word "lag" above should not necessarily be interpreted as a time lag. After all, even relatively simple linear systems such as an electric circuit containing resistors and capacitors exhibit a time lag between the input and the output. For most hysteretic systems, there is a very short time scale when its dynamic behavior and various related time dependences are observed. In magnetism, for example, the dynamic processes occurring on this very short time scale have been referred to as Barkhausen jumps. If observations are carried out over very long periods, creep or slow relaxation typically toward true thermodynamic equilibrium (or other types of equilibria that depend on the nature of the system) can be noticed. When observations are carried out without regard for very swift dynamic phenomena or very slow relaxation phenomena, the system appears to display irreversible behavior whose rate is practically independent of the driving force rate. This rate-independent irreversible behavior is the key feature that distinguishes hysteresis from most other dynamic processes in many systems.
If the displacement of a system with hysteresis is plotted on a graph against the applied force, the resulting curve is in the form of a loop. In contrast, the curve for a system without hysteresis is a single, not necessarily straight, line. Although the hysteresis loop depends on the material's physical properties, there is no complete theoretical description that explains the phenomenon. The family of hysteresis loops, from the results of different applied varying voltages or forces, form a closed space in three dimensions, called the hysteroid.
Hysteresis was initially seen as problematic, but is now thought to be of great importance in technology. For example, the properties of hysteresis are applied when constructing non-volatile storage for computers; as hysteresis allows most superconductors to operate at the high currents needed to create strong magnetic fields. Hysteresis is also important in living systems. Many critical processes occurring in living (or dying) cells use hysteresis to help stabilize them against the various effects of random chemical fluctuations.
Some early work on describing hysteresis in mechanical systems was performed by James Clerk Maxwell. Subsequently, hysteresis models have received significant attention in the works of Preisach (Preisach model of hysteresis), Neel and Everett in connection with magnetism and absorption. A simple parametric description of various hysteresis loops may be found in ref.^[1] (with the model, substitution of rectangle, triangle or trapezoidal pulses instead of the harmonic functions also allows to built piecewise-linear hysteresis loops frequently used in discrete automatics). More formal mathematical theory of systems with hysteresis was developed in 1970s, by a group of Russian mathematicians, which was led by Mark Krasnosel'skii, one of the founders of nonlinear analysis. He suggested an investigation of hysteresis phenomena using the theory of nonlinear operators.^{[citation needed]}

Informal definition

The phenomenon of hysteresis can conceptually be explained as follows: a system can be divided into subsystems or domains, much larger than an atomic volume, but still microscopic. Such domains occur in ferroelectric and ferromagnetic systems, since individual dipoles tend to group with each other, forming a small isotropic region. Each of the system's domains can be shown to have a metastable state. The metastable domains can in turn have two or more substates. Such a metastable state fluctuates widely from domain to domain, but the average represents the configuration of lowest energy. The hysteresis is simply the sum of all domains, or the sum of all metastable states.

Magnetic hysteresis

Hysteresis is well known in ferromagnetic materials. When an external magnetic field is applied to a ferromagnet, the atomic dipoles align themselves with the external field. Even when the external field is removed, part of the alignment will be retained: the material has become magnetized.

A family of B-H loops for grain-oriented electrical steel (B_R denotes remanence and H_C is the coercivity).

The relationship between magnetic field strength (H) and magnetic flux density (B) is not linear in such materials. If the relationship between the two is plotted for increasing levels of field strength, it will follow a curve up to a point where further increases in magnetic field strength will result in no further change in flux density. This condition is called magnetic saturation.
If the magnetic field is now reduced linearly, the plotted relationship will follow a different curve back towards zero field strength at which point it will be offset from the original curve by an amount called the remanent flux density or remanence.
If this relationship is plotted for all strengths of applied magnetic field the result is a sort of S- shaped loop. The 'thickness' of the middle bit of the S describes the amount of hysteresis, related to the coercivity of the material.
Its practical effects might be, for example, to cause a relay to be slow to release due to the remaining magnetic field continuing to attract the armature when the applied electric current to the operating coil is removed.

