Electric field

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In physics, the space surrounding an electric charge or in the presence of a time-varying magnetic field has a property called an electric field (that can also be equated to electric flux density). This electric field exerts a force on other electrically charged objects. The concept of electric field was introduced by Michael Faraday.

The electric field is a vector field with SI units of newtons per coulomb (N C⁻¹) or, equivalently, volts per meter (V m⁻¹). The strength of the field at a given point is defined as the force that would be exerted on a positive test charge of +1 coulomb placed at that point; the direction of the field is given by the direction of that force. Electric fields contain electrical energy with energy density proportional to the square of the field intensity. The electric field is to charge as gravitational acceleration is to mass and force density is to volume.

A moving charge has not just an electric field but also a magnetic field, and in general the electric and magnetic fields are not completely 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 mechanics, disturbances in the electromagnetic fields are called photons, and the energy of photons is quantized.

Definition

A stationary charged particle in an electric field experiences a force proportional to its charge given by the equation,

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{F} = q[- \nabla \phi - \frac { \partial \mathbf{A} } { \partial t }]

where the magnetic flux density is given by,

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{B} = \nabla \times \mathbf{A}

and where Failed to parse (Missing texvc executable; please see math/README to configure.): - \nabla \phi

is the Coulomb force. (See the section below). 

If the charged particle can be considered a point charge, the electric field is defined as the force it experiences per unit charge:

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{E} = \frac{\mathbf{F}}{q}

where

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{F}

is the electric force experienced by the particle

q is its charge

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{E}

is the electric field wherein the particle is located

Taken literally, this equation only defines the electric field at the places where there are stationary charges present to experience it. Furthermore, the force exerted by another charge Failed to parse (Missing texvc executable; please see math/README to configure.): q

will alter the source distribution, which means the electric field in the presence of Failed to parse (Missing texvc executable; please see math/README to configure.): q
differs from itself in the absence of Failed to parse (Missing texvc executable; please see math/README to configure.): q

. However, the electric field of a given source distribution remains defined in the absence of any charges with which to interact. This is achieved by measuring the force exerted on successively smaller test charges placed in the vicinity of the source distribution. By this process, the electric field created by a given source distribution is defined as the limit as the test charge approaches zero of the force per unit charge exerted thereupon.

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{E}=\lim_{q \to 0}\frac{\mathbf{F}}{q}

This allows the electric field to be dependent on the source distribution alone.

As is clear from the definition, the direction of the electric field is the same as the direction of the force it would exert on a positively-charged particle, and opposite the direction of the force on a negatively-charged particle. Since like charges repel and opposites attract (as quantified below), the electric field tends to point away from positive charges and towards negative charges.

Coulomb's law

The electric field surrounding a point charge is given by Coulomb's law:

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{E} =\frac{1}{4 \pi \varepsilon_0}\frac{Q}{r^2}\mathbf{\hat{r}} \qquad \mbox{(1)}

where

Q is the charge of the particle creating the electric field,

r is the distance from the particle with charge Q to the E-field evaluation point,

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{\hat{r}}

is the Unit vector pointing from the particle with charge Q to the E-field evaluation point,

Failed to parse (Missing texvc executable; please see math/README to configure.): \varepsilon_0

is the vacuum permittivity.

Coulomb's law is actually a special case of Gauss's Law, a more fundamental description of the relationship between the distribution of electric charge in space and the resulting electric field. Gauss's law is one of Maxwell's equations, a set of four laws governing electromagnetics.

Time-varying fields

Charges do not only produce electric fields. As they move, they generate magnetic fields, and if the magnetic field changes, it generates electric fields. A changing magnetic field gives rise to an electric field,

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{E} = - \nabla \phi - \frac { \partial \mathbf{A} } { \partial t }

which yields Faraday's law of induction,

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{\nabla} \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}

where

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{\nabla} \times \mathbf{E}

indicates the curl of the electric field,

Failed to parse (Missing texvc executable; please see math/README to configure.): -\frac{\partial \mathbf{B}} {\partial t}

represents the vector rate of decrease of magnetic field with time.

