# Electric field

In physics, an electric field or E-field is an effect produced by an electric charge that exerts a force on charged objects in its vicinity. The SI units of the electric field are newtons per coulomb or volts per meter (both are equivalent). Electric fields are composed of photons and contain electrical energy with energy density proportional to the square of the field intensity. In the static case, an electric field is composed of virtual photons being exchanged by the charged particle(s) creating the field. In the dynamic case the electric field is accompanied by a magnetic field, by a flow of energy, and by real photons.

The electric field is a vector quantity, and the electric field strength is the magnitude of this vector.

## Definition and derivation

The mathematical definition of the electric field is developed as follows. Coulomb's law gives the force between two point charges (infinitesimally small charged objects) as

${\displaystyle \mathbf {F} ={\frac {1}{4\pi \epsilon _{0}}}{\frac {q_{1}q_{2}}{r^{2}}}\mathbf {\hat {r}} (1)}$

where

• ${\displaystyle \epsilon _{0}}$ (pronounced epsilon-nought) is a physical constant, the permittivity of free space;
• ${\displaystyle q_{1}}$ and ${\displaystyle q_{2}}$ are the electric charges of the objects;
• ${\displaystyle r}$ is the magnitude of the separation vector between the objects;
• ${\displaystyle {\hat {r}}}$ is the unit vector representing the direction from one charge to the other.

In the SI system of units, force is given in newtons, charge in coulombs, and distance in metres. Thus, ${\displaystyle \epsilon _{0}}$ has units of C²/(N·m²).

This was known empirically. Suppose one of the charges is taken to be fixed, and the other one to be a moveable "test charge". Note that according to this equation, the force on the test object is proportional to its charge. The electric field is defined as the proportionality constant between charge and force:

${\displaystyle \mathbf {F} =q\mathbf {E} }$
${\displaystyle \mathbf {E} ={\frac {1}{4\pi \epsilon _{0}}}{\frac {Q}{r^{2}}}\mathbf {\hat {r}} }$

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 equation. When we speak of a "moveable test charge", this means that the charge can be moved to, and held at, any position.

Furthermore, Coulomb's law is actually a special case of Gauss's Law, a more fundamental description of the relationship between the distribution 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.

## Properties

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.

${\displaystyle E_{tot}=E_{1}+E_{2}+E_{3}\ldots \,\!}$

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

${\displaystyle \mathbf {E} ={\frac {1}{4\pi \epsilon _{0}}}\int {\frac {\rho }{r^{2}}}\mathbf {\hat {r}} \,d^{3}\mathbf {r} }$

where ${\displaystyle \rho }$ is the charge density, or the amount of charge per unit volume.

The electric field is equal to the negative gradient of the electric potential. In symbols,

${\displaystyle \mathbf {E} =-\mathbf {\nabla } \phi }$

Where Template:Phisymbol${\displaystyle (x,y,z)}$ 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 ${\displaystyle \varepsilon }$ which consists of the permittivity of free space ${\displaystyle \varepsilon _{0}}$ an the material dependent relative permittivity ${\displaystyle \varepsilon _{r}}$, yields to the Electric displacement field:

${\displaystyle {\vec {D}}=\varepsilon {\vec {E}}=\varepsilon _{0}\varepsilon _{r}{\vec {E}}}$

## Parallels between electrostatics and gravity

As explained above, electric field can be thought of as a proportionality constant when the force exerted on a test charge is proportional to the magnitude of the test charge. Put more simply, this is to say that an environment can be electrostatically quantified by the electric field in that environment; different physical facts of the environment combine to form this single number, and it is possible for different environments to have the same number for electric field. Any given object (that we are measuring the force on) has associated various "weights;" the electrostatic weight is the charge, and the gravitic weight is the mass. The electrostatic force on some object in the environment is then simply the strength of the environment (the electric field), times the magnitude of the electrostatic weight (the charge). This is similar to gravity, where any given environment has a gravitational acceleration, and the force on some object in that environment is simply the acceleration due to gravity (the environmental factor) times the mass of the object (the gravitic weight). For electrostatics, the factors that determine the electric field in an environment are:

1. The magnitude of the nearby charge
2. The square of the distance between that charge and the object being measured, and
3. Coulomb's constant.

The analogous factors for gravity are:

1. The mass of the nearby object
2. The square of the distance between that object and the object being measured, and
3. The Universal Gravitational Constant.

When measuring the force on a mass at sea level due to Earth's gravity, the first factor (mass of the nearby environment-determining object) is the mass of the Earth, while the second factor (square of the distance between the environment-determining object and the measured object) is the square of the Earth's radius.

The units of the electric field, newtons per coulomb, can thus by expressed as force per unit charge.