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Electrostatics is a branch of
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
that studies
electric charge Electric charge is the physical property of matter that causes charged matter to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons res ...
s at rest ( static electricity). Since classical times, it has been known that some materials, such as
amber Amber is fossilized tree resin that has been appreciated for its color and natural beauty since Neolithic times. Much valued from antiquity to the present as a gemstone, amber is made into a variety of decorative objects."Amber" (2004). In M ...
, attract lightweight particles after
rubbing A rubbing ('' frottage'') is a reproduction of the texture of a surface created by placing a piece of paper or similar material over the subject and then rubbing the paper with something to deposit marks, most commonly charcoal or pencil but ...
. The Greek word for amber, (), was thus the source of the word '
electricity Electricity is the set of physical phenomena associated with the presence and motion of matter that has a property of electric charge. Electricity is related to magnetism, both being part of the phenomenon of electromagnetism, as describe ...
'. Electrostatic phenomena arise from the forces that electric charges exert on each other. Such forces are described by
Coulomb's law Coulomb's inverse-square law, or simply Coulomb's 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 convention ...
. Even though electrostatically induced forces seem to be rather weak, some electrostatic forces are relatively large. The force between an
electron The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have n ...
and a proton, which together make up a
hydrogen Hydrogen is the chemical element with the symbol H and atomic number 1. Hydrogen is the lightest element. At standard conditions hydrogen is a gas of diatomic molecules having the formula . It is colorless, odorless, tasteless, non-to ...
atom Every atom is composed of a nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and a number of neutrons. Only the most common variety of hydrogen has no neutrons. Every solid, liquid, gas, a ...
, 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 A photocopier (also called copier or copy machine, and formerly Xerox machine, the generic trademark) is a machine that makes copies of documents and other visual images onto paper or plastic film quickly and cheaply. Most modern 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 Discharge may refer to Expel or let go * Discharge, the act of firing a gun * Discharge, or termination of employment, the end of an employee's duration with an employer * Military discharge, the release of a member of the armed forces from ser ...
. 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

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 r is the distance (in
meters The metre (British spelling) or meter (American spelling; see spelling differences) (from the French unit , from the Greek noun , "measure"), symbol m, is the primary unit of length in the International System of Units (SI), though its pr ...
) between two charges, then the force (in newtons) between two point charges q and Q (in coulombs) is: :F = \frac\frac= k_0\frac\, , where ''ε''0 is the vacuum permittivity, or permittivity of free space: :\varepsilon_0 \approx \mathrm. The SI units of ''ε''0 are equivalently A2s4 ⋅kg−1⋅m−3 or C2N−1⋅m−2 or F⋅m−1. The Coulomb constant is: : k_\text = \frac\approx \mathrm. A single proton has a charge of ''e'', and the
electron The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have n ...
has a charge of −''e'', where, : e = \mathrm. 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.


Electric field

The electric field, \vec, 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). It is defined as the electrostatic force \vec\, in newtons on a hypothetical small
test charge In physical theories, a test particle, or test charge, is an idealized model of an object whose physical properties (usually mass, charge, or size) are assumed to be negligible except for the property being studied, which is considered to be in ...
at the point due to
Coulomb's Law Coulomb's inverse-square law, or simply Coulomb's 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 convention ...
, divided by the magnitude of the charge q\, in coulombs :\vec =
Electric field lines Electricity is the set of physical phenomena associated with the presence and motion of matter that has a property of electric charge. Electricity is related to magnetism, both being part of the phenomenon of electromagnetism, as described by ...
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 N particles of charge Q_i, located at points \vec r_i (called ''source points''), the electric field at \vec r (called the ''field point'') is: : \vec(\vec r) =\frac\sum_^N \frac , where \vec\mathcal R_i = \vec r - \vec r_i , is the displacement vector from a ''source point'' \vec r_i to the ''field point'' \vec r, and \widehat\mathcal R_i = \vec\mathcal R_i / \left \, \vec\mathcal R_i \right \, 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 E =k_\textQ/\mathcal R^2, 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 In electromagnetism, charge density is the amount of electric charge per unit length, surface area, or volume. Volume charge density (symbolized by the Greek letter ρ) is the quantity of charge per unit volume, measured in the SI system i ...
\rho (\vec r) and can be obtained by converting this sum into a triple integral: :\vec(\vec r)= \frac \iiint \frac \rho (\vec r \,') \, \mathrm^3 r\,'


Gauss' law

Gauss' law In physics and electromagnetism, Gauss's law, also known as Gauss's flux theorem, (or sometimes simply called Gauss's theorem) is a law relating the distribution of electric charge to the resulting electric field. In its integral form, it s ...
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 Electric charge is the physical property of matter that causes charged matter to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons res ...
enclosed by the surface." Mathematically, Gauss's law takes the form of an integral equation: :\oint_S\vec \cdot\mathrm\vec = \frac\,Q_\text = \int_V\cdot \mathrm^3 r, where \mathrm^3 r =\mathrmx \ \mathrmy \ \mathrmz is a volume element. If the charge is distributed over a surface or along a line, replace \rho\,\mathrm^3r by \sigma \, \mathrmA or \lambda \, \mathrm\ell. The divergence theorem allows Gauss's Law to be written in differential form: :\vec\cdot\vec = . where \vec \cdot is the divergence operator.


