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**Electrostatics** is a branch of physics that studies electric charges at rest (static electricity).

- Coulomb's law
- Electric field
- Gauss' law
- Poisson and Laplace equations
- Electrostatic approximation
- Electrostatic potential
- Electrostatic energy
- Electrostatic pressure
- Triboelectric series
- Electrostatic generators
- Charge neutralization
- Electrostatic induction
- Static electricity
- Static electricity and chemical industry
- Electrostatic induction in commercial applications
- See also
- Footnotes
- References
- Further reading
- External links

Since classical physics, it has been known that some materials, such as amber, attract lightweight particles after rubbing. The Greek word for amber, or * electron*, was thus 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, some electrostatic forces such as the one between an electron and a proton, that together make up a hydrogen atom, is about 36 orders of magnitude stronger than the gravitational force acting between them.

There are many examples of electrostatic phenomena, from those as simple as the attraction of plastic wrap to one's hand after it is removed from a package, to the apparently spontaneous explosion of grain silos, the damage of electronic components during manufacturing, and photocopier & laser printer operation. 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 noticed only when at least one of the surfaces has a high resistance to electrical flow, because the charges that transfer are 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. The familiar phenomenon of a static "shock" is caused by the neutralization of charge built up in the body from contact with insulated surfaces.

Coulomb's law states that:

'The magnitude of the electrostatic force of attraction or repulsion between two point charges is directly proportional to the product of the magnitudes of charges and inversely proportional to the square of the distance between them.'

The force is along the straight line joining them. If the two charges have the same sign, the electrostatic force between them is repulsive; if they have different signs, the force between them is attractive.

If is the distance (in meters) between two charges, then the force (in newtons) between two point charges and (in coulombs) is:

where ε_{0} is the vacuum permittivity, or permittivity of free space:^{ [5] }

The SI units of ε_{0} are equivalently A ^{2} s ^{4} kg^{−1}m^{−3} or C ^{2} N ^{−1}m^{−2} or F m^{−1}. Coulomb's constant is:

A single proton has a charge of *e*, and the electron has a charge of −*e*, where,

These physical constants (ε_{0}, k_{0}, e) are currently defined so that *e* is exactly defined, and ε_{0} and k_{0} are measured quantities.

The electric field, , in units of newtons per coulomb or volts per meter, is a vector field that can be defined everywhere, except at the location of point charges (where it diverges to infinity).^{ [6] } It is defined as the electrostatic force in newtons on a hypothetical small test charge at the point due to Coulomb's Law, divided by the magnitude of the charge in coulombs

Electric field lines are useful for visualizing the electric field. Field lines begin on positive charge and terminate on negative charge. They are parallel to the direction of the electric field at each point, and the density of these field lines is a measure of the magnitude of the electric field at any given point.

Consider a collection of particles of charge , located at points (called *source points*), the electric field at (called the *field point*) is:^{ [6] }

where is the displacement vector from a *source point* to the *field point*, and is a unit vector that indicates the direction of the field. For a single point charge at the origin, the magnitude of this electric field is and points away from that charge if it is positive. The fact that the force (and hence the field) can be calculated by summing over all the contributions due to individual source particles is an example of the superposition principle. The electric field produced by a distribution of charges is given by the volume charge density and can be obtained by converting this sum into a triple integral:

Gauss' law states that "the total electric flux through any closed surface in free space of any shape drawn in an electric field is proportional to the total electric charge enclosed by the surface." Mathematically, Gauss's law takes the form of an integral equation:

where is a volume element. If the charge is distributed over a surface or along a line, replace by or . The divergence theorem allows Gauss's Law to be written in differential form:

where is the divergence operator.

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. In the absence of unpaired electric charge, the equation becomes Laplace's equation:

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. Electrostatics and magnetostatics can both be seen as Galilean limits for electromagnetism.^{ [7] }^{[ verification needed ]}

As 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 electric potential to regions of low electric potential, expressed mathematically as

The gradient theorem can be used to establish that the electrostatic potential is the amount of work per unit charge required to move a charge from point to point with the following line integral:

From these equations, we see that the electric potential is constant in any region for which the electric field vanishes (such as occurs inside a conducting object).

A single test particle's potential energy, , can be calculated from a line integral of the work, . We integrate from a point at infinity, and assume a collection of particles of charge , are already situated at the points . This potential energy (in Joules) is:

where is the distance of each charge from the test charge , which situated at the point , and is the electric potential that would be at if the test charge were not present. If only two charges are present, the potential energy is . The total electric potential energy due a collection of *N* charges is calculating by assembling these particles one at a time:

where the following sum from, *j* = 1 to *N*, excludes *i* = *j*:

This electric potential, is what would be measured at if the charge were missing. This formula obviously excludes the (infinite) energy that would be required to assemble each point charge from a disperse cloud of charge. The sum over charges can be converted into an integral over charge density using the prescription :

This second expression for electrostatic energy uses the fact that the electric field is the negative gradient of the electric potential, as well as vector calculus identities in a way that resembles integration by parts. These two integrals for electric field energy seem to indicate two mutually exclusive formulas for electrostatic energy density, namely and ; they yield equal values for the total electrostatic energy only if both are integrated over all space.

