# Electric field

Within the framework of the electromagnetism, the electric field is an object Physique which makes it possible to define and if required to measure in any point of the space the influence exerted remotely by particles electrically charged.

# Introduction

## Concept of electric field

The electric field is the expression of the forces which would result from the remote action of particles electrically charged on a particle test, divided by the value of the load of this particle test, just as the gravitational Champ for example results from the action of particles equipped with mass. This field has, in any point of space, a direction, a direction, and a size expressed in Volt by Mètre (V/m) or in newton by Coulomb (N/C) according to the International Système.

The expression of the electric field is directly resulting from the expression of the electrostatic force given by the Loi of Coulomb, and depends on the point of space where one places oneself. If one regards only one particle charged Q as source with the field, this one is directed sources towards the point considered and has as a value $E = \ frac \left\{1\right\} \left\{4 \ pi \ epsilon\right\} \ frac \left\{Q\right\} \left\{r^2\right\}$ where $Q$ is the electric charge, $\ epsilon \,$ the Permittivité of the medium and $r \,$ the distance between the source and the point considered.

## Effective measurement

To measure the influence of the sources, one can use another particle, it electrically so charged with a load $q \,$ (the unit IF of electric charge is the Coulomb, noted C), which is used as particle test . Then if one calls $\ vec \left\{E\right\} \,$ the electric field created by the sources at the place where the load test is (one will low give the details on the way of determining $\ vec \left\{E\right\}$) the latter undergoes a electric force $\ vec \left\{F\right\} _e$ given by
$\ vec \left\{F\right\} _e = Q \ cdot \ vec \left\{E\right\}$

## Effects

The electric field can thus put moving particles charged. With the difference of the Magnetic field it is able to accelerate them. Although negligible with large scales (such as for example in the majority of the planetary systems), the electric field has a dominating effect on microscopic scales, and is used for the study of the matter in the particle accelerator .

An electric field can be created relatively easily between two plates of condenser, i.e. two plates whose tension between the two is nonnull. See low for a detailed calculation.

## Analogy with the gravitational field

There exists a strong analogy between the electric field and the gravitational field: the expression of the field and the potential differ only from one constant, and the principal theorems of calculation (like that of superposition or Gauss) apply.

# Deepenings

## Particles creating a field

In the everyday life these sources of the electric field are most of the time electron S, negatively charged, or Proton S, positively charged.

## Dipole moment

### Definition

One generally calls electric dipole a unit made up of two of the same loads value, signs opposed, and placed close one to the other (from the point of view of the observer). The dipole moment is then the vector $\ vec \left\{p\right\} =q \ vec \left\{NP\right\}$, where $q$ is the value of the one of the load (positive) and $\ vec \left\{NP\right\}$ the going vector of the negative charge to the positive load.

### Application to the atomic nuclei

When the matter is presented in the form of Atome the electric charge of the electrons compensates for that of the protons which constitute the core of it. If one places oneself at an important distance from an atom compared to his size, one speaks about macroscopic scale, this last is thus comparable to a neutral body electrically. The electric field that it created is thus relatively very weak. In Astrophysical for example, the electric field created by the ordinary matter which constitutes the Planet S is negligible in front of the influence exerted by this same matter via the Gravitation. But although the Atom S and the Molécule S are Neutre S seen by far, the positive and negative loads are not localized at the same place. If one places oneself at a distance from the order of the size of the atom or molecule, it is what is called the microscopic scale, one realizes that this dissymmetry of provision of the loads generates a electric Dipole moment what is called. Such an electric dipole generates to him as an electric field but of intensity much weaker as that of an electric charge. One calls forces of van der Waals the forces exerted between the atoms and molecules because of the electric fields created by all these microscopic dipoles.

## Field and locality

The concept of electric field, although natural today, is actually rather subtle and is closely related to the concept of Localité in physics. It is interesting to be delayed above a moment.

If one considers an electric charge source $q_s \,$ and a load test $q_t \,$ placed in a point $P \,$ of space then the only measurable quantity indeed in experiments is the electric force $\ vec \left\{F\right\} _ \left\{S \ rightarrow T\right\} \,$ of the first over the second. It is important to realize that a priori the electric force is thus defined like an action remotely of a load on another. The conceptual projection of the concept of field is the following one: it is possible to replace this remote action of $q_s \,$ by the existence in any point of the space of a new quantity, of mathematically vectorial nature, called electric field and whose value $\ vec \left\{E\right\} \left(P\right) \,$ summarizes the influence of $q_s \,$ in each point of space. To determine the evolution of the load test $q_t \,$ it is not thus any more need to refer constantly to the load source located at far but only of reading the contained information locally in the electric field with the site of $q_t \,$. The force is then obtained according to the equation

$\ vec \left\{F\right\} _ \left\{S \ rightarrow T\right\} =q_t \ vec \left\{E\right\} \left(P\right) \,$

This principle of locality is absolutely not pain-killer. In particular a not-commonplace consequence of this one is that if two configurations of electric sources are considered and that in addition one can show that in a certain point of space the electric fields created by these two distributions are the same then necessarily effect from these two plays of source in this point are absolutely indistinguables.