Poisson and Laplace equations

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: :^2 \phi = - . This relationship is a form of
Poisson's equation 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 ...
. In the absence of unpaired electric charge, the equation becomes Laplace's equation: :^2 \phi = 0,


Electrostatic approximation

The validity of the electrostatic approximation rests on the assumption that the electric field is irrotational: :\vec\times\vec = 0. From Faraday's law, this assumption implies the absence or near-absence of time-varying magnetic fields: : = 0. 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.


Electrostatic potential

As the electric field is irrotational, it is possible to express the electric field as the
gradient In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...
of a scalar function, \phi, called the electrostatic potential (also known as the voltage). An electric field, E, points from regions of high electric potential to regions of low electric potential, expressed mathematically as :\vec = -\vec\phi. 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 a to point b with the following
line integral In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. The terms ''path integral'', ''curve integral'', and ''curvilinear integral'' are also used; '' contour integral'' is used as well, ...
: : -\int_a^b = \phi (\vec b) -\phi(\vec a). 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).


Electrostatic energy

A test particle's potential energy, U_\mathrm^, can be calculated from a
line integral In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. The terms ''path integral'', ''curve integral'', and ''curvilinear integral'' are also used; '' contour integral'' is used as well, ...
of the work, q_n\vec E\cdot\mathrm d\vec\ell. We integrate from a point at infinity, and assume a collection of N particles of charge Q_n, are already situated at the points \vec r_i. This potential energy (in
Joule The joule ( , ; symbol: J) is the unit of energy in the International System of Units (SI). It is equal to the amount of work done when a force of 1 newton displaces a mass through a distance of 1 metre in the direction of the force appli ...
s) is: : U_\mathrm^=q\phi(\vec r)=\frac\sum_^N \frac where \vec\mathcal = \vec r - \vec r_i is the distance of each charge Q_i from the
test charge In physical theories, a test particle, or test charge, is an idealized model of an object whose physical properties (usually mass, charge, or size) are assumed to be negligible except for the property being studied, which is considered to be in ...
q, which situated at the point \vec r, and \phi(\vec r) is the electric potential that would be at \vec r if the
test charge In physical theories, a test particle, or test charge, is an idealized model of an object whose physical properties (usually mass, charge, or size) are assumed to be negligible except for the property being studied, which is considered to be in ...
were not present. If only two charges are present, the potential energy is k_\textQ_1Q_2/r. The total electric potential energy due a collection of ''N'' charges is calculating by assembling these particles one at a time: :U_\mathrm^ = \frac\sum_^N Q_j \sum_^ \frac= \frac\sum_^N Q_i\phi_i , where the following sum from, ''j'' = 1 to ''N'', excludes ''i'' = ''j'': :\phi_i = \frac \sum_^N \frac. This electric potential, \phi_i is what would be measured at \vec r_i if the charge Q_i 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 \sum (\cdots) \rightarrow \int(\cdots)\rho \, \mathrm d^3r: :U_\mathrm^ = \frac \int\rho(\vec)\phi(\vec) \, \mathrm^3 r = \frac \int \left, \^2 \, \mathrm^3 r, This second expression for electrostatic energy uses the fact that the electric field is the negative
gradient In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...
of the electric potential, as well as vector calculus identities in a way that resembles
integration by parts In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivat ...
. These two integrals for electric field energy seem to indicate two mutually exclusive formulas for electrostatic energy density, namely \frac\rho\phi and \frac\varepsilon_0 E^2; they yield equal values for the total electrostatic energy only if both are integrated over all space.


Electrostatic pressure

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: : P = \frac E^2, This pressure tends to draw the conductor into the field, regardless of the sign of the surface charge.


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 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.


