Gauss’s Law
Gauss’s Law is one of the four fundamental equations that govern the behavior of electromagnetic fields. It is a very powerful way to calculate the electric field created by a distribution of charge, provided the distribution has sufficient symmetry to simplify calculations.
Gauss’s Law:
The electric flux through any closed surface is proportional to the net charge enclosed by that surface
\boxed{{\mathrm{\Phi }}_E=\frac{Q_{enc}}{{\varepsilon }_0}}
where $Q_{enc}$ denotes the charge enclosed by the closed surface which is often referred to as a Gaussian surface.
Important: in practice, the goal of Gauss’s Law is to derive the electric field $\overrightarrow{E}$ created by a charge distribution. Computing the electric flux is simply a means to an end and we will choose closed surfaces that are such that the electric field is constant in magnitude and direction over them, which will then allow us to write the electric flux as
\boxed{{\mathrm{\Phi }}_E=\overrightarrow{E}\cdot \overrightarrow{A}=EA{\mathrm{cos} \left(\theta \right)\ }}
where $A$ denotes the surface area of the Gaussian surface chosen. This simplification then allows us to extract $E$ from the expression of the electric flux and derive its expression easily. Thus, it is important to choose a Gaussian surface wisely for this simplification to be valid.
Method: how to choose a Gaussian surface?
In order to allow the above simplification of the electric flux integral, we must carefully choose a Gaussian surface with the following properties:
Fundamental Theorem: