Electrostatics

clinging to a cat's fur due to static electricity. The cat's fur becomes charged due to the triboelectricity. The electric field of the charged fur causes polarization of the molecules of the foam due to electrostatic induction, resulting in a slight attraction of the light plastic pieces to the fur.

Ch.30: Conductors, Insulators, and Charging by Induction

(2025). 9781119013846, John Wiley and Sons. . ISBN 9781119013846

This effect is also the cause of static cling in clothes.]]

Electrostatics is a branch of physics that studies slow-moving or stationary on macroscopic objects where quantum effects can be neglected. Under these circumstances the electric field, electric potential, and the charge density are related without complications from magnetic effects.

Since classical antiquity, it has been known that some materials, such as amber, attract lightweight particles after rubbing. The Greek language word (ἤλεκτρον), meaning 'amber', was thus the root of the word electricity. Electrostatic phenomena arise from the that electric charges exert on each other. Such forces are described by Coulomb's law.

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 and laser printing operation.

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Coulomb's law Coulomb's law states that:

Griffiths (2025). 9781108333511, Cambridge University Press. . ISBN 9781108333511

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) between two charges, then the force between two point charges $Q$ and $q$ is:

$F\; =\; \{1\backslash over\; 4\backslash pi\backslash varepsilon\_0\}$>

where ε0 = is the vacuum permittivity. Matthew Sadiku (2025). 9780195387759, Oxford University Press. ISBN 9780195387759 The SI unit of ε0 is equivalently ampere2⋅second4 ⋅kg−1⋅m−3 or coulomb2⋅N−1⋅m−2 or farad⋅m−1. Electric field The electric field, $\backslash mathbf\; E$, in units of newtons per coulomb or per meter, is a vector field that can be defined everywhere, except at the location of point charges (where it diverges to infinity). (2025). 9781107014022, Cambridge University Press. . ISBN 9781107014022 It is defined as the electrostatic force $\backslash mathbf\; F$ on a hypothetical small test charge at the point due to Coulomb's law, divided by the charge $q$ $\backslash mathbf\; E\; =\; \{\backslash mathbf\; F\; \backslash over\; q\}$ Field line 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. A collection of $n$ particles of charge $q\_i$, located at points $\backslash mathbf\; r\_i$ (called source points) generates the electric field at $\backslash mathbf\; r$ (called the field point) of: $\backslash mathbf\; E(\backslash mathbf\; r)\; =\; \{1\backslash over4\backslash pi\backslash varepsilon\_0\}\; \backslash sum\_\{i=1\}^n\; q\_i\; \{\backslash hat\backslash mathbf\; \{r-r\_i\}\backslash over\; \{|\backslash mathbf\; \{r-r\_i\}$

^2} = {1\over 4\pi\varepsilon_0} \sum_{i=1}^n q_i {\mathbf {r-r_i}\over ^3},

where $\backslash mathbf\; r-\backslash mathbf\; r\_i$ is the displacement vector from a source point $\backslash mathbf\; r\_i$ to the field point $\backslash mathbf\; r$, and $\backslash hat\backslash mathbf\; \{r-r\_i\}\; \backslash \; \backslash stackrel\{\backslash mathrm\{def\}\}\{=\}\backslash \; \backslash frac\{\backslash mathbf\; \{r-r\_i\}\}$ is the unit vector of the displacement vector that indicates the direction of the field due to the source at point $\backslash mathbf\{r\_i\}$. For a single point charge, $q$, at the origin, the magnitude of this electric field is $E\; =\; q/4\backslash pi\backslash varepsilon\_0\; r^2$ 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 $\backslash rho(\backslash mathbf\; r)$ and can be obtained by converting this sum into a triple integral:

$\backslash mathbf\; E(\backslash mathbf\; r)\; =\; \backslash frac\{1\}\{4\backslash pi\backslash varepsilon\_0\}\; \backslash iiint\; \backslash ,\; \backslash rho(\backslash mathbf\; r\text{'})\; \{\backslash mathbf\; \{r-r\text{'}\}\backslash over$^3} \mathrm{d}^3|\mathbf r'|

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Gauss's law Gauss's 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." Many numerical problems can be solved by considering a Gaussian surface around a body. Mathematically, Gauss's law takes the form of an integral equation:

$\backslash Phi\_E\; =\; \backslash oint\_S\backslash mathbf\; E\backslash cdot\; \backslash mathrm\{d\}\backslash mathbf\; A\; =\; \{Q\_\backslash text\{enclosed\}\backslash over\backslash varepsilon\_0\}\; =\; \backslash int\_V\{\backslash rho\backslash over\backslash varepsilon\_0\}\backslash mathrm\{d\}^3\; r,$

where $\backslash mathrm\{d\}^3\; r\; =\backslash mathrm\{d\}x\; \backslash \; \backslash mathrm\{d\}y\; \backslash \; \backslash mathrm\{d\}z$ is a volume element. If the charge is distributed over a surface or along a line, replace $\backslash rho\backslash ,\backslash mathrm\{d\}^3r$ by $\backslash sigma\; \backslash ,\; \backslash mathrm\{d\}A$ or $\backslash lambda\; \backslash ,\; \backslash mathrm\{d\}\backslash ell$. The divergence theorem allows Gauss's Law to be written in differential form:

$\backslash nabla\backslash cdot\backslash mathbf\; E\; =\; \{\backslash rho\backslash over\backslash varepsilon\_0\}.$

where $\backslash nabla\; \backslash cdot$ is the divergence.

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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:

$\{\backslash nabla\}^2\; \backslash phi\; =\; -\; \{\backslash rho\backslash over\backslash varepsilon\_0\}.$

This relationship is a form of Poisson's equation. In the absence of unpaired electric charge, the equation becomes Laplace's equation:

$\{\backslash nabla\}^2\; \backslash phi\; =\; 0,$

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Electrostatic approximation If the electric field in a system can be assumed to result from static charges, that is, a system that exhibits no significant time-varying magnetic fields, the system is justifiably analyzed using only the principles of electrostatics. This is called the "electrostatic approximation".

The validity of the electrostatic approximation rests on the assumption that the electric field is irrotational, or nearly so:

$\backslash nabla\backslash times\backslash mathbf\; E\; \backslash approx\; 0.$

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

$\{\backslash partial\backslash mathbf\; B\backslash over\backslash partial\; t\}\; \backslash approx\; 0.$

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 non-relativistic Galilean limits for electromagnetism. In addition, conventional electrostatics ignore quantum effects which have to be added for a complete description.

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Electrostatic potential As the electric field is irrotational, it is possible to express the electric field as the gradient of a scalar function, $\backslash phi$, 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

$\backslash mathbf\; E\; =\; -\backslash nabla\backslash phi.$

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:

$-\backslash int\_a^b\; \{\backslash mathbf\; E\backslash cdot\; \backslash mathrm\{d\}\backslash mathbf\; \backslash ell\}\; =\; \backslash phi\; (\backslash mathbf\; b)\; -\backslash phi(\backslash mathbf\; a).$

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).

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Electrostatic energy A test particle's potential energy, $U\_\backslash mathrm\{E\}^\{\backslash text\{single\}\}$, can be calculated from a line integral of the work, $q\_n\backslash mathbf\; E\backslash cdot\backslash mathrm\; d\backslash mathbf\backslash ell$. We integrate from a point at infinity, and assume a collection of $N$ particles of charge $Q\_n$, are already situated at the points $\backslash mathbf\; r\_i$. This potential energy (in ) is:

$U\_\backslash mathrm\{E\}^\{\backslash text\{single\}\}=q\backslash phi(\backslash mathbf\; r)=\backslash frac\{q\; \}\{4\backslash pi\; \backslash varepsilon\_0\}\backslash sum\_\{i=1\}^N\; \backslash frac\{Q\_i\}\{\backslash left\; \backslash |\backslash mathcal\{\backslash mathbf\; R\_i\}\; \backslash right\; \backslash |\}$

where $\backslash mathbf\backslash mathcal\; \{R\_i\}\; =\; \backslash mathbf\; r\; -\; \backslash mathbf\; r\_i$ is the distance of each charge $Q\_i$ from the test charge $q$, which situated at the point $\backslash mathbf\; r$, and $\backslash phi(\backslash mathbf\; r)$ is the electric potential that would be at $\backslash mathbf\; r$ if the test charge were not present. If only two charges are present, the potential energy is $Q\_1\; Q\_2/(4\backslash pi\backslash varepsilon\_0\; r)$. The total electric potential energy due a collection of N charges is calculating by assembling these particles one at a time:

