# 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 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.

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## 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

[itex]

\mathbf{F} = \frac{1}{4 \pi \epsilon_0}\frac{q_1 q_2}{r^2}\mathbf{\hat r} (1) [itex]

where

• [itex]\epsilon_0[itex] (pronounced epsilon-nought) is a physical constant, the permittivity of free space;
• [itex]q_1[itex] and [itex]q_2[itex] are the electric charges of the objects;
• [itex]r[itex] is the magnitude of the separation vector between the objects;
• [itex]\hat r[itex] 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, [itex]\epsilon_0[itex] 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:

[itex]

\mathbf{F} = q\mathbf{E} [itex]

[itex]

\mathbf{E} = \frac{1}{4 \pi \epsilon_0}\frac{Q}{r^2}\mathbf{\hat r} [itex]

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.

The above paragraph is perhaps misleading that the equation could refer to electrodynamics by referring to a "moveable test charge"; however, it is important to understand that by "moveable", this means that the charge can be moved to, and held at, any position.

In the case of electrostatics, where velocity of both particles is zero, the above equation holds.

## 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.

[itex]E_{tot} = E_1 + E_2 + E_3 \ldots \,\![itex]

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

[itex]

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

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

The electric field is equal to minus the gradient of the electric potential. If several spatially distributed charges generate such an electric potential, e.g. in a solid, an electric field gradient may be defined.

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