Electric Fields
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Claimed by ycho46
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Electric fields are caused by electric charges or varying magnetic fields. The former effect is described by Gauss's law, the latter by Faraday's law of induction, which together are enough to define the behavior of the electric field as a function of charge repartition and magnetic field. However, since the magnetic field is described as a function of electric field, the equations of both fields are coupled and together form Maxwell's equations that describe both fields as a function of charges and currents.
In the special case of a steady state (stationary charges and currents), the Maxwell-Faraday inductive effect disappears. The resulting two equations (Gauss's law [math]\displaystyle{ \nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0} }[/math] and Faraday's law with no induction term [math]\displaystyle{ \nabla \times \mathbf{E} = 0 }[/math]), taken together, are equivalent to Coulomb's law, written as [math]\displaystyle{ \boldsymbol{E}(\boldsymbol{r}) = {1\over 4\pi\varepsilon_0}\int d\boldsymbol{r'} \rho(\boldsymbol{r'}) {\boldsymbol{r} - \boldsymbol{r'} \over |\boldsymbol{r} - \boldsymbol{r'}|^3} }[/math] for a charge density [math]\displaystyle{ \mathbf{\rho}(\mathbf{r}) }[/math] ([math]\displaystyle{ \mathbf{r} }[/math] denotes the position in space). Notice that [math]\displaystyle{ \varepsilon_0 }[/math], the permittivity of vacuum, must be substituted if charges are considered in non-empty media.
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