Gauss’s Law states that the total electric flux through any closed surface (a surface enclosing a definite volume) is proportional to the total (net) electric charge inside the surface.
$$\Phi_{E} = \frac{q}{\epsilon_{0}}$$
Note that Gauss’s law is completely equivalent to Coulomb’s law.
It is good to introduce electric flux for the discussion of Gauss’s Law.
Electric Flux
Consider a flat area A perpendicular to a uniform electric field $\vec{E}$. The electric flux through the area is:
$$\Phi_{E} = EA$$
If the area A is flat but not perpendicular to the field $\vec{E}$,
$$\Phi_{E} = \vec{E}.\vec{A}$$
$$d \Phi_{E} = \vec{E} . d \vec{A}$$
In general, to find $\Phi_{E}$, you can use any of the formulas below:
$$\begin{aligned} \Phi_{E} &= \int \, d\Phi_{E} \\ &= \int \vec{E}.d\vec{A} \\ &= \int E \, \text{cos} \, \phi \, dA \\ &= \int E_{\perp} \, dA \\ &= \int E \, dA_{\perp} \end{aligned}$$
Note: In simple terms, electric flux through an area just means the “flow” of electric field through the area.
Next: Coulomb’s Law To Gauss’s Law
