
This article has multiple issues. Please help improve it or discuss these issues on the talk page. 
 This article needs attention from an expert on the subject. Please add a reason or a talk parameter to this template to explain the issue with the article. Consider associating this request with a WikiProject. (July 2013) 
 
An electric field is generated by electrically charged particles and timevarying magnetic fields. The electric field describes the electric force experienced by a motionless positively electrically charged test particle at any point in space relative to the source(s) of the field. The concept of an electric field was introduced by Michael Faraday.
Qualitative description
The electric field is a vector field. The field vector at a given point is defined as the force vector per unit charge that would be exerted on a stationary test charge at that point. An electric field is generated by electric charge, as well as by a timevarying magnetic field. Electric fields contain electrical energy with energy density proportional to the square of the field amplitude. The electric field is to charge as gravitational acceleration is to mass. The SI units of the field are newtons per coulomb (N⋅C^{−1}) or, equivalently, volts per metre (V⋅m^{−1}), which in terms of SI base units are kg⋅m⋅s^{−3}⋅A^{−1}.
An electric field that changes with time, such as due to the motion of charged particles producing the field, influences the local magnetic field. That is, the electric and magnetic fields are not separate phenomena; what one observer perceives as an electric field, another observer in a different frame of reference perceives as a mixture of electric and magnetic fields. For this reason, one speaks of "electromagnetism" or "electromagnetic fields". In quantum electrodynamics, disturbances in the electromagnetic fields are called photons, and the energy of photons is quantized.
Definition
Electric Field
Consider a point charge q with position (x,y,z). Now suppose the charge is subject to a force F_{on q} due to other charges. Since this force varies with the position of the charge and by Coulomb's Law it is defined at all points in space, F_{on q} is a continuous function of the charge's position. This suggests that there is some property of the space that causes the force which is exerted on the charge q. This property is called the electric field and it is defined by
 $\backslash mathbf\{E\}(x,y,z)\backslash equiv\; \backslash !\backslash frac\{\backslash mathbf\{F\}\_\backslash text\{on\; q\}(x,y,z)\}\{q\}$
Notice that the magnitude of the electric field has units of Force/Charge. Mathematically, the E field can be thought of as a function that associates a vector with every point in space. Each such vector's magnitude is proportional to how much force a charge at that point would "feel" if it were present and this force would have the same direction as the electric field vector at that point. It is also important to note that the electric field defined above is caused by a configuration of other electric charges. This means that the charge q in the equation above is not the charge that is creating the electric field, but rather, being acted upon by it. This definition does not give a means of computing the electric field caused by a group of charges.
Superposition
Array of discrete point charges
Electric fields satisfy the superposition principle. If more than one charge is present, the total electric field at any point is equal to the vector sum of the separate electric fields that each point charge would create in the absence of the others. That is,
 $\backslash mathbf\{E\}\; =\; \backslash sum\_i\; \backslash mathbf\{E\}\_i\; =\; \backslash mathbf\{E\}\_1\; +\; \backslash mathbf\{E\}\_2\; +\; \backslash mathbf\{E\}\_3\; \backslash cdots\; \backslash ,\backslash !$
where E_{i} is the electric field created by the ith point charge.
At any point of interest, the total Efield due to N point charges is simply the superposition of the Efields due to each point charge, given by
 $\backslash mathbf\{E\}\; =\; \backslash sum\_\{i=1\}^N\; \backslash mathbf\{E\}\_i\; =\; \backslash frac\{1\}\{4\backslash pi\backslash varepsilon\_0\}\; \backslash sum\_\{i=1\}^N\; \backslash frac\{Q\_i\}\{r\_i^2\}\; \backslash mathbf\{\backslash hat\{r\}\}\_i.$
where Q_{i} is the electric charge of the ith point charge,
$\backslash mathbf\{\backslash hat\{r\}\}\_i$ the corresponding unit vector of r_{i}, which is the position of charge Q_{i} with respect to the point of interest.
Continuum of charges
The superposition principle holds for an infinite number of infinitesimally small elements of charges – i.e. a continuous distribution of charge. By taking the limit as N approaches infinity in the previous equation, the electric field for a continuum of charges can be given by the integral:
 $\backslash mathbf\{E\}\; =\; \backslash int\_V\; d\backslash mathbf\{E\}\; =\; \backslash frac\{1\}\{4\backslash pi\backslash varepsilon\_0\}\; \backslash int\_V\backslash frac\{\backslash rho\}\{r^2\}\; \backslash mathbf\{\backslash hat\{r\}\}\backslash ,\backslash mathrm\{d\}V\; =\; \backslash frac\{1\}\{4\backslash pi\backslash varepsilon\_0\}\; \backslash int\_V\backslash frac\{\backslash rho\}\{r^3\}\; \backslash mathbf\{r\}\backslash ,\backslash mathrm\{d\}V\; \backslash ,\backslash !$
where ρ is the charge density (the amount of charge per unit volume), and dV is the differential volume element. This integral is a volume integral over the region of the charge distribution.
The equations above express the electric field of point charges as derived from Coulomb's law, which is actually a special case of Gauss's Law. While Coulomb's law is only true for stationary point charges, Gauss's law is true for all charges either in static form or in motion. Gauss's Law establishes a more fundamental relationship between the distribution of electric charge in space and the resulting electric field. It is one of Maxwell's equations governing electromagnetism.
Gauss's law allows the Efield to be calculated in terms of a continuous distribution of charge density. In differential form, it can be stated as
 $\backslash nabla\; \backslash cdot\; \backslash mathbf\{E\}\; =\; \backslash frac\{\backslash rho\}\{\backslash varepsilon\; \_0\}$
where ∇⋅ is the divergence operator, ρ is the total charge density, including free and bound charge, in other words all the charge present in the system (per unit volume).
