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Electric field

An electric field is a vector field that associates to each point in space the Coulomb force experienced by a unit test charge. The strength and direction of the Coulomb force F {\textstyle \mathbf {F} } on a charge q t {\textstyle q_{t}} depends on the electric field E {\textstyle \mathbf {E} } established by other charges that it finds itself in, such that F = q t E {\textstyle \mathbf {F} =q_{t}\mathbf {E} }. In the simplest case, the field is considered to be generated solely by a single source point charge. More generally, the field can be generated by a distribution of charges who contribute to the overall by the principle of superposition.

If the field is generated by a positive source point charge q {\textstyle q}, the direction of the electric field points along lines directed radially outwards from it, i.e. in the direction that a positive point test charge q t {\textstyle q_{t}} would move if placed in the field. For a negative point source charge, the direction is radially inwards.

The magnitude of the electric field E can be derived from Coulomb's law. By choosing one of the point charges to be the source, and the other to be the test charge, it follows from Coulomb's law that the magnitude of the electric field E created by a single source point charge Q at a certain distance from it r in vacuum is given by | E | = k e | q | r 2 {\displaystyle |\mathbf {E} |=k_{\text{e}}{\frac {|q|}{r^{2}}}}

A system of n discrete charges q i {\displaystyle q_{i}} stationed at r i = r − r i {\textstyle \mathbf {r} _{i}=\mathbf {r} -\mathbf {r} _{i}} produces an electric field whose magnitude and direction is, by superposition E (r) = 1 4 π ε 0 ∑ i = 1 n q i r ^ i | r i | 2 = 1 4 π ε 0 ∑ i = 1 n q i r i | r i | 3 {\displaystyle \mathbf {E} (\mathbf {r})={1 \over 4\pi \varepsilon _{0}}\sum _{i=1}^{n}q_{i}{{\hat {\mathbf {r} }}_{i} \over {|\mathbf {r} _{i}|}^{2}}={1 \over 4\pi \varepsilon _{0}}\sum _{i=1}^{n}q_{i}{\mathbf {r} _{i} \over {|\mathbf {r} _{i}|}^{3}}}