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# Electric field

In physics, an electric field is the effect produced by the existence of an electric charge, such as an electron, ion, or proton, in the volume of space or medium that surrounds it.

### Mathematical definition

The definition of the electric field is developed as follows. Coulomb's Law gives the force between two point charges:

$\mathbf{F} = \frac{q_1 q_2}{4 \pi \epsilon_0 \left|\mathbf{r}\right|^2}\mathbf{\hat r}$

This was known empirically (note - the equation is given for SI units). Suppose we take one of the charges to be fixed, and the other one to be a moveable "test object". We note that according to this equation, the force on the test object is proportional to its charge. We define the electric field to be the proportionality constant between charge and force:

$\mathbf{F} = q\mathbf{E}$

$\mathbf{E} = \frac{q} {4\pi\epsilon_0 \left|\mathbf{r}\right|^2}\mathbf{\hat r}$

Hence, electric field is dependent on position. A field, in this context, means a vector which is dependent on another vector - a vector valued vector function.

Another empirically known fact was that in the presence of a more complicated fixed object, the electric forces from the constituent charges can simply be added together. Hence, the electric field due to a composite object becomes

Etot = E1 + E2 + E3 + ...

where E1, E2, etc. are the electric field due to individual charges making up the object. This is what is meant when it is said that the electric field is "linear". For a continuous distribution of charge (rather than discrete points), we can define the electric field to be:

$\mathbf{E} = \int\frac{\rho} {4\pi\epsilon_0 \left|\mathbf{r}\right|^2}\mathbf{\hat r}\,d^{3}\mathbf{r}$

where ρ is the charge density - i.e. charge per unit volume.

See Maxwell's equations for the full set of equations governing electric fields.