# Electrostatics

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## Overview

For a less technical introduction see Static electricity

Electrostatics is the branch of science that deals with the phenomena arising from what seems to be stationary electric charges. Since [ancient history it is known that some materials attract light particles after rubbing. The greek word for amber, ήλεκτρον (electron), gave name for many areas of natural science. Electrostatic phenomena arise from the forces that electric charges carry out on each other. Such forces are described by Coulomb's law. Electrostatic phenomena include such as simple as the attraction of plastic wrap to your hand after you remove it from a package to 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.

The force F imposed by a charge Q on a probe q is proportional to the charge of the probe. That is, it can be described by the equation F=q·E, what defines the electric field E.

### The electrostatic approximation

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

${\vec {\nabla }}\times {\vec {E}}=0.$ From Faraday's law, this assumption implies the absence or near-absence of time-varying magnetic fields:

${\partial {\vec {B}} \over \partial t}=0.$ 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.

## 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

${\vec {E}}=-{\vec {\nabla }}\phi .$ ## Fundamental concepts

### Coulomb's law

Electric Potential is the amount of work done per unit charge, in bringing an unit positive charge from infinity to that point. 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.$Q_{1}$ and $Q_{2}$ :

${\vec {F}}={\frac {Q_{1}Q_{2}}{4\pi \varepsilon _{0}r^{2}}}{\hat {r}}\ ,$ where ε0 is the electric constant, a defined value:

$\varepsilon _{0}\ {\stackrel {\mathrm {def} }{=}}\ {\frac {1}{\mu _{0}{c_{0}}^{2}}}=8.854\ 187\ 817\ \times 10^{-12}$ 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) is defined as the force (in newtons) per unit charge (in coulombs). 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

${\vec {E}}={\frac {Q}{4\pi \varepsilon _{0}r^{2}}}{\hat {r}}.$ ### 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:

$\oint _{S}\varepsilon _{0}{\vec {E}}\cdot \mathrm {d} {\vec {A}}=\int _{V}\rho \cdot \mathrm {d} V.$ Alternatively, in differential form, the equation becomes

${\vec {\nabla }}\cdot \varepsilon _{0}{\vec {E}}=\rho .$ ### 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:

${\nabla }^{2}\phi =-{\rho \over \varepsilon _{0}}.$ This relationship is a form of Poisson's equation. Where ${\varepsilon _{0}}$ is Vacuum permittivity.

### Laplace's equation

In the absence of unpaired electric charge, the equation becomes

${\nabla }^{2}\phi =0,$ which is Laplace's equation.

## Triboelectric series

The triboelectric effect is a type of contact electrification in which certain materials become electrically charged when coming into contact with another, different, material, and are then separated. The polarity and strength of the charges produced differ according to the materials, surface roughness, temperature, strain, and other properties. It is therefore not very predictable, and only broad generalizations can be made. Amber, for example, can acquire an electric charge by friction with a material like wool. This property, first recorded by Thales of Miletus, suggested the word "electricity", from the Greek word for amber, èlectròn. 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

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 manufactures 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 within 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/cm, where pico siemens/cm is a measure of electrical conductivity), are called accumulators. Fluids having conductivities above 50 pico siemens/cm 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/cm 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 (203.2 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/cm, 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 