On a conductor, a surface charge will experience a force in the presence of an electric field. This force is the average of the discontinuous electric field at the surface charge. This average in terms of the field just outside the surface amounts to:

This pressure tends to draw the conductor into the field, regardless of the sign of the surface charge.

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 humans. Other examples of materials that can acquire a significant charge when rubbed together include glass rubbed with silk, and hard rubber rubbed with fur.

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 the amount of adhesive contact between the two surfaces. Usually insulators, i.e., 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 rarely generate charge imbalance, except 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. Electrostatic generators, devices which produce very high voltage at very low current and used for classroom physics demonstrations, rely on this effect.

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:
*Wimshurst machine*, and*Van de Graaff generator*.

The most familiar natural electrostatic phenomena, often regarded as an occasional annoyance in seasons of low humidity, is Static electricity. Static electricity is generally harmless, but it can be destructive and harmful in some situations (e.g. electronics manufacturing). When working in direct contact with integrated circuit electronics (especially delicate MOSFETs). In the presence of flammable gas, care must be taken to avoid accumulating and suddenly discharging a static charge (see Electrostatic discharge).

Electrostatic induction, discovered by British scientist John Canton in 1753 and Swedish professor Johan Carl Wilcke in 1762^{ [8] }^{ [9] }^{ [10] } is a redistribution of charges in an object caused by the electric field of a nearby charge. For example, if a positively charged object is brought near an uncharged metal object, the mobile negatively-charged electrons in the metal will be attracted by the external charge, and move to the side of the metal facing it, creating a negative charge on the surface. When the electrons move out of an area they leave a positive charge due to the metal atoms' nuclei, so the side of the metal object facing away from the charge acquires a positive charge. These *induced charges* disappear when the external charge is removed. Induction is also responsible for the attraction of light objects, such as balloons, paper scraps and foam packing peanuts to static charges. The surface charges induced in conductive objects exactly cancel external electric fields inside the conductor, so there is no electric field inside a metal object. This is the basis for the electric field shielding action of a Faraday cage. Since the electric field is the gradient of the voltage, electrostatic induction is also responsible for making the electric potential (voltage) constant throughout a conductive object.

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 coating processes used in manufacturing. Static electricity is a build-up of electric charges on two objects that have become separated from each other. Small electrical components can be damaged by static electricity, and component manufacturers use a number of antistatic devices to avoid this.

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), 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.

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.

- BS PD CLC/TR 50404:2003 Code of Practice for Control of Undesirable Static Electricity
- NFPA 77 (2007) Recommended Practice on Static Electricity
- API RP 2003 (1998) Protection Against Ignitions Arising Out of Static, Lightning, and Stray Currents

Electrostatic induction was used in the past to build high-voltage generators known as influence machines. The main component that emerged in these times is the capacitor. Electrostatic induction is also used for electro-mechanic precipitation or projection. In such technologies, charged particles of small sizes are collected or deposited intentionally on surfaces. Applications range from electrostatic precipitator to electrostatic coating and inkjet printing. Recently a new wireless power transfer technology has been based on electrostatic induction between oscillating distant dipoles.

- ↑ Ling, Samuel J.; Moebs, William; Sanny, Jeff (2019).
*University Physics, Vol. 2*. OpenStax. ISBN 9781947172210. Ch.30: Conductors, Insulators, and Charging by Induction - ↑ Bloomfield, Louis A. (2015).
*How Things Work: The Physics of Everyday Life*. John Wiley and Sons. p. 270. ISBN 9781119013846. - ↑ "Polarization".
*Static Electricity - Lesson 1 - Basic Terminology and Concepts*. The Physics Classroom. 2020. Retrieved 18 June 2021. - ↑ Thompson, Xochitl Zamora (2004). "Charge It! All About Electrical Attraction and Repulsion".
*Teach Engineering: Stem curriculum for K-12*. University of Colorado. Retrieved 18 June 2021. - ↑ Matthew Sadiku (2009).
*Elements of electromagnetics*. p. 104. ISBN 9780195387759. - 1 2 Purcell, Edward M. (2013).
*Electricity and Magnetism*. Cambridge University Press. pp. 16–18. ISBN 978-1107014022. - ↑ Heras, J. A. (2010). "The Galilean limits of Maxwell's equations".
*American Journal of Physics*.**78**(10): 1048–1055. arXiv: 1012.1068 . Bibcode:2010AmJPh..78.1048H. doi:10.1119/1.3442798. S2CID 118443242. - ↑ "Electricity".
*Encyclopaedia Britannica, 11th Ed*.**9**. The Encyclopaedia Britannica Co. 1910. p. 181. Retrieved 2008-06-23. - ↑ Heilbron, J. L. (1979).
*Electricity in the 17th and 18th Centuries: A Study of Early Modern Physics*. Univ. of California Press. ISBN 0520034783. - ↑ Sarkar, T. K.; Mailloux, Robert; Oliner, Arthur A., Ed. (2006).
*History of Wireless*. John Wiley and Sons. p. 9. ISBN 0471783013.