Electrostatic fields
Main article:
Electrostatics
Electrostatic fields are Efields which do not change with time, which happens when the charges are stationary.
The electric field at a point E(r) is equal to the negative gradient of the electric potential $\backslash scriptstyle\; \backslash mathbf\{\backslash Phi\}(\backslash mathbf\{r\})$, a scalar field at the same point:
 $\backslash mathbf\{E\}\; =\; \backslash nabla\; \backslash Phi$
where ∇ is the gradient operator. This is equivalent to the force definition above, since electric potential Φ is defined by the electric potential energy U per unit (test) positive charge:
 $\backslash Phi\; =\; \backslash frac\{U\}\{q\}$
and force is the negative of potential energy gradient:
 $\backslash mathbf\{F\}\; =\; \; \backslash nabla\; U$
If several spatially distributed charges generate such an electric potential, e.g. in a solid, an electric field gradient may also be defined.
Uniform fields
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 edge effects. Ignoring such effects, the equation for the magnitude of the electric field E is:
 $E\; =\; \; \backslash frac\{\backslash Delta\backslash phi\}\{d\}$
where Δϕ 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 nanoapplications, for instance in relation to semiconductors, a typical magnitude of an electric field is in the order of 1 volt/µm achieved by applying a voltage of the order of 1 volt between conductors spaced 1 µm apart.
Parallels between electrostatic and gravitational fields
Coulomb's law, which describes the interaction of electric charges:
 $$
\mathbf{F}=q\left(\frac{Q}{4\pi\varepsilon_0}\frac{\mathbf{\hat{r}}}{\mathbf{r}^2}\right)=q\mathbf{E}
is similar to Newton's law of universal gravitation:
 $$
\mathbf{F}=m\left(GM\frac{\mathbf{\hat{r}}}{\mathbf{r}^2}\right)=m\mathbf{g}
.
This suggests similarities between the electric field E and the gravitational field g, so sometimes mass is called "gravitational charge".
Similarities between electrostatic and gravitational forces:
 Both act in a vacuum.
 Both are central and conservative.
 Both obey an inversesquare law (both are inversely proportional to square of r).
Differences between electrostatic and gravitational forces:
 Electrostatic forces are much greater than gravitational forces for natural values of charge and mass. For instance, the ratio of the electrostatic force to the gravitational force between two electrons is about 10^{42}.
 Gravitational forces are attractive for like charges, whereas electrostatic forces are repulsive for like charges.
 There are not negative gravitational charges (no negative mass) while there are both positive and negative electric charges. This difference, combined with the previous two, implies that gravitational forces are always attractive, while electrostatic forces may be either attractive or repulsive.
Electrodynamic fields
Electrodynamic fields are Efields which do change with time, when charges are in motion.
An electric field can be produced not only by a static charge, but also by a changing magnetic field (in which case it is a nonconservative field). The electric field is then given by:
 $\backslash mathbf\{E\}\; =\; \; \backslash nabla\; \backslash varphi\; \; \backslash frac\; \{\; \backslash partial\; \backslash mathbf\{A\}\; \}\; \{\; \backslash partial\; t\; \}$
in which B satisfies
 $\backslash mathbf\{B\}\; =\; \backslash nabla\; \backslash times\; \backslash mathbf\{A\}$
and ∇× denotes the curl. The vector field B is the magnetic flux density and the vector A is the magnetic vector potential. Taking the curl of the electric field equation we obtain,
 $\backslash nabla\; \backslash times\; \backslash mathbf\{E\}\; =\; \backslash frac\{\backslash partial\; \backslash mathbf\{B\}\}\; \{\backslash partial\; t\}$
which is Faraday's law of induction, another one of Maxwell's equations.^{[1]}
Energy in the electric field
The electrostatic field stores energy. The energy density u (energy per unit volume) is given by^{[2]}
 $u\; =\; \backslash frac\{1\}\{2\}\; \backslash varepsilon\; \backslash mathbf\{E\}^2\; \backslash ,\; ,$
where ε is the permittivity of the medium in which the field exists, and E is the electric field vector (in newtons per coulomb).
The total energy U stored in the electric field in a given volume V is therefore
 $U\; =\; \backslash frac\{1\}\{2\}\; \backslash varepsilon\; \backslash int\_\{V\}\; \backslash mathbf\{E\}^2\; \backslash ,\; \backslash mathrm\{d\}V\; \backslash ,\; ,$
Further extensions
Definitive equation of vector fields
In the presence of matter, it is helpful in electromagnetism to extend the notion of the electric field into three vector fields, rather than just one:^{[3]}
 $\backslash mathbf\{D\}=\backslash varepsilon\_0\backslash mathbf\{E\}+\backslash mathbf\{P\}\backslash !$
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
The E and D fields are related by the permittivity of the material, ε.^{[4]}^{[5]}
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:
 $\backslash mathbf\{D(r)\}=\backslash varepsilon\backslash mathbf\{E(r)\}$
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:
 $D\_i=\backslash varepsilon\_\{ij\}E\_j$
For nonlinear media, E and D are not proportional. Materials can have varying extents of linearity, homogeneity and isotropy.
See also
References
External links
 Georgia State University
 University of Rochester
 University of Rochester
 [1] – 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 3D
 Electric Fields Applet – An applet that shows electric field lines as well as potential gradients.
 Interactive Flash simulation picturing the electric field of userdefined or preselected sets of point charges by field vectors, field lines, or equipotential lines. Author: David Chappell
This article was sourced from Creative Commons AttributionShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, EGovernment Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a nonprofit organization.