Editing Gauss's Law (Electric): (section)

= Gauss's Law (Electric): Definition and Mathematical Representation =

== Introduction ==
'''Gauss’s Law''' is a fundamental law in electrostatics that relates the electric flux through a closed surface to the total electric charge enclosed by that surface. It is one of the four equations in [[Maxwell's Equations]] and provides a powerful method for calculating electric fields, especially with high symmetry.

== Mathematical Formulation ==

=== Integral Form ===
<math>
\oint_{\text{closed surface}} \vec{E} \cdot d\vec{A} = \frac{Q_{\text{enc}}}{\varepsilon_0}
</math>

Where:
* <math>\vec{E}</math> is the electric field vector,
* <math>d\vec{A}</math> is an infinitesimal area vector on the closed surface (pointing outward),
* <math>Q_{\text{enc}}</math> is the total electric charge enclosed within the surface,
* <math>\varepsilon_0</math> is the vacuum permittivity (<math>\varepsilon_0 \approx 8.854 \times 10^{-12}\, \text{C}^2/\text{N·m}^2</math>).

=== Differential Form ===
Using the divergence theorem, Gauss's Law can also be expressed in differential form:

<math>
\nabla \cdot \vec{E} = \frac{\rho}{\varepsilon_0}
</math>

Where:
* <math>\nabla \cdot \vec{E}</math> is the divergence of the electric field,
* <math>\rho</math> is the volume charge density (charge per unit volume).

== Physical Meaning ==
Gauss's Law states that the total electric flux through a closed surface is proportional to the amount of electric charge enclosed within that surface. The law implies:

* A net outward flux occurs when positive charge is enclosed.
* A net inward flux occurs when negative charge is enclosed.
* If no charge is enclosed, the net electric flux is zero.

== Applications of Gauss’s Law ==

Gauss’s Law is especially useful when dealing with problems involving symmetry:

=== 1. Spherical Symmetry ===
For a point charge <math>q</math> at the center of a spherical surface of radius <math>r</math>:

<math>
E \cdot 4\pi r^2 = \frac{q}{\varepsilon_0} \quad \Rightarrow \quad E = \frac{1}{4\pi\varepsilon_0} \cdot \frac{q}{r^2}
</math>

=== 2. Cylindrical Symmetry ===
For an infinite line charge with linear charge density <math>\lambda</math>:

<math>
E \cdot (2\pi r L) = \frac{\lambda L}{\varepsilon_0} \quad \Rightarrow \quad E = \frac{\lambda}{2\pi \varepsilon_0 r}
</math>

=== 3. Planar Symmetry ===
For an infinite plane of surface charge density <math>\sigma</math>:

<math>
E = \frac{\sigma}{2 \varepsilon_0}
</math>

== Conditions for Application ==

* Symmetry is essential — spherical, cylindrical, or planar.
* The electric field must be constant in magnitude over the chosen Gaussian surface.
* The surface must be closed.

== Relation to Coulomb's Law ==

Gauss’s Law is consistent with [[Coulomb’s Law]] and can be derived from it for point charges. Conversely, Coulomb’s Law can also be derived from Gauss’s Law under the assumption of spherical symmetry.

== See Also ==
* [[Electric Field]]
* [[Electric Flux]]
* [[Coulomb's Law]]
* [[Maxwell's Equations]]
* [[Charge Density]]
* [[Vacuum Permittivity]]
* [[Electrostatics]]