Gauss's law for electrostatics states that the electric flux through any closed surface is equal to the net electric charge enclosed by the surface divided by the permittivity of free space. Mathematically, it can be expressed as follows:

∮ E ⋅ dA = q/ε0

where:

* ∮ means "the integral over"
* dA is a small area element on the closed surface
* q is the total charge enclosed by the surface
* ε0 is the permittivity of free space (8.854 × 10^−12 F/m)

The Gauss's law can be proved using the divergence theorem, which states that the integral of the divergence of a vector field over a closed surface is equal to the integral of the flux of the vector field through the surface. In the case of the electric field, the divergence is equal to the charge density.

The Gauss's law has many applications in electrostatics. For example, it can be used to calculate the electric field due to a point charge, a uniformly charged sphere, or a uniformly charged sheet. It can also be used to prove other electrostatics laws, such as Coulomb's law.

Here are some examples of the Gauss's law in daily life:

* The electric field around a charged hair comb can be calculated using the Gauss's law.
* The electric field inside a charged capacitor can also be calculated using the Gauss's law.
* The electric field around a charged lightning bolt can be calculated using the Gauss's law.