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An electric field is the force field that is the property of one electric charge (such as a single electron) that will affect a second charge by repelling or attracting that second charge. This is the very powerful physical force that tightly holds all matter together. Feynman: This **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 that point. Electric fields are created by electric charges plus are induced by time-varying magnetic fields. Then the electric field combines with the magnetic field to form the electromagnetic field.

## Definition of an electric field[edit]

An electric charge, such as an electron, creates and is surrounded by an electric force field. This field will exert a force, either repulsive or attractive, on any other electric charge placed in its vicinity. The electric field at a point is defined as the (vector) force, , that would be exerted on a stationary test particle of unit positive charge placed at that point. The force on a particle of charge will be, given by:. with SI units of newtons per coulomb (N⋅C) or, equivalently, volts per metre (V⋅m), which in terms of SI base units are kg⋅m⋅s⋅A. It should be noted that if the charge is moving, there is a second force field to consider: the magnetic field. The force on a test charge from both fields combined is called the Lorentz force.

## Sources of electric field[edit]

### Causes and description[edit]

Electric fields are caused by electric charges, described by Gauss's law, or varying magnetic fields, described by Faraday's law of induction. 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). Notice that , the permittivity of vacuum, must be substituted if charges are considered in non-empty media.

### Continuous vs. discrete charge representation[edit]

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.

## Superposition principle[edit]

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[edit]

Electrostatic fields are **E**-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.

### Electric potential[edit]

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 . This is analogous to the gravitational potential.

### Parallels between electrostatic and gravitational fields[edit]

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.

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

### Uniform fields[edit]

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*ϕ* 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 7006100000000000000♠10 V⋅m, achieved by applying a voltage of the order of 1 volt between conductors spaced 1 µm apart.

## Electrodynamic fields[edit]

Electrodynamic fields are **E**-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

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

## Energy in the electric field[edit]

The total energy per unit volume stored by the electromagnetic field is

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

## Further extensions[edit]

### Definitive equation of vector fields[edit]

In the presence of matter, it is helpful to extend the notion of the electric field into three vector fields:

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.

### Constitutive relation[edit]

The **E** and **D** fields are related by the permittivity of the material, *ε*.

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.

## See also[edit]

## References[edit]

Purcell, Edward, PhD. ELECTRICITY AND MAGNETISM, 3rd Edition. Cambridge University Press, Cambridge, UK. 2010 Browne, Michael, PhD. PHYSICS FOR ENGINEERING AND SCIENCE, 2nd edition. MGraw-Hill, Schaum. New York, 2011.

## External links[edit]

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- Electric field in "Electricity and Magnetism", R Nave – Hyperphysics, Georgia State University
- 'Gauss's Law' – Chapter 24 of Frank Wolfs's lectures at University of Rochester
- 'The Electric Field' – Chapter 23 of Frank Wolfs's lectures at University of Rochester
- MovingCharge.html – An applet that shows the electric field of a moving point charge
- Fields – a chapter from an online textbook
- Learning by Simulations Interactive simulation of an electric field of up to four point charges
- Java simulations of electrostatics in 2-D and 3-D
- Interactive Flash simulation picturing the electric field of user-defined or preselected sets of point charges by field vectors, field lines, or equipotential lines. Author: David Chappell