An example of situation where the concept of field, or an equivalent way the locality of the electromagnetic theory, becomes all its extensive appears when the question arises of determining the properties of transformation of an electrostatic field under the Transformations of Lorentz: let us consider a Boost of Lorentz given by a Flight Path Vector $\ vec \left\{v\right\} \,$ and the decomposition of the electric field $\ vec \left\{E\right\} \left(P_0\right) = \ vec \left\{E\right\} _ \left\{\ parallel\right\} \left(P_0\right) + \ vec \left\{E\right\} _ \left\{\ perp\right\} \left(P_0\right) \,$. This field is created by an arbitrary distribution of sources. By locality, while limiting oneself to the point $P_0 \,$ one can replace the charge distribution by a Condensateur plan container $P_0 \,$ and creating a uniform electric field equal to $\ vec \left\{E\right\} \left(P_0\right) \,$ in any point $P \,$ interior with his enclosure (one notes $\ sigma \,$ the surface Densité of associated load).

Let us suppose initially that $\ vec \left\{v\right\} \,$ is in the plan of this fictitious distribution surface (what is the case if the electric field is transverse with the movement) one from of deduced that in the new reference frame,

$\ sigma\text{'}= \ beta \ sigma \,$

by contraction lengths, with $\ beta= \left(1 \left\{v^2 \ over c^2\right\}\right) ^ \left\{- 1/2\right\}$, and thus

$\ vec \left\{E\right\} \text{'}\left(P_0\right) = \ beta \ vec \left\{E\right\} \left(P_0\right) \,$

So on the other hand the field is longitudinal then the surface distribution of the dummy loads is transverse and thus unallocated by the change of reference frame and then

$\ vec \left\{E\right\} \text{'}\left(P_0\right) = \ vec \left\{E\right\} \left(P_0\right) \,$

In the most general case of an unspecified direction one has then by Principe of superposition

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One thus deduced very simply the electric field in the new reference frame without never putting the question of the distribution of the real sources in the new reference frame (if the distribution of origin were complicated then to reproduce this result in a direct way would be very difficult in general). Let us insist finally once again on the absence of magnetic field in the original reference frame to derive this result.

## Simple examples of calculation of the electric field

The few examples which follow are simple applications of the theorem of Gauss.

That is to say a concentrated loading Q located in a point O . That is to say a point of space M . The electric field which causes Q in M is worth:

$\ vec \left\{E\right\} = \ frac \left\{Q\right\} \left\{4 \ pi \ varepsilon OM^2\right\} \ cdot \ frac \left\{\ overrightarrow \left\{OM\right\}\right\} \left\{OM\right\}$ with: $\ varepsilon_0$ the permitivity of the vacuum which is worth 8,85.10-12 C2N-1m-2
• the module of the electric field proportionally decrease with the square the distance D . Its direction passes by the point O (radial field ). The expression of its module at a distance D is:

$E = \ frac \left\{Q\right\} \left\{4 \ pi \ varepsilon d^2\right\}$
• the attenuation of the effect of a concentrated loading depends on the square of the distance. The effect of the charge$\ frac \left\{Q\right\} \left\{\ varepsilon\right\}$ in front of being distributed on the surface of a sphere $4 \ pi d^2$ which is all the more wide as one moves away from the load.

• It is interesting to note that if one considers the load created by a sphere uniformly charged in a point which is not interior for him (i.e.: the distance from the center point O of the sphere is higher than the ray of the sphere), the field created by this sphere is then identical to the field created by a concentrated loading placed out of O and of value the total load of the sphere.

### Field created by a wire infinitely long and uniformly charged

• One definite the linear load by:
$\ lambda = \ frac QL \,$ in C.m-1
Q being the load of a portion (element length) of the wire and L is the length of this portion
• the module electric field decrease proportionally with the distance D . Its direction is perpendicular to the wire and passes by the wire (radial field ). The expression of its module at a distance D is:

$E = \ frac \left\{\ lambda\right\} \left\{2 \ pi \ varepsilon D\right\} \,$
• the attenuation of the effect of an infinitely long wire depends on the distance. The effect of the charge$\ frac \left\{\ lambda\right\} \left\{\ varepsilon\right\} \,$ in front of being distributed on the perimeter of a circle $2 \ pi D \,$ which is all the more wide as one moves away from the load.

### Field created by an infinite plane plate, uniformly charged

• One defines the density of surface charge by:

$\ sigma = \ frac QA \,$ in C.m-2

Q being the load of an area (element of surface) of the plate and has is the surface of this area.

• the electric field created is uniform : its direction is a perpendicular in the plan and the expression of its module is the same one in any point of space and it is independent of the position:

$E = \ frac \left\{\ sigma\right\} \left\{2 \ varepsilon\right\} \,$

### Field created by a plane condenser

• the association of two plane plates identical, parallel and separated by a distance D constitutes a plane condenser of capacity:
$C = \ frac \left\{S \ varepsilon\right\} d$ out of F (Farad)
• the electric field inside checks:

$\ vec \left\{E\right\} = \ frac \left\{\ sigma\right\} \left\{\ varepsilon\right\} \ vec \left\{U\right\}$ with $\ sigma$ the density of surface charge carried by the reinforcements and $\ vec \left\{U\right\}$ an unit vector perpendicular to the plates in the Potential direction of the S decreasing.

# See too

• Condensing

• Electrostatic
• Law of Coulomb

## External bonds

• Topography of the electrostatic field (animation Flash)
• Traced lines of field and equipotential (animation Flash)

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