This means that a magnetic field changing in time produces a curled electric field, possibly also changing in time. The situation in which electric or magnetic fields change in time is no longer electrostatics, but rather electrodynamics or electromagnetics.

Properties (in electrostatics)

According to equation (1) above, electric field is dependent on position. The electric field due to any single charge falls off as the square of the distance from that charge.

Electric fields follow 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 respective electric fields that each object would create in the absence of the others.

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{E}_{\rm total} = \sum_i \mathbf{E}_i = \mathbf{E}_1 + \mathbf{E}_2 + \mathbf{E}_3 \ldots \,\!

If this principle is extended to an infinite number of infinitesimally small elements of charge, the following formula results:

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{E} = \frac{1}{4\pi\varepsilon_0} \int\frac{\rho}{r^2} \mathbf{\hat{r}}\,\mathrm{d}V

where

Failed to parse (Missing texvc executable; please see math/README to configure.): \rho

is the charge density, or the amount of charge per unit volume.

The electric field at a point is equal to the negative gradient of the electric potential there. In symbols,

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{E} = -\mathbf{\nabla}\phi

where

Failed to parse (Missing texvc executable; please see math/README to configure.): \phi(x, y, z)

is the scalar field representing the electric potential at a given point. 

If several spatially distributed charges generate such an electric potential, e.g. in a solid, an electric field gradient may also be defined.

Considering the permittivity Failed to parse (Missing texvc executable; please see math/README to configure.): \varepsilon

of a material, which may differ from the permittivity of free space Failed to parse (Missing texvc executable; please see math/README to configure.): \varepsilon_{0}

, the electric displacement field is:

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{D} = \varepsilon \mathbf{E}

Energy in the electric field

Main article: Electrical energy

The electric field stores energy. The energy density of the electric field is given by

Failed to parse (Missing texvc executable; please see math/README to configure.): u = \frac{1}{2} \varepsilon |\mathbf{E}|^2

where

Failed to parse (Missing texvc executable; please see math/README to configure.): \varepsilon

is the permittivity of the medium in which the field exists

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{E}

is the electric field vector.  

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

Failed to parse (Missing texvc executable; please see math/README to configure.): \int_{V} \frac{1}{2} \varepsilon |\mathbf{E}|^2 \, \mathrm{d}V

where

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathrm{d}V

is the differential volume element.

Parallels between electrostatics and gravity

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

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{F} = \frac{1}{4 \pi \varepsilon_0}\frac{Qq}{r^2}\mathbf{\hat{r}} = q\mathbf{E}

is similar to the Newtonian gravitation law:

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{F} = G\frac{Mm}{r^2}\mathbf{\hat{r}} = m\mathbf{g}

This suggests similarities between the electric field Failed to parse (Missing texvc executable; please see math/README to configure.): E

and the gravitational field Failed to parse (Missing texvc executable; please see math/README to configure.): g , so sometimes mass is called "gravitational charge".

Similarities between electrostatic and gravitational forces:

1. Both act in a vacuum.
2. Both are central and conservative.
3. Both obey an inverse-square law (both are inversely proportional to square of r).
4. Both propagate with finite speed c.
5. Both the Electric charge and the mass are invariant under Lorentz transformations.

Differences between electrostatic and gravitational forces:

1. Electrostatic forces are much greater than gravitational forces (by about 10³⁶ times).
2. Gravitational forces are attractive for like charges, whereas electrostatic forces are repulsive for like charges.
3. There are no negative gravitational charges (no negative mass) while there are both positive and negative electric charges. This difference combined with previous implies that gravitational forces are always attractive, while electrostatic forces may be either attractive or repulsive.

See also

• Electrostatics
• Electrodynamics
• Maxwell's equations are the full set of equations governing electric fields.
• Magnetism
• Teltron Tube

External links

• 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-Dar:حقل كهربائي

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