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 Contact electrification is a phrase that describes a phenomenon whereby surfaces become electrically charged, via a number of possible mechanisms, when two or more objects come within close proximity of one another. When two objects are "touched" ...
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 Plastics are a wide range of synthetic or semi-synthetic materials that use polymers as a main ingredient. Their plasticity makes it possible for plastics to be moulded, extruded or pressed into solid objects of various shapes. This adapta ...
,
glass Glass is a non- crystalline, often transparent, amorphous solid that has widespread practical, technological, and decorative use in, for example, window panes, tableware, and optics. Glass is most often formed by rapid cooling (quenchin ...
, 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 A corona discharge is an electrical discharge caused by the ionization of a fluid such as air surrounding a conductor carrying a high voltage. It represents a local region where the air (or other fluid) has undergone electrical breakdown ...
, or other phenomena. Both phenomena can exist simultaneously in the same system. :See also: '' Wimshurst machine'', and '' Van de Graaff generator''.


Charge neutralization

The most familiar natural electrostatic phenomenon, 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

Electrostatic induction, discovered by British scientist
John Canton John Canton FRS (31 July 1718 – 22 March 1772) was a British physicist. He was born in Middle Street Stroud, Gloucestershire, to a weaver, John Canton (b. 1687) and Esther (née Davis). As a schoolboy, he became the first person to determi ...
in 1753 and Swedish professor
Johan Carl Wilcke Johan Carl Wilcke was a Swedish physicist. Biography Wilcke was born in Wismar, son of a clergyman who in 1739 was appointed second pastor of the German Church in Stockholm. He went to the German school in Stockholm and enrolled at the Univers ...
in 1762 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
electron The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have n ...
s 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 In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...
of the voltage, electrostatic induction is also responsible for making the
electric potential The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as the amount of work energy needed to move a unit of electric charge from a reference point to the specific point in ...
( voltage) constant throughout a conductive object.


Static electricity

Before the year 1832, when
Michael Faraday Michael Faraday (; 22 September 1791 – 25 August 1867) was an English scientist who contributed to the study of electromagnetism and electrochemistry. His main discoveries include the principles underlying electromagnetic inducti ...
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 Xerography is a dry photocopying technique. Originally called electrophotography, it was renamed xerography—from the roots el, ξηρός, label=none ''xeros'', meaning "dry" and -γραφία ''-graphia'', meaning "writing"—to emphasiz ...
, 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 device An antistatic device is any device that reduces, dampens, or otherwise inhibits electrostatic discharge, or ESD, which is the buildup or discharge of static electricity. ESD can damage electrical components such as computer hard drives, and even i ...
s 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 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 A resistor–capacitor circuit (RC circuit), or RC filter or RC network, is an electric circuit composed of resistors and capacitors. It may be driven by a voltage or current source and these will produce different responses. A first order RC ...
. 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 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.


Applicable standards

*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 in commercial applications

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 A capacitor is a device that stores electrical energy in an electric field by virtue of accumulating electric charges on two close surfaces insulated from each other. It is a passive electronic component with two terminals. The effect of ...
. 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 Electrostatic coating is a manufacturing process that employs charged particles to more efficiently paint a workpiece. Paint, in the form of either powdered particles or atomized liquid, is initially projected towards a conductive workpiece using no ...
and inkjet printing. Electrostatic actuators have recently been attracting interest in the soft robotics research area. Electrostatic actuators can be employed as clutches for wearable devices which can exhibit mechanical impedance tuning and improved energy efficiency. Other relevant applications include but not limited to multimode hydraulically amplified electrostatic actuators for wearable haptics and robots driven by electrostatic actuator.


See also

*
Electromagnetism In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions o ...
* Electronegativity * Electrostatic discharge * Electrostatic separator * Electrostatic voltmeter * Ionic bond * Permittivity and relative permittivity * Quantisation of charge


Footnotes


References

* * * *


Further reading

;Essays *William J. Beaty (1997), "
Humans and sparks: The Cause, Stopping the Pain, and 'Electric People
'". ;Books *
William Cecil Dampier Sir William Cecil Dampier FRS (born William Cecil Dampier Whetham) (27 December 1867 – 11 December 1952) was a British scientist, agriculturist, and science historian who developed a method of extracting lactose (milk sugar) from whey. ...
(1905), ''The Theory of Experimental Electricity'',
Cambridge University Press Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by King Henry VIII in 1534, it is the oldest university press in the world. It is also the King's Printer. Cambridge University Pr ...
, (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 1Second edition, 1916, volume 2
via
Internet Archive The Internet Archive is an American digital library with the stated mission of "universal access to all knowledge". It provides free public access to collections of digitized materials, including websites, software applications/games, music, ...
.


External links

*
The Feynman Lectures on Physics Vol. II Ch. 4: ElectrostaticsIntroduction to Electrostatics
Point charges can be treated as a distribution using the Dirac delta function {{Authority control