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An **electric field** surrounds an electric charge, and exerts force on other charges in the field, attracting or repelling them.^{ [1] }^{ [2] } Electric field is sometimes abbreviated as **E-field**. Mathematically the electric field is a vector field that associates to each point in space the force, called the Coulomb force, that would be experienced per unit of charge by an infinitesimal test charge at rest at that point.^{ [3] }^{ [4] }^{ [5] } The SI unit for electric field strength is volt per meter (V/m).^{ [6] } Newtons per coulomb (N/C) is also used as a unit of electric field strengh. Electric fields are created by electric charges, or by time-varying magnetic fields. Electric fields are important in many areas of physics, and are exploited practically in electrical technology. On an atomic scale, the electric field is responsible for the attractive force between the atomic nucleus and electrons that holds atoms together, and the forces between atoms that cause chemical bonding. Electric fields and magnetic fields are both manifestations of the electromagnetic force, one of the four fundamental forces (or interactions) of nature.

**Electric charge** is the physical property of matter that causes it to experience a force when placed in an electromagnetic field. There are two types of electric charges; *positive* and *negative*. Like charges repel and unlike attract. An object with an absence of net charge is referred to as *neutral*. Early knowledge of how charged substances interact is now called classical electrodynamics, and is still accurate for problems that do not require consideration of quantum effects.

In vector calculus and physics, a **vector field** is an assignment of a vector to each point in a subset of space. A vector field in the plane, can be visualised as: a collection of arrows with a given magnitude and direction, each attached to a point in the plane. Vector fields are often used to model, for example, the speed and direction of a moving fluid throughout space, or the strength and direction of some force, such as the magnetic or gravitational force, as it changes from one point to another point.

**Coulomb's law**, or **Coulomb's inverse-square law**, is a law of physics for quantifying Coulomb's force, or electrostatic force. Electrostatic force is the amount of force with which stationary, electrically charged particles either repel, or attract each other. This force and the law for quantifying it, represent one of the most basic forms of force used in the physical sciences, and were an essential basis to the study and development of the theory and field of classical electromagnetism. The law was first published in 1785 by French physicist Charles-Augustin de Coulomb.

- Definition
- Sources
- Causes and description
- Continuous vs. discrete charge representation
- Superposition principle
- Electrostatic fields
- Electric potential
- Parallels between electrostatic and gravitational fields
- Uniform fields
- Electrodynamic fields
- Energy in the electric field
- Further extensions
- Definitive equation of vector fields
- Constitutive relation
- See also
- References
- External links

From Coulomb's law a particle with electric charge at position exerts a force on a particle with charge at position of

- where is the unit vector in the direction from point to point , and
*ε*_{0}is the electric constant (also known as "the absolute permittivity of free space") in C^{2}m^{−2}N^{−1}

When the charges and have the same sign this force is positive, directed away from the other charge, indicating the particles repel each other. When the charges have unlike signs the force is negative, indicating the particles attract. To make it easy to calculate the Coulomb force on any charge at position this expression can be divided by , leaving an expression that only depends on the other charge (the *source* charge)^{ [4] }^{ [5] }

This is the *electric field* at point due to the point charge ; it is a vector equal to the Coulomb force per unit charge that a positive point charge would experience at the position . Since this formula gives the electric field magnitude and direction at any point in space (except at the location of the charge itself, , where it becomes infinite) it defines a vector field. From the above formula it can be seen that the electric field due to a point charge is everywhere directed away from the charge if it is positive, and toward the charge if it is negative, and its magnitude decreases with the inverse square of the distance from the charge.

In mathematics, physics, and engineering, a **Euclidean vector** is a geometric object that has magnitude and direction. Vectors can be added to other vectors according to vector algebra. A Euclidean vector is frequently represented by a line segment with a definite direction, or graphically as an arrow, connecting an *initial point**A* with a *terminal point**B*, and denoted by

If there are multiple charges, the resultant Coulomb force on a charge can be found by summing the vectors of the forces due to each charge. This shows the electric field obeys the * superposition principle *: the total electric field at a point due to a collection of charges is just equal to the vector sum of the electric fields at that point due to the individual charges.^{ [5] }^{ [7] }

The **superposition principle**, also known as **superposition property**, states that, for all linear systems, the net response caused by two or more stimuli is the sum of the responses that would have been caused by each stimulus individually. So that if input *A* produces response *X* and input *B* produces response *Y* then input produces response.

- where is the unit vector in the direction from point to point .

This is the definition of the electric field due to the point *source charges*. It diverges and becomes infinite at the locations of the charges themselves, and so is not defined there.

