Electric Flux

Before Gauss's theorem can be stated, one more quantity must be defined precisely: electric flux, the measure of how much of an electric field passes through a given surface.

Flux through a flat surface in a uniform field. For a flat surface of area $A$ placed in a uniform field $\vec{E}$, with $\hat{n}$ the unit vector normal (perpendicular) to the surface, the electric flux through it is defined as

where $\vec{A}=A\hat{n}$ is the "area vector" (a vector of magnitude $A$ pointing along the surface's outward normal), and $\theta$ is the angle between $\vec{E}$ and $\hat{n}$. Flux is a scalar quantity (the dot product of two vectors is a number, not a vector), with SI unit $\text{N}\,\text{m}^2/\text{C}$.

Three special cases illustrate the formula: if the field is parallel to the surface's normal ($\theta=0^\circ$), $\Phi=EA$, the maximum possible flux for that area and field strength; if the field runs exactly along the surface, i.e. perpendicular to the normal ($\theta=90^\circ$), $\Phi=0$ -- no field lines actually cross the surface, they merely graze it; and for a general angle in between, only the component of $\vec{E}$ along the normal, $E\cos\theta$, contributes to the flux.

Flux through a curved or non-uniform surface. When the surface is not flat, or the field is not uniform, the surface is divided into a very large number of infinitesimally small area elements $d\vec{A}$, each small enough that $\vec{E}$ can be treated as constant and the element as flat; the flux through each element is $d\Phi=\vec{E}\cdot d\vec{A}$, and the total flux is the sum (integral) of all these contributions:

For a closed surface (one that completely encloses a volume, such as a sphere or a cube, with no gaps), this is written with a circle through the integral sign, $\Phi=\oint\vec{E}\cdot d\vec{A}$, and the outward normal is always used at every point of the surface by convention -- so that flux due to field lines leaving the enclosed volume counts as positive, and flux due to field lines entering the enclosed volume counts as negative. …