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PowerPedia:Electric field

Lasted edited by Andrew Munsey, updated on June 15, 2016 at 1:43 am.

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In There was an error working with the wiki: Code[7] on charged objects. The concept of electric field was introduced by There was an error working with the wiki: Code[8] quantity, and the There was an error working with the wiki: Code[10] is the There was an error working with the wiki: Code[11] magnitude of this vector.

Introduction

The SI units of the electric field are There was an error working with the wiki: Code[12]s per There was an error working with the wiki: Code[13] (N C-1) or Volts per Meter (V m-1) (which are equivalent). Electric fields contain Electrical energy with There was an error working with the wiki: Code[14] proportional to the square of the field intensity. Electric fields exist around all charges the direction of field lines at a point is defined by the direction of the electric force exerted on a positive test charge placed at that point. The strength of the field is defined by the ratio of the electric force on a charge at a point to the magnitude of the charge placed at that point. In the dynamic case the electric field is accompanied by a Magnetic field (if charges producing electric field move with constant velocity), or by Electromagnetic field (when charges move with acceleration). Both electric and magnetic fields have Energy associated with them, and e/m field being a field in motion also has There was an error working with the wiki: Code[15]. Energy of e/m field is quantized (and quanta are called There was an error working with the wiki: Code[16]s).

Definition (for electrostatics)

Electric field is defined as the Electric force per unit charge. The direction of the field is taken to be the direction of the force it would exert on a positive test charge. The electric field is radially outward from a positive charge and radially in toward a negative point charge.

The electric field is defined as the proportionality constant between charge and force (in other words, the force per unit of test charge):

:

{E} = \fracThere was an error working with the wiki: Code[1]{q} = \frac{1}{4 \pi \epsilon_0}\frac{Q}{r^2}{\hat r}

:::where

:::: {F} is the electric force given by There was an error working with the wiki: Code[17],

:::: q is the charge of a "test charge",

:::: 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,

:::: {\hat{r} } is the There was an error working with the wiki: Code[18] pointing from the particle with charge Q to the E-field evaluation point, and

:::: \epsilon_0 is the There was an error working with the wiki: Code[9].

However, note that this equation is only true in the case of Electrostatics, that is to say, when there is nothing moving. The more general case of moving charges causes this equation to become the Lorentz force equation. When we speak of a "moveable test charge", this means only that the above equations hold regardless of the position of the (stationary) test charge.

Furthermore, There was an error working with the wiki: Code[19] is actually a special case of There was an error working with the wiki: Code[20], 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 There was an error working with the wiki: Code[21], a set of four laws governing 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 There was an error working with the wiki: Code[22]. If more than one charge is present, the total electric field at any point is equal to the There was an error working with the wiki: Code[23] of the respective electric fields that each object would create in the absence of the others.

:{E}_{\rm total} = \sum_i {E}_i = {E}_1 + {E}_2 + {E

}_3 \ldots \,\!

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

:

{E} = \frac{1}{4\pi\epsilon_0} \int\frac{\rho}{r^2} {\hat r}\,d^{3}{r}

where \rho is the There was an error working with the wiki: Code[24], or the amount of charge per unit There was an error working with the wiki: Code[25].

The electric field at a point is equal to the negative There was an error working with the wiki: Code[26] of the There was an error working with the wiki: Code[27] there. In symbols,

:

{E} = -{\nabla}\phi

Where \phi(x, y, z) is the There was an error working with the wiki: Code[28] representing the electric potential at a given point. If several spatially distributed charges generate such an There was an error working with the wiki: Code[29], e.g. in a There was an error working with the wiki: Code[30], an There was an error working with the wiki: Code[31] may also be defined.

Considering the There was an error working with the wiki: Code[32] \varepsilon of a material, which is the product of the permittivity of free space \varepsilon_{0} and the material-dependent There was an error working with the wiki: Code[33] \varepsilon_{r}, yields the There was an error working with the wiki: Code[34]:

:{D} = \varepsilon {E} = \varepsilon_{0} \varepsilon_{r} {E}

Energy in the electric field

There was an error working with the wiki: Code[2]

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

: u = \frac{1}{2} \epsilon |E|^2

where \epsilon is the There was an error working with the wiki: Code[35] of the medium in which the field exists, and E is the electric field vector. The total energy stored in the electric field in a given volume V is therefore

: \int_{V} \frac{1}{2} \epsilon |E|^2 \, d\tau

where d\tau is the differential volume element.

Parallels between electrostatics and gravity

There was an error working with the wiki: Code[36], which describes the interaction of electric charges:

:

{F} = \frac{1}{4 \pi \epsilon_0}\frac{Qq}{r^2}{\hat r} = q{E}

is similar to the Newtonian gravitation law:

:

{F} = G\frac{Mm}{r^2}{\hat r} = m{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 inverse-square law (both are inversely proprotional to square of r).

# Both propagate with finite speed c.

Differences between electrostatic and gravitational forces:

# Electrostatic forces are much greater than gravitational forces (by about 1036 times).

# Gravitational forces are always attractive in nature, whereas electrostatic forces may be either attractive or repulsive.

# Gravitational forces are independent of the medium whereas electrostatic forces depend on the medium. This is due to the fact that a medium contains charges the fast motion of these charges, in response to an external electromagnetic field, produces a large secondary electromagnetic field which should be accounted for. While slow motion of ordinary masses in response to changing gravitational field produces extremely weak secondary "gravimagnetic field" which may be neglected in most cases (except, of course, when mass moves with relativistic speeds).

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. This "secondary" electric field can be computed using There was an error working with the wiki: Code[37],

:\nabla \times {E} = -\frac{\partial {B}} {\partial t}

where \nabla \times {E} indicates the There was an error working with the wiki: Code[38] of the electric field, and -\frac{\partial {B}} {\partial t} represents the vector rate of decrease of There was an error working with the wiki: Code[39] 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 There was an error working with the wiki: Code[40] or There was an error working with the wiki: Code[41]. In this case, There was an error working with the wiki: Code[42] no longer provides a useful definition of electric field as given above. Instead, the more general There was an error working with the wiki: Code[43], along with Faraday's law, determines the electric field.

related articles

There was an error working with the wiki: Code[44] are the full set of equations governing electric fields.

Electromagnetism

Electrostatics

There was an error working with the wiki: Code[45]

Magnetism

External articles and references

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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

There was an error working with the wiki: Code[1], Wikipedia: The Free Encyclopedia. Wikimedia Foundation.