The Coulomb force on a charge of magnitude at any point in space is equal to the product of the charge and the electric field at that point

The units of the electric field in the SI system are newtons per coulomb (N/C), or volts per meter (V/m); in terms of the SI base units they are kg⋅m⋅s^{−3}⋅A^{−1}

The **newton** is the International System of Units (SI) derived unit of force. It is named after Isaac Newton in recognition of his work on classical mechanics, specifically Newton's second law of motion.

The **volt** is the derived unit for electric potential, electric potential difference (voltage), and electromotive force. It is named after the Italian physicist Alessandro Volta (1745–1827).

The International System of Units defines seven units of measure as a basic set from which all other SI units can be derived. The **SI base units** and their physical quantities are the metre for measurement of length, the kilogram for mass, the second for time, the ampere for electric current, the kelvin for temperature, the candela for luminous intensity, and the mole for amount of substance.

The electric field due to a continuous distribution of charge in space (where is the charge density in coulombs per cubic meter) can be calculated by considering the charge in each small volume of space at point as a point charge, and calculating its electric field at point

where is the unit vector pointing from to , then adding up the contributions from all the increments of volume by integrating over the volume of the charge distribution

Electric fields are caused by electric charges, described by Gauss's law,^{ [8] } or varying magnetic fields, described by Faraday's law of induction.^{ [9] } Together, these laws are enough to define the behavior of the electric field as a function of charge repartition and magnetic field. However, since the magnetic field is described as a function of electric field, the equations of both fields are coupled and together form Maxwell's equations that describe both fields as a function of charges and currents.

In the special case of a steady state (stationary charges and currents), the Maxwell-Faraday inductive effect disappears. The resulting two equations (Gauss's law and Faraday's law with no induction term ), taken together, are equivalent to Coulomb's law, written as for a charge density ( denotes the position in space).^{ [10] } Notice that , the permitivity of vacuum, must be substituted if charges are considered in non-empty media.

The equations of electromagnetism are best described in a continuous description. However, charges are sometimes best described as discrete points; for example, some models may describe electrons as point sources where charge density is infinite on an infinitesimal section of space.

A charge located at can be described mathematically as a charge density , where the Dirac delta function (in three dimensions) is used. Conversely, a charge distribution can be approximated by many small point charges.

Electric fields satisfy the superposition principle, because Maxwell's equations are linear. As a result, if and are the electric fields resulting from distribution of charges and , a distribution of charges will create an electric field ; for instance, Coulomb's law is linear in charge density as well.

This principle is useful to calculate the field created by multiple point charges. If charges are stationary in space at , in the absence of currents, the superposition principle proves that the resulting field is the sum of fields generated by each particle as described by Coulomb's law:

Electrostatic fields are electric fields which do not change with time, which happens when charges and currents are stationary. In that case, Coulomb's law fully describes the field.^{ [11] }

If a system is static, such that magnetic fields are not time-varying, then by Faraday's law, the electric field is curl-free. In this case, one can define an electric potential, that is, a function such that .^{ [12] } This is analogous to the gravitational potential.

Coulomb's law, which describes the interaction of electric charges:

is similar to Newton's law of universal gravitation:

(where ).

This suggests similarities between the electric field **E** and the gravitational field **g**, or their associated potentials. Mass is sometimes called "gravitational charge" because of that similarity.^{[ citation needed ]}

Electrostatic and gravitational forces both are central, conservative and obey an inverse-square law.

A uniform field is one in which the electric field is constant at every point. It can be approximated by placing two conducting plates parallel to each other and maintaining a voltage (potential difference) between them; it is only an approximation because of boundary effects (near the edge of the planes, electric field is distorted because the plane does not continue). Assuming infinite planes, the magnitude of the electric field *E* is:

where Δ*V* is the potential difference between the plates and *d* is the distance separating the plates. The negative sign arises as positive charges repel, so a positive charge will experience a force away from the positively charged plate, in the opposite direction to that in which the voltage increases. In micro- and nano-applications, for instance in relation to semiconductors, a typical magnitude of an electric field is in the order of ^{6} V⋅m^{−1}, achieved by applying a voltage of the order of 1 volt between conductors spaced 1 10 µm apart.

Electrodynamic fields are electric fields which do change with time, for instance when charges are in motion.

The electric field cannot be described independently of the magnetic field in that case. If **A** is the magnetic vector potential, defined so that , one can still define an electric potential such that:

One can recover Faraday's law of induction by taking the curl of that equation

^{ [13] }

which justifies, a posteriori, the previous form for **E**.

The total energy per unit volume stored by the electromagnetic field is^{ [14] }

where *ε* is the permittivity of the medium in which the field exists, its magnetic permeability, and **E** and **B** are the electric and magnetic field vectors.