Hysteresis loop: magnetization (M) as function of magnetic field strength (H)

This curve for a particular material influences the design of a magnetic circuit,
This is also a very important effect in magnetic tape and other magnetic storage media like hard disks. In these materials it would seem obvious to have one polarity represent a bit, say north for 1 and south for 0. However, to change the storage from one to the other, the hysteresis effect requires the knowledge of what was already there, because the needed field will be different in each case. In order to avoid this problem, recording systems first overdrive the entire system into a known state using a process known as bias. Analog magnetic recording also uses this technique. Different materials require different biasing, which is why there is a selector switch for this on the front of most cassette recorders.
In order to minimize this effect and the energy losses associated with it, ferromagnetic substances with low coercivity and low hysteresis loss are used, like permalloy.
In many applications small hysteresis loops are driven around points in the B-H plane. Loops near the origin have a higher µ. The smaller loops the more they have a soft magnetic (lengthy) shape. As a special case, a damped AC field demagnetizes any material.
Magnetic field hysteresis loss causes heating. This effect is used in induction cooking, where an alternating magnetic field causes a ferrite container to heat directly rather than being heated by an external heat-source.

Electrical hysteresis

Electrical hysteresis typically occurs in ferroelectric material, where domains of polarization contribute to the total polarization. Polarization is the electrical dipole moment (either C·m^-2 or C·m).

Elastic hysteresis

Elastic hysteresis of an idealized rubber band. The area in the centre of the hysteresis loop is the energy dissipated as heat

Elastic hysteresis is analogous to magnetic hysteresis and was one of the first types of hysteresis to be examined.^[2][3]
A simple way to understand it is in terms of a rubber band with weights attached to it. If the top of a rubber band is hung on a hook and small weights are attached to the bottom of the band one at a time, it will get longer. As more weights are loaded onto it, the band will continue to extend because the force the weights are exerting on the band is increasing. When each weight is taken off, or unloaded, it will get shorter as the force is reduced. As the weights are taken off, each weight that produced a specific length as it was loaded onto the band now produces a slightly longer length as it is unloaded. This is because the band does not obey Hooke's law perfectly.
In one sense the rubber band was harder to stretch when it was being loaded than when it was being unloaded. In another sense, as one unloads the band, the cause (the force of the weights) lags behind the effect (the length) because a smaller value of weight produces the same length. In another sense more energy was required during the loading than the unloading; that energy must have gone somewhere, it was dissipated or "lost" as heat.
Elastic hysteresis is more pronounced when the loading and unloading is done quickly than when it is done slowly.^[4] Some materials such as hard metals don't show elastic hysteresis under a moderate load, whereas other hard materials like granite and marble do. Materials such as rubber exhibit a high degree of elastic hysteresis.

Liquid-solid phase transitions

Hysteresis manifests itself in state transitions when melting temperature and freezing temperature do not agree. For example, agar melts at 85 °C and solidifies from 32 to 40 °C. This is to say that once agar is melted at 85 °C, it retains a liquid state until cooled to 40 °C. Therefore, from the temperatures of 40 to 85 °C, agar can be either solid or liquid, depending on which state it was before.

Contact angle hysteresis

The contact angle formed between a liquid and solid phase can be measured dynamically. When the maximum liquid volume is removed from the drop without the interfacial area decreasing the receding contact angle is thus measured. When volume is added to the maximum before the interfacial area increases, this is the advancing contact angle. The difference between the advancing and receding contact angles is referred to as the contact angle hysteresis.

Adsorption hysteresis

Hysteresis can also occur during physical adsorption processes. In this type of hysteresis, the quantity adsorbed is different when gas is being added than it is when being removed. The specific causes of adsorption hysteresis are still an active area of research, but it linked to differences in the nucleation and evaporation mechanisms inside mesopores. These mechanisms are further complicated by effects such as cavitation and pore blocking.
In physical adsorption, hysteresis is evidence of mesoporosity-indeed, the definition of mesopores (2-50 nm) is associated with the appearance (50 nm) and disappearance (2 nm) of mesoporosity in nitrogen adsorption isotherms as a function of Kelvin radius.^[5] An adsorption isotherm showing hysteresis is said to be of Type IV (for a wetting adsorbate) or Type V (for a non-wetting adsorbate), and hysteresis loops themselves are classified according to how symmetric the loop is.^[6] Adsorption hysteresis loops also have the unusual property that it is possible to scan within a hysteresis loop by reversing the direction of adsorption while on a point on the loop. The resulting scans are called "crossing," "converging," or "returning," depending on the shape of the isotherm at this point.^[7]

