Gauss’ Law

Symmetry

There are three fundamental symmetrires:

• Planar
• Cylindrical
• Spherical

Electric Flux

The concept of flux describes how much of something goes through a given area.

• There is a outward flux through a closed surface containing a positive charge
• There is a negative flux through a closed surface containing a negative charge
• There is no net flux through a closed surface containing no net charge

Formally, it is the dot product of a vector field with an area. Electric flux is represented by the symbol $\Phi$, and its standard units are $\textrm{N}\cdot\textrm{m}^2/\textrm{C}$.

The electric flux of a constant electric field is

\[\Phi = \vec E\cdot\vec A\]

where $\vec A$ is the area vector, a vector whose magnitude is the area of the surface and whose direction is $\hat n$, the unit vector normal to the surface.

Calculating Flux

Flux may be expressed as the continuous sum of the dot product of an electric field over an area.

\[\Phi=\oint{\vec E\cdot d\vec A}\]

Properties of Conductors

• The electric field is zero inside a conductor
• Any excess charge must reside entirely on the surface(s)