As **E** and **B** fields are coupled, it would be misleading to split this expression into "electric" and "magnetic" contributions. However, in the steady-state case, the fields are no longer coupled (see Maxwell's equations). It makes sense in that case to compute the electrostatic energy per unit volume:

The total energy *U* stored in the electric field in a given volume *V* is therefore

In the presence of matter, it is helpful to extend the notion of the electric field into three vector fields:^{ [15] }

where **P** is the electric polarization – the volume density of electric dipole moments, and **D** is the electric displacement field. Since **E** and **P** are defined separately, this equation can be used to define **D**. The physical interpretation of **D** is not as clear as **E** (effectively the field applied to the material) or **P** (induced field due to the dipoles in the material), but still serves as a convenient mathematical simplification, since Maxwell's equations can be simplified in terms of free charges and currents.

The **E** and **D** fields are related by the permittivity of the material, *ε*.^{ [16] }^{ [15] }

For linear, homogeneous, isotropic materials **E** and **D** are proportional and constant throughout the region, there is no position dependence: For inhomogeneous materials, there is a position dependence throughout the material:

For anisotropic materials the **E** and **D** fields are not parallel, and so **E** and **D** are related by the permittivity tensor (a 2nd order tensor field), in component form:

For non-linear media, **E** and **D** are not proportional. Materials can have varying extents of linearity, homogeneity and isotropy.

- Classical electromagnetism
- Field strength
- Signal strength in telecommunications
- Magnetism
- Teltron tube
- Teledeltos, a conductive paper that may be used as a simple analog computer for modelling fields

An **electric potential** is the amount of work needed to move a unit of positive charge from a reference point to a specific point inside the field without producing an acceleration. Typically, the reference point is the Earth or a point at infinity, although any point beyond the influence of the electric field charge can be used.

In physics, **Gauss's law**, also known as **Gauss's flux theorem**, is a law relating the distribution of electric charge to the resulting electric field. The surface under consideration may be a closed one enclosing a volume such as a spherical surface.

In physics, specifically electromagnetism, the **Biot–Savart law** is an equation describing the magnetic field generated by a constant electric current. It relates the magnetic field to the magnitude, direction, length, and proximity of the electric current. The Biot–Savart law is fundamental to magnetostatics, playing a role similar to that of Coulomb's law in electrostatics. When magnetostatics does not apply, the Biot–Savart law should be replaced by Jefimenko's equations. The law is valid in the magnetostatic approximation, and is consistent with both Ampère's circuital law and Gauss's law for magnetism. It is named after Jean-Baptiste Biot and Félix Savart, who discovered this relationship in 1820.

**Synchrotron radiation** is the electromagnetic radiation emitted when charged particles are accelerated radially, i.e., 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, then the emission is called cyclotron emission. If, on the other hand, 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

In mathematics, **Poisson's equation** is a partial differential equation of elliptic type with broad utility in mechanical engineering and theoretical physics. It arises, for instance, to describe the potential field caused by a given charge or mass density distribution; with the potential field known, one can then calculate gravitational or electrostatic field. It is a generalization of Laplace's equation, which is also frequently seen in physics. The equation is named after the French mathematician, geometer, 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 physics, the **electric displacement field**, denoted by **D**, is a vector field that appears in Maxwell's equations. It accounts for the effects of free and bound charge within materials. "**D**" stands for "displacement", as in the related concept of displacement current in dielectrics. In free space, the electric displacement field is equivalent to flux density, a concept that lends understanding to Gauss's law. In the International System of Units (SI), it is expressed in units of coulomb per meter squared (C⋅m^{−2}).

**Electric potential energy**, or **electrostatic 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 **Coulomb constant**, the **electric force constant**, or the **electrostatic constant** (denoted *k*_{e}, *k* or *K*) is a proportionality constant in electrodynamics equations. In SI units, it is exactly equal to 8987551787.3681764 N·m^{2}·C^{−2}, or approximately 8.99×10^{9} N·m^{2}·C^{−2}. It was named after the French physicist Charles-Augustin de Coulomb (1736–1806) who introduced Coulomb's law.

The **Larmor formula** is used to calculate the total power radiated by a non relativistic point charge as it accelerates or decelerates. This is used in the branch of physics known as electrodynamics and is not to be confused with the Larmor precession from classical nuclear magnetic resonance. It was first derived by J. J. Larmor in 1897, in the context of the wave theory of light.

In electromagnetism, **charge density** is the amount of electric charge per unit length, surface area, or volume. *Volume charge density* is the quantity of charge per unit volume, measured in the SI system in coulombs per cubic meter (C•m^{−3}), at any point in a volume. *Surface charge density* (σ) is the quantity of charge per unit area, measured in coulombs per square meter (C•m^{−2}), at any point on a surface charge distribution on a two dimensional surface. *Linear charge density* (λ) is the quantity of charge per unit length, measured in coulombs per meter (C•m^{−1}), at any point on a line charge distribution. Charge density can be either positive or negative, since electric charge can be either positive or negative.