An **electric field** is the physical field that surrounds electrically-charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field for a system of charged particles. Electric fields originate from electric charges, or from time-varying magnetic fields. Electric fields and magnetic fields are both manifestations of the electromagnetic force, one of the four fundamental forces of nature.

In electromagnetism, the **absolute permittivity**, often simply called **permittivity** and denoted by the Greek letter *ε* (epsilon), is a measure of the electric polarizability of a dielectric. A material with high permittivity polarizes more in response to an applied electric field than a material with low permittivity, thereby storing more energy in the material. In electrostatics, the permittivity plays an important role in determining the capacitance of a capacitor.

The **electric potential** is the amount of work energy needed to move a unit of electric charge from a reference point to the specific point in an electric field with negligible acceleration of the test charge to avoid producing kinetic energy or radiation by test charge. Typically, the reference point is the Earth or a point at infinity, although any point can be used. More precisely it is the energy per unit charge for a small test charge that does not disturb significantly the field and the charge distribution producing the field under consideration.

In physics and electromagnetism, **Gauss's law**, also known as **Gauss's flux theorem**, is a law relating the distribution of electric charge to the resulting electric field. In its integral form, it states that the flux of the electric field out of an arbitrary closed surface is proportional to the electric charge enclosed by the surface, irrespective of how that charge is distributed. Even though the law alone is insufficient to determine the electric field across a surface enclosing any charge distribution, this may be possible in cases where symmetry mandates uniformity of the field. Where no such symmetry exists, Gauss's law can be used in its differential form, which states that the divergence of the electric field is proportional to the local density of charge.

**Synchrotron radiation** is the electromagnetic radiation emitted when charged particles are accelerated radially, e.g., when they are subject to an acceleration perpendicular to their velocity. It is produced, for example, in synchrotrons using bending magnets, undulators and/or wigglers. If the particle is non-relativistic, the emission is called cyclotron emission. If the particles are relativistic, sometimes referred to as ultrarelativistic, the emission is called synchrotron emission. Synchrotron radiation may be achieved artificially in synchrotrons or storage rings, or naturally by fast electrons moving through magnetic fields. The radiation produced in this way has a characteristic polarization and the frequencies generated can range over the entire electromagnetic spectrum, which is also called continuum radiation.

In physics, **screening** is the damping of electric fields caused by the presence of mobile charge carriers. It is an important part of the behavior of charge-carrying fluids, such as ionized gases, electrolytes, and charge carriers in electronic conductors . In a fluid, with a given permittivity *ε*, composed of electrically charged constituent particles, each pair of particles interact through the Coulomb force as

**Poisson's equation** is an elliptic partial differential equation of broad utility in theoretical physics. For example, the solution to Poisson's equation is the potential field caused by a given electric charge or mass density distribution; with the potential field known, one can then calculate electrostatic or gravitational (force) field. It is a generalization of Laplace's equation, which is also frequently seen in physics. The equation is named after French mathematician and physicist Siméon Denis Poisson.

In the calculus of variations, a field of mathematical analysis, the **functional derivative** relates a change in a Functional to a change in a function on which the functional depends.

**Classical electromagnetism** or **classical electrodynamics** is a branch of theoretical physics that studies the interactions between electric charges and currents using an extension of the classical Newtonian model. The theory provides a description of electromagnetic phenomena whenever the relevant length scales and field strengths are large enough that quantum mechanical effects are negligible. For small distances and low field strengths, such interactions are better described by quantum electrodynamics.

In electromagnetism, **displacement current density** is the quantity ∂* D*/∂

In plasmas and electrolytes, the **Debye length**, is a measure of a charge carrier's net electrostatic effect in a solution and how far its electrostatic effect persists. A **Debye sphere** is a volume whose radius is the Debye length. With each Debye length, charges are increasingly electrically screened. Every Debye‐length , the electric potential will decrease in magnitude by 1/e. Debye length is an important parameter in plasma physics, electrolytes, and colloids. The corresponding Debye screening wave vector for particles of density , charge at a temperature is given by in Gaussian units. Expressions in MKS units will be given below. The analogous quantities at very low temperatures are known as the Thomas–Fermi length and the Thomas–Fermi wave vector. They are of interest in describing the behaviour of electrons in metals at room temperature.

