Gauß's Law for Electric Flux

Gauß's law states that the number of field lines passing through the surface of a solid $Ω$ (the electric flux) is proportional to the charge enclosed within.

Φ = \subset \supset \iint 𝜕 Ω ⃗ 𝐄 \cdot 𝑑 ⃗ 𝐚 = 𝑄 e n c 𝜖 0 = 1 𝜖 0 ∭ Ω 𝜌 𝑑 𝜏'

Using Острогра́дский's divergence theorem, we see

⃗ \nabla \cdot ⃗ 𝐄 = 𝜌 𝜖 0