One of the important aspects of electromagnetism is understanding the patterns of the electric and magnetic fields that the charges and current produce. These patterns can often suggest a symmetry – a regular order to the pattern that helps us deal with the field. Symmetries are a deep part of physics. In this case, we are discussing geometric symmetries, which suggest that there's a particular kind of charge or current producing the field. These symmetries can be used to reformulate the calculations to find the electric field (Gauss' Law) or magnetic field (Ampere's Law) more simply.

Gauss' Law helps us to calculate the electric field when there is sufficient symmetry to use it. That is, Gauss' Law, as mathematical statement, is always true, but it's only useful in limited cases (namely, for planes, cylinders and spheres of charge). The total electric flux is always proportional to the enclosed charge, but that doesn't mean we can always calculate the electric field from Gauss' Law.

$$\Phi_E = \oint \vec{E} \bullet d\vec{A} = \dfrac{q_{enc}}{\varepsilon_0}$$

For a chosen Gaussian surface, if the electric field points the same direction as the area vector everywhere on the surface and we can argue that the electric field is constant in magnitude over the surface, then we can simplify Gauss' Law sufficiently where it becomes useful,

$$\oint \vec{E} \bullet d\vec{A} = E \oint dA = \dfrac{q_{enc}}{\varepsilon_0}$$

The example we have seen a number of times is the point charge, $q$. If we encapsulate the point charge with an imaginary spherical surface of radius $r$, such that the point charge is at the center, we can easily find the electric field of the charge,

$$\oint \vec{E} \cdot d\vec{A} = \dfrac{q_{enc}}{\varepsilon_0}$$ $$E \oint dA = \dfrac{q}{\varepsilon_0}$$ $$E 4\pi r^2 = \dfrac{q}{