Matric potential hysteresis

The relationship between matric water potential and water content is the basis of the water retention curve. Matric potential measurements (Ψ_m) are converted to volumetric water content (θ) measurements based on a site or soil specific calibration curve. Hysteresis is a source of water content measurement error. Matric potential hysteresis arises from differences in wetting behaviour causing dry medium to re-wet; that is, it depends on the saturation history of the porous medium. Hysteretic behaviour means that, for example, at a matric potential (Ψ_m) of 5 kPa, the volumetric water content (θ) of a fine sandy soil matrix could be anything between 8% to 25%.^[8]
Tensiometers are directly influenced by this type of hysteresis. Two other types of sensors used to measure soil water matric potential are also influenced by hysteresis effects within the sensor itself. Resistance blocks, both nylon and gypsum based, measure matric potential as a function of electrical resistance. The relation between the sensor’s electrical resistance and sensor matric potential is hysteretic. Thermocouples measure matric potential as a function of heat dissipation. Hysteresis occurs because measured heat dissipation depends on sensor water content, and the sensor water content–matric potential relationship is hysteretic. As of 2002^[update], only desorption curves are usually measured during calibration of soil moisture sensors. Despite the fact that it can be a source of significant error, the sensor specific effect of hysteresis is generally ignored.^[9]

Energy

When hysteresis occurs with extensive and intensive variables, the work done on the system is the area under the hysteresis graph.

Economics

Some economic systems show signs of hysteresis. For example, export performance is subject to strong hysteresis effects: it may take a big push (i.e. sizable changes in incentives) to start a country's exports, but once the transition is made, not much may be required to keep them going.
Hysteresis is a hypothesized property of unemployment rates: that there is a ratchet effect, so a short-term rise in unemployment rates tends to persist. An example is the notion that inflationary policy leads to a permanently higher 'natural' rate of unemployment (NAIRU), due to the proposition that inflationary expectations are 'sticky' downward because of wage rigidities and imperfections in the labour market.
Many economists also argue that unemployment itself is subject to hysteresis effects. Unemployment persistence is argued to arise from various factors that include demand deficiency and labour market institutions.
Hysteresis shows in game theory, for example, applied to quality, honesty or corruption. Slightly different initial conditions can lead to opposite results, stable "good" and "bad" equilibria.
Behavioral economists attempt to measure the utility gain from obtaining an item, and the utility loss from losing the same item. With great regularity, the utility loss is greater than the utility gain, meaning that if a person goes through a complete cycle of gaining and losing, the person may be worse off than if he or she had never received the initial gain.

User interface design

The field of user interface design has borrowed the term hysteresis to refer to times when the state of the user interface intentionally lags behind the apparent user input. For example, a menu that was drawn in response to a mouse-over event may remain on-screen for a brief moment after the mouse has moved out of the trigger region and the menu region. This allows the user to move the mouse directly to an item on the menu, even if part of that direct mouse path is outside of both the trigger region and the menu region. For instance, right-clicking on the desktop in most Windows interfaces will create a menu that exhibits this behavior.

Electronics

Sharp hysteresis loop of a Schmitt trigger

Hysteresis can be used to filter signals so that the output reacts slowly by taking recent history into account. For example, a thermostat controlling a heater may turn the heater on when the temperature drops below A degrees, but not turn it off until the temperature rises above B degrees. Thus the on/off output of the thermostat to the heater when the temperature is between A and B depends on the history of the temperature. This prevents rapid switching on and off as the temperature drifts around the set point.
A Schmitt trigger is a simple electronic circuit that also exhibits this property. Often, some amount of hysteresis is intentionally added to an electronic circuit (or digital algorithm) to prevent unwanted rapid switching. This and similar techniques are used to compensate for contact bounce in switches, or noise in an electrical signal.
A latching relay uses a solenoid to actuate a ratcheting motion that keeps the relay closed even if power to the relay is terminated.
Hysteresis is essential to the workings of the memristor, a circuit component which "remembers" changes in the current passing through it by changing its resistance.^[10]

Cell biology

Cells undergoing cell division exhibit hysteresis in that it takes a higher concentration of cyclins to switch them from G2 phase into mitosis than to stay in mitosis once begun.^[11]

Neuroscience

The property by which some neurons do not return to their basal conditions from a stimulated condition immediately after removal of the stimulus is an example of hysteresis. See also: Refractory period.

