Editing Permittivity (section)

=== Gauss's law ===
Permittivity is connected to electric flux (and by extension electric field) through [[Gauss's law]]. Gauss's law states that for a closed [[Gaussian surface]], {{mvar|S}},

<math display="block">\Phi_E = \frac{Q_\text{enc}}{\varepsilon_0} = \oint_S \mathbf{E} \cdot \mathrm{d} \mathbf{A}\ ,</math>
where <math>\Phi_E</math> is the net electric flux passing through the surface, <math>Q_\text{enc}</math> is the charge enclosed in the Gaussian surface, <math>\mathbf{E}</math> is the electric field vector at a given point on the surface, and <math>\mathrm{d} \mathbf{A}</math> is a differential area vector on the Gaussian surface.

If the Gaussian surface uniformly encloses an insulated, symmetrical charge arrangement, the formula can be simplified to

<math display="block">E\ A\ \cos \theta = \frac{\; Q_\text{enc} }{\ \varepsilon_0\ }\ ,</math>
where <math>\ \theta\ </math> represents the angle between the electric field lines and the normal (perpendicular) to {{mvar|S}}.

If all of the electric field lines cross the surface at 90°, the formula can be further simplified to

<math display="block">\ E = \frac{\; Q_\text{enc} }{\ \varepsilon_0\ A\ } ~.</math>

Because the surface area of a sphere is <math>\ 4 \pi r^2\ ,</math> the electric field a distance <math>r</math> away from a uniform, spherical charge arrangement is

<math display="block">\ E = \frac{ Q }{\ \varepsilon_0 A\ } = \frac{ Q }{\ \varepsilon_0\ \left(4\ \pi\ r^2\right)\ } = \frac{ Q }{\ 4 \pi\ \varepsilon_0\ r^2\ } ~.</math>

This formula applies to the electric field due to a point charge, outside of a conducting sphere or shell, outside of a uniformly charged insulating sphere, or between the plates of a spherical capacitor.