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.

In atomic, molecular, and optical physics and quantum chemistry, the **molecular Hamiltonian ** is the Hamiltonian operator representing the energy of the electrons and nuclei in a molecule. This operator and the associated Schrödinger equation play a central role in computational chemistry and physics for computing properties of molecules and aggregates of molecules, such as thermal conductivity, specific heat, electrical conductivity, optical, and magnetic properties, and reactivity.

There are various **mathematical descriptions of the electromagnetic field** that are used in the study of electromagnetism, one of the four fundamental interactions of nature. In this article, several approaches are discussed, although the equations are in terms of electric and magnetic fields, potentials, and charges with currents, generally speaking.

The **Liénard–Wiechert potentials** describe the classical electromagnetic effect of a moving electric point charge in terms of a vector potential and a scalar potential in the Lorenz gauge. Built directly from Maxwell's equations, these potentials describe the complete, relativistically correct, time-varying electromagnetic field for a point charge in arbitrary motion, but are not corrected for quantum-mechanical effects. Electromagnetic radiation in the form of waves can be obtained from these potentials. These expressions were developed in part by Alfred-Marie Liénard in 1898 and independently by Emil Wiechert in 1900.

In continuum mechanics, a **compatible** deformation **tensor field** in a body is that *unique* tensor field that is obtained when the body is subjected to a continuous, single-valued, displacement field. **Compatibility** is the study of the conditions under which such a displacement field can be guaranteed. Compatibility conditions are particular cases of integrability conditions and were first derived for linear elasticity by Barré de Saint-Venant in 1864 and proved rigorously by Beltrami in 1886.

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, the most commonly used unit in atomic physics and chemistry is the debye (D).

- ↑ Purcell, Edward M.; Morin, David J. (2013).
*Electricity and Magnetism**,*(3rd ed.). New York: Cambridge University Press. pp. 14–20. ISBN 978-1-107-01402-2. - ↑ Browne, p 225: "... around every charge there is an aura that fills all space. This aura is the electric field due to the charge. The electric field is a vector field... and has a magnitude and direction."
- ↑ Richard Feynman (1970).
*The Feynman Lectures on Physics Vol II*. Addison Wesley Longman. ISBN 978-0-201-02115-8. - 1 2 Purcell, Edward (2011).
*Electricity and Magnetism, 2nd Ed*. Cambridge University Press. pp. 8–9, 15–16. ISBN 1139503553. - 1 2 3 Serway, Raymond A.; Vuille, Chris (2014).
*College Physics, 10th Ed*. Cengage Learning. pp. 532–533. ISBN 1305142829. - ↑ International Bureau of Weights and Measures (2018-02-05),
*SI Brochure: The International System of Units (SI)*(PDF) (Draft) (9th ed.), p. 23 - ↑ Purcell (2011)
*Electricity and Magnetism*, 2nd Ed., p. 20-21 - ↑ Purcell, p 25: "Gauss's Law: the flux of the electric field E through any closed surface... equals 1/e times the total charge enclosed by the surface."
- ↑ Purcell, p 356: "Faraday's Law of Induction."
- ↑ Purcell, p7: "... the interaction between electric charges
*at rest*is described by Coulomb's Law: two stationary electric charges repel or attract each other with a force proportional to the product of the magnitude of the charges and inversely proportional to the square of the distance between them. - ↑ Purcell, p5-7.
- ↑ gwrowe (8 October 2011). "Curl & Potential in Electrostatics".
*physicspages.com*. Retrieved 21 January 2017. - ↑ Huray, Paul G. (2009).
*Maxwell's Equations*. Wiley-IEEE. p. 205. ISBN 0-470-54276-4. - ↑ Introduction to Electrodynamics (3rd Edition), D.J. Griffiths, Pearson Education, Dorling Kindersley, 2007, ISBN 81-7758-293-3
- 1 2 Electromagnetism (2nd Edition), I.S. Grant, W.R. Phillips, Manchester Physics, John Wiley & Sons, 2008, ISBN 978-0-471-92712-9
- ↑
*Electricity and Modern Physics (2nd Edition)*, G.A.G. Bennet, Edward Arnold (UK), 1974, ISBN 0-7131-2459-8

- Purcell, Edward; Morin, David (2010).
*ELECTRICITY AND MAGNETISM*(3rd ed.). Cambridge University Press, New York. ISBN 978-1-107-01402-2.CS1 maint: Multiple names: authors list (link) - Browne, Michael (2011).
*PHYSICS FOR ENGINEERING AND SCIENCE*(2nd ed.). McGraw-Hill, Schaum, New York. ISBN 978-0-07-161399-6.

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- Electric field in "Electricity and Magnetism", R Nave – Hyperphysics, Georgia State University
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