Respiratory physiology

The transpulmonary pressure vs Volume curve of inhalation is different from the Pressure vs Volume curve of exhalation, the difference being described as hysteresis. Lung volume at any given pressure during inhalation is less than the lung volume at any given pressure during exhalation.^[12]

Applications

Hysteresis represents states, and the characteristic curve shape is sometimes remiscent of a two-value state, also called a bistable state. The hysteresis curve really contains infinitely many states, but a simple application is to let the threshold regions (usually to the left and to the right) represent respectively the on and off states. In this way, the system can be regarded as bistable. Note that even if no external field is applied, the position of the hysteresis curve might change with time: it is not necessarily stationary; i.e. the system may not stay in the exact same state as it had previously. The system might need new energy transfer to be stationary.
The hysteresis effect can be used when connecting complex circuits with the so-called passive matrix addressing. This scheme is praised as a technique that can be used in modern nanoelectronics, electrochrome cells, memory effect, etc. In this scheme, shortcuts are made between adjacent components (see crosstalk) and the hysteresis helps to keep the components in a particular state while the other components change states. That is, one can address all rows at the same time instead of doing each individually.
In economics, hysteresis is used extensively in the area of Labour markets. According to theories based on hysteresis, Economic downturns (Recession) result in an individual becoming unemployed, losing his/her skills (commonly developed 'on the job'), demotivated/disillusioned, and employers may use time spent in unemployment as a screen. In times of an Economic upturn or 'boom', the workers affected will not share in the prosperity, remaining Long-Term Unemployed (>52 weeks). Hysteresis has been put forward as a possible explanation for the poor unemployment performance of many economies in the 1990s. Labour market reform, and/or strong economic growth, may not therefore aid this pool of long-term unemployed, and thus specific targeted training programs are presented as a possible policy solution.
In the field of audio electronics, a noise gate often implements hysteresis intentionally to prevent the gate from "chattering" when signals close to its threshold are applied.
Small vehicle suspensions using rubber (or other elastomers) can achieve the dual function of springing and damping because rubber, unlike metal springs, has pronounced hysteresis and does not return all the absorbed compression energy on the rebound. Mountain bikes have frequently made use of elastomer suspension, as did the original Mini car.

Electret

Electret (formed of elektr- from "electricity" and -et from "magnet") is a dielectric material that has a quasi-permanent electric charge or dipole polarisation. An electret generates internal and external electric fields, and is the electrostatic equivalent of a permanent magnet. Oliver Heaviside coined this term in 1885. Materials with electret properties were, however, already studied since the early 18th century. One particular example is the electrophorus, a device consisting of a slab with electret properties and a separate metal plate. The electrophorus was originally invented by Johan Carl Wilcke in Sweden and again by Alessandro Volta in Italy.

Similarity to capacitors

There is a similarity between electrets and the dielectric layer used in capacitors; the difference is that dielectrics in capacitors possess an induced polarization that is only transient, dependent on the potential applied on the dielectric, while dielectrics with electret properties exhibit quasi-permanent charge storage or dipole polarization in addition. Some materials also display ferroelectricity; i.e. they react to the external fields with a hysteresis of the polarization; ferroelectrics can retain the polarization permanently because they are in thermodynamic equilibrium, and are used in ferroelectric capacitors. Although electrets are only in a metastable state, those fashioned from very low leakage materials can retain excess charge or polarization for many years.An electret microphone is a type of condenser microphone, which eliminates the need for a power supply by using a permanently-charged material.
An electret is a stable dielectric material with a permanently-embedded static electric charge (which, due to the high resistance of the material, will not decay for hundreds of years). The name comes from electrostatic and magnet; drawing analogy to the formation of a magnet by alignment of magnetic domains in a piece of iron. Electrets are commonly made by first melting a suitable dielectric material such as a plastic or wax that contains polar molecules, and then allowing it to re-solidify in a powerful electrostatic field. The polar molecules of the dielectric align themselves to the direction of the electrostatic field, producing a permanent electrostatic 'bias'.