**Electric potential energy**, is a potential energy that results from conservative Coulomb forces and is associated with the configuration of a particular set of point charges within a defined system. An *object* may have electric potential energy by virtue of two key elements: its own electric charge and its relative position to other electrically charged *objects*.

The chemists Peter Debye and Erich Hückel noticed that solutions that contain ionic solutes do not behave ideally even at very low concentrations. So, while the concentration of the solutes is fundamental to the calculation of the dynamics of a solution, they theorized that an extra factor that they termed gamma is necessary to the calculation of the activity coefficients of the solution. Hence they developed the **Debye–Hückel equation** and **Debye–Hückel limiting law**. The activity is only proportional to the concentration and is altered by a factor known as the activity coefficient . This factor takes into account the interaction energy of ions in solution.

A **Gaussian surface** is a closed surface in three-dimensional space through which the flux of a vector field is calculated; usually the gravitational field, the electric field, or magnetic field. It is an arbitrary closed surface *S* = ∂*V* used in conjunction with Gauss's law for the corresponding field by performing a surface integral, in order to calculate the total amount of the source quantity enclosed; e.g., amount of gravitational mass as the source of the gravitational field or amount of electric charge as the source of the electrostatic field, or vice versa: calculate the fields for the source distribution.

The **method of image charges** is a basic problem-solving tool in electrostatics. The name originates from the replacement of certain elements in the original layout with imaginary charges, which replicates the boundary conditions of the problem.

**Spherical multipole moments** are the coefficients in a series expansion of a potential that varies inversely with the distance R to a source, *i.e.*, as 1/*R*. Examples of such potentials are the electric potential, the magnetic potential and the gravitational potential.

In general relativity, a point mass deflects a light ray with impact parameter by an angle approximately equal to

In mathematics, **potential flow around a circular cylinder** is a classical solution for the flow of an inviscid, incompressible fluid around a cylinder that is transverse to the flow. Far from the cylinder, the flow is unidirectional and uniform. The flow has no vorticity and thus the velocity field is irrotational and can be modeled as a potential flow. Unlike a real fluid, this solution indicates a net zero drag on the body, a result known as d'Alembert's paradox.

**Coulomb's law**, or **Coulomb's inverse-square law**, is an experimental law of physics that quantifies the amount of force between two stationary, electrically charged particles. The electric force between charged bodies at rest is conventionally called *electrostatic force* or **Coulomb force**. Although the law was known earlier, it was first published in 1785 by French physicist Charles-Augustin de Coulomb, hence the name. Coulomb's law was essential to the development of the theory of electromagnetism, maybe even its starting point, as it made it possible to discuss the quantity of electric charge in a meaningful way.

The **electric dipole moment** is a measure of the separation of positive and negative electrical charges within a system, that is, a measure of the system's overall polarity. The SI units for electric dipole moment are coulomb-meter (C⋅m); however, a commonly used unit in atomic physics and chemistry is the debye (D).

This article includes a list of general references, but it remains largely unverified because it lacks sufficient corresponding inline citations .(September 2019) |

- Faraday, Michael (1839).
*Experimental Researches in Electricity*. London: Royal Inst. - Michael Faraday.
*Experimental Researches in Electricity, Volume 1*at Project Gutenberg - Halliday, David; Robert Resnick; Kenneth S. Krane (1992).
*Physics*. New York: John Wiley & Sons. ISBN 0-471-80457-6. - Griffiths, David J. (1999).
*Introduction to Electrodynamics*. Upper Saddle River, NJ: Prentice Hall. ISBN 0-13-805326-X. - Hermann A. Haus; James R. Melcher (1989).
*Electromagnetic Fields and Energy*. Englewood Cliffs, NJ: Prentice-Hall. ISBN 0-13-249020-X.

- Essays

- William J. Beaty (1997) "
*Humans and sparks: The Cause, Stopping the Pain, and 'Electric People*"

- Books

- William Cecil Dampier (1905)
*The Theory of Experimental Electricity*, Cambridge University Press, (Cambridge physical series). xi, 334 p. illus., diagrs. 23 cm. LCCN 05040419 //r33 - William Thomson Kelvin (1872) Reprint of Papers on Electrostatics and Magnetism By William Thomson Kelvin, Macmillan
- Alexander McAulay (1893)
*The Utility of Quaternions in Physics*, Electrostatics—General Problem. Macmillan - Alexander Russell (1904)
*A Treatise on the Theory of Alternating Currents*, Cambridge University Press, Second edition, 1914, volume 1. Second edition, 1916, volume 2 via Internet Archive

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