$U\_\backslash mathrm\{E\}^\{\backslash text\{total\}\}\; =\; \backslash frac\{1\; \}\{4\backslash pi\backslash varepsilon\; \_0\}\backslash sum\_\{j=1\}^N\; Q\_j\; \backslash sum\_\{i=1\}^\{j-1\}\; \backslash frac\{Q\_i\}\{r\_\{ij\}\}=\; \backslash frac\{1\}\{2\}\backslash sum\_\{i=1\}^N\; Q\_i\backslash phi\_i\; ,$

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

$\backslash phi\_i\; =\; \backslash frac\{1\}\{4\backslash pi\; \backslash varepsilon\; \_0\}\; \backslash sum\_\{\backslash stackrel\{j=1\}\{j\; \backslash ne\; i\}\}^N\; \backslash frac\{Q\_j\}\{r\_\{ij\}\}.$

This electric potential, $\backslash phi\_i$ is what would be measured at $\backslash mathbf\; r\_i$ 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 $\backslash sum\; (\backslash cdots)\; \backslash rightarrow\; \backslash int(\backslash cdots)\backslash rho\; \backslash ,\; \backslash mathrm\; d^3r$:

$U\_\backslash mathrm\{E\}^\{\backslash text\{total\}\}\; =\; \backslash frac\{1\}\{2\}\; \backslash int\backslash rho(\backslash mathbf\; r)\backslash phi(\backslash mathbf\; r)\; \backslash ,\; \backslash mathrm\{d\}^3\; r\; =\; \backslash frac\{\backslash varepsilon\_0\; \}\{2\}\; \backslash int\; \backslash left|\{\backslash mathbf\{E\}\}\backslash right|^2\; \backslash ,\; \backslash mathrm\{d\}^3\; r,$

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 $\backslash frac\{1\}\{2\}\backslash rho\backslash phi$ and $\backslash frac\{1\}\{2\}\backslash varepsilon\_0\; E^2$; they yield equal values for the total electrostatic energy only if both are integrated over all space.

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Electrostatic pressure Inside of an electrical conductor, there is no electric field.

Edward M. Purcell (2025). 9781107014022, Cambridge Univ. Press. . ISBN 9781107014022

The external electric field has been balanced by surface charges due to movement of charge carriers, either to or from other parts of the material, known as electrostatic induction. The equation connecting the field just above a small patch of the surface and the surface charge is $$\backslash mathbf\{E\backslash cdot\; \backslash hat\{n\}\}\; =\; \backslash frac\; \{\backslash sigma\}\; \{\backslash epsilon\_0\}$$ where

• $\backslash mathbf\{\backslash hat\{n\}\}$ = the surface unit normal vector,
• $\backslash mathbf\{\backslash sigma\}$ = the surface charge density.

The average electric field, half the external value, also exerts a force (Coulomb's law) on the conductor patch where the force $\backslash mathbf\{f\}$ is given by

$\backslash mathbf\{f\}\; =\; \backslash frac\; \{1\}\; \{2\; \backslash epsilon\_0\}\; \backslash sigma^2\; \backslash mathbf\{\backslash hat\{n\}\}$.

In terms of the field just outside the surface, the force is equivalent to a pressure given by:

$P\; =\; \backslash frac\{\; \backslash varepsilon\_0\; \}\{2\}\; (\backslash mathbf\{E\backslash cdot\; \backslash hat\{n\}\})^2,$

This pressure acts normal to the surface of the conductor, independent of whether: the mobile charges are electrons, Electron hole or Proton conductor; the sign of the surface charge; or the sign of the surface normal component of the electric field.

David J. Griffiths (2023). 9781009397735, Cambridge University Press. . ISBN 9781009397735

Note that there is a similar form for electrostriction in a dielectric.

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See also

• Electrostatic generator, machines that create static electricity.
• Electrostatic induction, separation of charges due to electric fields.
• Permittivity and relative permittivity, the electric polarizability of materials.
• Quantization of charge, the charge units carried by electrons or protons.
• Static electricity, stationary charge accumulated on a material.
• Triboelectric effect, separation of charges due to sliding or contact.

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Further reading

• (1989). 013249020X, Prentice-Hall. 013249020X
• (1992). 9780471804574, John Wiley & Sons. . ISBN 9780471804574
• Griffiths, David J. (1999). 013805326X, Prentice Hall. . 013805326X

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External links

• The Feynman Lectures on Physics Vol. II Ch. 4: Electrostatics
• Introduction to Electrostatics: Point charges can be treated as a distribution using the Dirac delta function

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