Physics with Calculus/Electromagnetism/Gauss' Law

Basic Theory
Gauss' Law relates the amount of charge contained within a volume to the electric flux passing through the surface of that volume. It was developed by Carl Fredric Gauss, an 18th century mathematician. Electric flux can only truly be described by mathematics, but has an intuitive meaning as well. Essentially, electric flux is the amount of 'electric field' going through a surface. Think of a cage which is immersed in a flowing river, where the cage is meshed so that the water runs through it fairly unabated. The 'flux' is the amount of water passing through the cage's surface at any one moment. For this discussion we shall not concern ourselves with electric fields which change in time, so consider the electric flux to be constant in time.

When using Gauss' law we create something called a "Gaussian surface", or just the "Gaussian". A Gaussian is an imaginary surface which is completely enclosed.

Electric Flux (Flux: field line flow) Φ: how much field passes through a surface S. Uniform Field (parallel field lines) Φ=E A cos (theta) Where theta is the angle measured between direction of field lines and direction for the normal to the surface.

$$\Phi_f = \int_S \mathbf{F} \cdot \mathbf{dA}$$ where F is a vector field, dA is the area element of the surface S, directed as the surface normal, and Φf is the resulting flux. For a surface that encloses the field generator

Gauss' Law

$$\Phi = \oint_S \mathbf{E} \cdot d\mathbf{A} = {1 \over \epsilon_o} \int_V \rho \cdot dV = \frac{Q_A}{\epsilon_o}$$

where $$\mathbf{E}$$ is the electric field, $$d\mathbf{A}$$ is the area of a differential square on the closed surface $$S$$ with an outward facing surface normal defining its direction, $$\mathrm{Q}_A$$ is the charge enclosed by the surface, $$\rho$$ is the charge density at a point in $$V$$, $$\epsilon_o$$ is the permittivity of free space and $$\oint_S$$ is the integral over the surface $$S$$ enclosing volume $$V$$.