Electret types

There are two types of electrets:

• Real-charge electrets which contain excess charge of one or both polarities, either
  • on the dielectric's surfaces (a surface charge)
  • within the dielectric's volume (a space charge)
• Oriented-dipole electrets contain oriented (aligned) dipoles. Ferroelectric materials are one variant of these.

Cellular space charge electrets with internal bipolar charges at the voids provide a new class of electret materials, that mimic ferroelectrics, hence they are known as ferroelectret. Ferroelectrets display strong piezoelectricity, comparable to ceramic piezoelectric materials.
Some dielectric materials are capable of acting both ways.

Materials

Electret materials are quite common in nature. Quartz and other forms of silicon dioxide, for example, are naturally occurring electrets. Today, most electrets are made from synthetic polymers, e.g. fluoropolymers, polypropylene, polyethyleneterephthalate, etc. Real-charge electrets contain either positive or negative excess charges or both, while oriented-dipole electrets contain oriented dipoles. The quasi-permanent internal or external electric fields created by electrets can be exploited in various applications.

Manufacture

Bulk electrets can be prepared by cooling a suitable dielectric material within a strong electric field, after heating it above its melting temperature. The field repositions the charge carriers or aligns the dipoles within the material. When the material cools, solidification freezes them in position. Materials used for electrets are usually waxes, polymers or resins. One of the earliest recipes consists of 45% carnauba wax, 45% white rosin, and 10% white beeswax, melted, mixed together, and left to cool in a static electric field of several kilovolts/cm. The thermo-dielectric effect, related to this process, was first described by the Brazilian researcher Joaquim Costa Ribeiro.
Electrets can also be manufactured by embedding excess negative charge within a dielectric using a particle accelerator, or by stranding charges on, or near, the surface using high voltage corona discharges, a process called corona charging. Excess charge within an electret decays exponentially. The decay constant is a function of the material's relative dielectric constant and its bulk resistivity. Materials with extremely high resistivity, such as Teflon, may retain excess charge for many hundreds of years. Most commercially produced electrets are based on fluoropolymers (eg. amorphous Teflon) machined to thin films.

Applications

Electret materials have recently found commercial and technical interest. For example, they are used in electret microphones and in copy machines. They are also used in some types of air filters, for electrostatic collection of dust particles, and in electret ion chambers for measuring ionizing radiation or radon. See U.S. Patent 6,969,484 for "Manufacturing Method and Device for Electret Processed Product"

Electrostatics

Electrostatics is the branch of science that deals with the phenomena arising from stationary or slow-moving electric charges.
Since classical antiquity it was known that some materials such as amber attract light particles after rubbing. The Greek word for amber, ήλεκτρον (electron), 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.

Fundamental concepts

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 is directly proportional to the product of the magnitudes of each charge and inversely proportional to the square of the distance between the charges.Q₁ and Q₂:

where ε₀ is the electric constant, a defined value:

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

The 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:

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:

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". The constant of proportionality is the permittivity of free space.
Mathematically, Gauss's law takes the form of an integral equation:

Alternatively, in differential form, the equation becomes

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. Where is Vacuum permittivity.

Laplace's equation

In the absence of unpaired electric charge, the equation becomes

which is Laplace's equation.

The 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:

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.

Electrostatic potential

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, E, 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.

Triboelectric series

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.

Electrostatic generators

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.

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).

Charge 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.

'Static' electricity

Lightning over Oradea in Romania

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.

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 pico siemens/m, where pico siemens/m is a measure of electrical conductivity), are called accumulators. Fluids having conductivities above 50 pico siemens/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 pico siemens/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 pico siemens /cm 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.
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 pico siemens/m, bonding and earthing are not adequate for charge dissipation, and anti-static additives may be required.
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

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